NP Sets Research Articles

Minimization and NP multifunctions

The implicit characterizations of the polynomial-time computable functions FP given by Bellantoni–Cook and Leivant suggest that this class is the complexity-theoretic analog of the primitive recursive functions. Hence, it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's definition of FP by bounded recursion and for Bellantoni–Cook's safe recursion and prove that the resulting classes capture exactly NPMV , the nondeterministic polynomial-time computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion defined by Leivant and show that the latter characterizes the total functions of NPMV . We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. This view is reinforced by earlier results of Spreen and Stahl who show that for many of the relationships between partial recursive functions and r.e. sets, analogous relationships hold between NPMV and NP sets. Furthermore, since NPMV is obtained from FP in the same way as the recursive functions are obtained from the primitive recursive functions (when defined via function schemes), this also gives further evidence that FP is properly seen as playing the role of primitive recursion.

Lower bounds and the hardness of counting properties

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan (Math. Logic Quart. 46 (4) (2000) 489–504) started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rothe (Theoret. Comput. Sci. 244 (1–2) (2000) 205–217) improved the UP-hardness lower bound to UP O(1)-hardness. The present paper raises the lower bound for nontrivial counting properties from UP O(1)-hardness to FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. Furthermore, we prove that no relativizable technique can raise this lower bound to FewP-⩽ 1- tt p -hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and for a broad class of leaf-language classes we prove a sufficient condition for the natural analog of Rice's Theorem to hold.

Average-case intractability vs. worst-case intractability

We show that not all sets in NP (or other levels of the polynomial-time hierarchy) have efficient average-case algorithms unless the Arthur-Merlin classes MA and AM can be derandomized to NP and various subclasses of P /poly collapse to P . Furthermore, other complexity classes like P ( PP ) and PSPACE are shown to be intractable on average unless they are easy in the worst case.

Minimization and [formula omitted] multifunctions

The implicit characterizations of the polynomial-time computable functions FP given by Bellantoni–Cook and Leivant suggest that this class is the complexity-theoretic analog of the primitive recursive functions. Hence, it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's definition of FP by bounded recursion and for Bellantoni–Cook's safe recursion and prove that the resulting classes capture exactly NPMV , the nondeterministic polynomial-time computable partial multifunctions. We also consider the relationship between our schemes and a notion of nondeterministic recursion defined by Leivant and show that the latter characterizes the total functions of NPMV . We view these results as giving evidence that NPMV is the appropriate analog of partial recursive. This view is reinforced by earlier results of Spreen and Stahl who show that for many of the relationships between partial recursive functions and r.e. sets, analogous relationships hold between NPMV and NP sets. Furthermore, since NPMV is obtained from FP in the same way as the recursive functions are obtained from the primitive recursive functions (when defined via function schemes), this also gives further evidence that FP is properly seen as playing the role of primitive recursion.

On reducibility and symmetry of disjoint NP pairs

We consider some problems about pairs of disjoint NP sets. The theory of these sets with a natural concept of reducibility is, on the one hand, closely related to the theory of proof systems for propositional calculus, and, on the other, it resembles the theory of NP completeness. Furthermore, such pairs are important in cryptography. Among others, we prove that the Broken Mosquito Screen pair of disjoint NP sets can be polynomially reduced to Clique–Coloring pair and thus is polynomially separable and we show that the pair of disjoint NP sets canonically associated with the Resolution proof system is symmetric.

Separability and one-way functions

We settle all relativized questions of the relationships between the following five propositions: • P=NP. • P=UP. • P=NP ∩ coNP. • All disjoint pairs of NP sets are P-separable. • All disjoint pairs of coNP sets are P-separable. We make the first widespread use of variations of generic oracles to achieve the necessary relativized worlds.

Analytic sets in descriptive set theory and NP sets in complexity theory

Motivated by the analogy (NP/Poly) ∼ analytic, we propose a co-analytic set W whose finite equivalent Wfinite is coNP-complete. The complement of W is in fact a variant of infinite clique. A co...

