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 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)

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 t ::= c | x  ::= R(t, …, t) | t=t |  Æ  |  Ç  | :  | 9 x.  | 8 x.  t ::= c | x  ::= R(t, …, t) | t=t |  Æ  |  Ç  | :  | 9 x.  | 8 x.  Vocabulary:  = {R 1, …, R n, c 1, …, c m } Variables: x 1, x 2, … In the future: Second Order Logic = SO Add:  ::= 9 R.  | 8 R.  This is SYNTAX

8 Model or  -Structure A = STRUCT[  ] = all  -structures

9 Interpretation Given: –a  -structure A –A formula  with free variables x 1, …, x n –N constants a 1, …, a n 2 A Define A ²  (a 1, …, a n ) –Inductively on 

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

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

13 Logical Inference We write  ²  if: 8 A, if A ²  then A ²  Note: ² is a semantic operation

14 Godel’s Completeness Result Theorem (soundness) If  `  then  ²  Theorem (completeness) If  ²  then  `  Which one is easy / hard ? It follows that  ²  is r.e. Note: we always assume that  is r.e.

15 Godel’s Completness Result  is inconsistent if  ` false Otherwise it is called consistent  has a model if there exists A s.t. A ²  Theorem (Godel’s extended theorem)  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  0 µ ,  0 is satisfiable, then  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  has “arbitrarily large finite models” –This means that 8 n, there exists a finite model A with |A| ¸ n s.t. A ²  Then show that  has an infinite model A [in class]

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

20 Decidability CN(  ) = {  |  ²  } A theory T is a set s.t. CN(T) = T  is complete if 8  either  ²  or  ² :  If T is finitely axiomatizable and complete then it is decidable. Los-Vaught test: if T has no finite models and is -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  = {R}; a  -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 , i.e. G ²  iff G is connected Let  ’=  [ {s,t}  k = : 9 x 1, …, x k R(s,x 1 ) Æ … Æ R(x k,t) The set  = {  } [ {  1,  2, …} is satisfiable (by compactness) Let G be a model: G ²  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  = ; then EVEN is not in FO Proof [in class] But what do we do if   ; ?