1. 1 Finite Model Theory Lecture 9 Logics and Complexity Classes (cont’d)

2. 2 Outline Proof of Immerman and Vardi’s theorems Normal form for lfp Datalog Other logics capturing complexity classes

3. 3 Proof: LFP = PTIME LFP + < µ PTIME [why ?] To show PTIME µ LFP + < Let M = (Q, , , , q 0, Q a, Q r ) be a PTIME Turing machine Input: Enc( A ) = 01 n Enc(R 1 A )…Enc(R m A ) Output: accept or reject Need: formula  s.t. A ²  iff T accepts A

4. 4 Proof: LFP = PTIME Time t and space p vary from 0 to n k -1 –Represent them as k-tuples Can define t < t’ [ how ? ] Can define succ(t, t’) [ how ?] Goal: express the following: T 0 (p, t) = true iff tape has 0 on p at time t T 1 (p, t) = true iff tape has 1 on p at time t H q (p, t) = true iff head is on p at time t, state q All these can be expressed by simultaneous fixpoint [ how ? ]

5. 5 Proof: PFP = PSPACE PFP + < µ PSPACE [ why ? ] To show PSPACE µ PFP + < proceed as before, but can’t define T 0 (p, t) because we can’t express t Instead define T 0 (p), T 1 (p), H q (p) using a PFP Each new step recomputes these tables Make sure to define T 0 (p+1) = T 0 (p) etc whenever reaching an accepting state

6. 6 Remark Somewhere in the proof we need to use + and £, defined in terms of < Where ?

7. 7 Simultaneous Fixpoints U 1, U 2, …, U k = finite sets Definition Let F : P(U 1 ) £ … £ P(U k ) ! P(U 1 ) £ … £ P(U k ). A fixpoint for F is X = (X 1, …, X k ), X 1 µ U 1, …, X k µ U k s.t. F(X) = X lfp(F), ifp(F), pfp(F) defined as before

8. 8 Simultaneous Fixpoints Given  = (  1 (R 1, …, R k, x 1 ), …,  k (R 1, …, R k, x k )) lfp R i simult [  ](t), ifp R i simult [  ](t), pfp R i simult [  ](t) Mean the following: compute the lfp/ifp/pfp extract the i’th component (since index is R i )

9. 9 Simultaneous Fixpoints LFP simult = LFP IFP simult = IFP PFP simult = PFP

10. 10 Normal Form Every IFP or LFP formula is equivalent to one formula “with only one LFP, at the top”: 9 x 1 … 9 x k. lfp R [  ](x) where  is in FO (i.e. without fixpoints) Proof: in the book (not hard), but better see next

11. 11 Datalog Recall: a conjunctive query has the form: q(x) = 9 y 1 … 9 y k.  (x, y) where  is a conjunction of positive atomic formulas Example: stands for 9 u. 9 v.(R(x,u) Æ R(u,v) Æ R(v,y)) q(x,y) :- R(x,u), R(u,v), R(v,y)

12. 12 Datalog Definition. EDB Extensional relations = are those in  Definition. IDB Intensional relations = m new relation names R 1, …, R m Definition. A datalog rule is: where R =intensional  = conjunctive query R(x) :- 

13. 13 Datalog Definition. A datalog program is a set of rules: R i 1 :-  1 R i 2 :-  2.... R i m :-  m

14. 14 Examples in Datalog Transitive closure: Same generation: T(x,y) :- R(x,y) T(x,y) :- R(x,z), T(z,y) S(x,y) :- Root(u), R(u,x), R(u,y) S(x,y) :- S(u,v), R(u,x), R(v,y)

15. 15 Datalog Semantics Simultaneous least fixpoints: Given a program P, define the operators F P as follows: F P (R 1, …, R m ) = (  1, …,  m ) Next, take the lfp(F P )

16. 16 Adding negation Semipositive datalog: Allow negation only on the EDBs If we allow negation on IDBs, what is the semantics ?

17. 17 Semantics 1: Stratified Datalog Split datalog program into “strata”. Each stratum may use negation of IDBs defined by previous strata. Evaluate program in stages Example: find pairs x,y that are not connected: T(x,y) :- R(x,y) T(x,y) :- R(x,z), T(z,y) Answer(x,y) :- : T(x,y) here we have 2 strata

18. 18 Semantics 2: Inflationary Datalog We know what this is… Find all pairs x, y that are not connected: T(x,y) :- R(x,y) T(x,y) :- R(x,z), T(z,y) oldT(x,y) :- T(x,y) oldTexceptFinalT(x,y) :- T(x,y), R(x’,z’),T(z’,y’), : T(x’,y’) Answ(x,y) :- : T(x,y), oldT(x’,y’), : oldTexceptFinalT(x’,y’)

19. 19 Datalog : s ½ Datalog : i Every stratified datalog program can be expressed as an inflationary datalog program The “winning game” query can be expressed in inflationary datalog but not in stratified datalog [ show the query in class ] [ inexpressibility: Kolaitis’91, in Information and Control]

20. 20 Datalog : i = LFP Theorem Datalog : i = LFP Corollary Datalog :,< i = PTIME

21. 21 Semantics 3: Partial Fixpoint Datalog Partial fixpoint semantics Theorem Datalog : p = PFP Corollary Datalog :,< p = PSPACE

22. 22 FO+TrCl Definition Transitive closure. Let  (x, y, z) be a formula, where |x| = |y| = k, and t 1, t 2 are tuples of length k. Then: [trcl x, y  (x, y, z)](t 1, t 2 ) is a formula with free variables z.

23. 23 TrCl The logic FO extended with trcl is denoted TrCl Note: may combing trcl arbitrarily with Æ, Ç, :, 8, 9.

24. 24 Examples in TrCl Is the graph connected: Is there a path from u to v where all edges are labeled with the same color: here c is a free variable in trcl Do we need free variables ? 8 u. 8 v.[trcl x,y (R(x,y))](u,v) 9 c..[trcl x,y (R(x,c,y))](u,v)

25. 25 Homework Problem Let TrCl r denote the logic where trcl is applied to tuples x, y with |x| = |y| · r Consider structures that represent strings over, say  = {a,b}. I.e.  = {<, A, B}. E.g. abbab is represented by ({1,2,3,4,5}, <, A(1),A(4),B(2),B(3),B(5)) Prove that on strings TrCl 1  TrCl 2

26. 26 Immerman’s Theorem Theorem TrCl + < = NLOGSPACE Corollary NLOGSPACE is closed under complementation.

27. 27 Proof Sketch 1. TrCl \subseteq NLOGSPACE Lemma. Denote PosTrCl the language where trcl is used only in positive positions (under even number of negations). Then: PosTrCL + < = TrCl + < You proved it already in 531 ! Plus: recall how inflationary datalog expresses not TC.

28. 28 Proof Sketch NLOGSPACE \subseteq TrCl + < Standard simulation of a Turing Machine

29. 29 Logics and Complexity Classes DTrCl = transitive closure on deterministic graphs (i.e. where 8 x. 9 · 1 y.  (x, y)). Theorem DTrCl+< = LOGSPACE ATrCl = alternating transitive closure [ define in class ] Theorem ATrCl + < = PTIME

30. 30 Major Open Problem What happens without < ? Open problem. Find a “logic” that captures precisely the order invariant PTIME properties Read 10.7 in the book (pp. 204)