Sparse Sets and Collapse of Complexity Classes

In this paper a new upward separation technique is developed. It is applied to prove that for a class of functions F a separation DTIME(F)≠NTIME(F) can be characterized by the existence of (not only polynomially) sparse sets in certain complexity classes. As a consequence, a solution of an open question of J. Hartmanis (1983, Inform. Process. Lett.16, 55–60) is obtained: There is an nO(log n)–sparse set in NP−P iff ∪c>1NTIME(2c□n)≠∪c>1DTIME(2c□n). Further we prove that there is an nO(log n)–sparse set in NP−coNP iff ∪c>1NTIME(2c□n)≠∪c>1coNTIME(2c□n). The technique also allows us to characterize the existence of sets of different densities in NP−P by the existence of slightly denser sets in NTIME(F)−DTIME(F) for certain classes of functions F. For example, there is an nO(log n)–sparse set in NP−P iff there is an nO((log n)3)–sparse set in ∪c>1NTIME(nclog n)−∪c>1DTIME(nclog n). The end of the paper is devoted to limitations of the technique.

Separation of NP-Completeness Notions

We use hypotheses of structural complexity theory to separate various NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is $\mbox{${\leq}^{\rm P}_{\rm T}$}$-complete but not $\mbox{${\leq}^{\rm P}_{tt}$}$-complete. We provide fairly thorough analyses of the hypotheses that we introduce.

The counting complexity of group-definable languages

A group family is a countable family B={B n} n>0 of finite black-box groups, i.e., the elements of each group B n are uniquely encoded as strings of uniform length (polynomial in n) and for each B n the group operations are computable in time polynomial in n. In this paper we study the complexity of NP sets A which has the following property: the set of solutions for every x∈ A is a subgroup (or is the right coset of a subgroup) of a group B i(| x|) from a given group family B , where i is a polynomial. Such an NP set A is said to be defined over the group family B . Decision problems like Graph Automorphism, Graph Isomorphism, Group Intersection, Coset Intersection, and Group Factorization for permutation groups give natural examples of such NP sets defined over the group family of all permutation groups. We show that any such NP set defined over permutation groups is low for PP and C =P. As one of our main results we prove that NP sets defined over abelian black-box groups are low for PP. The proof of this result is based on the decomposition theorem for finite abelian groups. As an interesting consequence of this result we obtain new lowness results: Membership Testing, Group Intersection, Group Factorization, and some other problems for abelian black-box groups are low for PP and C =P. As regards the corresponding counting problem for NP sets over any group family of arbitrary black-box groups, we prove that exact counting of number of solutions is in FP AM. Consequently, none of these counting problems can be #P-complete unless PH collapses.

The two faces of privatisation: political and economic logics in transitional South Africa

Privatisation has both an economic and a political face. In South Africa, this Janus-faced character is revealed by the juxtaposition of economic and political arguments for and against the process of privatisation initiated by the National Party, during the transition to majority rule. This paper argues that the NP set out to fundamentally reorganise the structure of South Africa's political economy as an exit strategy. Although it justified privatisation by employing economic arguments, this ostensible depoliticisation masked political motives. Ironically, the weak economic case reveals the political face of privatisation in South Africa. The timing and context of South Africa's privatisation process is difficult to defend on economic grounds, but it did make political sense.

Tally NP Sets and Easy Census Functions

We study the question of whether every P set has an easy (i.e., polynomial-time computable) census function. We characterize this question in terms of unlikely collapses of language and function classes such as #P 1 ⊆FP, where #P 1 is the class of functions that count the witnesses for tally NP sets. We prove that every #P 1 PH function can be computed in FP #P 1 #P 1 . Consequently, every P set has an easy census function if and only if every set in the polynomial hierarchy does. We show that the assumption #P 1 ⊆FP implies P=BPP and PH⊆MOD k P for each k ⩾2. We also relate a set's property of having an easy census function to other well-studied properties of sets, such as rankability and scalability (the closure of the rankable sets under P-isomorphisms). Finally, we prove that it is no more likely that the census function of any set in P can be approximated (more precisely, can be n α -enumerated in time n β for fixed α and β ) than that it can be precisely computed in polynomial time.

