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 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 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 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 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 Degree-theoretic results about (d) A < P B ~ A < p B. Definition: Given any Z reducibilities, < and Yn ~h I- < , we say"< transcends < " if there exist (e) A polynomial p, x ¢ A is decidable in time Theorem Z: < transcends following Lemma: (g) If A < 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 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. diagonalization construction is very "loose, " Otherwise, we define x cA and ¢i(x) ¢ A. in that there are few constraints on our choice Go on to stage y + 1. of x at each stage. Thus, by choosing the values of x to be sufficiently separated (a END OF CONSTRUCTION technique due to Machtey) it requires A is clearly recursive, and the reader sufficiently less time to simulate the computation~ nay verify that for pairing functions chosen of preceding stages to bring the complexity down as in [9], for example, A is infinite and to Z Ixl. (Strings not used in thediagonalization eoninfinite. Now if A M i (x) runs in _< P(I xl ) steps ] But for x computable sets. The same technique could also sufficiently long, b(x) > p (ix I ), and for some be applied to all transcendence results in 113 Sections III and IV, reducing the complexity of all computation of ~, must be polynomial- I l relevant sets to Z Ixl . time-bounded. If we were to restrict our The technique used in the proof of the attention to a specific representation of Boolean Lemma actually yields results more powerful functions, say one using only the symbols than stated. First, the function b may be /~ , V and "~ , then the polynomial bound chosen as large as we like; for example, if we on the generation of the set and function is a choose b so that b is eventually greater than sufficient requirement for our defintion, as it each primitive recursive function of the length implies a polynomial bound on the evaluation of its argument, then a set A is produced time. However, we wlsh to leave the represent- with A not many-one reducible to A in ation of the function arbitrary, so both primitive recursive time. Second, we see that restrictions are needed. It is unknown whether it is not only restriction to ~ , V and "I. very simple form of polynomial-bounded We let A be a fixed finite alphabet, for the procedure, involving asking only a single oracle encoding of Boolean functions, and let question. Since this is an obvious analogue to c~ ~u{0,1t. < (one-question truth-table reducibility), l-+tt Definition: A tt-condition is a member of we are led to define polynomial-bounded truth- table reducibilitie s: A;:" c(c{ 0, I}*)* A tt-condition generator is a recursive mapping from { 0, II* into IIl Polynomial-time Truth-table Reducibilities * c(c { 0,1 } *)*. We recall that A is tt - reducible A tt-condition evaluator is a recursive mapping (truth-table reducible) to B if, given any x, from A* C{0,1}* into {0, I}. one can effectively compute both a finite set of Let e be a tt-condition evaluator. arguments Xl, XZ,...,Xk, and a Boolean A it-condition ~cc XlCXzC'''cx k is e-satisfied function c~ such that: by B._C {0,1}* iff e(Otc CB(Xl)CB(X~'''CB(X~)=I. xe A ~ c~ (CB(Xl), CB(X Z) ..... CB(Xk)) = I, A 114 bounded-truth-table reducible to B) provided (c) All are transitive. A number of c's. (The same is true for btt and k-tt. ) A E if A For implications not given in Theorem 3, we A reducible to B) if the evaluator e has the (k+l-c and k+l-d refer to k+l-question property that conjuctive and disj~mctive reducibilities e(~c ~l~Z "..C;k) = I ¢~ crI =~2 ..... Crk = i. respectively, defined by the obvious restrictions on the generator and evaluator. ) (b) A 115 construct a pair of sets A and B, preserving a The reader may complete the proof and Z-sided reducibility of the first type, while ve rification. conducting a Z-sided diagonalization over Again, judicious choice of arguments on reducibilities of the second type. For example, which to diagonalize will bring the complexity A in constructing sets A and B for (d), we ofA and B downto Z Ix[ . Also, asbefore, preserve the conditions: all the results in Theorem 4 may be strengthened by