Discrete mathematics 1From Wikipedia, the free encyclopedia

Contents

1 Antimatroid 11.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Paths and basic words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Convex geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Join-distributive lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.6 Supersolvable antimatroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 Join operation and convex dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Enumeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Closure (mathematics) 82.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 P closures of binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Closure operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Cutting sequence 123.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 DIMACS 134.1 The DIMACS Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Discrete mathematics 155.1 Grand challenges, past and present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2 Topics in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

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5.2.1 Theoretical computer science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.2.2 Information theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2.3 Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2.4 Set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.6 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.2.7 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.8 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215.2.9 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.10 Calculus of finite differences, discrete calculus or discrete analysis . . . . . . . . . . . . . . 225.2.11 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.12 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.13 Operations research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2.14 Game theory, decision theory, utility theory, social choice theory . . . . . . . . . . . . . . 235.2.15 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.2.16 Discrete analogues of continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . 245.2.17 Hybrid discrete and continuous mathematics . . . . . . . . . . . . . . . . . . . . . . . . . 24

5.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.5 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6 Discrete Mathematics (journal) 276.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.2 Abstracting and indexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.3 Notable publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Formal language 297.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.2 Words over an alphabet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.3 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

7.4.1 Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.5 Language-specification formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317.6 Operations on languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

7.7.1 Programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327.7.2 Formal theories, systems and proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

CONTENTS iii

7.9.1 Citation footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347.9.2 General references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

7.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

8 Greedoid 368.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.2 Classes of greedoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.4 Greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Inversion (discrete mathematics) 399.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.2 Weak order of permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 409.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9.4.1 Source bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.2 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429.4.3 Presortedness measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

10 Outline of discrete mathematics 4310.1 Subjects in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4310.2 Discrete mathematical disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.3 Concepts in discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

10.3.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4410.3.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3.3 Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3.4 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4510.3.5 Elementary algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3.6 Mathematical relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4610.3.7 Mathematical phraseology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3.8 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4710.3.9 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.3.10 Propositional logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

10.4 Discrete mathematicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4810.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

11 Portal:Discrete mathematics 5011.1 Discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

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11.2 Selected article . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5111.3 WikiProjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5211.4 Selected picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.5 Did you know? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.6 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5311.7 Topics in Discrete mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.8 Related portals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5411.9 Wikimedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

12 Trinomial triangle 5712.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5712.2 Recursion formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.3 The middle entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.4 Chess mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5812.5 Importance in combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5912.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6012.8 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 61

12.8.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6112.8.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6212.8.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Chapter 1

Antimatroid

{a,b}

{a,b,c}

{a,c} {b,c}

{a} {c}

Ø

«abc»«ab»«»«a»

«ac»«acb»«c»«ca»«cab»«cb»«cba»

{a} {c}

{a,b} {b,c}

{a,b,c,d}

«abcd»

«acbd»

«cabd»

«cbad»

{a,b,c,d}

Three views of an antimatroid: an inclusion ordering on its family of feasible sets, a formal language, and the corresponding pathposet.

In mathematics, an antimatroid is a formal system that describes processes in which a set is built up by includingelements one at a time, and in which an element, once available for inclusion, remains available until it is included.Antimatroids are commonly axiomatized in two equivalent ways, either as a set system modeling the possible statesof such a process, or as a formal language modeling the different sequences in which elements may be included.Dilworth (1940) was the first to study antimatroids, using yet another axiomatization based on lattice theory, andthey have been frequently rediscovered in other contexts;[1] see Korte et al. (1991) for a comprehensive survey ofantimatroid theory with many additional references.The axioms defining antimatroids as set systems are very similar to those of matroids, but whereas matroids are definedby an exchange axiom (e.g., the basis exchange, or independent set exchange axioms), antimatroids are defined insteadby an anti-exchange axiom, from which their name derives. Antimatroids can be viewed as a special case of greedoidsand of semimodular lattices, and as a generalization of partial orders and of distributive lattices. Antimatroids areequivalent, by complementation, to convex geometries, a combinatorial abstraction of convex sets in geometry.

1

2 CHAPTER 1. ANTIMATROID

Antimatroids have been applied to model precedence constraints in scheduling problems, potential event sequencesin simulations, task planning in artificial intelligence, and the states of knowledge of human learners.

1.1 Definitions

An antimatroid can be defined as a finite family F of sets, called feasible sets, with the following two properties:

• The union of any two feasible sets is also feasible. That is, F is closed under unions.

• If S is a nonempty feasible set, then there exists some x in S such that S \ {x} (the set formed by removing xfrom S) is also feasible. That is, F is an accessible set system.

Antimatroids also have an equivalent definition as a formal language, that is, as a set of strings defined from a finitealphabet of symbols. A language L defining an antimatroid must satisfy the following properties:

• Every symbol of the alphabet occurs in at least one word of L.

• Each word of L contains at most one copy of any symbol.

• Every prefix of a string in L is also in L.

• If s and t are strings in L, and s contains at least one symbol that is not in t, then there is a symbol x in s suchthat tx is another string in L.

If L is an antimatroid defined as a formal language, then the sets of symbols in strings of L form an accessible union-closed set system. In the other direction, if F is an accessible union-closed set system, and L is the language of stringss with the property that the set of symbols in each prefix of s is feasible, then L defines an antimatroid. Thus, thesetwo definitions lead to mathematically equivalent classes of objects.[2]

1.2 Examples

• A chain antimatroid has as its formal language the prefixes of a single word, and as its feasible sets the setsof symbols in these prefixes. For instance the chain antimatroid defined by the word “abcd” has as its formallanguage the strings {ε, “a”, “ab”, “abc”, “abcd"} and as its feasible sets the sets Ø, {a}, {a,b}, {a,b,c}, and{a,b,c,d}.[3]

• A poset antimatroid has as its feasible sets the lower sets of a finite partially ordered set. By Birkhoff’s rep-resentation theorem for distributive lattices, the feasible sets in a poset antimatroid (ordered by set inclusion)form a distributive lattice, and any distributive lattice can be formed in this way. Thus, antimatroids can beseen as generalizations of distributive lattices. A chain antimatroid is the special case of a poset antimatroidfor a total order.[3]

• A shelling sequence of a finite set U of points in the Euclidean plane or a higher-dimensional Euclidean spaceis an ordering on the points such that, for each point p, there is a line (in the Euclidean plane, or a hyperplanein a Euclidean space) that separates p from all later points in the sequence. Equivalently, p must be a vertexof the convex hull of it and all later points. The partial shelling sequences of a point set form an antimatroid,called a shelling antimatroid. The feasible sets of the shelling antimatroid are the intersections of U with thecomplement of a convex set.[3]

• A perfect elimination ordering of a chordal graph is an ordering of its vertices such that, for each vertex v,the neighbors of v that occur later than v in the ordering form a clique. The prefixes of perfect eliminationorderings of a chordal graph form an antimatroid.[3]

1.3. PATHS AND BASIC WORDS 3

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A shelling sequence of a planar point set. The line segments show edges of the convex hulls after some of the points have beenremoved.

1.3 Paths and basic words

In the set theoretic axiomatization of an antimatroid there are certain special sets called paths that determine thewhole antimatroid, in the sense that the sets of the antimatroid are exactly the unions of paths. If S is any feasible setof the antimatroid, an element x that can be removed from S to form another feasible set is called an endpoint of S,and a feasible set that has only one endpoint is called a path of the antimatroid. The family of paths can be partiallyordered by set inclusion, forming the path poset of the antimatroid.For every feasible set S in the antimatroid, and every element x of S, one may find a path subset of S for which xis an endpoint: to do so, remove one at a time elements other than x until no such removal leaves a feasible subset.Therefore, each feasible set in an antimatroid is the union of its path subsets. If S is not a path, each subset in thisunion is a proper subset of S. But, if S is itself a path with endpoint x, each proper subset of S that belongs to theantimatroid excludes x. Therefore, the paths of an antimatroid are exactly the sets that do not equal the unions oftheir proper subsets in the antimatroid. Equivalently, a given family of sets P forms the set of paths of an antimatroidif and only if, for each S in P, the union of subsets of S in P has one fewer element than S itself. If so, F itself is thefamily of unions of subsets of P.In the formal language formalization of an antimatroid we may also identify a subset of words that determine the wholelanguage, the basic words. The longest strings in L are called basic words; each basic word forms a permutation ofthe whole alphabet. For instance, the basic words of a poset antimatroid are the linear extensions of the given partialorder. If B is the set of basic words, L can be defined from B as the set of prefixes of words in B. It is often convenientto define antimatroids from basic words in this way, but it is not straightforward to write an axiomatic definition of

4 CHAPTER 1. ANTIMATROID

antimatroids in terms of their basic words.

1.4 Convex geometries

See also: Convex set, Convex geometry and Closure operator

If F is the set system defining an antimatroid, with U equal to the union of the sets in F, then the family of sets

G = {U \ S | S ∈ F}

complementary to the sets in F is sometimes called a convex geometry, and the sets in G are called convex sets. Forinstance, in a shelling antimatroid, the convex sets are intersections of U with convex subsets of the Euclidean spaceinto which U is embedded.Complementarily to the properties of set systems that define antimatroids, the set system defining a convex geometryshould be closed under intersections, and for any set S in G that is not equal to U there must be an element x not in Sthat can be added to S to form another set in G.A convex geometry can also be defined in terms of a closure operator τ that maps any subset of U to its minimalclosed superset. To be a closure operator, τ should have the following properties:

• τ(∅) = ∅: the closure of the empty set is empty.

• Any set S is a subset of τ(S).

• If S is a subset of T, then τ(S) must be a subset of τ(T).

• For any set S, τ(S) = τ(τ(S)).

The family of closed sets resulting from a closure operation of this type is necessarily closed under intersections. Theclosure operators that define convex geometries also satisfy an additional anti-exchange axiom:

• If neither y nor z belong to τ(S), but z belongs to τ(S ∪ {y}), then y does not belong to τ(S ∪ {z}).

A closure operation satisfying this axiom is called an anti-exchange closure. If S is a closed set in an anti-exchangeclosure, then the anti-exchange axiom determines a partial order on the elements not belonging to S, where x ≤ y inthe partial order when x belongs to τ(S ∪ {y}). If x is a minimal element of this partial order, then S ∪ {x} is closed.That is, the family of closed sets of an anti-exchange closure has the property that for any set other than the universalset there is an element x that can be added to it to produce another closed set. This property is complementary tothe accessibility property of antimatroids, and the fact that intersections of closed sets are closed is complementaryto the property that unions of feasible sets in an antimatroid are feasible. Therefore, the complements of the closedsets of any anti-exchange closure form an antimatroid.[4]

1.5 Join-distributive lattices

Any two sets in an antimatroid have a unique least upper bound (their union) and a unique greatest lower bound(the union of the sets in the antimatroid that are contained in both of them). Therefore, the sets of an antimatroid,partially ordered by set inclusion, form a lattice. Various important features of an antimatroid can be interpreted inlattice-theoretic terms; for instance the paths of an antimatroid are the join-irreducible elements of the correspondinglattice, and the basic words of the antimatroid correspond to maximal chains in the lattice. The lattices that arise fromantimatroids in this way generalize the finite distributive lattices, and can be characterized in several different ways.

• The description originally considered by Dilworth (1940) concerns meet-irreducible elements of the lattice.For each element x of an antimatroid, there exists a unique maximal feasible set Sx that does not contain x (Sxis the union of all feasible sets not containing x). Sx is meet-irreducible, meaning that it is not the meet of any

1.6. SUPERSOLVABLE ANTIMATROIDS 5

two larger lattice elements: any larger feasible set, and any intersection of larger feasible sets, contains x and sodoes not equal Sx. Any element of any lattice can be decomposed as a meet of meet-irreducible sets, often inmultiple ways, but in the lattice corresponding to an antimatroid each element T has a unique minimal familyof meet-irreducible sets Sx whose meet is T ; this family consists of the sets Sx such that T ∪ {x} belongs to theantimatroid. That is, the lattice has unique meet-irreducible decompositions.

