Logic: Lecture 19, 27 October 2015
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(Reference: Applied Automata Theory by Wolfgang Thomas, Chapters 0,1)

Goedel: (N,0,+,*) is incomplete (and hence undecidable)
Presburger: (N,0,+) is decidable

Can we extend theory of (N,0,succ) in another direction?

First order vs second order logic
  FO: quantify over individual elements of a set
  SO: quantify over functions, relations

Monadic second order logic (MSO): 

  Quantify over unary relations
  Equivalent to adding set quantifiers:

Focus on MSO over (N,0,succ)

Examples:

  x < y is definable
  Singleton(X), Finite(X)  
  Principle of Mathematical Induction 

Restrict to finite models and add labels to positions

  Additional constants min, max
  Interpret over words over a finite alphabet

MSO definable languages

  Given a sentence phi : L(phi) = { w | w |= phi }

  Examples: Fix alphabet = {a,b,c}
  - Words containing an a
  - Words with even length

Regular expressions vs MSO definability

  - Words with a pair of b's with no c's in between
  - Words without a pair of b's with no c's in between
  - Words where every pair of b's has no c's in between

  Regular expressions are good for "positive" requirements, not
  so natural for "negative" conditions.  Not easy to transform
  regular expressions to account for small changes in the
  requirement.

  Logical specifications can be complemented, combined using
  and/or.  Easy to inductively combine constraints on language.

Finite automata

  DFAs and NFAs
  Complete and incomplete DFAs
  Checking emptiness