Logics, automata and algorithms for graphs

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Logics, automata and algorithms for graphs

• p. madhusudan
• (madhu)
• University of Illinois
• at Urbana-Champaign, USA

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What is logic?

• Logic is a precise mathematical language where
• every sentence has a precise meaning.
• Example FOL (N, , lt, 0, 1)
• 8x 9y (x y) -- every number has some
  
• number greater than
  it
• Example Regular expressions
• Non-example English law books

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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

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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

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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, lt) is decidable
• using automata theory

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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

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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)

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Overview

• Graphs of bounded tree width (Lec 7)
• Courcelles 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)