Notes

Outline

Finite Model Theory Lecture 8 Logics and Complexity Classes

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

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 ?

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

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     9k. 8A.  A ² f iff Aadomk ² f PROBLEM.  IS WEAK DOMAIN INDEPENDENCE DECIDIABLE ? CAN ONE COMPUTE K (OR AN UPPER BOUND) ?

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

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

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.)

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)

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

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 ?

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

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

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

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

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:

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

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)

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

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.

Example 1 s = {E} (graphs) What does this formula say ?

Examples What does this say ?

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 !

Example 3 (cont’d)

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

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:

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

Discussion 2 Simultaneous fixpoints Is there a path of even length from a to b ?

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

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

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