Theory of Languages and Automatace.sharif.edu/courses/97-98/2/ce415-1/resources/root... · 2020. 9....

Theory of Languages and Automata

Chapter 0 - Introduction

Sharif University of Technology

ReferencesO Main Reference

§ M. Sipser, ”Introduction to the Theory of Computation,� 3nd Ed., Cengage Learning , 2013.

O Additional References§ P. Linz, �An Introduction to Formal Languages and Automata,� 3rd

Ed., Jones and Barlett Publishers, Inc., 2001.§ J.E. Hopcroft, R. Motwani and J.D. Ullman, �Introduction to

Automata Theory, Languages, and Computation,� 2nd Ed., Addison-Wesley, 2001.

§ P.J. Denning, J.B. Dennnis, and J.E. Qualitz, �Machines, Languages, and Computation,� Prentice-Hall, Inc., 1978.

§ P.J. Cameron, �Sets, Logic and Categories,� Springer-Verlag, London limited, 1998.

Prof. MovagharTheory of Languages and Automata 2

Grading Policy

O Assignments 20%O First Quiz 15-20%O Second Quiz 15-20%O Final Exam 40-50%


Teaching Assistants

O Teaching Assistants:

O Khashayar GatmiryO E-mail: [email protected]

O Farbod EkbataniO E-mail: [email protected]

Main Topics

O Complexity TheoryO Computability TheoryO Automata TheoryO Mathematical NotionsO AlphabetO StringsO Languages


Complexity Theory

O The scheme for classifying problems according to their computational difficulty

O Options:ü Alter the problem to a more easily solvable one by

understanding the root of difficultyü Approximate the perfect solutionü Use a procedure that occasionally is slow but usually

runs quicklyü Consider alternatives, such as randomized computation

O Applied area:ü Cryptography


Computability Theory

O Determining whether a problem is solvable by Computers

O Classification of problems as solvable ones and unsolvable ones


Automata Theory

O Definitions and properties of mathematical models of computationü Finite automaton

Usage: Text Processing, Compilers, Hardware Designü Context-free grammar

Usage: Programming Languages, Artificial Intelligence


Logic

O The science of the formal principles of reasoning


History of Logic

O Logic was known as 'dialectic' or 'analytic' in Ancient Greece. The word 'logic' (from the Greek logos, meaning discourse or sentence) does not appear in the modern sense until the commentaries of Alexander of Aphrodisias, writing in the third century A.D.

O While many cultures have employed intricate systems of reasoning, and logical methods are evident in all human thought, an explicit analysis of the principles of reasoning was developed only in three traditions: those of China, India, and Greece.


History of Logic (cont.)O Although exact dates are uncertain, particularly in the case of

India, it is possible that logic emerged in all three societies by the 4th century BC.

O The formally sophisticated treatment of modern logic descends from the Greek tradition, particularly Aristotelian logic, which was further developed by Islamic Logicians and then medieval European logicians.

O The work of Frege in the 19th century marked a radical departure from the Aristotlian leading to the rapid development of symbolic logic, later called mathematical logic.


Modern Logic

O Descartes proposed using algebra, especially techniques for solving for unknown quantities in equations, as a vehicle for scientific exploration.

O The idea of a calculus of reasoning was also developed by Leibniz. He was the first to formulate the notion of a broadly applicable system of mathematical logic.

O Frege in his 1879 work extended formal logic beyond propositional logic to include quantificationto represent the "all", "some" propositions of Aristotelian logic.


Modern Logic (cont.)

O A logic is a language of formulas.O A formula is a finite sequence of symbols with a

syntax and semantics.O A logic can have a formal system.O A formal system consists of a set of axioms and

rules of inference.


Logic Types of Interest

O Propositional LogicO Predicate Logic


Syntax of Propositional Logic

O Let {p0, p1, ..} be a countable set of propositional variables,

O {Ù, Ú, ¬, →, ↔} be a finite set of connectives,

O Also, there are left and right brackets,O A propositional variable is a formula,O If φ and ψ are formulas, then so are (¬φ), (φ Ù ψ),

(φ Ú ψ), (φ → ψ), and (φ ↔ ψ).