• A second characterization concerns the intervals in the lattice, the sublattices defined by a pair of lattice elementsx ≤ y and consisting of all lattice elements z with x ≤ z ≤ y. An interval is atomistic if every element in it is thejoin of atoms (the minimal elements above the bottom element x), and it is Boolean if it is isomorphic to thelattice of all subsets of a finite set. For an antimatroid, every interval that is atomistic is also boolean.

• Thirdly, the lattices arising from antimatroids are semimodular lattices, lattices that satisfy the upper semimod-ular law that for any two elements x and y, if y covers x ∧ y then x ∨ y covers x. Translating this conditioninto the sets of an antimatroid, if a set Y has only one element not belonging to X then that one element maybe added to X to form another set in the antimatroid. Additionally, the lattice of an antimatroid has the meet-semidistributive property: for all lattice elements x, y, and z, if x ∧ y and x ∧ z are both equal then they alsoequal x ∧ (y ∨ z). A semimodular and meet-semidistributive lattice is called a join-distributive lattice.

These three characterizations are equivalent: any lattice with unique meet-irreducible decompositions has booleanatomistic intervals and is join-distributive, any lattice with boolean atomistic intervals has unique meet-irreducibledecompositions and is join-distributive, and any join-distributive lattice has unique meet-irreducible decompositionsand boolean atomistic intervals.[5] Thus, we may refer to a lattice with any of these three properties as join-distributive.Any antimatroid gives rise to a finite join-distributive lattice, and any finite join-distributive lattice comes from anantimatroid in this way.[6] Another equivalent characterization of finite join-distributive lattices is that they are graded(any two maximal chains have the same length), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.[7] The antimatroid representing a finite join-distributive lattice can be recoveredfrom the lattice: the elements of the antimatroid can be taken to be the meet-irreducible elements of the lattice, andthe feasible set corresponding to any element x of the lattice consists of the set of meet-irreducible elements y suchthat y is not greater than or equal to x in the lattice.This representation of any finite join-distributive lattice as an accessible family of sets closed under unions (that is, asan antimatroid) may be viewed as an analogue of Birkhoff’s representation theorem under which any finite distributivelattice has a representation as a family of sets closed under unions and intersections.

1.6 Supersolvable antimatroids

Motivated by a problem of defining partial orders on the elements of a Coxeter group, Armstrong (2007) studied an-timatroids which are also supersolvable lattices. A supersolvable antimatroid is defined by a totally ordered collectionof elements, and a family of sets of these elements. The family must include the empty set. Additionally, it musthave the property that if two sets A and B belong to the family, the set-theoretic difference B \ A is nonempty, and xis the smallest element of B \ A, then A ∪ {x} also belongs to the family. As Armstrong observes, any family of setsof this type forms an antimatroid. Armstrong also provides a lattice-theoretic characterization of the antimatroidsthat this construction can form.

1.7 Join operation and convex dimension

If A and B are two antimatroids, both described as a family of sets, and if the maximal sets in A and B are equal, wecan form another antimatroid, the join of A and B, as follows:

A ∨B = {S ∪ T | S ∈ A ∧ T ∈ B}.

Note that this is a different operation than the join considered in the lattice-theoretic characterizations of antimatroids:it combines two antimatroids to form another antimatroid, rather than combining two sets in an antimatroid to formanother set. The family of all antimatroids that have a given maximal set forms a semilattice with this join operation.

6 CHAPTER 1. ANTIMATROID

Joins are closely related to a closure operation that maps formal languages to antimatroids, where the closure of alanguage L is the intersection of all antimatroids containing L as a sublanguage. This closure has as its feasible setsthe unions of prefixes of strings in L. In terms of this closure operation, the join is the closure of the union of thelanguages of A and B.Every antimatroid can be represented as a join of a family of chain antimatroids, or equivalently as the closure ofa set of basic words; the convex dimension of an antimatroid A is the minimum number of chain antimatroids (orequivalently the minimum number of basic words) in such a representation. If F is a family of chain antimatroidswhose basic words all belong to A, then F generates A if and only if the feasible sets of F include all paths of A. Thepaths of A belonging to a single chain antimatroid must form a chain in the path poset of A, so the convex dimensionof an antimatroid equals the minimum number of chains needed to cover the path poset, which by Dilworth’s theoremequals the width of the path poset.[8]

If one has a representation of an antimatroid as the closure of a set of d basic words, then this representation canbe used to map the feasible sets of the antimatroid into d-dimensional Euclidean space: assign one coordinate perbasic word w, and make the coordinate value of a feasible set S be the length of the longest prefix of w that is asubset of S. With this embedding, S is a subset of T if and only if the coordinates for S are all less than or equal tothe corresponding coordinates of T. Therefore, the order dimension of the inclusion ordering of the feasible sets isat most equal to the convex dimension of the antimatroid.[9] However, in general these two dimensions may be verydifferent: there exist antimatroids with order dimension three but with arbitrarily large convex dimension.

1.8 Enumeration

The number of possible antimatroids on a set of elements grows rapidly with the number of elements in the set. Forsets of one, two, three, etc. elements, the number of distinct antimatroids is

1, 3, 22, 485, 59386, 133059751, ... (sequence A119770 in OEIS).

1.9 Applications

Both the precedence and release time constraints in the standard notation for theoretic scheduling problems maybe modeled by antimatroids. Boyd & Faigle (1990) use antimatroids to generalize a greedy algorithm of EugeneLawler for optimally solving single-processor scheduling problems with precedence constraints in which the goal isto minimize the maximum penalty incurred by the late scheduling of a task.Glasserman & Yao (1994) use antimatroids to model the ordering of events in discrete event simulation systems.Parmar (2003) uses antimatroids to model progress towards a goal in artificial intelligence planning problems.In mathematical psychology, antimatroids have been used to describe feasible states of knowledge of a human learner.Each element of the antimatroid represents a concept that is to be understood by the learner, or a class of problems thathe or she might be able to solve correctly, and the sets of elements that form the antimatroid represent possible sets ofconcepts that could be understood by a single person. The axioms defining an antimatroid may be phrased informallyas stating that learning one concept can never prevent the learner from learning another concept, and that any feasiblestate of knowledge can be reached by learning a single concept at a time. The task of a knowledge assessment systemis to infer the set of concepts known by a given learner by analyzing his or her responses to a small and well-chosenset of problems. In this context antimatroids have also been called “learning spaces” and “well-graded knowledgespaces”.[10]

1.10 Notes

[1] Two early references are Edelman (1980) and Jamison (1980); Jamison was the first to use the term “antimatroid”.Monjardet (1985) surveys the history of rediscovery of antimatroids.

[2] Korte et al., Theorem 1.4.

1.11. REFERENCES 7

[3] Gordon (1997) describes several results related to antimatroids of this type, but these antimatroids were mentioned earliere.g. by Korte et al. Chandran et al. (2003) use the connection to antimatroids as part of an algorithm for efficiently listingall perfect elimination orderings of a given chordal graph.

[4] Korte et al., Theorem 1.1.

[5] Adaricheva, Gorbunov & Tumanov (2003), Theorems 1.7 and 1.9; Armstrong (2007), Theorem 2.7.

[6] Edelman (1980), Theorem 3.3; Armstrong (2007), Theorem 2.8.

[7] Monjardet (1985) credits a dual form of this characterization to several papers from the 1960s by S. P. Avann.

[8] Edelman & Saks (1988); Korte et al., Theorem 6.9.

[9] Korte et al., Corollary 6.10.

[10] Doignon & Falmagne (1999).

1.11 References• Adaricheva, K. V.; Gorbunov, V. A.; Tumanov, V. I. (2003), “Join-semidistributive lattices and convex ge-

ometries”, Advances in Mathematics 173 (1): 1–49, doi:10.1016/S0001-8708(02)00011-7.

• Armstrong, Drew (2007), The sorting order on a Coxeter group, arXiv:0712.1047.

• Birkhoff, Garrett; Bennett, M. K. (1985), “The convexity lattice of a poset”,Order 2 (3): 223–242, doi:10.1007/BF00333128.

• Björner, Anders; Ziegler, Günter M. (1992), “8 Introduction to greedoids”, in White, Neil, Matroid Appli-cations, Encyclopedia of Mathematics and its Applications 40, Cambridge: Cambridge University Press, pp.284–357, doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537

• Boyd, E. Andrew; Faigle, Ulrich (1990), “An algorithmic characterization of antimatroids”, Discrete AppliedMathematics 28 (3): 197–205, doi:10.1016/0166-218X(90)90002-T.

• Chandran, L. S.; Ibarra, L.; Ruskey, F.; Sawada, J. (2003), “Generating and characterizing the perfect elimina-tion orderings of a chordal graph” (PDF), Theoretical Computer Science 307 (2): 303–317, doi:10.1016/S0304-3975(03)00221-4.

• Dilworth, Robert P. (1940), “Lattices with unique irreducible decompositions”, Annals of Mathematics 41 (4):771–777, doi:10.2307/1968857, JSTOR 1968857.

• Doignon, Jean-Paul; Falmagne, Jean-Claude (1999), Knowledge Spaces, Springer-Verlag, ISBN 3-540-64501-2.

• Edelman, Paul H. (1980), “Meet-distributive lattices and the anti-exchange closure”, Algebra Universalis 10(1): 290–299, doi:10.1007/BF02482912.

• Edelman, Paul H.; Saks, Michael E. (1988), “Combinatorial representation and convex dimension of convexgeometries”, Order 5 (1): 23–32, doi:10.1007/BF00143895.

• Glasserman, Paul; Yao, David D. (1994), Monotone Structure in Discrete Event Systems, Wiley Series in Prob-ability and Statistics, Wiley Interscience, ISBN 978-0-471-58041-6.

• Gordon, Gary (1997), “A β invariant for greedoids and antimatroids”, Electronic Journal of Combinatorics 4(1): Research Paper 13, MR 1445628.

• Jamison, Robert (1980), “Copoints in antimatroids”, Proceedings of the Eleventh Southeastern Conference onCombinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1980), Vol. II, Con-gressus Numerantium 29, pp. 535–544, MR 608454.

• Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Springer-Verlag, pp. 19–43, ISBN3-540-18190-3.

• Monjardet, Bernard (1985), “A use for frequently rediscovering a concept”,Order 1 (4): 415–417, doi:10.1007/BF00582748.

• Parmar, Aarati (2003), “Some Mathematical Structures Underlying Efficient Planning”, AAAI Spring Sympo-sium on Logical Formalization of Commonsense Reasoning (PDF).

Chapter 2

Closure (mathematics)

For other uses, see Closure (disambiguation).

A set has closure under an operation if performance of that operation on members of the set always produces amember of the same set; in this case we also say that the set is closed under the operation. For example, the integersare closed under subtraction, but the positive integers are not: 1 and 2 are both positive integers, but the result ofsubtracting 2 from 1 is not a positive integer. Another example is the set containing only the number zero, which isclosed under addition, subtraction and multiplication.Similarly, a set is said to be closed under a collection of operations if it is closed under each of the operationsindividually.

2.1 Basic properties

A set that is closed under an operation or collection of operations is said to satisfy a closure property. Often aclosure property is introduced as an axiom, which is then usually called the axiom of closure. Modern set-theoreticdefinitions usually define operations as maps between sets, so adding closure to a structure as an axiom is superfluous;however in practice operations are often defined initially on a superset of the set in question and a closure proof isrequired to establish that the operation applied to pairs from that set only produces members of that set. For example,the set of even integers is closed under addition, but the set of odd integers is not.When a set S is not closed under some operations, one can usually find the smallest set containing S that is closed.This smallest closed set is called the closure of S (with respect to these operations). For example, the closure undersubtraction of the set of natural numbers, viewed as a subset of the real numbers, is the set of integers. An importantexample is that of topological closure. The notion of closure is generalized by Galois connection, and further bymonads.The set S must be a subset of a closed set in order for the closure operator to be defined. In the preceding example, itis important that the reals are closed under subtraction; in the domain of the natural numbers subtraction is not alwaysdefined.The two uses of the word “closure” should not be confused. The former usage refers to the property of being closed,and the latter refers to the smallest closed set containing one that may not be closed. In short, the closure of a setsatisfies a closure property.

