Chicago Journal of Theoretical Computer Science The MIT Press Volume 1999, Article 2 Complements of Multivalued Functions ISSN 1073–0486. MIT Press Journals, Five Cambridge Center, Cambridge, MA 02142-1493 USA; (617)253-2889; [email protected], journals- [email protected]. Published one article at a time in LATEX source form on the Internet. Pagination varies from copy to copy. For more information and other articles see: http://mitpress.mit.edu/CJTCS/ • http://www.cs.uchicago.edu/publications/cjtcs/ • ftp://mitpress.mit.edu/pub/CJTCS • ftp://cs.uchicago.edu/pub/publications/cjtcs • Fenner, et al. Complements of Multivalued Functions (Info) The Chicago Journal of Theoretical Computer Science is abstracted or indexed in R R R Research Alert, SciSearch, Current Contents /Engineering Computing & Tech- R nology, and CompuMath Citation Index. c 1999 The Massachusetts Institute of Technology. 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Kurtz, Raimund Seidel Editors: Martin Abadi Greg Frederickson John Mitchell Pankaj Agarwal Andrew Goldberg Ketan Mulmuley Eric Allender Georg Gottlob Gil Neiger Tetsuo Asano Vassos Hadzilacos David Peleg Laszlo´ Babai Juris Hartmanis Andrew Pitts Eric Bach Maurice Herlihy James Royer Stephen Brookes Ted Herman Alan Selman Jin-Yi Cai Stephen Homer Nir Shavit Anne Condon Neil Immerman Eva Tardos Cynthia Dwork Howard Karloff Sam Toueg David Eppstein Philip Klein Moshe Vardi Ronald Fagin Phokion Kolaitis Jennifer Welch Lance Fortnow Stephen Mahaney Pierre Wolper Steven Fortune Michael Merritt Managing Editor: Michael J. O’Donnell Electronic Mail: [email protected] [ii] Chicago Journal of Theoretical Computer Science 1999-3 Complements of Multivalued Functions Stephen Fenner Frederic Green Steven Homer Alan L. Selman Thomas Thierauf Heribert Vollmer 19 March, 1999 Abstract We study the class coNPMV of complements of NPMV functions. Though defined symmetrically to NPMV, this class exhibits very different properties. We clarify the complexity of coNPMV by showing that it is es- sentially the same as that of NPMVNP. Complete functions for coNPMV are exhibited and central complexity-theoretic properties of this class are stud- ied. We show that computing maximum satisfying assignments can be done in coNPMV, which leads us to a comparison of NPMV and coNPMV with Krentel’s classes MaxP and MinP. The difference hierarchy for NPMV is related to the query hierarchy for coNPMV. Finally, we examine a func- tional analogue of Chang and Kadin’s relationship between a collapse of the Boolean hierarchy over NP and a collapse of the polynomial-time hierarchy. 1 Introduction 1-1 Consider the complexity class NPMV of partial multivalued functions that are com- puted nondeterministically in polynomial time. As this class captures the complex- ity of computing witnesses of sets in NP,by studying this class, and more generally, by studying relations among complexity classes of partial multivalued functions, we directly contribute to understanding the complexity of computing witnesses. It is well known that a partial multivalued function f belongs to NPMV if and only if it is polynomial length-bounded and graph(f)= x;y : y is a value of f(x) belongs to NP. {h i } 1-2 Now consider the class coNPMV. We will give a formal definition in the pre- liminaries section below. It will follow from the definition that a partial multivalued 1 Chicago Journal of Theoretical Computer Science 1999-3 Fenner, et al. Complements of Multivalued Functions 1 § function f belongs to coNPMV if and only if it is polynomial length-bounded and graph(f) belongs to coNP. Given this symmetry, graphs of functions in NPMV are in NP while graphs of functions in coNPMV are in coNP, and given what we know about NP and coNP, one might expect that coNPMV has essentially the same complexity as NPMV. Indeed, it is easy to see that coNPMV = NPMV if and only if NP = coNP. However, the point of this paper is to show that in many ways coNPMV is a more powerful class than is NPMV. One can derive more information from computing the complement of a function in NPMV than from computing the function. For one example of this phenomenon, we prove here that coNPMV is not included in FPNPMV unless the polynomial hierarchy collapses. (This is an extension of a result of Fenner et al [FHOS97].) Thus, a coNPMV oracle provides more information than an NPMV oracle. This is surprising since function oracles, just as set oracles, provide knowledge about both their domains and their co-domains. 