Semantics of Propositional Logic

O Any formula, which involves the propositional variables p0,...,pn , can be used to define a functionof n variables, that is, a function from the set {T, F}n

to {T, F}. This function is often represented as the truth table of the formula and is defined to be the semantics of that formula.


Examples

ψ φ (¬φ) (φ Ù ψ) (φ Ú ψ) (φ → ψ) (φ ↔ψ)

T T F T T T T

F T F F T F F

T F T F T T F

F F T F F T T


Other Definitions

O A formulae is a tautology if it is always true.§ (P Ú (¬P)) is a tautology.

O A formulae is a contradiction if it is never true.§ (P Ù (¬P)) is a contradiction.

O A formulae is a contingency if it is sometimes true.§ (P → (¬P)) is a contingency.


Formal System

A formal system includes the following:O An alphabet A, a set of symbols.O A set of formulae, each of which is a string of

symbols from A.O A set of axioms, each axiom being a formula. O A set of rules of inference, each of which takes as �input� a finite sequence of formulae and produces as output a formula.


Proof

O A proof in a formal system is just a finite sequence of formulae such that each formula in the sequence either is an axiom or is obtained from earlier formulae by applying a rule of inference.


Theorem

O A theorem of the formal system is just the last formula in a proof.

O Example:For any formula φ, the formula (φ→φ) is a theorem of the propositional logic.


A Formal System for Propositional Logic

O There are three �schemes� of axioms, namely:§ (A1) (φ→(ψ → φ))§ (A2) (φ→(ψ → θ))→ ((φ→ ψ)→ (φ → θ))§ (A3) (((¬φ)→(¬ψ))→ (ψ → φ))

O Each of these formulas is an axiom, for all choices of formulae φ, ψ, θ.

O There is only one rule of inference, namely Modus Ponens: From φ and (φ→ ψ), infer ψ.


Example of a ProofO Using (A2), taking φ, ψ, θ to be φ, (φ→φ) and φ

respectively ((φ→((φ→φ)→φ))→((φ→(φ→φ))→(φ →φ)))

O Using (A1), taking φ, ψ to be φ and (φ→φ) respectively(φ→((φ→φ)→φ))

O Using Modus Ponens((φ→(φ→φ))→(φ →φ))

O Using (A1), taking φ, ψ to be φ and φ respectively(φ→(φ→φ))

O Using Modus Ponens(φ→φ)


Soundness & Completeness

O A formal system is said to be sound if all theorems in that system are tautology.

O A formal system is said to be complete if all tautologies in that system are theorems.


Predicate

O A proposition involving some variables, functions and relations

O ExampleP(x) = “x > 3”Q(x,y,z) = “x2 + y2 = z2”


Quantifier

O Universal: “for all” �

�x P(x) � P(x1) � P(x2) � P(x3) �…

O Existential: “there exists” ��x P(x) � P(x1) � P(x2) � P(x3) �…

O Combinations:�x �y y > x


Quantifiers: Negation

O ¬ (�x P(x)) ��x ¬ P(x)O ¬ (�x P(x)) ��x ¬ P(x)O ¬ �x �y P(x,y) ��x �y ¬ P(x,y)O ¬ �x �y P(x,y) ��x �y ¬ P(x,y)


Rules of Inference

1. Direct Proof2. Indirect Proof3. Proof by Contradiction4. Proof by Cases 5. Induction


Direct Proof

O If the two propositions (premises) p and p → q are theorems, we may deduce that the proposition q is also a theorem.

O This fundamental rule of inference is called modus ponens by logicians.

p → q p q


Indirect Proof

O Proves p→ q by instead proving the contra-positive, ~q→ ~p

p → q~q ~p


Proof by Contradiction

O The rule of inference used is that from theorems p and ¬q → ¬p, we may deduce theorem q.

p~q → ~p q


Proof by Cases

O You want to prove p→ qO P can be decomposed to some cases:

p ↔ p1 ν p2 ν…ν pn

O Independently prove the n implications given by

pi → q for 1 ≤ i ≤ n.