2.2 Closed sets

A set is closed under an operation if that operation returns a member of the set when evaluated on members of theset. Sometimes the requirement that the operation be valued in a set is explicitly stated, in which case it is known asthe axiom of closure. For example, one may define a group as a set with a binary product operator obeying severalaxioms, including an axiom that the product of any two elements of the group is again an element. However themodern definition of an operation makes this axiom superfluous; an n-ary operation on S is just a subset of Sn+1. By

8

2.3. P CLOSURES OF BINARY RELATIONS 9

its very definition, an operator on a set cannot have values outside the set.Nevertheless, the closure property of an operator on a set still has some utility. Closure on a set does not necessarilyimply closure on all subsets. Thus a subgroup of a group is a subset on which the binary product and the unaryoperation of inversion satisfy the closure axiom.An operation of a different sort is that of finding the limit points of a subset of a topological space (if the space isfirst-countable, it suffices to restrict consideration to the limits of sequences but in general one must consider at leastlimits of nets). A set that is closed under this operation is usually just referred to as a closed set in the context oftopology. Without any further qualification, the phrase usually means closed in this sense. Closed intervals like [1,2]= {x : 1 ≤ x ≤ 2} are closed in this sense.A partially ordered set is downward closed (and also called a lower set) if for every element of the set all smallerelements are also in it; this applies for example for the real intervals (−∞, p) and (−∞, p], and for an ordinal numberp represented as interval [ 0, p); every downward closed set of ordinal numbers is itself an ordinal number.Upward closed and upper set are defined similarly.

2.3 P closures of binary relations

The notion of a closure can be applied for an arbitrary binary relation R ⊆ S×S, and an arbitrary property P in thefollowing way: the P closure of R is the least relation Q ⊆ S×S that contains R (i.e. R ⊆ Q) and for which propertyP holds (i.e. P(Q) is true). For instance, one can define the symmetric closure as the least symmetric relationcontaining R. This generalization is often encountered in the theory of rewriting systems, where one often uses more“wordy” notions such as the reflexive transitive closure R*—the smallest preorder containing R, or the reflexivetransitive symmetric closure R≡—the smallest equivalence relation containing R, and therefore also known as theequivalence closure. When considering a particular term algebra, an equivalence relation that is compatible withall operations of the algebra [note 1] is called a congruence relation. The congruence closure of R is defined as thesmallest congruence relation containing R.For arbitrary P and R, the P closure of R need not exist. In the above examples, these exist because reflexivity,transitivity and symmetry are closed under arbitrary intersections. In such cases, the P closure can be directly definedas the intersection of all sets with property P containing R.[1]

Some important particular closures can be constructively obtained as follows:

• clᵣₑ (R) = R ∪ { ⟨x,x⟩ : x ∈ S } is the reflexive closure of R,

• cl (R) = R ∪ { ⟨y,x⟩ : ⟨x,y⟩ ∈ R } is its symmetry closure,

• cl ᵣ (R) = R ∪ { ⟨x1,xn⟩ : n >1 ∧ ⟨x1,x2⟩, ..., ⟨xn−₁,xn⟩ ∈ R } is its transitive closure,

• clₑ ,Σ(R) = R ∪ { ⟨f(x1,…,xi−₁,xi,xi₊₁,…,xn), f(x1,…,xi−₁,y,xi₊₁,…,xn)⟩ : ⟨xi,y⟩ ∈ R ∧ f ∈ Σ n-ary ∧ 1 ≤ i ≤n ∧ x1,...,xn ∈ S } is its embedding closure with respect to a given set Σ of operations on S, each with a fixedarity.

The relation R is said to have closure under some clₓₓₓ, if R = clₓₓₓ(R); for example R is called symmetric if R =cl (R).Any of these four closures preserves symmetry, i.e., if R is symmetric, so is any clₓₓₓ(R). [note 2] Similarly, all fourpreserve reflexivity. Moreover, cl ᵣ preserves closure under clₑ ,Σ for arbitrary Σ. As a consequence, the equivalenceclosure of an arbitrary binary relationR can be obtained as cl ᵣ (cl (clᵣₑ (R))), and the congruence closure with respectto some Σ can be obtained as cl ᵣ (clₑ ,Σ(cl (clᵣₑ (R)))). In the latter case, the nesting order does matter; e.g. if S isthe set of terms over Σ = { a, b, c, f } and R = { ⟨a,b⟩, ⟨f(b),c⟩ }, then the pair ⟨f(a),c⟩ is contained in the congruenceclosure cl ᵣ (clₑ ,Σ(cl (clᵣₑ (R)))) of R, but not in the relation clₑ ,Σ(cl ᵣ (cl (clᵣₑ (R)))).

2.4 Closure operator

Main article: closure operator

10 CHAPTER 2. CLOSURE (MATHEMATICS)

Given an operation on a set X, one can define the closure C(S) of a subset S of X to be the smallest subset closedunder that operation that contains S as a subset, if any such subsets exist. Consequently, C(S) is the intersection ofall closed sets containing S. For example, the closure of a subset of a group is the subgroup generated by that set.The closure of sets with respect to some operation defines a closure operator on the subsets of X. The closed sets canbe determined from the closure operator; a set is closed if it is equal to its own closure. Typical structural propertiesof all closure operations are: [2]

• The closure is increasing or extensive: the closure of an object contains the object.

• The closure is idempotent: the closure of the closure equals the closure.

• The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closureof some object.These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of allsubsets of a set.If X is contained in a set closed under the operation then every subset of X has a closure.

2.5 Examples• In topology and related branches, the relevant operation is taking limits. The topological closure of a set is the

corresponding closure operator. The Kuratowski closure axioms characterize this operator.

• In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of thevector space that includes X and is closed under the operation of linear combination. This subset is a subspace.

• In matroid theory, the closure of X is the largest superset of X that has the same rank as X.

• In set theory, the transitive closure of a set.

• In set theory, the transitive closure of a binary relation.

• In algebra, the algebraic closure of a field.

• In commutative algebra, closure operations for ideals, as integral closure and tight closure.

• In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.

• In the theory of formal languages, the Kleene closure of a language can be described as the set of strings thatcan be made by concatenating zero or more strings from that language.

• In group theory, the conjugate closure or normal closure of a set of group elements is the smallest normalsubgroup containing the set.

• In mathematical analysis and in probability theory, the closure of a collection of subsets of X under countablymany set operations is called the σ-algebra generated by the collection.

2.6 See also• Open set

• Clopen set

2.7 Notes[1] that is, such that e.g. xRy implies f(x,x2) R f(y,x2) and f(x1,x) R f(x1,y) for any binary operation f and arbitrary x1,x2 ∈ S

[2] formally: if R = cl (R), then clₓₓₓ(R) = cl (clₓₓₓ(R))

2.8. REFERENCES 11

2.8 References[1] Baader, Franz; Nipkow, Tobias (1998). Term Rewriting and All That. Cambridge University Press. pp. 8–9.

[2] Birkhoff, Garrett (1967). Lattice Theory. Colloquium Publications 25. Am. Math. Soc. p. 111.

Chapter 3

Cutting sequence

The Fibonacci word is an example of a Sturmian word. The start of the cutting sequence shown here illustrates the start of the word0100101001.

In digital geometry, a cutting sequence is a sequence of symbols whose elements correspond to the individual gridlines crossed (“cut”) as a curve crosses a square grid.[1]

Sturmian words are a special case of cutting sequences where the curves are straight lines of irrational slope.[2]

3.1 References[1] Monteil, T. (2011). “The complexity of tangent words”. Electronic Proceedings in Theoretical Computer Science 63: 152.

doi:10.4204/EPTCS.63.21.

[2] Pytheas Fogg (2002) p.152

• Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathemat-ics 1794. Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. Berlin: Springer-Verlag.ISBN 3-540-44141-7. Zbl 1014.11015.

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Chapter 4

DIMACS

TheCenter for DiscreteMathematics and Theoretical Computer Science (DIMACS) is a collaboration betweenRutgers University, Princeton University, and the research firms AT&T, Bell Labs, Applied Communication Sciences,and NEC. It was founded in 1989 with money from the National Science Foundation. Its offices are located on theRutgers campus, and 250 members from the six institutions form its permanent members.DIMACS is devoted to both theoretical development and practical applications of discrete mathematics and theoret-ical computer science. It engages in a wide variety of evangelism including encouraging, inspiring, and facilitatingresearchers in these subject areas, and sponsoring conferences and workshops.Fundamental research in discrete mathematics has applications in diverse fields including Cryptology, Engineering,Networking, and Management Decision Support.The current director of DIMACS is Rebecca Wright. Past directors were Fred S. Roberts, Daniel Gorenstein andAndrás Hajnal.[1]

4.1 The DIMACS Challenges

DIMACS sponsors implementation challenges to determine practical algorithm performance on problems of interest.There have been eleven DIMACS challenges so far.

• 1990-1991: Network Flows and Matching

• 1992-1992: NP-Hard Problems: Max Clique, Graph Coloring, and SAT

• 1993-1994: Parallel Algorithms for Combinatorial Problems

• 1994-1995: Computational Biology: Fragment Assembly and Genome Rearrangement

• 1995-1996: Priority Queues, Dictionaries, and Multidimensional Point Sets

• 1998-1998: Near Neighbor Searches

• 2000-2000: Semidefinite and Related Optimization Problems

• 2001-2001: The Traveling Salesman Problem

• 2005-2005: The Shortest Path Problem

• 2011-2012: Graph Partitioning and Graph Clustering

• 2013-2014: Steiner Tree Problems

4.2 References[1] A history of mathematics at Rutgers, Charles Weibel.

13

14 CHAPTER 4. DIMACS

4.3 External links• DIMACS Website

Chapter 5

Discrete mathematics

For the mathematics journal, see Discrete Mathematics (journal).Discretemathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

1

23

54

6

Graphs like this are among the objects studied by discrete mathematics, for their interesting mathematical properties, their usefulnessas models of real-world problems, and their importance in developing computer algorithms.

In contrast to real numbers that have the property of varying “smoothly”, the objects studied in discrete mathematics– such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separatedvalues.[2] Discrete mathematics therefore excludes topics in “continuous mathematics” such as calculus and analysis.Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterizedas the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of thenatural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term“discrete mathematics.”[4] Indeed, discrete mathematics is described less by what is included than by what is excluded:continuously varying quantities and related notions.The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimesapplied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant tobusiness.

15

16 CHAPTER 5. DISCRETE MATHEMATICS

Research in discrete mathematics increased in the latter half of the twentieth century partly due to the developmentof digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations fromdiscrete mathematics are useful in studying and describing objects and problems in branches of computer science,such as computer algorithms, programming languages, cryptography, automated theorem proving, and software de-velopment. Conversely, computer implementations are significant in applying ideas from discrete mathematics toreal-world problems, such as in operations research.Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuousmathematics are often employed as well.In the university curricula, “Discrete Mathematics” appeared in the 1980s, initially as a computer science supportcourse; its contents was somewhat haphazard at the time. The curriculum has thereafter developed in conjunction toefforts by ACM and MAA into a course that is basically intended to develop mathematical maturity in freshmen; assuch it is nowadays a prerequisite for mathematics majors in some universities as well.[5][6] Some high-school-leveldiscrete mathematics textbooks have appeared as well.[7] At this level, discrete mathematics it is sometimes seen apreparatory course, not unlike precalculus in this respect.[8]

The Fulkerson Prize is awarded for outstanding papers in discrete mathematics.

