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Co~Nparison of Polynomial- Time Reducibilities Richard Ladner University of Washington Seattle, Wash. Nanay Lynch University Of

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Co~Nparison of Polynomial- Time Reducibilities Richard Ladner University of Washington Seattle, Wash. Nanay Lynch University Of Co~nparison of Polynomial- Time Reducibilities interesting relationships. Richard Ladner University of Washington I. Introduction Seattle, Wash. Computation-resource-bounded reducibil- Nanay Lynch University of Southern California ities play a role in the theory of computational Los Angeles, Cal. complexity which is analogous to, and perhaps AlaD Selman Florida State University as important as, the various kinds of effective Tallahas see, Fla. reducibilities used in recursive function theory. Abstract Just as the effective reducibilities are used to Comparison of the polynomial-time-bounded classify problems according to their degrees reducibilities introduced by Cook [1] and of unsolvability, [9] space- and time- bou6ded t<arp [4] leads naturally to the definition of reducibilities may be used to classify problems several intermediate truth-table reducibilities. according to their complexity level. We give definitions and comparisons for these Tne most fruitful resource-bounded reducibilities ; we note, in particular, that reducibilities thus far have been the polynomial- all reducibilities of this type which do not time-bounded reducibilities of Cook [I] and have obvious implication relationships are I<arp [4], corresponding respectively to in fact distinct in a strong sense. Proofs Turing and many-one reducibilities in recursive are by simultaneous diagonalization and function theory. Other resource-bounded encoding constructions. reducibilities have been defined and used as well Work of Meyer and Stockmeyer [7] and [3] [5] [6] [8]; they differ from Cook's or Karp's Gill [Z] then leads u.,~ to define nondeterministic only in the bound on time or space allowed versions of all of our reducibilities. Although for the reduction, and thus they also correspond many of the definitions degenerate, comparison to Turing or many-one reducibility. of the remaining nondeterministic reducibilities We begin by comparing Cook's and Karp's among themselves and with the corresponding reducibilities in Section II; as examination of o lr deterministic reducibilities yields some proof that they are distinct shows that a simple ii0 kind of poynomial-bounded truth-table Of course, this strengthening would require a reducibility is actually involved. This leads prior demonstration that P ~ NP. us to study, in Section Ill, polynomial-time- In Section IV, we study nondeterministic bounded analogues of all the usual kinds of versions of polynomial truth-table reducibilities. truth-table reducibilities. Thus, we &re study- This investigation is influenced by Gill's work irLg restrictions on the form of allowable using nondeterministic polynomial Turing reduction procedures, as well as on their reducibility [z], which gives important evidence complexity. Besides basic results about the for the uselessness of two common techniques individual reducibilities, the primary type (simulation and diagonalization) for the solution 9 of result we show is a strong form of distinct- of P -NP. The structure of the nondeterministic ness among the various relations. reducibilities turns out to be interesting in itself. Further impetus for studying polynomial- Finally, in Section V, we present open ti,~e truth-table reducibilities is the hope that questions arising from this work. exploiting the analogy between recursive function theory and the theory of polynomial- II Polynomial-time Turing and many-one computable functions may help to sove R educibilitie s 9 problems such as P = NP. Various results We define < p and <P (polynomial- suggest a parallel between the class of recur- -# ?n time Turing reducibility and polynomial-time sire sets and P(the class of polynomial- many-one reducibility) to be the reducibilities computable sets), as well as between the class used by Cook and Karp respectively. Specifically, of recursively enumerable sets and NP (the we restrict the sets involved in our reducibilities clan s of nond eterministic polynomial- computable to be reeursive sets of strings over the alphabet sets). For example, we note Cook's charac- {0, i}. We write I xl for the length of string terization of N1° [I] using existential quantific- x. Then we write : . ation. Although the usual argument for show- in,,~ "recursively enumerable ~=> recursive" P A < B iff there is an oracle Turing machine l~4 does not seem to apply in showing P # NP, we T and a polynomial p such that x ¢ A hope that further study of the analogy may exactly if Iv[ accepts x with B as its provide useful insight into the problem. oracle, within P(I xl ) steps. This analogy also leads us to wonder We write: whether our results could be strengthened to show that any of the defined reducibilities A <PB iff there is a function f{ 0, I} ;'.