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Finite Model Theory

Finite Model Theory. Lecture 1: Overview and Background. Motivation. Applications: DB, PL, KR, complexity theory, verification Results in FMT often claimed to be known Sometimes people confuse them Hard to learn independently Yet intellectually beautiful

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Finite Model Theory

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  1. Finite Model Theory Lecture 1: Overview and Background

  2. Motivation • Applications: • DB, PL, KR, complexity theory, verification • Results in FMT often claimed to be known • Sometimes people confuse them • Hard to learn independently • Yet intellectually beautiful • In this course we will learn FMT together

  3. Organization • Powerpoint lectures in class • Some proofs on the whiteboard • No exams • Most likely no homeworks • But problems to “think about” • Come to class, participate

  4. Resources www.cs.washington.edu/599ds Books • Leonid Libkin, Elements of Finite Model Theorymain text • H.D. Ebbinghaus, J. Flum, Finite Model Theory • Herbert Enderton A mathematical Introduction to Logic • Barwise et al. Model Theory (reference model theory book; won't really use it)

  5. Today’s Outline • Background in Model Theory • A taste of what’s different in FMT

  6. Classical Model Theory • Universal algebra + Logic = Model Theory • Note: the following slides are not representative of the rest of the course

  7. First Order Logic = FO Vocabulary: s = {R1, …, Rn, c1, …, cm} Variables: x1, x2, … t ::= c | x f ::= R(t, …, t) | t=t | fÆf | fÇf | :f | 9 x. f | 8 x.f In the future:Second Order Logic = SO Add: f ::= 9 R. f | 8 R.f This is SYNTAX

  8. Model or s-Structure A = <A, R1A, …, RnA, c1A, …, cmA> STRUCT[s] = all s-structures

  9. Interpretation • Given: • a s-structure A • A formula f with free variables x1, …, xn • N constants a1, …, an2 A • Define A ²f(a1, …, an) • Inductively on f

  10. Classical Results • Godel’s completeness theorem • Compactness theorem • Lowenheim-Skolem theorem • [Godel’s incompleteness theorem] We discuss these in some detail next

  11. Satisfiability/Validity • f is satisfiable if there exists a structure A s.t. A ²f • f is valid if for all structures A, A ²f • Note: f is valid iff :f is not satisfiable

  12. Logical Inference • Let G be a set of formulas • There exists a set of inference rules that define G`f [white board…] Proposition Checking G`f is recursively enumerable. Note: ` is a syntactic operation

  13. Logical Inference • We write G²f if: 8 A, if A ²G then A ²f • Note: ² is a semantic operation

  14. Godel’s Completeness Result Theorem (soundness) If G`f then G²f Theorem (completeness) If G²f then G`f Which one is easy / hard ? It follows that G²f is r.e. Note: we always assume that G is r.e.

  15. Godel’s Completness Result • G is inconsistent if G` false • Otherwise it is called consistent • G has a model if there exists A s.t. A ²G Theorem (Godel’s extended theorem) G is consistent iff it has a model This formulation is equivalent to the previous one [why ? Note: when proving it we need certain properties of `]

  16. Compactness Theorem Theorem If for any finite G0µG, G0 is satisfiable, then G is satisfiable Proof: [in class]

  17. Completeness v.s. Compactness • We can prove the compactness theorem directly, but it will be hard. • The completeness theorem follows from the compactness theorem [in class] • Both are about constructing a certain model, which almost always is infinite

  18. Application • Suppose G has “arbitrarily large finite models” • This means that 8 n, there exists a finite model A with |A| ¸ n s.t. A ²G • Then show that G has an infinite model A [in class]

  19. Lowenheim-Skolem Theorem Theorem If G has a model, then G has an enumerable model Upwards-downwards theorem: Theorem [Lowenheim-Skolem-Tarski] Let l be an infinite cardinal. If G has a model then it has a model of cardinality l

  20. Decidability • CN(G) = {f | G²f} • A theory T is a set s.t. CN(T) = T • T is complete if 8f either T²f or T²:f • If T is finitely axiomatizable and complete then it is decidable. • Los-Vaught test: if T has no finite models and is l-categorical then T is complete

  21. Some Great Theories • Dense linear orders with no endpoints [in class] • (N, 0, S) [in class] • (N, 0, S, +) Pressburger Arithmetic • (N, +, £) : Godel’s incompleteness theorem

  22. Summary of Classical Results • Completeness, Compactness, LS

  23. A Taste of FMT Example 1 • Let s = {R}; a s-structure A is a graph • CONN is the property that the graph is connected Theorem CONN is not expressible in FO

  24. A taste of FMT • Proof Suppose CONN is expressed by f, i.e. G ²f iff G is connected • Let s’=s[ {s,t}yk = :9 x1, …, xk R(s,x1) Æ … Æ R(xk,t) • The set G = {f} [ {y1, y2, …} is satisfiable (by compactness) • Let G be a model: G ²f but there is no path from s to t, contradiction THIS PROOF IS INSSUFFICIENT OF US. WHY ?

  25. A taste of FMT Example 2 • EVEN is the property that |A| = even Theorem If s = ; then EVEN is not in FO • Proof [in class] But what do we do if s¹; ?

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