Proof by Induction

O To prove

1. Proof P(0),2. Proof

( )xP x"

[ ( ) ( 1)]x P x P x" ® +


Constructive Existence Proof

O Want to prove that

O Find an a and then prove that P(a) is true

)(xPx$


Non-Constructive Existence Proof

O Want to prove that

O You cannot find an a that P(a) is trueO Then, you can use proof by contradiction:

)(xPx$

FxPxxPx ®"«$ )(~)(~


Intuitive (Naïve) Set Theory

There are three basic concepts in set theory:

O MembershipO ExtensionO Abstraction


Membership

O Membership is a relation that holds between a set and an object

O x�A to mean �the object x is a member of the set A�, or �x belongs to A�. The negation of this assertion is written x�A as an abbreviation for the proposition ¬(x�A)

O One way to specify a set is to list its elements. For example, the set A = {a, b, c} consists of three elements. For this set A, it is true that a�A but d�A


Extension

O The concept of extension is that two sets are identical if and if only if they contain the same elements. Thus we write A=B to mean

�x [x�A ⟺x�B]


Abstraction

Each property defines a set, and each set defines a property

O If p(x) is a property then we can define a set AA = {x � p(x)}

O If A is a set then we can define a predicate p(x) p(x) = x�A


Intuitive versus axiomatic set theory

O The theory of set built on the intuitive concept of membership, extension, and abstraction is known as intuitive (naïve) set theory.

O As an axiomatic theory of sets, it is not entirely satisfactory, because the principle of abstraction leads to contradictions when applied to certain simple predicates.


Russell’s Paradox

Let p(X) be a predicate defined asP(X) = (XÏX)

Define set R asR={X|P(X)}

O Is P(R) true?O Is P(R) false?


Gottlob Frege�s Comments

O The logician Gottlob Frege was the first to develop mathematics on the foundation of set theory. He learned of Russell Paradox while his work was in press, and wrote, �A scientist can hardly meet with anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr. Bertrand Russell as the work was nearly through the press.�


Power Set

O The set of all subsets of a given set A is known as the power set of A, and is denoted by P(A):

P(A) = {B | B Í A}


Set Operation-Union

O The union of two sets A and B isA È B = {x | (x Î A) Ú (x Î B)}

and consists of those elements in at least one of Aand B.

O If A1, …, An constitute a family of sets, their union is

= (A1 È … ÈAn)= {x| x Î Ai for some i, 1≤ i ≤ n}

Aii=1

n


Set Operation-Intersection

O The intersection of two sets A and B isA Ç B = {x | (x Î A) � (x Î B)}

and consists of those elements in at least one of Aand B.

O If A1, …, An constitute a family of sets, their intersection is

= (A1 Ç … Ç An) = {x| x Î Ai for all i, 1≤ i ≤ n}

Aii=1

n


Set Operation-Complement

O The complement of a set A is a set Ac defined as:Ac = {x | x � A}

O The complement of a set B with respect to A, also denoted as A-B, is defined as:

A-B = {x � A | x � B}


Ordered Pairs and n-tuples

O An ordered pair of elements is written (x,y)

where x is known as the first element, and y is known as the second element.

O An n-tuple is an ordered sequence of elements(x1, x2, …, xn)

And is a generalization of an ordered pair.


Ordered Sets and Set Products

O By Cartesian product of two sets A and B, we mean the set

A×B = {(x,y) | xÎA , yÎB}

O Similarly,A1×A2×…An = {x1ÎA1, x2ÎA2, …, xnÎAn}


Relations

A relation ρ between sets A and B is a subset of A×B:ρ Í A×B

O The domain of ρ is defined asDρ = {x Î A | for some y Î B, (x,y) Î ρ}

O The range of ρ is defined asRρ = {y Î B | for some x Î A, (x,y) Î ρ}

O If ρ Í A×A, then ρ is called a �relation on A�.