~~{ 0,i }* ~n are distinct on NP. Similar resursive function computable in polynomial time such theory results generally show distinctness on that x e A exactly if f(x)¢ B. the class of recursively enumerable sets [9]. iii In all our notation for reducibilities, the subscript (n) A<PIs, B cNP=A cNi~. m (T or m, for example) will indicate the form of All of these results have elementary proofs, the reduction procedure, while the superscript and many have been previously noted. The key (P, for example) refers to the time bound. We idea in (a), (f), (g) and (h) is direct simulation note that our definitions are independent of stand- of the oracle. (g) and (h) are not known to hold ard Turing machine conventions, a fact which is for <P . ((h) for <P would imply that NP is often convenient in our proofs; we can use closed under complement. ) This fact, together simpler machine models in a diagonalization with the following, provides good reason for and more complex models in a simulation con- P considering reducibilities other that < : struction. T We may easily obtain the following basic Proposition: (Meyer) Let A be any < - facts, similar to basic results about < and < m complete set in NP. Then A and 7% are in [9]: -T m m-comparable iff NP is closed under Theorem h (a) A <:PB, B ¢ P ~ A cP. complement. Degree-theoretic results about <P and < P (b) A <PB = A < p B. Yn T are explored in [5]. We now wish to show that (c) <P and < p are reflexive and transitive ~n <P and < p are distinct; to do so we use the T ?n relations. following notion of distinctness: (d) A < P B ~ A < p B. Definition: Given any Z reducibilities, < and Yn ~h I- < , we say"< transcends < " if there exist (e) A <PB ¢~ A <P B ~ A <PB ¢.2~<PB -~ -T T recursive sets A and B such that A < B, i- B < A, A _~ B and B _~ A. (That is, A (f) If A <P B and for each string x, xsB -T I- z z and B are 1-equivalent but Z-incomparable. ) is decidable in time < t (] x I ), then for some polynomial p, x ¢ A is decidable in time Theorem Z: < transcends <P T m _< P (I xl ) + P(I xI ) max {t([ y[ ) I I Yl _< p(I x[) }. Proof: Immediate from Theorem 1 and the following Lemma: (g) If A <PB and for each string x, x s B is Frl decidable nondeterministically in time Lemma: There exists an infinite, coinfinite < t (I xl ) , then for some polynomial p, x sA recursive set A such that A ~ A. m is decidable nondeterministically in time < Proof of the Lemma: The set A is constructed _< p(Ixl) + max [t(lyl) I I yl<p(Ixl )}. in stages numbered 0,1, Z .... At each stage, we attempt to diagonalize over a many-one 112 reduction procedure. Tnus, we need an effective y sufficiently large, ~l(y ) = i, causing the enumeration of polynomial-time-bounded condition ix eA ~ ¢i(x) eA] to be falsified at reduction procedures; it is sufficient to select stage y. some recursive function b which is eventally The above proof is simple but is presented greater than each polynomial (that is, b: 10, 1 }~"-¢N, since many of the results to follow can be and (Vp, a polynomial)(:~x)(Vx I [x[_>~)[b(x)2 proved by essentially similar ideas. This p([ xl )] ),and use it as a bound on the number of Lemma shows that Theorem 1 (d) cannot be steps in the computation of Turing machines in strengthened analogously to (e). some ordinary G'ddelnumbering {M i} • Since As noted in Section I, it would be desirable we only know that b(x) is eventually greater to know on what complexity classes of sets the than p(Ixl ) , we return to consider each machine reducibilities can be shown to differ For M i infinitely often Let ~l and TrT_~ be the example, can a set A as in the Lemma be orojection functions for some pairing function constructed with A sNP? More tractably, N ×N-*N (e.g , see[9]) can we show that P ~ NP would imply the Stage ~ Let x be the first string (in a natural existence of such a set A in NP? If we ordering of binary strings) whose membership naively measure the complexity of the set A in A is not yet determined. Let i = vl(y). constructed in the Lemma, we note that it is See if M. on input x converges within 1 roughly Z zlx] on argument x, since this much b(x) steps, if not, define x cA ¢*y is even, time is required to simulate and keep track and go on to stage y + 1. Otherwise, let ¢i(x)be of the results of enough stages in the construction the output. We wish to falsify: xtA ¢~ ¢i(x) ¢ A. to determine if x sA. However, the If ¢i(x)'s membership in A is already determined, we define: xsA ~ ¢i(x) cA.
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