5.1 Grand challenges, past and present

The history of discrete mathematics has involved a number of challenging problems which have focused attentionwithin areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem,first stated in 1852, but not proved until 1976 (by Kenneth Appel and Wolfgang Haken, using substantial computerassistance).[9]

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axiomsof arithmetic are consistent. Gödel’s second incompleteness theorem, proved in 1931, showed that this was notpossible – at least not within arithmetic itself. Hilbert’s tenth problem was to determine whether a given polynomialDiophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that thiscould not be done.The need to break German codes in World War II led to advances in cryptography and theoretical computer science,with the first programmable digital electronic computer being developed at England’s Bletchley Park with the guid-ance of Alan Turing and his seminal work, On Computable Numbers.[10] At the same time, military requirementsmotivated advances in operations research. The Cold War meant that cryptography remained important, with fun-damental advances such as public-key cryptography being developed in the following decades. Operations researchremained important as a tool in business and project management, with the critical path method being developedin the 1950s. The telecommunication industry has also motivated advances in discrete mathematics, particularlyin graph theory and information theory. Formal verification of statements in logic has been necessary for softwaredevelopment of safety-critical systems, and advances in automated theorem proving have been driven by this need.Computational geometry has been an important part of the computer graphics incorporated into modern video gamesand computer-aided design tools.Several fields of discrete mathematics, particularly theoretical computer science, graph theory, and combinatorics,are important in addressing the challenging bioinformatics problems associated with understanding the tree of life.[11]

Currently, one of the most famous open problems in theoretical computer science is the P = NP problem, whichinvolves the relationship between the complexity classes P and NP. The Clay Mathematics Institute has offered a $1million USD prize for the first correct proof, along with prizes for six other mathematical problems.[12]

5.2 Topics in discrete mathematics

5.2.1 Theoretical computer science

Main article: Theoretical computer scienceTheoretical computer science includes areas of discrete mathematics relevant to computing. It draws heavily on

graph theory and mathematical logic. Included within theoretical computer science is the study of algorithms forcomputing mathematical results. Computability studies what can be computed in principle, and has close ties to logic,

5.2. TOPICS IN DISCRETE MATHEMATICS 17

Much research in graph theory was motivated by attempts to prove that all maps, like this one, could be colored using only four colorsso that no areas of the same color touched. Kenneth Appel and Wolfgang Haken proved this in 1976.[9]

while complexity studies the time taken by computations. Automata theory and formal language theory are closelyrelated to computability. Petri nets and process algebras are used to model computer systems, and methods fromdiscrete mathematics are used in analyzing VLSI electronic circuits. Computational geometry applies algorithmsto geometrical problems, while computer image analysis applies them to representations of images. Theoreticalcomputer science also includes the study of various continuous computational topics.

18 CHAPTER 5. DISCRETE MATHEMATICS

Complexity studies the time taken by algorithms, such as this sorting routine.

5.2.2 Information theory

Main article: Information theoryInformation theory involves the quantification of information. Closely related is coding theory which is used to designefficient and reliable data transmission and storage methods. Information theory also includes continuous topics suchas: analog signals, analog coding, analog encryption.

5.2.3 Logic

Main article: Mathematical logic

Logic is the study of the principles of valid reasoning and inference, as well as of consistency, soundness, andcompleteness. For example, in most systems of logic (but not in intuitionistic logic) Peirce’s law (((P→Q)→P)→P)is a theorem. For classical logic, it can be easily verified with a truth table. The study of mathematical proof is par-ticularly important in logic, and has applications to automated theorem proving and formal verification of software.Logical formulas are discrete structures, as are proofs, which form finite trees[13] or, more generally, directed acyclicgraph structures[14][15] (with each inference step combining one or more premise branches to give a single conclusion).The truth values of logical formulas usually form a finite set, generally restricted to two values: true and false, butlogic can also be continuous-valued, e.g., fuzzy logic. Concepts such as infinite proof trees or infinite derivation treeshave also been studied,[16] e.g. infinitary logic.

5.2.4 Set theory

Main article: Set theory

5.2. TOPICS IN DISCRETE MATHEMATICS 19

101 0111110 1001110 1011110 1001111 0000110 0101110 0100110 1001110 0001

WikipediaThe ASCII codes for the word “Wikipedia”, given here in binary, provide a way of representing the word in information theory, aswell as for information-processing algorithms.

Set theory is the branch of mathematics that studies sets, which are collections of objects, such as {blue, white, red}or the (infinite) set of all prime numbers. Partially ordered sets and sets with other relations have applications inseveral areas.In discrete mathematics, countable sets (including finite sets) are the main focus. The beginning of set theory as abranch of mathematics is usually marked by Georg Cantor's work distinguishing between different kinds of infiniteset, motivated by the study of trigonometric series, and further development of the theory of infinite sets is outsidethe scope of discrete mathematics. Indeed, contemporary work in descriptive set theory makes extensive use oftraditional continuous mathematics.

20 CHAPTER 5. DISCRETE MATHEMATICS

5.2.5 Combinatorics

Main article: Combinatorics

Combinatorics studies the way in which discrete structures can be combined or arranged. Enumerative combinatoricsconcentrates on counting the number of certain combinatorial objects - e.g. the twelvefold way provides a unifiedframework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumer-ation (i.e., determining the number) of combinatorial structures using tools from complex analysis and probabilitytheory. In contrast with enumerative combinatorics which uses explicit combinatorial formulae and generating func-tions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Design theory is a studyof combinatorial designs, which are collections of subsets with certain intersection properties. Partition theory studiesvarious enumeration and asymptotic problems related to integer partitions, and is closely related to q-series, specialfunctions and orthogonal polynomials. Originally a part of number theory and analysis, partition theory is now con-sidered a part of combinatorics or an independent field. Order theory is the study of partially ordered sets, both finiteand infinite.

5.2.6 Graph theory

Main article: Graph theoryGraph theory, the study of graphs and networks, is often considered part of combinatorics, but has grown large enough

Graph theory has close links to group theory. This truncated tetrahedron graph is related to the alternating group A4.

and distinct enough, with its own kind of problems, to be regarded as a subject in its own right.[17] Graphs are one ofthe prime objects of study in discrete mathematics. They are among the most ubiquitous models of both natural andhuman-made structures. They can model many types of relations and process dynamics in physical, biological andsocial systems. In computer science, they can represent networks of communication, data organization, computationaldevices, the flow of computation, etc. In mathematics, they are useful in geometry and certain parts of topology, e.g.knot theory. Algebraic graph theory has close links with group theory. There are also continuous graphs, however

5.2. TOPICS IN DISCRETE MATHEMATICS 21

for the most part research in graph theory falls within the domain of discrete mathematics.

5.2.7 Probability

Main article: Discrete probability theory

Discrete probability theory deals with events that occur in countable sample spaces. For example, count observationssuch as the numbers of birds in flocks comprise only natural number values {0, 1, 2, ...}. On the other hand, continuousobservations such as the weights of birds comprise real number values and would typically be modeled by a continuousprobability distribution such as the normal. Discrete probability distributions can be used to approximate continuousones and vice versa. For highly constrained situations such as throwing dice or experiments with decks of cards,calculating the probability of events is basically enumerative combinatorics.

5.2.8 Number theory

The Ulam spiral of numbers, with black pixels showing prime numbers. This diagram hints at patterns in the distribution of primenumbers.

Main article: Number theory

22 CHAPTER 5. DISCRETE MATHEMATICS

Number theory is concerned with the properties of numbers in general, particularly integers. It has applications tocryptography, cryptanalysis, and cryptology, particularly with regard to modular arithmetic, diophantine equations,linear and quadratic congruences, prime numbers and primality testing. Other discrete aspects of number theoryinclude geometry of numbers. In analytic number theory, techniques from continuous mathematics are also used.Topics that go beyond discrete objects include transcendental numbers, diophantine approximation, p-adic analysisand function fields.

5.2.9 Algebra

Main article: Abstract algebra

Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: booleanalgebra used in logic gates and programming; relational algebra used in databases; discrete and finite versions ofgroups, rings and fields are important in algebraic coding theory; discrete semigroups and monoids appear in thetheory of formal languages.

5.2.10 Calculus of finite differences, discrete calculus or discrete analysis

Main article: finite difference

A function defined on an interval of the integers is usually called a sequence. A sequence could be a finite sequencefrom a data source or an infinite sequence from a discrete dynamical system. Such a discrete function could be definedexplicitly by a list (if its domain is finite), or by a formula for its general term, or it could be given implicitly by arecurrence relation or difference equation. Difference equations are similar to a differential equations, but replacedifferentiation by taking the difference between adjacent terms; they can be used to approximate differential equationsor (more often) studied in their own right. Many questions and methods concerning differential equations havecounterparts for difference equations. For instance where there are integral transforms in harmonic analysis forstudying continuous functions or analog signals, there are discrete transforms for discrete functions or digital signals.As well as the discrete metric there are more general discrete or finite metric spaces and finite topological spaces.

5.2.11 Geometry

Main articles: discrete geometry and computational geometry

Discrete geometry and combinatorial geometry are about combinatorial properties of discrete collections of geomet-rical objects. A long-standing topic in discrete geometry is tiling of the plane. Computational geometry appliesalgorithms to geometrical problems.

5.2.12 Topology

Although topology is the field of mathematics that formalizes and generalizes the intuitive notion of “continuous defor-mation” of objects, it gives rise to many discrete topics; this can be attributed in part to the focus on topological invari-ants, which themselves usually take discrete values. See combinatorial topology, topological graph theory, topologicalcombinatorics, computational topology, discrete topological space, finite topological space, topology (chemistry).

5.2.13 Operations research

Main article: Operations researchOperations research provides techniques for solving practical problems in business and other fields — problems suchas allocating resources to maximize profit, or scheduling project activities to minimize risk. Operations researchtechniques include linear programming and other areas of optimization, queuing theory, scheduling theory, network

5.2. TOPICS IN DISCRETE MATHEMATICS 23

Computational geometry applies computer algorithms to representations of geometrical objects.

theory. Operations research also includes continuous topics such as continuous-time Markov process, continuous-time martingales, process optimization, and continuous and hybrid control theory.

5.2.14 Game theory, decision theory, utility theory, social choice theory

Decision theory is concerned with identifying the values, uncertainties and other issues relevant in a given decision,its rationality, and the resulting optimal decision.Utility theory is about measures of the relative economic satisfaction from, or desirability of, consumption of variousgoods and services.Social choice theory is about voting. A more puzzle-based approach to voting is ballot theory.Game theory deals with situations where success depends on the choices of others, which makes choosing the bestcourse of action more complex. There are even continuous games, see differential game. Topics include auctiontheory and fair division.

24 CHAPTER 5. DISCRETE MATHEMATICS

PERT charts like this provide a business management technique based on graph theory.

5.2.15 Discretization

Main article: Discretization

Discretization concerns the process of transferring continuous models and equations into discrete counterparts, oftenfor the purposes of making calculations easier by using approximations. Numerical analysis provides an importantexample.

5.2.16 Discrete analogues of continuous mathematics

There are many concepts in continuous mathematics which have discrete versions, such as discrete calculus, discreteprobability distributions, discrete Fourier transforms, discrete geometry, discrete logarithms, discrete differentialgeometry, discrete exterior calculus, discrete Morse theory, difference equations, discrete dynamical systems, anddiscrete vector measures.In applied mathematics, discrete modelling is the discrete analogue of continuous modelling. In discrete modelling,discrete formulae are fit to data. A common method in this form of modelling is to use recurrence relation.In algebraic geometry, the concept of a curve can be extended to discrete geometries by taking the spectra ofpolynomial rings over finite fields to be models of the affine spaces over that field, and letting subvarieties or spectra ofother rings provide the curves that lie in that space. Although the space in which the curves appear has a finite numberof points, the curves are not so much sets of points as analogues of curves in continuous settings. For example, everypoint of the form V (x− c) ⊂ SpecK[x] = A1 for K a field can be studied either as SpecK[x]/(x− c) ∼= SpecK, a point, or as the spectrum SpecK[x](x−c) of the local ring at (x-c), a point together with a neighborhood aroundit. Algebraic varieties also have a well-defined notion of tangent space called the Zariski tangent space, making manyfeatures of calculus applicable even in finite settings.

5.2.17 Hybrid discrete and continuous mathematics

The time scale calculus is a unification of the theory of difference equations with that of differential equations, whichhas applications to fields requiring simultaneous modelling of discrete and continuous data. Another way of modelingsuch a situation is the notion of hybrid dynamical system.

5.3. SEE ALSO 25

5.3 See also• Outline of discrete mathematics

• CyberChase, a show that teaches Discrete Mathematics to children

5.4 References[1] Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008.

[2] Weisstein, Eric W., “Discrete mathematics”, MathWorld.

[3] Biggs, Norman L. (2002), Discrete mathematics, Oxford Science Publications (2nd ed.), New York: The Clarendon PressOxford University Press, p. 89, ISBN 9780198507178, MR 1078626, Discrete Mathematics is the branch of Mathematicsin which we deal with questions involving finite or countably infinite sets.

[4] Brian Hopkins, Resources for Teaching Discrete Mathematics, Mathematical Association of America, 2008.

