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Finite Model Theory Lecture Overview: Applications, Results, and Organization

This lecture provides an introduction to Finite Model Theory (FMT) covering topics like background in model theory, differences from Classical Model Theory, and key results such as Godel’s completeness and Lowenheim-Skolem theorems. The course includes PowerPoint lectures, whiteboard proofs, and encourages class participation with minimal exams or homework. Resources like textbooks are recommended, and the course aims to tackle the intellectually beautiful but challenging aspects of FMT together.

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Finite Model Theory Lecture Overview: Applications, Results, and Organization

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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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