1. 1 Finite Model Theory Lecture 16 L  1  Summary and 0/1 Laws

2. 2 Outline Summary on L  1  –All you need to know in 5 slides ! Start 0/1 Laws: Fagin’s theorem –Will continue next time Infinitary Logics and 0-1 Laws, Kolaitis&Vardi, 1992 New paper:

3. 3 Summary on L  1  Notation Comes from in classical logic L  = formulas where: –Conjunctions/disjunctions of ordinal <  Ç i 2   i, Æ  2 , where  <  –Quantifier chains of ordinal <  9 i 2  x i. , where  <  Hence, L 1  = [  L 

4. 4 Summary on L  1  Motivation Any algorithmic computation that applies FO formulas is expressible in L  1  Relational machines While-programs with statements R :=  Fixpoint logics: LFP, IFP, PFP, etc, etc Consequence: cannot express EVEN, HAMILTONEAN

5. 5 Summary on L  1  Canonical Structure Any algorithmic computation on A can be decomposed Compute the ¼ k equivalence relation on k-tuples, and order the equivalence classes ) in LFP [how do we choose k ???] Then compute on ordered structure ) any complexity Consequence: PTIME=PSPACE iff IFP=PFP But note that DTC  TC yet L  ? NL [ why ?]

6. 6 Summary on L  1  Pebble Games: with k pebbles Notation: A  1  k B if duplicator wins Theorem 1. For any two structures A, B: A, B are L k 1  equivalent iff A  1  k B Theorem 2. If A, B are finite: A, B are FO k equivalent iff A, B are L k 1  equivalent iff A  1  k B

7. 7 Summary on L  1  Definability of FO k types FO k types are the same as L k 1  types [ why ?] Theorem [Dawar, Lindell, Weinstein] The type of A (or of (A, a)) can be expressed by some  2 FO k B ²  [b] iff Tp k (A,a) = Tp k (B,b) Difficult result: was unknown to Kolaitis&Vardi

8. 8 0/1 Laws in Logic Motivation: random graphs 0/1 law for FO proven by Glebskii et al., then rediscovered by Fagin (and with nicer proof) –Only for constant probability distribution Later extended to other logics, and other probability distributions Why we care: applications in degrees of belief, probabilistic databases, etc.

9. 9 Definitions Let  = a vocabulary Let n ¸ 0, and A n µ STRUCT[  ] be all models over domain {0, 1, …, n-1} Uniform probability distribution on A n Given sentence , denote  n (  ) its probability

10. 10 Definition Denote  (  ) = lim n ! 1  n (  ) if it exists Definition A logic L has a convergence law if for every sentence ,  (  ) exists Definition A logic L has a 0/1 law if for every sentence ,  (  ) exists and is 0 or 1

11. 11 Theorems Suppose  has no constants Theorem [Fagin 76, Glebskii et al. 69] FO admits a 0/1 law Theorem [Kolaitis and Vardi 92] L  1  admits a 0/1 law

12. 12 Application What does this tell us for database query processing ? Don’t bother evaluating a query: it’s either true or false, with high probability

13. 13 Examples [ in class ] Compute  n (  ), then  (  ): R(0,1) /* I’m using constants here */ R(0,1) Æ R(0,3) Æ : R(1,3) 9 x.R(2,x) : ( 9 x. 9 y.R(x,y)) 8 x. 8 y.( 9 z.R(x,z) Æ R(z,y))

14. 14 Types We only need rank-0 types (i.e. no quantifiers) Recall the definition Definition A type t(x) over variables (x 1, …, x m ) is conjunction of a maximally consistent set of atomic formulas over x 1, …, x m

15. 15 Types The type t(x) says: For each i, j whether x i = x j or x i  x j For each R and each x i 1, …, x i p whether R(x i 1, …, x i p ) or : R(x i 1, …, x i p )

16. 16 Extension Axioms Definition Type s(x, z) extends the type t(x) if {s, t} is consistent; Equivalently: every conjunct in t occurs in s Definition The extension axiom for types t, s is the formula  t,s = 8 x 1 … 8 x k (t(x) ) 9 z.s(x, z))

17. 17 Example of Extension Axiom t(x 1, x 2, x 3 ) = x 1  x 2 Æ x 2  x 3 Æ x 1  x 3 Æ R(x 1,x 2 ) Æ R(x 2,x 3 ) Æ R(x 2,x 2 ) Æ : R(x 1, x 1 ) Æ : R(x 2, x 1 ) Æ … x1x1 x2x2 x3x3 s(x 1, x 2, x 3, z) = t(x 1, x 2, x 3 ) Æ z  x 1 Æ z  x 2 Æ z  x 3 Æ R(z,x 1 ) Æ R(x 3,z) Æ R(z,z) Æ : R(x 1, z) Æ : (z, x 2 ) Æ … z

18. 18 Example of Extension Axiom 8 x 1. 8 x 2. 8 x 3. (t(x 1, x 2, x 3 ) ) 9 z. s(x 1, x 2, x 3, z))  t,s =

19. 19 The Theory T Let T be the set of all extension axioms –Studied by Gaifman Is T consistent ? –In a model of T the duplicator always wins [ why ? ] Does it have finite models ? Does it have infinite models ?