[5] Ken Levasseur; Al Doerr. Applied Discrete Structures. p. 8.

[6] Albert Geoffrey Howson, ed. (1988). Mathematics as a Service Subject. Cambridge University Press. pp. 77–78. ISBN978-0-521-35395-3.

[7] Joseph G. Rosenstein. Discrete Mathematics in the Schools. American Mathematical Soc. p. 323. ISBN 978-0-8218-8578-9.

[8] http://ucsmp.uchicago.edu/secondary/curriculum/precalculus-discrete/

[9] Wilson, Robin (2002). Four Colors Suffice. London: Penguin Books. ISBN 978-0-691-11533-7.

[10] Hodges, Andrew. Alan Turing: the enigma. Random House, 1992.

[11] Trevor R. Hodkinson; John A. N. Parnell (2007). Reconstruction the Tree of Life: Taxonomy And Systematics of Large AndSpecies Rich Taxa. CRC PressINC. p. 97. ISBN 978-0-8493-9579-6.

[12] “Millennium Prize Problems”. 2000-05-24. Retrieved 2008-01-12.

[13] A. S. Troelstra; H. Schwichtenberg (2000-07-27). Basic Proof Theory. Cambridge University Press. p. 186. ISBN978-0-521-77911-1.

[14] Samuel R. Buss (1998). Handbook of Proof Theory. Elsevier. p. 13. ISBN 978-0-444-89840-1.

[15] Franz Baader; Gerhard Brewka; Thomas Eiter (2001-10-16). KI 2001: Advances in Artificial Intelligence: Joint Ger-man/Austrian Conference on AI, Vienna, Austria, September 19-21, 2001. Proceedings. Springer. p. 325. ISBN 978-3-540-42612-7.

[16] Brotherston, J.; Bornat, R.; Calcagno, C. (January 2008). “Cyclic proofs of program termination in separation logic”. ACMSIGPLAN Notices 43 (1). CiteSeerX: 10 .1 .1 .111 .1105.

[17] Graphs on Surfaces, Bojan Mohar and Carsten Thomassen, Johns Hopkins University press, 2001

5.5 Further reading• Norman L. Biggs (2002-12-19). Discrete Mathematics. Oxford University Press. ISBN 978-0-19-850717-8.

• John Dwyer (2010). An Introduction to Discrete Mathematics for Business & Computing. ISBN 978-1-907934-00-1.

• Susanna S. Epp (2010-08-04). Discrete Mathematics With Applications. Thomson Brooks/Cole. ISBN 978-0-495-39132-6.

• Ronald Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics.

• Ralph P. Grimaldi (2004). Discrete and CombinatorialMathematics: An Applied Introduction. Addison Wesley.ISBN 978-0-201-72634-3.

26 CHAPTER 5. DISCRETE MATHEMATICS

• Donald E. Knuth (2011-03-03). The Art of Computer Programming, Volumes 1-4a Boxed Set. Addison-WesleyProfessional. ISBN 978-0-321-75104-1.

• Jiří Matoušek; Jaroslav Nešetřil (1998). Discrete Mathematics. Oxford University Press. ISBN 978-0-19-850208-1.

• Obrenic, Bojana (2003-01-29). Practice Problems in Discrete Mathematics. Prentice Hall. ISBN 978-0-13-045803-2.

• Kenneth H. Rosen; John G. Michaels (2000). Hand Book of Discrete and Combinatorial Mathematics. CRCPressI Llc. ISBN 978-0-8493-0149-0.

• Kenneth H. Rosen (2007). Discrete Mathematics: And Its Applications. McGraw-Hill College. ISBN 978-0-07-288008-3.

• Andrew Simpson (2002). Discrete Mathematics by Example. McGraw-Hill Incorporated. ISBN 978-0-07-709840-7.

• Veerarajan, T.(2007), Discrete mathematics with graph theory and combinatorics, Tata Mcgraw Hill

5.6 External links• Discrete mathematics at the utk.edu Mathematics Archives, providiing links to syllabi, tutorials, programs, etc.

• Iowa Central: Electrical Technologies Program Discrete mathematics for Electrical engineering.

Chapter 6

Discrete Mathematics (journal)

For the area of mathematics, see Discrete mathematics.

Discrete Mathematics is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics,combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-HollandPublishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition,the journal publishes a number of special issues each year dedicated to a particular topic. Although originally itpublished articles in French and German, it now allows only English language articles. The editor-in-chief is DouglasWest (University of Illinois, Urbana).

6.1 History

The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on topublish a total of 84 papers in the journal.

6.2 Abstracting and indexing

The journal is abstracted and indexed in:

• ACM Computing Reviews

• Cambridge Scientific Abstracts

• Current Contents/Physics, Chemical, & Earth Sciences

• International Abstracts in Operations Research

• Mathematical Reviews

• PASCAL

• Science Citation Index

• Zentralblatt MATH

• Scopus

According to the Journal Citation Reports, the journal has a 2012 impact factor of 0.578.[1]

27

28 CHAPTER 6. DISCRETE MATHEMATICS (JOURNAL)

6.3 Notable publications• The 1972 paper by László Lovász on the study of perfect graphs (Lovász, László (1972). “Normal hypergraphs

and the perfect graph conjecture”. Discrete Mathematics 2 (3): 253–267. doi:10.1016/0012-365X(72)90006-4.)

• The 1973 short note “Acyclic orientations of graphs” by Richard Stanley on the study of the chromatic poly-nomial and its generalizations (Stanley, R. P. (1973). “Acyclic orientations of graphs”. Disc. Math. 5 (2):171–178. doi:10.1016/0012-365X(73)90108-8.)

• Václav Chvátal introduced graph toughness in 1973 (Chvátal, Václav (1973). “Tough graphs and Hamiltoniancircuits”. Discrete Mathematics 5 (3): 215–228. doi:10.1016/0012-365X(73)90138-6. MR 0316301.)

• The 1975 paper by László Lovász on the linear programming relaxation for the set cover problem.

• The 1980 paper by Philippe Flajolet on the combinatorics of continued fractions. (P. Flajolet, Combinatorialaspects of continued fractions, Discrete Math. 32 (1980), 125–161.)

• The 1985 paper by Bressoud and Zeilberger proved Andrews's q-Dyson conjecture (Zeilberger, Doron; Bressoud,David M. (1985). “A proof of Andrews’ q-Dyson conjecture”. Discrete Mathematics 54 (2): 201–224.doi:10.1016/0012-365X(85)90081-0. ISSN 0012-365X. MR 791661.)

6.4 References[1] “Discrete Mathematics”. 2012 Journal Citation Reports. Web of Science (Science ed.). Thomson Reuters. 2013.

6.5 External links• Official website

Chapter 7

Formal language

This article is about a technical term in mathematics and computer science. For related studies about natural lan-guages, see Formal semantics (linguistics). For formal modes of speech in natural languages, see Register (sociolin-guistics).

In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols that may be

S

NP

N'

AdjP

Adj'

Adj

Colorless

N'

N

ideas

N'

AdjP

Adj'

Adj

green

VP

V'

V'

V

sleep

AdvP

Adv'

Adv

furiously

Structure of a syntactically well-formed, although nonsensical English sentence (historical example from Chomsky 1957).

constrained by rules that are specific to it.The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language maybe formed; frequently it is required to be finite.[1] The strings formed from this alphabet are called words, and the

29

30 CHAPTER 7. FORMAL LANGUAGE

words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas.A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar,also called its formation rule.The field of formal language theory studies primarily the purely syntactical aspects of such languages—that is,their internal structural patterns. Formal language theory sprang out of linguistics, as a way of understanding thesyntactic regularities of natural languages. In computer science, formal languages are used among others as thebasis for defining the grammar of programming languages and formalized versions of subsets of natural languagesin which the words of the language represent concepts that are associated with particular meanings or semantics. Incomputational complexity theory, decision problems are typically defined as formal languages, and complexity classesare defined as the sets of the formal languages that can be parsed by machines with limited computational power. Inlogic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems,and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulationof formal languages in this way.

7.1 History

The first formal language is thought be the one used by Gottlob Frege in his Begriffsschrift (1879), literally meaning“concept writing”, and which Frege described as a “formal language of pure thought.”[2]

Axel Thue's early Semi-Thue system which can be used for rewriting strings was influential on formal grammars.

7.2 Words over an alphabet

An alphabet, in the context of formal languages, can be any set, although it often makes sense to use an alphabet in theusual sense of the word, or more generally a character set such as ASCII or Unicode. Alphabets can also be infinite;e.g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses,contains infinitely many elements x0, x1, x2, … that play the role of variables. The elements of an alphabet are calledits letters.A word over an alphabet can be any finite sequence, or string, of characters or letters, which sometimes may includespaces, and are separated by specified word separation characters. The set of all words over an alphabet Σ is usuallydenoted by Σ* (using the Kleene star). The length of a word is the number of characters or letters it is composedof. For any alphabet there is only one word of length 0, the empty word, which is often denoted by e, ε or λ. Byconcatenation one can combine two words to form a new word, whose length is the sum of the lengths of the originalwords. The result of concatenating a word with the empty word is the original word.In some applications, especially in logic, the alphabet is also known as the vocabulary and words are known asformulas or sentences; this breaks the letter/word metaphor and replaces it by a word/sentence metaphor.

7.3 Definition

A formal language L over an alphabet Σ is a subset of Σ*, that is, a set of words over that alphabet. Sometimesthe sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of'well-formed expressions’.In computer science and mathematics, which do not usually deal with natural languages, the adjective “formal” isoften omitted as redundant.While formal language theory usually concerns itself with formal languages that are described by some syntacticalrules, the actual definition of the concept “formal language” is only as above: a (possibly infinite) set of finite-lengthstrings composed from a given alphabet, no more nor less. In practice, there are many languages that can be describedby rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to theintuitive concept of a “language,” one described by syntactic rules. By an abuse of the definition, a particular formallanguage is often thought of as being equipped with a formal grammar that describes it.

7.4. EXAMPLES 31

7.4 Examples

The following rules describe a formal language L over the alphabet Σ = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, = }:

• Every nonempty string that does not contain "+" or "=" and does not start with “0” is in L.

• The string “0” is in L.

• A string containing "=" is in L if and only if there is exactly one "=", and it separates two valid strings of L.

• A string containing "+" but not "=" is in L if and only if every "+" in the string separates two valid strings of L.

• No string is in L other than those implied by the previous rules.

Under these rules, the string “23+4=555” is in L, but the string "=234=+" is not. This formal language expressesnatural numbers, well-formed addition statements, and well-formed addition equalities, but it expresses only what theylook like (their syntax), not what they mean (semantics). For instance, nowhere in these rules is there any indicationthat “0” means the number zero, or that "+" means addition.

7.4.1 Constructions

For finite languages one can explicitly enumerate all well-formed words. For example, we can describe a language Las just L = {"a”, “b”, “ab”, “cba"}. The degenerate case of this construction is the empty language, which containsno words at all (L = ∅).However, even over a finite (non-empty) alphabet such as Σ = {a, b} there are infinitely many words: “a”, “abb”,“ababba”, “aaababbbbaab”, …. Therefore formal languages are typically infinite, and describing an infinite formallanguage is not as simple as writing L = {"a”, “b”, “ab”, “cba"}. Here are some examples of formal languages:

• L = Σ*, the set of all words over Σ;

• L = {"a"}* = {"a”n}, where n ranges over the natural numbers and “a”n means “a” repeated n times (this is theset of words consisting only of the symbol “a”);

• the set of syntactically correct programs in a given programming language (the syntax of which is usuallydefined by a context-free grammar);

• the set of inputs upon which a certain Turing machine halts; or

• the set of maximal strings of alphanumeric ASCII characters on this line, i.e., the set {"the”, “set”, “of”,“maximal”, “strings”, “alphanumeric”, “ASCII”, “characters”, “on”, “this”, “line”, “i”, “e"}.

7.5 Language-specification formalisms

Formal language theory rarely concerns itself with particular languages (except as examples), but is mainly concernedwith the study of various types of formalisms to describe languages. For instance, a language can be given as

• those strings generated by some formal grammar;

• those strings described or matched by a particular regular expression;

• those strings accepted by some automaton, such as a Turing machine or finite state automaton;

• those strings for which some decision procedure (an algorithm that asks a sequence of related YES/NO ques-tions) produces the answer YES.

