1. 1 Finite Model Theory Lecture 1: Overview and Background

2. 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. 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. 4 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)

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

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

7. 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. 8 Model or  -Structure A = STRUCT[  ] = all  -structures

9. 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. 10 Classical Results Godel’s completeness theorem Compactness theorem Lowenheim-Skolem theorem [Godel’s incompleteness theorem] We discuss these in some detail next

11. 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. 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. 13 Logical Inference We write  ²  if: 8 A, if A ²  then A ²  Note: ² is a semantic operation

14. 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. 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. 16 Compactness Theorem Theorem If for any finite  0 µ ,  0 is satisfiable, then  is satisfiable Proof: [in class]

17. 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. 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. 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. 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. 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. 22 Summary of Classical Results Completeness, Compactness, LS

23. 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. 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. 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   ; ?