Types of Relations on Sets

O A relation ρ is reflexive if(x,x) Îρ, for each x ÎA

O A relation ρ is symmetric if, for all x,y ÎA(x,y) Î ρ implies (y,x)Î ρ

O A relation ρ is antisymmetric if, for all x,yÎA(x,y) Î ρ and (y,x) Î ρ implies x=y

O A relation ρ is transitive if, for all x,y,z ÎA(x,y) Î ρ and (y,z) Î ρ implies (x,z) Î ρ


Partial Order and Equivalence Relations

O A relation ρ on a set A is called a partial ordering of A if ρ is reflexive, anti-symmetric, and transitive.

O A relation ρ on a set A is called an equivalencerelation if ρ is reflexive, symmetric, and transitive.


Total Ordering

O A relation ρ on a set A is a total ordering if ρis a partial ordering and, for each pair of elements (x,y) in A×A at least one of (x,y)Îρor (y,x)Îρ is true.


Inverse Relation

O For any relation ρ Í A×B, the inverse of ρ is defined by

ρ-1 = {(y,x) | (x,y) Î ρ}

O If Dρ and Rρ are the domain and range of ρ, thenDρ-1 = Rρ and Rρ-1 = Dρ


Equivalence

O Let ρ Í A×A be an equivalence relation on A. The equivalence class of an element x is defined as

[x] = {y Î A | (x,y) Î ρ}

O An equivalence relation on a set partitions the set.


Functions

A relation f Í A× B is a function if it has the property:for all x,y,z, (x,y) Î f and (x,z) Î f implies y = z

O If f Í A× B is a function, we write f: A→ B

and say that f maps A into B. We use the common notation

Y = f(x)to mean (x,y) Î f.


Functions (cont.)

O As before, the domain of f is the set Df = {xÎ A | for some yÎ B, (x,y) Î f}

and the range of f is the setRf = {yÎ B | for some x Î A, (x,y) Î f}

O If Df Í A, we say the function is a partial function; if Df = A, we say that f is a total function.


Functions (cont.)

O If x Î Df, we say that f is defined at x; otherwise f isundefined at x.

O If Rf = B, we say that f maps Df onto B.O If a function f has the property

for all x,y,z, f(x) = z and f(y) = z implies x = ythen f is a one-to-one function. If f: A→ B is a one-to-one function, f gives a one-to

one correspondence between elements of its domain and range.


Functions (cont.)

O Let X be a set and suppose A Í X. Define functionCA: X→ {0,1}

such that CA(x) = 1, if xÎ A; CA(x) = 0, otherwise. CA(x) is called the characteristic function of set A with respect to set X.

O If f Í A× B is a function, then the inverse of f is the set

f--1 = {(y,x) | (x,y) Î f} f--1 is a function if and only if f is one-to-one.


Functions (cont.)

O Let f: A→ B be a function, and suppose that X Í A. Then the set

Y = f(X) = {y Î B | y = f(x) for some x Î X}is known as the image of X under f.

O Similarly, the inverse image of a set Y included in the range of f is

f -1(Y) = {x Î A | y = f(x) for some y Î Y}


CardinalityO Two sets A and B are of equal cardinality, written as

|A| = |B|if and only if there is a one-to-one function f: A→ Bthat maps A onto B.

O We write |A| ≤ |B|

if B includes a subset C such that |A| = |C|.O If |A| ≤ |B| and |A| ≠ |B|, then A has cardinality less

than that of B, and we write|A| < |B|


Cardinality (cont.)O Let J = {1, 2, …} and Jn = {1, 2, …, n}.O A sequence on a set X is a function f: J→ X. A sequence

may be written as f(1), f(2), f(3), …

However, we often use the simpler notationx1, x2, x3, …., xi Î X

O A finite sequence of length n on X is a function f: Jn → X, usually written as

x1, x2, x3, …., xn, xi Î XO The sequence of length zero is the function f: → X.


Finite and Infinite Sets

O A set A is finite if |A| = |Jn| for some integer n ≥ 0, in which case we say that A has cardinality n.

