Logics, automata and algorithms for

graphs

p. madhusudan(madhu)

University of Illinois at Urbana-Champaign, USA

What is logic? Logic is a precise mathematical language where every sentence has a precise meaning.

Example: FOL (N, +, <, 0, 1) 8x 9y (x · y) -- every number has some

number greater than it

Example: Regular expressions

Non-example: English; law books

Logics on graphs

A graph G=(V,E) can be treated as a logical structure.

The set of vertices V is the universe; E is a relation

Eg. Every vertex is adjacent to some vertex: 8x 9y E(x,y)

Logics on graphs can hence state algorithmic problems on graphs

Logics on graphs Two main problems:

Membership: Given a graph G, solve the algorithmic problem for G. i.e. does G satisfy ? Eg. Is the graph 3-colorable?

Emptiness: Given is there some graph that satisfies it?Eg. planar and 5-colorable but not 4-colorable -- 4 color theorem

What are automata? Automata are machines that process structures

(graphs), and accept or reject them.

Eg. Automata on words, automata on trees

Automata usually have a decidable membership/emptiness problems

(unlike Turing machines).

Hence give decidable algorithms for structures Logic Automata Eg. FOL (N, +, 0, <) is decidable using automata theory

Algorithms

Algorithms on structures.

Algorithmic problem Logic eg. 3 colorability of graphs can be expressed using logic

Leads to designing linear time algorithms for problems on particular classes of graphs

ThemeAlgorithmic problem Logic Automata Algorithm

Overview Overview (Lec 1) Automata on words: (Lec 2)

Closure properties, monadic second order logic (MSO), equivalence of MSO and regular languages

Decidability of Presburger arithmetic using automata (Lec 3) Automata on trees: (Lec 4) Closure properties, top-down vs bottom-up tree automata,

MSO, equivalence of MSO and regular tree languages.

Deciding MSO on Series parallel graphs using tree interpretations (Lec 5)

Deciding MSO on Nested words using tree interpretations Applications to XML stream processing (Lec 6)

Overview Graphs of bounded tree width (Lec 7); Courcelle’s thm: MSO on BTW graphs is solvable in linear time (Lec 8)

Decidability of satisfiability for logics on graphs; and Finite model theory (Lec 9 and Lec 10)