Typical questions asked about such formalisms include:

• What is their expressive power? (Can formalism X describe every language that formalism Y can describe?Can it describe other languages?)

32 CHAPTER 7. FORMAL LANGUAGE

• What is their recognizability? (How difficult is it to decide whether a given word belongs to a language describedby formalism X?)

• What is their comparability? (How difficult is it to decide whether two languages, one described in formalismX and one in formalism Y, or in X again, are actually the same language?).

Surprisingly often, the answer to these decision problems is “it cannot be done at all”, or “it is extremely expen-sive” (with a characterization of how expensive). Therefore, formal language theory is a major application area ofcomputability theory and complexity theory. Formal languages may be classified in the Chomsky hierarchy based onthe expressive power of their generative grammar as well as the complexity of their recognizing automaton. Context-free grammars and regular grammars provide a good compromise between expressivity and ease of parsing, and arewidely used in practical applications.

7.6 Operations on languages

Certain operations on languages are common. This includes the standard set operations, such as union, intersection,and complement. Another class of operation is the element-wise application of string operations.Examples: suppose L1 and L2 are languages over some common alphabet.

• The concatenation L1L2 consists of all strings of the form vw where v is a string from L1 and w is a string fromL2.

• The intersection L1 ∩ L2 of L1 and L2 consists of all strings which are contained in both languages

• The complement ¬L of a language with respect to a given alphabet consists of all strings over the alphabet thatare not in the language.

• The Kleene star: the language consisting of all words that are concatenations of 0 or more words in the originallanguage;

• Reversal:

• Let e be the empty word, then eR = e, and• for each non-empty word w = x1…xn over some alphabet, let wR = x …x1,• then for a formal language L, LR = {wR | w ∈ L}.

• String homomorphism

Such string operations are used to investigate closure properties of classes of languages. A class of languages is closedunder a particular operation when the operation, applied to languages in the class, always produces a language in thesame class again. For instance, the context-free languages are known to be closed under union, concatenation, andintersection with regular languages, but not closed under intersection or complement. The theory of trios and abstractfamilies of languages studies the most common closure properties of language families in their own right.[3]

7.7 Applications

7.7.1 Programming languages

Main articles: Syntax (programming languages) and Compiler compiler

A compiler usually has two distinct components. A lexical analyzer, generated by a tool like lex, identifies the tokensof the programming language grammar, e.g. identifiers or keywords, which are themselves expressed in a simplerformal language, usually by means of regular expressions. At the most basic conceptual level, a parser, usually

7.7. APPLICATIONS 33

generated by a parser generator like yacc, attempts to decide if the source program is valid, that is if it belongs to theprogramming language for which the compiler was built. Of course, compilers do more than just parse the sourcecode—they usually translate it into some executable format. Because of this, a parser usually outputs more than ayes/no answer, typically an abstract syntax tree, which is used by subsequent stages of the compiler to eventuallygenerate an executable containing machine code that runs directly on the hardware, or some intermediate code thatrequires a virtual machine to execute.

7.7.2 Formal theories, systems and proofs

Symbols andstrings of symbols

Well-formed formulas

Theorems

This diagram shows the syntactic divisions within a formal system. Strings of symbols may be broadly divided into nonsense andwell-formed formulas. The set of well-formed formulas is divided into theorems and non-theorems.

Main articles: Theory (mathematical logic) and Formal system

In mathematical logic, a formal theory is a set of sentences expressed in a formal language.A formal system (also called a logical calculus, or a logical system) consists of a formal language together with adeductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformationrules which may be interpreted as valid rules of inference or a set of axioms, or have both. A formal system is usedto derive one expression from one or more other expressions. Although a formal language can be identified withits formulas, a formal system cannot be likewise identified by its theorems. Two formal systems FS and FS ′ mayhave all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntacticconsequence of a formula B in one but not another for instance).

34 CHAPTER 7. FORMAL LANGUAGE

A formal proof or derivation is a finite sequence of well-formed formulas (which may be interpreted as propositions)each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference. The lastsentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can beinterpreted as true propositions.

Interpretations and models

Main articles: Formal semantics (logic), Interpretation (logic) and Model theory

Formal languages are entirely syntactic in nature but may be given semantics that give meaning to the elements of thelanguage. For instance, in mathematical logic, the set of possible formulas of a particular logic is a formal language,and an interpretation assigns a meaning to each of the formulas—usually, a truth value.The study of interpretations of formal languages is called formal semantics. In mathematical logic, this is oftendone in terms of model theory. In model theory, the terms that occur in a formula are interpreted as mathematicalstructures, and fixed compositional interpretation rules determine how the truth value of the formula can be derivedfrom the interpretation of its terms; a model for a formula is an interpretation of terms such that the formula becomestrue.

7.8 See also• Combinatorics on words

• Grammar framework

• Formal method

• Mathematical notation

• Associative array

• String (computer science)

7.9 References

7.9.1 Citation footnotes[1] See e.g. Reghizzi, Stefano Crespi (2009), Formal Languages and Compilation, Texts in Computer Science, Springer, p. 8,

ISBN 9781848820500, An alphabet is a finite set.

[2] Martin Davis (1995). “Influences of Mathematical Logic on Computer Science”. In Rolf Herken. The universal Turingmachine: a half-century survey. Springer. p. 290. ISBN 978-3-211-82637-9.

[3] Hopcroft & Ullman (1979), Chapter 11: Closure properties of families of languages.

7.9.2 General references

• A. G. Hamilton, Logic for Mathematicians, Cambridge University Press, 1978, ISBN 0-521-21838-1.

• Seymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975,ISBN 0-7204-2506-9.

• Michael A. Harrison, Introduction to Formal Language Theory, Addison-Wesley, 1978.

• John E. Hopcroft and Jeffrey D. Ullman, Introduction to Automata Theory, Languages, and Computation,Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 81-7808-347-7.

• Rautenberg, Wolfgang (2010). A Concise Introduction to Mathematical Logic (3rd ed.). New York: SpringerScience+Business Media. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6.

7.10. EXTERNAL LINKS 35

• Grzegorz Rozenberg, Arto Salomaa, Handbook of Formal Languages: Volume I-III, Springer, 1997, ISBN3-540-61486-9.

• Patrick Suppes, Introduction to Logic, D. Van Nostrand, 1957, ISBN 0-442-08072-7.

7.10 External links• Hazewinkel, Michiel, ed. (2001), “Formal language”, Encyclopedia of Mathematics, Springer, ISBN 978-1-

55608-010-4

• Alphabet at PlanetMath.org.

• Language at PlanetMath.org.

• University of Maryland, Formal Language Definitions

• James Power, “Notes on Formal Language Theory and Parsing”, 29 November 2002.

• Drafts of some chapters in the “Handbook of Formal Language Theory”, Vol. 1-3, G. Rozenberg and A.Salomaa (eds.), Springer Verlag, (1997):

• Alexandru Mateescu and Arto Salomaa, “Preface” in Vol.1, pp. v-viii, and “Formal Languages: AnIntroduction and a Synopsis”, Chapter 1 in Vol. 1, pp.1-39

• Sheng Yu, “Regular Languages”, Chapter 2 in Vol. 1• Jean-Michel Autebert, Jean Berstel, Luc Boasson, “Context-Free Languages and Push-Down Automata”,

Chapter 3 in Vol. 1• Christian Choffrut and Juhani Karhumäki, “Combinatorics of Words”, Chapter 6 in Vol. 1• Tero Harju and Juhani Karhumäki, “Morphisms”, Chapter 7 in Vol. 1, pp. 439 - 510• Jean-Eric Pin, “Syntactic semigroups”, Chapter 10 in Vol. 1, pp. 679-746• M. Crochemore and C. Hancart, “Automata for matching patterns”, Chapter 9 in Vol. 2• Dora Giammarresi, Antonio Restivo, “Two-dimensional Languages”, Chapter 4 in Vol. 3, pp. 215 - 267

Chapter 8

Greedoid

In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originallyintroduced by Whitney in 1935 to study planar graphs and was later used by Edmonds to characterize a class of opti-mization problems that can be solved by greedy algorithms. Around 1980, Korte and Lovász introduced the greedoidto further generalize this characterization of greedy algorithms; hence the name greedoid. Besides mathematicaloptimization, greedoids have also been connected to graph theory, language theory, poset theory, and other areas ofmathematics.

8.1 Definitions

A set system (F, E) is a collection F of subsets of a ground set E (i.e. F is a subset of the power set of E). Whenconsidering a greedoid, a member of F is called a feasible set. When considering a matroid, a feasible set is alsoknown as an independent set.An accessible set system (F, E) is a set system in which every nonempty feasible set X contains an element x suchthat X\{x} is feasible. This implies that any nonempty, finite accessible set system necessarily contains the empty set∅.A greedoid (F, E) is an accessible set system that satisfies the exchange property:

• for all X,Y ∈ F with |X| > |Y|, there is some x ∈ X\Y such that Y∪{x} ∈ F

(Note: Some people reserve the term exchange property for a condition on the bases of a greedoid, and prefer to callthe above condition the “Augmentation Property”.)A basis of a greedoid is a maximal feasible set, meaning it is a feasible set but not contained in any other one. Abasis of a subset X of E is a maximal feasible set contained in X.The rank of a greedoid is the size of a basis. By the exchange property, all bases have the same size. Thus, the rankfunction is well defined. The rank of a subset X of E is the size of a basis of X.

8.2 Classes of greedoids

Most classes of greedoids have many equivalent definitions in terms of set system, language, poset, simplicial complex,and so on. The following description takes the traditional route of listing only a couple of the more well-knowncharacterizations.An interval greedoid (F, E) is a greedoid that satisfies the Interval Property:

• if A, B, C ∈ F with A ⊆ B ⊆ C, then, for all x ∈ E\C, (A∪{x} ∈ F and C∪{x} ∈ F) implies B∪{x} ∈ F

Equivalently, an interval greedoid is a greedoid such that the union of any two feasible sets is feasible if it is containedin another feasible set.

36

8.3. EXAMPLES 37

An antimatroid (F, E) is a greedoid that satisfies the Interval Property without Upper Bounds:

• if A, B ∈ F with A ⊆ B, then, for all x ∈ E\B, A∪{x} ∈ F implies B∪{x} ∈ F

Equivalently, an antimatroid is (i) a greedoid with a unique basis; or (ii) an accessible set system closed under union.It is easy to see that an antimatroid is also an interval greedoid.A matroid (F, E) is a greedoid that satisfies the Interval Property without Lower Bounds:

• if B, C ∈ F with B ⊆ C, then, for all x ∈ E\C, C∪{x} ∈ F implies B∪{x} ∈ F

It is easy to see that a matroid is also an interval greedoid.

8.3 Examples

• Consider an undirected graph G. Let the ground set be the edges of G and the feasible sets be the edge set ofeach forest (i.e. subgraph containing no cycle) of G. This set system is called the cycle matroid. A set systemis said to be a graphic matroid if it is the cycle matroid of some graph. (Originally cycle matroid was definedon circuits, or minimal dependent sets. Hence the name cycle.)

• Consider a finite, undirected graph G rooted at the vertex r. Let the ground set be the vertices of G and thefeasible sets be the vertex subsets containing r that induce connected subgraphs of G. This is called the vertexsearch greedoid and is a kind of antimatroid.

• Consider a finite, directed graph D rooted at r. Let the ground set be the (directed) edges of D and the feasiblesets be the edge sets of each directed subtree rooted at r with all edges pointing away from r. This is called theline search greedoid, or directed branching greedoid. It is an interval greedoid, but neither an antimatroidnor a matroid.

• Consider an m-by-n matrix M. Let the ground set E be the indices of the columns from 1 to n and the feasiblesets be F = {X ⊆ E: submatrix M{₁,...,|X|},X is an invertible matrix}. This is called the Gaussian eliminationgreedoid because this structure underlies the Gaussian elimination algorithm. It is a greedoid, but not aninterval greedoid.