20. 20 The Theory T Let  k be the conjunction of all extension axioms for types with up to k variables There exists a finite model for  k [why ?] Hence any finite subset of T has a model Hence T has a model. [can it be finite ?]

21. 21 The Model(s) of T T has no finite models, hence it must have some infinite model By Lowenheim-Skolem, it has a countable model

22. 22 The Theory T Theorem T is  -categorical Proof: let A, B be two countable model. Idea: use a back-and-forth argument to find an isomorphism f : A ! B

23. 23 The Theory T Theorem T is  -categorical Proof: (cont’d) A = {a 1, a 2, a 3, ….} B = {b 1, b 2, b 3, ….} Build partial isomorphisms f 1 µ f 2 µ f 3 µ … such that: 8 n. 9 m. a n 2 dom(f m ) and 8 n. 9 m. b n 2 rng(f m ) [in class] Then f = ( [ m ¸ 1 f m ) : A ! B is an isomorphism

24. 24 The Theory T Corollary T has a unique countable model R R = the Rado graph = the “random” graph Corollary The theory Th(T) is complete

25. 25 0/1 Law for FO Lemma For every extension axiom ,  (  ) = lim n  n (  ) = 1 Proof: later Corollary For any m extension axioms  1, …,  m :  (  1 Æ … Æ  m ) = 1 Proof  n ( : (  1 Æ … Æ  m )) =  n ( :  1 Ç … Ç :  m ) ·  n ( :  1 ) + … +  n ( :  m ) ! 0

26. 26 Fagin’s 0/1 Law for FO Theorem For every  2 FO, either  (  ) = 0 or  (  ) = 1. Proof. Case 1: R ² . Then there exists m extension axioms s.t.  1, …,  m ² . Then  n (  ) ¸  n (  1 Æ … Æ  m ) ! 1 Case 2: R 2 . Then R ² : , hence  ( :  ) = 1, and  (  ) = 0

27. 27 Proof for the Extension Axioms Let  = 8 x. t(x) ) 9 z.s(x, z) Assume wlog that t asserts x i  x j forall i  j. Denote  (x) the formula Æ i < j x i  x j –Hence t(x) =  (x) Æ t’(x) Similarly, s asserts z  x i forall i. Denote  (x, z) = Æ i x i  z –Hence s(x, z) = t(x) Æ  (x, z) Æ s’(x, z) where all atomic predicates in s’(x, z) contain z Hence:  = 8 x.(  (x) Æ t’(x) ) 9 z.  (x,z) Æ s’(x, z))

28. 28 Proof for the Extension Axioms :  = 9 x.(  (x) Æ t’(x) Æ 8 z.(  (x, z) ) : s’(x, z)))  n ( :  ) ·  n ( 9 x.(  (x) Æ 8 z.(  (x, z) ) : s’(x, z))))

29. 29 Proof for the Extension Axioms  n ( :  ) ·  n ( 9 x.(  (x) Æ 8 z.(  (x, z) ) : s’(x, z)))) ·  a 1,..., a k 2 {1, …, n}  n ( 8 z. (  (x, z) ) : s’(a 1, …, a k, z))) = n(n-1)…(n-k+1)  n ( 8 z.  (x, z) ) : s’(1, 2, …, k, z)) · n k  n ( 8 z.  (x, z) ) : s’(1, 2, …, k, z)) = = n k  z=k+1, n : s’(1,2,…,k,z) /* by independence !! */ = n k ( 1 - 1 / 2 2k+1 ) n-k /* since s’ is about 2k+1 edges */ ! 0 when n ! 1

30. 30 Complexity Theorem [Grandjean] The problem whether  (  ) = 0 or 1 is PSPACE complete

31. 31 Discussion Old way to think about formulas and models: finite satsfiability/ validity FO  unsatisfiable  valid Undecidable

32. 32 Discussion New way to think about formulas and models: probability FO  unsatisfiable  valid PSPACE  )=0  )=1