Adaptive versus nonadaptive queries to NP and p-selective sets

We prove that P = NP follows if for some \( k > 0 \), the class \( {\rm PF}^{\rm NP}_{tt} \) of functions that are computable in polynomial time by nonadaptively accessing an oracle in NP is effectively included in PFNP[k⌈log n⌉— 1], the class of functions that are computable in polynomial k⌈log n⌉— 1 queries to an oracle in NP.¶We draw several observations and relationships between the following two properties of a complexity class \( \cal C \): whether there exists a truthtable hard p-selective language for \( \cal C \), and whether polynomially-many nonadaptive queries to \( \cal C \) can be answered by making O(log n)-many adaptive queries to \( \cal C \) (in symbols, whether \( {\rm PF}^{\cal C}_{tt} \subseteq {\rm PF}^{\cal C}[O({\rm log}\,n)] \)). Among these, we show that if there exists an NP-hard p-selective set under truth-table reductions, then \( {\rm PF}^{\rm NP}_{tt} \subseteq {\rm PF}^{\rm NP}[O({\log}\,n)] \). However, if \( {\cal C} \supseteq {\rm ZPP}^{\rm NP} \), then these two properties are equivalent.

SELF-SPECIFYING MACHINES

We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that [Formula: see text]-reduce to NP sets. A natural variant accepts exactly the languages that [Formula: see text]-reduce to P sets. We show that these two classes coincide if and only if [Formula: see text], where the latter class denotes the sets acceptable via at most one question to #P followed by at most a constant number of questions to NP.

Elastic unitarity and the K-matrix in a Lorentz covariant representation of the NN interaction

A Lorentz covariant representation of the NN t-matrix has been obtaained over the energy range from 150 to 500 MeV by solving the integral equation that connects the t-matrix with the K-matrix. The K-matrix is expanded in a complete set of on-shell Lorentz invariant amplitudes represented phenomenologically by isoscalar and isovector “meson” exchanges. The real part of the K-matrix is fit over the energy range from 150 to 500 MeV using coupling strengths that are allowed to vary quadratically with energy. Above the pion production threshold at T lab = 280 MeV, the real K-matrix is supplemented by an imaginary part with linear energy dependence. The K-matrix parameters are fit to thesmost recent (January 1999) Arndt amplitudes [R.A. Arndt, D. Roper, VPI and SU Scattering Analysis Interactive Dial-in Program and Data Base]. Direct and exchange contributions to the K-matrix are handled explicitly in the formalism. The resulting t-matrix satisfies elastic unitary below the pion production threshold and contains non-local terms that are not present in direct Love-Franey parameterizations of the t-matrix. Results are given for the NN amplitudes and compared with both the Arndt amplitudes and amplitudes obtained from a direct fit of the t-matrix [O.V. Maxwell, Nucl. Phys. A 600 (1996) 509]. Results are also given for a selected set of np and pp observables.

Deterministic and Randomized Bounded Truth-Table Reductions of P, NL, and L to Sparse Sets

We prove that there is no sparse hard set for P under logspace computable bounded truth-table reductions unless P=L. In case of reductions computable in NC1, the collapse goes down to P=NC1. We parameterize this result and obtain a generic theorem allowing us to vary the sparseness condition, the space bound and the number of queries of the truth-table reduction. Another instantiation yields that there is no quasipolynomially dense hard set for P under polylog-space computable truth-table reductions using polylogarithmically many queries unless P is in polylog-space. We also apply the proof technique to NL and L. We establish that there is no sparse hard set for NL under logspace computable bounded truth-table reductions unless NL=L and that there is no sparse hard set for L under NC1-computable bounded truth-table reductions unless L=NC1. We show that all these results carry over to the randomized setting: If we allow two-sided error randomized reductions with confidence at least inversely polynomial, we obtain collapses to the corresponding randomized classes in the multiple access model. In addition, we prove that there is no sparse hard set for NP under two-sided error randomized polynomial-time bounded truth-table reductions with confidence at least inversely polynomial unless NP=RP.