8.4 Greedy algorithm

In general, a greedy algorithm is just an iterative process in which a locally best choice, usually an input of minimumweight, is chosen each round until all available choices have been exhausted. In order to describe a greedoid-basedcondition in which a greedy algorithm is optimal, we need some more common terminologies in greedoid theory.Without loss of generality, we consider a greedoid G = (F, E) with E finite.A subset X of E is rank feasible if the largest intersection of X with any feasible set has size equal to the rank of X.In a matroid, every subset of E is rank feasible. But the equality does not hold for greedoids in general.A function w: E → ℝ is R-compatible if {x ∈ E: w(x) ≥ c} is rank feasible for all real numbers c.An objective function f: 2S → ℝ is linear over a set S if, for all X ⊆ S, we have f(X) = Σₓ ∈ X w(x) for some weightfunction w: S → ℜ.Proposition. A greedy algorithm is optimal for every R-compatible linear objective function over a greedoid.The intuition behind this proposition is that, during the iterative process, each optimal exchange of minimum weightis made possible by the exchange property, and optimal results are obtainable from the feasible sets in the underlyinggreedoid. This result guarantees the optimality of many well-known algorithms. For example, a minimum spanningtree of a weighted graph may be obtained using Kruskal’s algorithm, which is a greedy algorithm for the cycle matroid.

38 CHAPTER 8. GREEDOID

8.5 See also• Matroid

• Polymatroid

8.6 References• Björner, Anders; Ziegler, Günter M. (1992), White, Neil, ed., “Matroid Applications”, Matroid applications,

Encyclopedia of Mathematics and its Applications (Cambridge: Cambridge University Press) 40: 284–357,doi:10.1017/CBO9780511662041.009, ISBN 0-521-38165-7, MR 1165537, Zbl 0772.05026 |chapter= ig-nored (help)

• Edmonds, Jack (1971), “Matroids and the greedy algorithm”,Mathematical Programming 1: 127–113, doi:10.1007/BF01584082,Zbl 0253.90027.

• Helman, Paul; Moret, Bernard M. E.; Shapiro, Henry D. (1993), “An exact characterization of greedy struc-tures”, SIAM Journal on Discrete Mathematics 6 (2): 274–283, doi:10.1137/0406021, Zbl 0798.68061.

• Korte, Bernhard; Lovász, László (1981), “Mathematical structures underlying greedy algorithms”, in Gecseg,Ferenc, Fundamentals of Computation Theory: Proceedings of the 1981 International FCT-Conference, Szeged,Hungaria, August 24–28, 1981, Lecture Notes in Computer Science 117, Berlin: Springer-Verlag, pp. 205–209, doi:10.1007/3-540-10854-8_22, Zbl 0473.68019.

• Korte, Bernhard; Lovász, László; Schrader, Rainer (1991), Greedoids, Algorithms and Combinatorics 4, NewYork, Berlin: Springer-Verlag, ISBN 3-540-18190-3, Zbl 0733.05023.

• Oxley, James G. (1992), Matroid theory, Oxford Science Publications, Oxford: Oxford University Press, ISBN0-19-853563-5, Zbl 0784.05002.

• Whitney, Hassler (1935), “On the abstract properties of linear independence”, American Journal of Mathe-matics 57 (3): 509–533, doi:10.2307/2371182, JSTOR 2371182, Zbl 0012.00404.

8.7 External links• Introduction to Greedoids

• Theory of Greedy Algorithms

• Submodular Functions and Optimization

• Matchings, Matroids and Submodular Functions

Chapter 9

Inversion (discrete mathematics)

In computer science and discrete mathematics, an inversion is a pair of places of a sequence where the elements onthese places are out of their natural order.

9.1 Definitions

Formally, let (A(1), . . . , A(n)) be a sequence of n distinct numbers. If i < j and A(i) > A(j) , then the pair (i, j)is called an inversion of A .[1][2]

The inversion number of a sequence is one common measure of its sortedness.[3][2] Formally, the inversion numberis defined to be the number of inversions, that is,

inv(A) = |{(A(i), A(j)) | i < j and A(i) > A(j)}| .[3]

Equivalently, it is the Kendall tau distance from the identity permutation. Other measures of (pre-)sortedness includethe minimum number of elements that can be deleted from the sequence to yield a fully sorted sequence, the numberand lengths of sorted “runs” within the sequence, the Spearman footrule (sum of distances of each element from itssorted position), and the smallest number of exchanges needed to sort the sequence.[4] Standard comparison sortingalgorithms can be adapted to compute the inversion number in time O(n log n).The inversion vector V(i) of the sequence is defined for i = 2, ..., n as V [i] = |{k | k < i and A(k) > A(i)}| . Inother words each element is the number of elements preceding the element in the original sequence that are greaterthan it. Note that the inversion vector of a sequence has one less element than the sequence, because of course thenumber of preceding elements that are greater than the first is always zero. Each permutation of a sequence has aunique inversion vector and it is possible to construct any given permutation of a (fully sorted) sequence from thatsequence and the permutation’s inversion vector.[5]

9.2 Weak order of permutations

The set of permutations on n items can be given the structure of a partial order, called the weak order of permuta-tions, which forms a lattice.To define this order, consider the items being permuted to be the integers from 1 to n, and let Inv(u) denote the setof inversions of a permutation u for the natural ordering on these items. That is, Inv(u) is the set of ordered pairs (i,j) such that 1 ≤ i < j ≤ n and u(i) > u(j). Then, in the weak order, we define u ≤ v whenever Inv(u) ⊆ Inv(v).The edges of the Hasse diagram of the weak order are given by permutations u and v such that u < v and such that vis obtained from u by interchanging two consecutive values of u. These edges form a Cayley graph for the group ofpermutations that is isomorphic to the skeleton of a permutohedron.The identity permutation is the minimum element of the weak order, and the permutation formed by reversing theidentity is the maximum element.

39

40 CHAPTER 9. INVERSION (DISCRETE MATHEMATICS)

The permutation (4,1,5,2,6,3) has the inversion vector (0,1,0,2,0,3) and the inversion set {(1,2),(1,4),(3,4),(1,6),(3,6),(5,6)}. Theinversion vector converted to decimal is 373.

9.3 See also

• Factorial number system (a factorial number is a reflected inversion vector)

9.3. SEE ALSO 41

Inversion set of the permutation(0,15, 14,1, 13,2, 3,12,11,4, 5,10, 6,9, 8,7)showing the pattern of theThue–Morse sequence

• Transpositions, simple transpositions, inversions and sorting

• Damerau–Levenshtein distance

• Parity of a permutation

Sequences in the OEIS:

• Index entries for sequences related to factorial numbers

• Reflected inversion vectors: A007623 and A108731

• Sum of inversion vectors, cardinality of inversion sets: A034968

• Inversion sets of finite permutations interpreted as binary numbers: A211362 (related permutation: A211363)

• Finite permutations that have only 0s and 1s in their inversion vectors: A059590 (their inversion sets: A211364)

42 CHAPTER 9. INVERSION (DISCRETE MATHEMATICS)

• Numbers of permutations of n elements with k inversions; Mahonian numbers: A008302 (their row maxima;Kendall-Mann numbers: A000140)

• Number of connected labeled graphs with n edges and n nodes: A057500

• Arrays of permutations with similar inversion sets and inversion vectors: A211365, A211366, A211367,A211368, A211369, A100630, A211370, A051683

9.4 References[1] Cormen et al. 2001, pp. 39.

[2] Vitter & Flajolet 1990, pp. 459.

[3] Barth & Mutzel 2004, pp. 183.

[4] Mahmoud 2000, pp. 284.

[5] Pemmaraju & Skiena 2003, pp. 69.

9.4.1 Source bibliography

• Barth, Wilhelm; Mutzel, Petra (2004). “Simple and Efficient Bilayer Cross Counting”. Journal of GraphAlgorithms and Applications 8 (2): 179–194. doi:10.7155/jgaa.00088.

• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algo-rithms (2nd ed.). MIT Press and McGraw-Hill. ISBN 0-262-53196-8.

• Mahmoud, Hosam Mahmoud (2000). “Sorting Nonrandom Data”. Sorting: a distribution theory. Wiley-Interscience series in discrete mathematics and optimization 54. Wiley-IEEE. ISBN 978-0-471-32710-3.

• Pemmaraju, Sriram V.; Skiena, Steven S. (2003). “Permutations and combinations”. Computational discretemathematics: combinatorics and graph theory with Mathematica. Cambridge University Press. ISBN 978-0-521-80686-2.

• Vitter, J.S.; Flajolet, Ph. (1990). “Average-Case Analysis of Algorithms and Data Structures”. In van Leeuwen,Jan. Algorithms and Complexity 1 (2nd ed.). Elsevier. ISBN 978-0-444-88071-0.

9.4.2 Further reading

• Margolius, Barbara H. (2001). “Permutations with Inversions”. Journal of Integer Sequences 4.

9.4.3 Presortedness measures

• Mannila, Heikki (1984). “Measures of presortedness and optimal sorting algorithms”. Lecture Notes in Com-puter Science 172: 324–336. doi:10.1007/3-540-13345-3_29.

• Estivill-Castro, Vladimir; Wood, Derick (1989). “A new measure of presortedness”. Information and Com-putation 83 (1): 111–119. doi:10.1016/0890-5401(89)90050-3.

• Skiena, Steven S. (1988). “Encroaching lists as a measure of presortedness”. BIT 28 (4): 755–784. doi:10.1007/bf01954897.

Chapter 10

Outline of discrete mathematics

The following outline is provided as an overview of and topical guide to discrete mathematics:Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.In contrast to real numbers that have the property of varying “smoothly”, the objects studied in discrete mathematics– such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separatedvalues.[2] Discrete mathematics therefore excludes topics in “continuous mathematics” such as calculus and analysis.Included below are many of the standard terms used routinely in university-level courses and in research papers. Thisis not, however, intended as a complete list of mathematical terms; just a selection of typical terms of art that may beencountered.

10.1 Subjects in discrete mathematics

• Logic – a study of reasoning

• Set theory – a study of collections of elements

• Number theory –

• Combinatorics – a study of counting

• Graph theory –

• Digital geometry and digital topology

• Algorithmics – a study of methods of calculation

• Information theory –

• Computability and complexity theories – dealing with theoretical and practical limitations of algorithms

• Elementary probability theory and Markov chains

• Linear algebra – a study of related linear equations

• Functions –

• Partially ordered set –

• Probability –

• Proofs –

• Counting –

• Relation –

43

44 CHAPTER 10. OUTLINE OF DISCRETE MATHEMATICS

10.2 Discrete mathematical disciplines

For further reading in discrete mathematics, beyond a basic level, see these pages. Many of these disciplines areclosely related to computer science.