O A set is infinite if it is not finite.O A set X is denumerable if |X| = |J|. O A set is countable if it is either finite or

denumerable.O A set is uncountable if it is not countable.


Some Properties

O Proposition: Every subset of J is countable. Consequently, each subset of any denumerable set is countable.

O Proposition: A function f: J→ Y has a countable range. Hence any function on a countable domain has a countable range.

O Proposition: The set J × J is denumerable. Therefore, A × B is countable for arbitrary countable sets A and B.


Some Properties (cont.)

O Proposition: The set A È B is countable whenever Aand B are countable sets.

O Proposition: Every infinite set X is at least denumerable ; that is |X| ≥ |J|.

O Proposition: The set of all infinite sequence on {0, 1} is uncountable.


Some Properties (cont.)

O Proposition: (Schröder-Bernestein Theorem) For any set A and B, if |A| ≥ |B| and |B| ≥ |A|, then|A| = |B|.

O Proposition: (Cantor’s Theorem)For any set X,

|X| < |P(X)|.


1-1 correspondence Q↔NO Proof (dove-tailing):


Countable Sets

O Any subset of a countable setO The set of integers, algebraic/rational numbersO The union of two/finite number of countable setsO Cartesian product of a finite number of countable sets O The set of all finite subsets of NO Set of binary strings


Diagonal Argument


Uncountable Sets

O R, R2, P(N)O The intervals [0,1), [0, 1], (0, 1)O The set of all real numbersO The set of all functions from N to {0, 1}O The set of functions N→ NO Any set having an uncountable subset


Transfinite Cardinal Numbers

O Cardinality of a finite set is simply the number of elements in the set.

O Cardinalities of infinite sets are not natural numbers, but are special objects called transfinite cardinal numbers.

O À0:º|N|, is the first transfinite cardinal number.O continuum hypothesis claims that |R|=À1, the second

transfinite cardinal.


Alphabet

O A finite and nonempty set of symbols (usually shown by ∑ or Г).

O Example: ∑ = {$, &, ', … , )}��


Strings

O A finite list of symbols from an alphabetO Example: houseO If ω is a string over ∑, the length of ω, written |ω|, is

the number of symbols that it contains. O The empty string (ε or λ) is the string of length zero.

εω = ωε = ωO String z is a substring of ω if z appears consecutively

within ω.


Operations on StringsO Reverse of a string:

w = a1a2…an ababaaabbbwR = an…a2a1 bbbaaababa

O Concatenation:w = a1a2…an abbav = b1b2…bm bbbaaawv = a1a2…an b1b2…bm abbabbbaaa |wv| = |w| + |v|


Operations on AlphabetO * :

§ The set of all strings that can be produced from ∑.

O + :§ The set of all strings, excluding λ, that can be produced from ∑.§ Suppose that λ={λ}. Then:


LanguagesO A set of stringsO Any language on the alphabet ∑ is a subset of ∑*.O Examples:


Operations on LanguagesLanguages are a special kind of sets and operations on sets can be defined on them as well.

O Union

O Intersection

O Relative Complement

O Complementλ, b, aa, ab, bb, aaa, …}


Operations on Languages (cont.)

O Reverse of a language

O Example:


Operations on Languages (cont.)

O Concatenation

O Example:


Operations on Languages (cont.)O *: Kleene *

§ The set of all strings that can be produced by concatenation of strings of a language.

§ Example:

O +: § The set of all strings, excluding λ, that can be produced by

concatenation of strings of a language.


Gödel Numbering

O Let ∑ be an alphabet containing n objects. Let h: ∑→ Jn be an arbitrary one-to-one correspondence. Define function f as:

f: ∑* → Nsuch that f(ε) =0; f(w.v) = n f(w) + h(v), for wÎ ∑* and vÎ ∑.f is called a Gödel Numbering of ∑*.

O Proposition: ∑* is denumerable.O Proposition: Any language on an alphabet is countable.


Theory of Languages and Automatace.sharif.edu/courses/97-98/2/ce415-1/resources/root... · 2020. 9....

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