Finite Model Theory Lecture 6 Complexity of FO

Outline Data Complexity Query Complexity and Combined Complexity

Complexities Data Complexity the complexity of {A | A ² f} for fixed f Query Complexity the complexity of {f | A ² f} for fixed A Combined complexity the complexity of {(A, f) | A ² f}

Data Complexity For every f, the complexity of {A | A ² f} is in PTIME [Why ?] However, it is much lower than PTIME (next)

Data Complexity Theorem The data complexity of FO is uniform AC0 What is AC0 ? What is uniform AC0 ? We will review next, but most importantly: uniform-AC0 µ LOGSPACE µ PTIME

AC0 Fix n ¸ 0. A boolean circuit with n inputs, C, is a rooted DAG with nodes labeled with labels from: {Æ, Ç, :, x1, …, xn} size(C) = number of gates depth(C) = length of longest path

AC0 Definition A language L µ {0,1}* is in non-uniform AC0 if there exists d > 0, a polynomial p(n), and a family of circuits (Cn)n ¸ 0 s.t.: size(Cn) · p(n) depth(Cn) · d, and 8 w 2 {0,1}*: w 2 L iff Cn(w) = true, where n=|w|

FO and AC0 Let All be a vocabulary consisting of all relations on N All = P(N) [ P(N2) [ P(N3) [ … In All we have names for <, +, /, … Definition FO(All) = FO over vocabulary All Interpretation: consider only ordered domains, assimilate with {0, 1, 2, …, n-1}. Each relation R on N is interpreted as its restriction to {0, 1, …, n-1}. Note: we can express EVEN in FO(All) [why ?]

FO and AC0 Theorem FO(All) µ non-uniform AC0 Proof sketch in class (hint: it’s simple…)

PARITY Let s = {U} (a unary table) The property PARITY is true on models A where |UA| is even PARITY and EVEN are very related, but…

Parity Theorem [Furst-Saxe-Sipser, Ajtai] PARITY is not expressible in non-uniform AC0 Corollary PARITY is not expressible in FO(All) Comment: i.e. there is no formula in FO over vocabulary {U} [ All that checks if |U| is even. Corollary Graph connectivity is not expressible in FO(All) [why ?]

Discussion EVEN is not expressible in FO But EVEN is expressible in FO(<, +) PARITY is not expressible in FO(<, +, exp, …, any-relation-on-N, ….) !

Uniform AC0 Non-uniform AC0 can express non-computable properties ! Need to restrict the association n ! Cn to something easily computable Complex definitions in complexity theory textbooks… Better: define uniform AC0 = FO(+, £) Alternatively FO(+, £) = FO(<, BIT)

Combined Complexity Theorem The combined complexity is in PSPACE [proof: in classs] Note: proof in book is wrong.

Combined Complexity Recursive function: function Eval(f) if f = 9 x.y then for all a 2 A do if Eval(y[a/x]) = true then return true return false if f = 8 x.y then for all a 2 A do if Eval(y[a/x]) = false then return false return true if f = y1 Æ y2 then … if f = R(a1, …, ak) then …

Query Complexity Theorem There exists a structure A s.t. the query complexity {f | A ² f} is PSPACE complete

Query Complexity Proof. Recall the Satisfiability of Quantified Boolean Formulas problem (QBF): is a QB formula F true ? Example: F = 9 X1 8 X2 8 X3 (X1 Æ X2 Ç : X1 Æ X3) is it true ? Note: boolean satisfiability (the NP-hard problem) is in QBF [why ?]

Query Complexity A = (A, UA), s.t. A = {0,1}, UA = {1} Theorem [Stockmeyer] QBF is PSPACE complete Return to our proof: reduction from QBF. Given QBF formula F, let s = {U}, where U = unary A = (A, UA), s.t. A = {0,1}, UA = {1} Translate F (a QB formula) to f (an FO formula) s.t. F is true iff A ² f [how ?]