Notes

Outline

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 In this course we will learn FMT together

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

Resources www.cs.washington.edu/599ds Books Leonid Libkin, Elements of Finite Model Theory main 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)

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

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

First Order Logic = FO

Model or s-Structure

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

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

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

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

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

Godel’s Completeness Result Theorem (soundness) If G ` f then G ² f Theorem (completeness) If G ² f then G ` f

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 ² G 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 `]

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

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

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 ² G Then show that G has an infinite model A   [in class]

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

Decidability CN(G) = {f | G ² f} A theory T is a set s.t. CN(T) = T  T is complete if 8 f either T ² f or T ² : 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

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

Summary of Classical Results Completeness, Compactness, LS

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

A taste of FMT Proof  Suppose CONN is expressed by f, i.e. G ² f 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 ² f but there is no path from s to t, contradiction THIS PROOF IS INSSUFFICIENT OF US. WHY ?

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 s ¹ ; ?