making the bound b as large as desired. zll¢ A ¢*(zll0 s B^z ii00¢ B)~(zlI000¢B ^ zll0000¢B) The same is not true for the following result: z01s B¢~ (z010¢A ^z0100cA)v(z01000¢ A^z010000¢ A) Theorem 5: z010k¢ A ¢~ z010ks BJI because of the following: for all strings z. All membership questions not Note: A These conditions are strong enough to force analogously to Thus, we let b(x) = Z Ix] - 1. We will procedures. For ins'~tance, to show A~PB, we C obtain If we already have, or if it is possible to . yes define q0 ~ B for some q0¢Q, then we define no As\ yes q0 ~ B, x cA, q ¢ B for all undetermined q eQ, q ~ q0' zll0ksB for 1 116 If we U zll ~9,1} k, then wsAc~we B. Definition: A If M i on input x converges within b(x) A to B within polynomial time, and for this steps, consider the it-condition ¢i(x). Note computation, on mput x, M only asks questions that at most b(x) = zIxI-i questions are that are on the list f(x). represented in the truth table, so some We note that this last definition describes w cxlll{0,1} Ixl is not in the truth table. We a sort of weak truth-table reducibility. define membership in B of elements of IV Nondeterministic Reducibilities A natural way to generalize the definitions (J xlll {0, I Ik and of values in the truth 0 _< k Say ~(x) = c~CCXlCX 2c...cx n. Then if nondeterministic reducibility appears is in Mj on input c~c CB(Xl)CB(Xz)..-CB(Xn) G ll's paper [Z]. He shows, for reducibility, that we Bc*xeA ¢~ ¢j(~CCB(Xl)CB(Xz)... CB(Xn) ) = 0, (i) there exist recursive sets B with and other values as required to preserve the A We complete our consideration of oracle case, these results seem to show that deterministic tt-reducibilities by noting the neither a diagonalization nor a simulation will following two equivalent formulations of our probably be useful in deciding whether P=NP. definition of 117 Definition: A polynomial-time Turing reducible to B) iff Theorem 6: A machine M and a polynomial p such that with A oracle B, M runs in time bounded by p for A In part, the following theorem suggests that Lemma: There exist recursive sets A, B,C nondeterminism recovers the power of using 118 with A N<__PB, B Proof of Lemma: We preserve the conditions (The corresponding statement is false for (Z Ix [-i, for example), a necessity because of (e) Again, remaining details are left to the A with A ~NPA. For (d), we construct m reader. an infinite, coinfinite set A with A ~NP~- and P Note: We may also easily show A We conject,tre, but have not yet proved, Finally, parallel to the existential quantifier that For nondeterminlstic reducibilities whose Definition: A 119 differ on the exponential-comput able sets. We a maximal transitive reducibility. would like to show that they differ on NP (for Degree-theoretic questions about all the example, that there exist A, B ¢ NP with reducibilities remain, as well as questions A < PB but A~PB). This, of course, T m about complete sets at various complexity levels. would imply P ~ NP. Perhaps we can These may someday prove relevant to a show: classification of natural problems by their P~NP = More strongly, perhaps we can show: P~NP =T-completeness and m-complete- Acknowledgments: We v~ uld like to thank Albert Meyer and Michael Machtey for some ness differ. Same questions for the other deterministic reducibilities. very valuable suggestions on this work. We would like to develop stronger notions R eferences: of distinctness between reducibilities, than i. Cook, S.A. The complexity of theorem- proving procedures. Third annual ACM "transcendence.,, For example, can we show Symposium on Theory of Computing (197[). that specific representation (such as using A , v, -~ 4. Harp, R.M. Reducibility among combinatorial problems. Complexity of only), do we obtain as less general reducibility? Computer Computations. Miller and Thatcher (eds.) Plenum Press(1973). We may define,, a new reducibility, 5. Ladner, R. E. Polynomial time reducibility, analogous to enumeration reducibility, as Fifth Annual ACM Symposium on Theory of Computing (1973). follows: 6. Lynch, N.A. Relativization of the Theorv of computational complexity, Ph. D thesis A 120 Addendum Contemporary with Gill [2], parallel results have been independently obtained by T. Baker (Computational Complexity and Nondeterminism in Flowchart Programs, Ph.D. thesis, Cornell University, 1973). 121