• Automata theory –

• Combinatorics –

• Combinatorial geometry –

• Computational geometry –

• Digital geometry –

• Discrete geometry –

• Graph theory –

• Mathematical logic –

• Combinatorial optimization –

• Set theory –

• Combinatorial topology –

• Number theory –

• Information theory –

• Game theory –

10.3 Concepts in discrete mathematics

10.3.1 Sets

• Set (mathematics) –

• Element (mathematics) –• Venn diagram –• Empty set –• Subset –• Union (set theory) –

• Disjoint union –• Intersection (set theory) –

• Disjoint sets –• Complement (set theory) –• Symmetric difference –

• Ordered pair –

• Cartesian product –

• Power set –

• Simple theorems in the algebra of sets –

• Naive set theory –

• Multiset –

10.3. CONCEPTS IN DISCRETE MATHEMATICS 45

10.3.2 Functions• Function –• Domain of a function –• Codomain –• Range of a function –• Image (mathematics) –• Injective function –• Surjection –• Bijection –• Function composition –• Partial function –• Multivalued function –• Binary function –• Floor function –• Sign function –• Inclusion map –• Pigeonhole principle –• Relation composition –• Permutations –• Symmetry –

10.3.3 OperationsBinary operator –

• Associativity –• Commutativity –• Distributivity

10.3.4 ArithmeticDecimal –

• Binary numeral system –• Divisor –• Division by zero –• Indeterminate form –• Empty product –• Euclidean algorithm –• Fundamental theorem of arithmetic –• Modular arithmetic –• Successor function

46 CHAPTER 10. OUTLINE OF DISCRETE MATHEMATICS

10.3.5 Elementary algebra

Main article: Elementary algebra

Left-hand side and right-hand side of an equation –

• Linear equation –

• Quadratic equation –

• Solution point –

• Arithmetic progression –

• Recurrence relation –

• Finite difference –

• Difference operator –

• Groups –

• Group isomorphism –

• Subgroups –

• Fermat’s little theorem –

• Cryptography –

• Faulhaber’s formula –

10.3.6 Mathematical relations

• Binary relation –

• Mathematical relation –

• Reflexive relation –

• Reflexive property of equality –

• Symmetric relation –

• Symmetric property of equality –

• Antisymmetric relation –

• Transitivity (mathematics) –

• Transitive closure –• Transitive property of equality –

• Equivalence and identity

• Equivalence relation –• Equivalence class –• Equality (mathematics) –

• Inequation –• Inequality (mathematics) –

• Similarity (geometry) –• Congruence (geometry) –

10.3. CONCEPTS IN DISCRETE MATHEMATICS 47

• Equation –• Identity (mathematics) –

• Identity element –• Identity function –

• Substitution property of equality –• Graphing equivalence –• Extensionality –• Uniqueness quantification –

10.3.7 Mathematical phraseology

If and only if –

• Necessary and sufficient (Sufficient condition) –

• Distinct –

• Difference –

• Absolute value –

• Up to –

• Modular arithmetic –

• Characterization (mathematics) –

• Normal form –

• Canonical form –

• Without loss of generality –

• Vacuous truth –

• Contradiction, Reductio ad absurdum –

• Counterexample –

• Sufficiently large –

• Pons asinorum –

• Table of mathematical symbols –

• Contrapositive –

• Mathematical induction –

10.3.8 Combinatorics

Main article: Combinatorics

• Permutations and combinations –

• Permutation –

• Combination –

• Factorial –

48 CHAPTER 10. OUTLINE OF DISCRETE MATHEMATICS

• Empty product –

• Pascal’s triangle –

• Combinatorial proof –

• Bijective proof –• Double counting (proof technique) –

10.3.9 Probability

Main article: Probability

• Average –

• Expected value –

• Discrete random variable –

• Sample space –

• Event –

• Conditional Probability –

• Independence –

• Random variables –

10.3.10 Propositional logic

Logical operator –

• Truth table –

• De Morgan’s laws –

• Open sentence –

• List of topics in logic –

10.4 Discrete mathematicians

• Paul Erdős –

• Ronald Graham –

10.5 See also

10.6 References[1] Richard Johnsonbaugh, Discrete Mathematics, Prentice Hall, 2008.

[2] Weisstein, Eric W., “Discrete mathematics”, MathWorld.

10.7. EXTERNAL LINKS 49

10.7 External links• Archives

• Jonathan Arbib & John Dwyer, Discrete Mathematics for Cryptography, 1st Edition ISBN 978-1-907934-01-8.

• John Dwyer & Suzy Jagger, Discrete Mathematics for Business & Computing, 1st Edition 2010 ISBN 978-1-907934-00-1.

Chapter 11

Portal:Discrete mathematics

• Wikipedia portalsCulture

• Geography

• Health

• History

• Mathematics

• Natural sciences

• People

• Philosophy

• Religion

• Society

• Technology

edit

11.1 Discrete mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete in the sense of notsupporting or requiring the notion of continuity. Discrete objects can be enumerated by integers. Topics in discretemathematics include number theory (which deals mainly with the properties of integers), combinatorics, logic, graphs,algorithms, and formal languages.Discrete mathematics has become popular in recent decades because of its applications to computer science. Discretemathematics is the mathematical language of computer science. Concepts and notations from discrete mathemat-ics are useful in studying and describing objects and problems in all branches of computer science, such as com-puter algorithms, programming languages, cryptography, automated theorem proving, and software development.Conversely, computer implementations are tremendously significant in applying ideas from discrete mathematics toreal-world applications, such as in operations research.The set of objects studied in discrete mathematics can be finite or infinite. In real-world applications, the set of objectsof interest are mainly finite, the study of which is often called finite mathematics. In some mathematics curricula,the term “finite mathematics” refers to courses that cover discrete mathematical concepts for business, while “discretemathematics” courses emphasize discrete mathematical concepts for computer science majors.edit

50

11.2. SELECTED ARTICLE 51

Discrete mathematics icon

11.2 Selected article

In mathematics, computing, linguistics and related disciplines, an algorithm is a type of effective method in whicha definite list of well-defined instructions for completing a task, when given an initial state, will proceed through awell-defined series of successive states, eventually terminating in an end-state. The transition from one state to thenext is not necessarily deterministic; some algorithms, known as probabilistic algorithms, incorporate randomness.A partial formalization of the concept began with attempts to solve the Entscheidungsproblem (the “decision prob-lem”) that David Hilbert posed in 1928. Subsequent formalizations were framed as attempts to define "effectivecalculability" (Kleene 1943:274) or “effective method” (Rosser 1939:225); those formalizations included the Gödel-Herbrand-Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's“Formulation I” of 1936, and Alan Turing's Turing machines of 1936-7 and 1939.edit

52 CHAPTER 11. PORTAL:DISCRETE MATHEMATICS

Lamp doesn't work

Lampplugged in?

Bulbburned out?

Repair lamp

No

Replace bulb

Plug in lamp

Yes

No

Yes

11.3 WikiProjects

The Mathematics WikiProject is the center for mathematics-related editing on Wikipedia. Join the discussion onthe project’s talk page.

11.4. SELECTED PICTURE 53

Project pages

• Participants• Current activity• Manual of style• Conventions

Essays

• Advice on using Wikipedia for mathematics self-study• Proofs

Subprojects

• PlanetMath Exchange (Talk)• Graphics (Talk)• Article assessment (Talk)

Related projectsComputer science | Cryptography | Game theory | Numbers | Physics | Science | Statisticsedit

11.4 Selected picture

A Penrose tiling, an example of a tiling that can completely cover an infinite plane, but only in a pattern which isnon-repeating (aperiodic).edit

11.5 Did you know?

• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonicsolids known to the ancient Greeks.

• ...that the Catalan numbers solve a number of problems in combinatorics such as the number of ways to com-pletely parenthesize an algebraic expression with n+1 factors?

edit

11.6 Categories

Coding theory • Combinatorics • Digital systems • Discrete geometry • Factorial and binomial topics • Finitedifferences • Graph theory • Permutations • Symmetric functions

edit

54 CHAPTER 11. PORTAL:DISCRETE MATHEMATICS

Penrose tiling

11.7 Topics in Discrete mathematics

edit

11.8 Related portals

ScienceHistory of science Philosophy of science Emergence

Complex Adaptive Behaviour

Simple Self-Organized Local Relationships

Changing External

Environment

Info Out

Info In

Pos

itive

Fee

dbac

k (A

mpl

ifyin

g)

Negative Feedback

(Dam

pening)

Info Out

Info In

Changing External

Environment

Changing External

Environment

Changing External

Environment Systems science Mathematics Biology Chemistry

Physics Earth sciences Technology

• What are portals?

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11.9. WIKIMEDIA 55

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56 CHAPTER 11. PORTAL:DISCRETE MATHEMATICS

Category puzzle

Chapter 12

Trinomial triangle

The trinomial triangle is a variation of Pascal’s triangle. The difference between the two is that an entry in thetrinomial triangle is the sum of the three (rather than the two in Pascal’s triangle) entries above it:

11 1 1

1 2 3 2 11 3 6 7 6 3 1

1 4 10 16 19 16 10 4 1

The k -th entry of the n -th row is denoted by

(n

k

)2

Rows are counted starting from 0. The entries of the n -th row are indexed starting with −n from the left, and themiddle entry has index 0. The symmetry of the entries of a row about the middle entry is expressed by the relationship

(n

k

)2

=

(n

−k

)2

12.1 Properties

Then -th row corresponds to the coefficients in the polynomial expansion of the expansion of the trinomial (1+x+x2)raised to the n -th power:[1]

(1 + x+ x2

)n=

2n∑j=0

(n

j − n

)2

xj =n∑

k=−n

(n

k

)2

xn+k

or, symmetrically,

(1 + x+ 1/x)n=

n∑k=−n

(n

k

)2

xk

hence the alternative name trinomial coefficients because of their relationship to the multinomial coefficients:

(n

k

)2

=∑

0≤µ,ν≤nµ+2ν=n+k

n!

µ! ν! (n− µ− ν)!

57

58 CHAPTER 12. TRINOMIAL TRIANGLE

Furthermore, the diagonals have interesting properties, such as their relationship to the triangular numbers.The sum of the elements of n -th row is 3n .

12.2 Recursion formula

The trinomial coefficients can be generated using the following recursion formula:[1]

(0

0

)2

= 1

(n+1k

)2=

(n

k−1

)2+(nk

)2+

(n

k+1

)2

for n ≥ 0 ,

where(nk

)2= 0 for k < −n and k > n .

12.3 The middle entries

The middle entries of the trinomial triangle (sequence A002426 in OEIS)

1, 1, 3, 7, 19, 51, 141, 393, 1107, 3139, …

were studied by Euler. The middle entry for the n -th row is given by

(n

0

)2

=n∑

k=0

n(n− 1) · · · (n− 2k + 1)

(k!)2=

n∑k=0

(n

2k

)(2k

k

).

The corresponding generating function is[2]

1 + x+ 3x2 + 7x3 + 19x4 + . . . =1√

(1 + x)(1− 3x).

Euler also noted the following exemplum memorabile inductionis fallacis (“notable example of fallacious induction”):

3(n+10

)2−(n+20

)2= fn(fn + 1) for 0 ≤ n ≤ 7

where fn stands for the Fibonacci sequence. For larger n , however, this relationship is incorrect. George Andrewsexplained this fallacy using the general identity[3]

2∑k∈Z

[(n+ 1

10k

)2

−(

n+ 1

10k + 1

)2

]= fn(fn + 1).

12.4 Chess mathematics

The triangle corresponds to the number of possible paths that can be taken by the king in a game of chess. The entryin a cell represents the number of different paths (using a minimum number of moves) the king can take to reach thecell.

12.5. IMPORTANCE IN COMBINATORICS 59

Number of ways to reach a cell with the minimum number of moves

12.5 Importance in combinatorics

The coefficient of xk in the polynomial expansion of(1 + x+ x2

)n specifies the number of different ways of ran-domly drawing k cards from two sets of n identical playing cards.[4] For example, in such a card game with two setsof the three cards A, B, C, the choices look like this:In particular, this results in

(24

12−24

)2=

(24−12

)2=

(2412

)2

as the number of different hands in a game of Doppelkopf.Alternatively, it is also possible to arrive at this number by considering the number of ways of choosing p pairs ofidentical cards from the two sets, which is

(np

). The remaining k − 2p cards can then be chosen in

(n−pk−2p

)ways,[4]

which can be written in terms of the binomial coefficients as

(n

k − n

)2

=

min(n,[k/2])∑p=max(0,k−n)

(n

p

)(n− p

k − 2p

)

For example,

60 CHAPTER 12. TRINOMIAL TRIANGLE

6 =

(3

2− 3

)2

=

(3

0

)(3

2

)+

(3

1

)(2

0

)= 1 · 3 + 3 · 1

The example above corresponds to the three ways of selecting two cards without pairs of identical cards (AB, AC,BC) and the three ways of selecting a pair of identical cards (AA, BB, CC).

12.6 References[1] Weisstein, Eric W., “Trinominal Coefficient”, MathWorld.

[2] Weisstein, Eric W., “Central Trinomial Coefficient”, MathWorld.

[3] George Andrews, Three Aspects for Partitions. Séminaire Lotharingien de Combinatoire, B25f (1990) Online copy

[4] Andreas Stiller: Pärchenmathematik. Trinomiale und Doppelkopf. (“Pair mathematics. Trinomials and the game of Dop-pelkopf"). c't Issue 10/2005, p. 181ff

12.7 Further reading• Leonhard Euler, Observationes analyticae (“Analytical observations”). Novi Commentarii academiae scien-tiarum Petropolitanae 11 (1767) 124–143 PDF

12.8. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 61

12.8 Text and image sources, contributors, and licenses

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62 CHAPTER 12. TRINOMIAL TRIANGLE

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64 CHAPTER 12. TRINOMIAL TRIANGLE

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