Functions Computable with Nonadaptive Queries to NP

We study FP|| NP , the class of functions that can be computed in polynomial time with nonadaptive queries to an NP oracle. This is motivated by the question of whether it is possible to compute witnesses for NP sets within FP|| NP . The known algorithms for this task all require sequential access to the oracle. On the other hand, there is no evidence known yet that this should not be possible with parallel queries. We define a class of optimization problems based on NP sets, where the optimum is taken over a polynomially bounded range (NPbOpt). We show that if such an optimization problem is based on one of the known NP-complete sets, then it is hard for FP|| NP . Moreover, we characterize FP|| NP as the class of functions that reduces to such optimization functions. We call this property strong hardness. The main question is whether these function classes are complete for FP|| NP . That is, whether it is possible to compute an optimal value for a given optimization problem in FP|| NP . We show that these optimization problems are complete for FP|| NP , if and only if one can compute membership proofs for NP sets in FP|| NP . This indicates that the completeness question is a hard one.

Polynomial-Time Multi-Selectivity

We introduce a generalization of Selman's P-selectivity that yields a more flexible notion of selectivity, called (polynomial-time) multi-selectivity, in which the selector is allowed to operate on multiple input strings. Since our introduction of this class, it has been used to prove the first known (and optimal) lower bounds for generalized selectivity-like classes in terms of EL_2, the second level of the extended low hierarchy. We study the resulting selectivity hierarchy, denoted by SH, which we prove does not collapse. In particular, we study the internal structure and the properties of SH and completely establish, in terms of incomparability and strict inclusion, the relations between our generalized selectivity classes and Ogihara's P-mc (polynomial-time membership-comparable) classes. Although SH is a strictly increasing infinite hierarchy, we show that the core results that hold for the P-selective sets and that prove them structurally simple also hold for SH. In particular, all sets in SH have small circuits; the NP sets in SH are in Low_2, the second level of the low hierarchy within NP; and SAT cannot be in SH unless P = NP. Finally, it is known that P-Sel, the class of P-selective sets, is not closed under union or intersection. We provide an extended selectivity hierarchy that is based on SH and that is large enough to capture those closures of the P-selective sets, and yet, in contrast with the P-mc classes, is refined enough to distinguish them.

NP-hard sets are superterse unless NP is small

We show that the class P- mc( n) of polynomial time n-membership comparable sets (defined by Ogihara, 1995) has p-measure 0. Using this result, we will show that the class of sets which are ⩽ tt P -reducible to some P-selective set has p-measure 0. Furthermore, we will show that if NP does not have p-measure 0 then no ⩽ tt p -hard set for NP is P-approximable.

UP and the low and high hierarchies: A relativized separation

The low and high hierarchies within NP were introduced by Schoning in order to classify sets in NP. It is not known whether the low and high hierarchies include all sets in NP. In this paper, using the circuit lower-bound techniques of Hastad and Ko, we construct an oracle set relative to which UP contains a language that is not in any level of the low hierarchy and such that no language in UP is in any level of the high hierarchy. Thus, in order to prove that UP contains languages that are in the high hierarchy or that UP is contained in the low hierarchy, it is necessary to use nonrelativizable proof techniques. Since it is known that UPA is low for PPA for all setsA, our result also shows that the interaction between UP and PP is crucial for the lowness of UP for PP; changing the base class to any level of the polynomial-time hierarchy destroys the lowness of UP.