1. Finite Model Theory Lecture 8
Logics and Complexity Classes

2. Reading Assignment
R. Fagin, Finite-Model Theory – a Personal Perspective
May skip Sec. 6 for now, since we will cover this in class

3. Problem 1
Let s ¹ ;.
[Fagin’74] A class C µ STRUCT[s] is in 9 SO iff it is in NP
Now let s = ;. A structure = a number
[Fagin’74, Jones&Selman’74] A set of numbers C µ N is in 9 SO iff it is in NE (nondeterministic exponential time)
PROVE IT. WHY THE JUMP IN COMPLEXITY ?

4. Problem 2 Let A = (A, R1A, …, RmA) be a structure.
The active domain, adom(A) = the set of all constants in s, and all constants occurring in R1A, …, RmA.
Let Aadom = (adom(A), RA1, …, RAm)
A formula f is domain independent if 8 A: A ² f iff Aadom ² f
Undecidable if f is domain independent

5. Problem 2 (con’t)
Let k ¸ 0. The k’th active domain of A adomk(A) = adom(A)[{some k constants in A -adom(A)} adomk(A) = A if |A - adom(A)| · k
Aadomk = (adomk(A), RA1, …, RAm)
Say that f is weak domain independent if k. 8A. A ² f iff Aadomk ² f
PROBLEM. IS WEAK DOMAIN INDEPENDENCE DECIDIABLE ? CAN ONE COMPUTE K (OR AN UPPER BOUND) ?

6. Problem 3
f(R, x) is monotone in R if for any two structures A, B s.t. RA µ RB, A=B, and SA=SB for any other relation name S, {a | A ² f} µ {a | B ² f}
Prove that it is undecidable if a formula f(R, x) is monotone in R

7. Outline Fixpoints Fixpoint Logics
Immerman and Vardi’s results: FO+LFP+< = FO+IFP+< = PTIME FO+PFP+< = PSPACE

8. Motivation Recall Fagin’s result: 9 SO = NP
Are there other logics that capture complexity classes ?
Yes ! But those logics are not as elegant as 9 SO. (Or, it takes more arguing to present them as elegant.)

9. Least Fixpoint Fix U = a finite set
Definition Let F : P(U) ! P(U). A fixpoint for F is X µ U s.t. F(X) = X
Definition The least fixpoint is some X s.t. for every other fixpoint Y, X µ Y
Notation: lfp(F)

10. Fixpoints Define: X0 = ; Xi+1 = F(Xi) X1 = [i ¸ 0 Xi
Is X1 the least fixpoint of F ?

11. Least Fixpoint F is monotone if X µ Y ) F(X) µ F(Y)
Theorem [Tarski-Knaster] If F is monotone, then lfp(F) exists and is given by any of the following two expressions: lfp(F) = Å {Y | F(Y) µ Y} lfp(F) = X1
Note: one of the above fails when U = infinite. Which one ?

12. Inflationary Fixpoint
F is inflationary if Y µ F(Y) for all Y
Fact. If F is inflationary, then X1 is a fixpoint
[is X1 the least fixpoint ?]
Denote ifp(F) = X1

13. Inflationary Fixpoint
Let F be arbitrary.
Define G(Y) = Y [ F(Y).
G is inflationary [why ?], hence X1 is its inflationary fixpoint
Define ifp(F) = X1

14. Partial Fixpoint Let F be arbitrary, X0 = ;, Xi+1 = F(Xi)
Define: pfp(F) = Xn if Xn = Xn+1 pfp(F) = ; if there is no such n

15. Fixpoints
The distinction is necessary only when F is not monotone. Otherwise:
Fact If F is monotone, then: lfp(F) = ifp(F) = pfp(F)

16. Fixed Point Logics s = vocabulary, R Ï s
Let f(R, x) be a formula over s [ {R} here |x| = arity(R)
[that is, x are all the free variables in f]
Define three new formulas:
[lfpR, x f(R, x)][t]
[ifpR, x f(R, x)][t]
[pfpR, x f(R, x)][t]

17. Fixed Point Logics Hence, three new logics: FO + LFP or LFP
FO + IFP or IFP
FO + PFP of PFP
Need semantics (next)

18. Fixed Point Logics
Given a structure A and formula f(R, x), define the following operator Ff : P(A) ! P(A): Ff(X) = { a | A ² f(X/R, a) }
The meaning of [lfpR, x f(R, x)][t] is lfp(Ff)
The meaning of [ifpR, x f(R, x)][t] is ifp(Ff)
The meaning of [pfpR, x f(R, x)][t] is pfp(Ff)

19. LFP WAIT ! lfp is defined only if Ff is monotone.
Perhaps restrict lfp to monotone formulas f only ?

20. LFP Theorem Checking if f is monontone is undecidable
[proof: in class]
Hence in lfp we require R to occur under an even number of negations.

21. Example 1 s = {E} (graphs) What does this formula say ?
8 u.8 v.[lfpR, x, y f(R, x, y)](u,v) where f(R, x, y) = (E(x,y) Ç 9 z.(E(x,z) Æ R(z,y))

22. Examples What does this say ?
8 u.[lfpS,x f(S, x)](u), where: f(S, x) = 8 y.(E(y,x) ! S(y))

23. Example 3 Consider the following game with one pebble on a graph G
Player 1 places pebble on node a
Player 2 moves the pebble to a successor
Player 1 moves the pebble to a successor …
Whoever cannot move, looses
WRITE A FORMULA THAT CHECKS IF PLAYER 1 WINS !

24. Example 3 (cont’d) [lfpS,xf(S,x)](a)
where: f(S,x) = 8 y(E(x,y) ! 9 z.(E(y,z) Æ S(z))

25. Fixed Point Logics LFP, IFP, PFP look very complicated and procedural
Nothing one can do; deeper analysis brings back some of the elegance (future lectures)
We made several arbitrary choices.
Let’s analyze some of the choices

26. Discussion 1 Should we allow free variables in lfp, ifp, pfp ? Example
Colored graph: E(x, c, y)
Check if every pair of nodes is connected by a path where all edges have the same color:
8 x.8 y.9 c.[lfpR,x,yf(R, x,y,c)](x,y)
Where
f(R,x,y,c) = E(x,c,y) Ç 9 z.(R(x,z) Æ E(z,c,y))

27. Discussion 1 (cont’d)
We can express the same formula without free variables in lfp
[show in class]
This holds in general

28. Discussion 2 Simultaneous fixpoints
Is there a path of even length from a to b ?
[lfpR,xf(R, S, x),y(R, S, x)](b)
Where
f(R,S,x) = 9 z.(S(x,z) Æ E(z,x)) y(R,S,x) = E(a,x) Ç 9 z.(R(x,z) Æ E(z,x))
EXPRESS THE SAME THING WITH ONLY ONE LFP !

29. Discussion 3 Nested Fixpoints
Example: have < express +, £ (in class)
How can we remove the nested fixpoint ?

30. LFP = IFP Theorem [Gurevitch and Shelah] LFP = IFP
Technically involved proof, see the book

31. Main Results
Theorem [Immeman86, Vardi82] LFP + < = IFP + < = PTIME
Theorem [Vardi82] PFP + < = PSPACE