PP is Closed Under Intersection z y y Daniel Spielman Nick Reingold Richard Beigel Yale College Yale University Yale University Abstract In his seminal pap er on probabilistic Turing machines Gill asked whether the class PP is closed under intersection and union We give a p ositive answer to this question We also show that PP is closed un der a variety of p olynomialtime truthtable reductions Consequences in complexity theory include the denite collapse and assuming P 6 PP separation of certain query hierarchies over PP Similar techniques allow us to combine several threshold gates into a single threshold gate Consequences in the study of circuits include the simulation of circuits with a small numb er of threshold gates by circuits having only a single threshold gate at the ro ot p erceptrons and a lower b ound on the numb er of threshold gates needed to compute the parity function Intro duction The class PP was dened in by John Gill and indep endently by Janos Simon in PP is the class of languages accepted by a p olynomialtime b ounded nondeterministic Turing machine that accepts when more than half of its paths are accepting and rejects when more than half of its paths are rejecting this denition is from and is slightly dierent from but equivalent to the usual denition see Section Gill noted that PP is closed under complementation but stated that it was not known if PP is closed under intersection and union Since Gills pap er PP and related counting classes have b een studied exten sively by numerous researchers though few closure prop erties have b een shown for the class In Russo showed that the sym metric dierence of two sets in PP is also in PP and in Beigel Hemachandra The authors may b e reached by writing to Department of Computer Science PO Box New Haven CT or by sending electronic mail to lastnamerstnamecsyaleedu y Supp orted in part by NSF grants CCR and CCR z Supp orted in part by NSF grant CCR under an REU supplement and Wechsung showed that PP is closed under p olynomialtime parity reduc tions Gills question remained op en however and it was widely conjectured that PP was not closed under intersection or union We prove that PP is in fact closed under intersection and union and even under p olynomialtime conjunctive and disjunctive reductions Consequently PP is closed under p olynomialtime truthtable reductions in which the truth table predicate is computed by a b oundeddepth Bo olean formula and hence under p olynomialtime Turing reductions that make O log n queries That is PP P PP Relative to oracles this collapse cannot b e extended to a larger O log n T numb er of queries For functions computed with a b ounded numb er of queries the PP X b ehavior is quite dierent PF PF for any oracle X unless P PP k tt k T Our strongest closure prop erty is that PP is closed under p olynomialtime truthtable reductions in which the truth predicate is computed by an explicitly pro duced multilinear p olynomial this includes all symmetric functions as a sp e cial case The techniques presented here have b een extended by Fortnow and Reingold to show that PP is closed under general p olynomialtime truthtable reductions The technique for combining PP machines can also b e applied to threshold gates with p olynomialsized weights For example we show how to compute the AND of k threshold gates as the threshold of ANDs We also show that any constant depth circuit with AND OR NOT and threshold gates can b e simulated by a circuit with a single threshold gate at the ro ot with depth greater by a constant and only a limited increase in size If the original circuit p olylog n has size and only O log log n threshold gates then the new circuit still p olylog n has size As an application we prove that no constant depth circuit with olog n o n AND OR and NOT gates in arbitrary p ositions and threshold gates o n wires can compute parity This is the rst natural example of a function that is known to require more than a constant numb er of threshold gates in such a circuit Previous lower b ounds had b een obtained for circuits consisting en tirely of threshold gates Ha jnal et al have shown that inner pro duct mo d cannot b e computed by any p olynomial size depth circuit of threshold gates Paturi and Saks have shown that a depth circuit of threshold gates which computes the parity on n inputs requires n log n threshold gates Siu et al have shown that a depthd circuit of threshold gates which com d putes the parity on n inputs requires dn log n threshold gates Recently Beigel extending our techniques has shown that that no constant depth cir o o o n n cuit with n threshold gates AND OR and NOT gates and