1. Logics, automata and algorithms for graphs p. madhusudan (madhu) University of Illinois at Urbana-Champaign, USA

2. What is logic?  Logic is a precise mathematical language where every sentence has a precise meaning.  Example: FOL (N, +, <, 0, 1) 8 x 9 y (x · y) -- every number has some number greater than it  Example: Regular expressions  Non-example: English; law books

3. 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: 8 x 9 y E(x,y)  Logics on graphs can hence state algorithmic problems on graphs

4. 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

5. 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

6. 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 Theme Algorithmic problem  Logic  Automata  Algorithm

7. 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)

8. 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)