1-3 We will define many-one reductions between multivalued functions. This will be a straightforward adaptation of the many-one metric reducibility of Krentel [Kre88]. In Section 3, we will consider many-one complete functions for coNPMV. 1-4 Consider the partial multivalued function sat, defined so that y is a value of sat(') if and only if y is a satisfying assignment of Boolean formula '. The function sat is complete for NPMV. Nevertheless, in Section 4 we will see that sat and similar functions belong to coNPMV. Even the seemingly more powerful FPNP-complete function maxsat, that gives the maximum satisfying assignment of a formula, is contained in coNPMV. However, we will see that neither NPMV nor FPNP are contained in coNPMV, and hence coNPMV is not closed under metric many-one reductions, unless the polynomial-time hierarchy collapses. Clearly, these function classes have strange closure properties, which we describe below. 1-5 As an upper bound on the complexity of coNPMV,we show that, for any k 2, ≥ coNPMV NPMV(2) NPMV(k) ⊆ ⊆ ⊆ NPMV(k +1) NPMV(Poly) NPMVNP; ⊆ ⊆ where NPMV(k) is the k-th level of the difference hierarchy for NPMV as defined by Fenner et al. [FHOS97]. 1-6 On the other hand, even though there is an infinite hierarchy of complexity classes between coNPMV and NPMVNP (the difference hierarchy over NPMV does not collapse unless the polynomial-time hierarchy collapses [FHOS97]), our results suggest that the complexity of coNPMV is essentially the same as the complexity 2 Chicago Journal of Theoretical Computer Science 1999-3 Fenner, et al. Complements of Multivalued Functions 2 § NP NP 1 1 of NPMV : We prove in Section 5 that NPMV = π2 coNPMV (where π2 is the projection function that maps a pair of strings to its first◦ component). It follows that NPMVNP is the closure of coNPMV under metric many-one reductions. 1-7 In Section 6, we show that if the difference hierarchy for NPMV collapses, then the NPMV oracle hierarchy collapses. This is the functional analogue of the well-known result by Chang and Kadin relating a collapse of the Boolean hierarchy over NP to a collapse of the polynomial-time hierarchy. 1-8 Finally, we remark that the phenomenon that universal quantification seems to lead to larger function classes was previously observed by Toda. We show in Section 7 how this observation follows from our results. 2 Preliminaries 2-1 We fix Σ to be the finite alphabet 0;1 . Let < denote the standard lexicographic { n } order on Σ∗.Forn 0 we define Σ = x Σ∗ x = n .By ; we denote a standard pairing function≥ on Σ Σ . { ∈ || | } h· ·i ∗ × ∗ 2-2 We use the standard complexity classes P and NP for (nondeterministic) poly- p p p Σk 1 nomial time, Σk and ∆k = P − for the levels of the polynomial-time hierarchy, and NP(k) for the levels of the Boolean hierarchy, for k 1. ≥ 2-3 Let f be a relation on Σ Σ . We will call f a (partial) multivalued function ∗ × ∗ from Σ∗ to Σ∗.Byf(x) y we denote that (x;y) f and say that f maps x to y. By set-f(x) we denote the7→ set of outcomes of f on ∈x, set-f(x)= y : f(x) y . The graph of f is graph(f)= x;y : f(x) y . The domain of{ f, dom(7→f),is} the set of x where set-f(x) is{h nonempty.i We7→ will} say that f is undefined at x if x dom(f). The domain of a class of functions is dom( )= dom(f) f . 2-4 6∈ F F { | ∈F} 2-5 Given partial multivalued functions f and g, define g to be a refinement of f if dom(g) = dom(f) and graph(g) graph(f). Let and be classes of partial multivalued functions. Purely as⊆ a convention, if f Fis a partialG multivalued function, we define f c if contains a refinement of f, and we define c if, for every f , f∈ G .G This notation is consistent with our intuitionF⊆ thatG ∈F ∈c G c should entail that the complexity of computing values of functions in is notF⊆ greaterG than the complexity of computing values of functions in . F G 2-6 A transducer T is a nondeterministic Turing machine with a read-only input tape, a write-only output tape, read-write work tapes, and accepting states in the usual manner. T computes a value y on an input string x if there is an accepting computation of T on x for which y is the final content of T ’s output tape.