wires can compute parity The remainder of the pap er is organized as follows In Section we dene PrTIMEtn and PP and we show how to combine nondeterministic Turing machines according to a sequence of rational functions In Section we con struct the rational functions appropriate for our closure prop erties The closure prop erties are proved in Section In Section we show how the techniques of Section can b e mo died to apply to threshold circuits and in Section we ap ply these techniques to obtain the parity lower b ound mentioned ab ove Finally in Section we consider query hierarchies over PP Notation Throughout this pap er we will use X rather than the customary x to denote an input to a Turing machine We will use x to denote a variable ranging over the reals We will use jxj to mean the absolute value of the real numb er x To avoid confusion we will never use jX j to denote the length of the input X All logarithms are base two logarithms Building Turing machines from rational func tions Beigel and Gill and Gundermann Nasser and Wechsung have used p oly nomials to prove closure prop erties of various counting classes In this section we extend the techniques of where they used a single p olynomial we use a sequence of rational functions These new twists app ear to b e crucial to obtaining our closure prop erties for PP Fenner et al provide a convenient notation for studying counting classes like PP Denition For a nondeterministic Turing machine N and input X let Gap N X denote the numb er of accepting paths of N on input X minus the numb er of rejecting paths of N on input X Denition A language L is in PrTIMEtn if there exists a tntime b ounded nondeterministic Turing machine N such that for all inputs X X L Gap N X X L Gap N X Gill shows that NTIMEtn PrTIMEtn It should b e noted that in our denition of PrTIMEtn all accepting reject ing paths are counted equally regardless of length as in Other denitions of PrTIMEtn either insist that all paths have the same length or weight the paths according to length For timeconstructible tn these denitions are equivalent O Denition PP PrTIMEn For any nondeterministic Turing machine N let the complement machine N b e the machine which runs N and then rejects if N accepts and denoted accepts if N rejects Clearly Gap N X Gap N X Let N and N b e two nondeterministic Turing machines Consider the Turing machine N which nondeterministically cho oses to run either N or N It is easy to see that for all inputs X Gap N X Gap N X GapN X Let N b e a nondeterministic Turing machine which runs as follows First run N if N accepts then run N otherwise run N It is not hard to verify that for all X GapN X GapN X Gap N X Combining these observations we obtain Lemma b elow Denition A sequence of p olynomials fp x x g is snuniform if each n k co ecient of each p is an integer and sn is a b ound on the time needed to n compute the degree of p or to compute the co ecient of any monomial in p n n Lemma Let N N be tntime bounded nondeterministic Turing ma k chines Let fp x x g be an snuniform sequence of polynomials Sup n k pose p has degree d and each coecient of p is bounded in absolute value n n n by M Then there exists a nondeterministic Turing machine N that runs in n d k n time O log M d tn sn such that for al l X Gap N X n n k p y y where n is the length of X and y is Gap N X n k i i k x Pro of N rst nondeterministically cho oses a monomial of p say cx n k d k d k n n with c Since p has at most monomials this requires time O log sn n k k Then N nondeterministically cho oses one of c branches computes the pro duct as describ ed ab ove and complements if necessary This takes an additional time O log M d tn sn Figure shows the computation tree for machine n n N Note that in the denition of PrTIMEtn only the sign of the Gap is essen tial Although we cannot directly apply Lemma to a rational function since we cannot divide we can build a Turing machine such that the sign of the Gap is given by the sign of a rational function We dene the degree of a rational function to b e the maximum of the degrees of its numerator and denominator A sequence fr x x g of rational functions is snuniform if b oth the se n k quence of numerators and the sequence of denominators are snuniform Lemma Let N N be tntime bounded nondeterministic Turing ma k chines Let fr x x g be an snuniform sequence of rational functions n k where the degree of r is d and both the numerator and denominator of r have n n n integer coecients bounded in absolute value by M Then there exists a nonde n d k n d tn sn terministic Turing machine N that runs in time O log M n n k such that Gap N X and r y y have the same sign