Polynomial-time Many-one Research Articles

Towards the exact complexity of realizability for Safety LTL

We study the realizability and strong satisfiability problems for SafetyLTL, a syntactic fragment of Linear Temporal Logic (▪) capturing safe formulas. While it is well-known that realizability for this fragment lies in ▪, the best-known lower bound is ▪-hardness. Surprisingly, closing this gap has proven an elusive task. Previous works have claimed first ▪-completeness [1] and later ▪-completeness [2] for this problem, but both of these proofs turned out to be incorrect.We revisit the problem of the exact classification of the complexity of realizability for ▪ through the lens of seemingly weaker fragments. While we cannot settle the question for ▪, we study a subfragment of it consisting of formulas of the form ▪, where α is a present formula over system variables and ψ contains Next as the only temporal operator. We prove that the realizability problem for this new fragment, which we call ▪, is ▪-complete, and observe that this fragment is equirealizable to existing more expressive fragments, such as the class ▪[3].Furthermore, we revisit the techniques used in the purported proof of ▪-completeness of Arteche and Hermo [1], and observe that, while incorrect in their original claims, their proofs can be modified to classify the complexity of strong satisfiability, a necessary condition for realizability introduced by Kupferman, Sadigh, and Seshia [4]. We prove that, with regards to strong satisfiability, the fragments ▪ and ▪ are in fact equivalent under polynomial-time many-one reductions.

Direct product primality testing of graphs is GI-hard

We investigate the computational complexity of the graph primality testing problem with respect to the direct product (also known as Kronecker, cardinal or tensor product). In [1] Imrich proves that both primality testing and a unique prime factorization can be determined in polynomial time for (finite) connected and nonbipartite graphs. The author states as an open problem how results on the direct product of nonbipartite, connected graphs extend to bipartite connected graphs and to disconnected ones. In this paper we partially answer this question by proving that the graph isomorphism problem is polynomial-time many-one reducible to the graph compositeness testing problem (the complement of the graph primality testing problem). As a consequence of this result, we prove that the graph isomorphism problem is polynomial-time Turing reducible to the primality testing problem. Our results show that connectedness plays a crucial role in determining the computational complexity of the graph primality testing problem.

Typical forcings, NP search problems and an extension of a theorem of Riis

We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such forcings to produce models of the universal variant of relativized T21. We apply this result to study the relative complexity of total (type 2) NP search problems associated to finitary combinatorial principles.Complexity theory compares such problems with respect to polynomial time many-one or Turing reductions. From a logical perspective such problems are graded according to the bounded arithmetic theories that prove their totality. The logical analogue of a reduction is to prove the totality of one problem from the totality of another. The link between the two perspectives is tight for what we call universal variants of relativized bounded arithmetics. We strengthen a theorem of Buss and Johnson (2012) that infers relative bounded depth Frege proofs of totality from polynomial time Turing reducibility.As an application of our general forcing method we derive a strong form of Riis' finitization theorem (1993). We extend it by exhibiting a simple model-theoretic property that implies independence from the universal variant of relativized T21 plus the weak pigeonhole principle. More generally, we show that the universal variant of relativized T21 does not prove (the totality of the total NP search problem associated to) a strong finitary combinatorial principle from a weak one. Being weak or strong are simple model-theoretic properties based on the behavior of the principles with respect to finite structures that are only partially defined.

Complexity of existential positive first-order logic

Let Γ be a (not necessarily finite) structure with a finite relational signature. We prove that deciding whether a given existential positive sentence holds in Γ is in LOG SPACE or complete for the class CSP(Γ)NP under deterministic polynomial-time many-one reductions. Here, CSP(Γ)NP is the class of problems that can be reduced to the constraint satisfaction problem of Γ under non-deterministic polynomial-time many-one reductions.

On the Complexity of Matroid Isomorphism Problem

We study the complexity of testing if two given matroids are isomorphic. The problem is easily seen to be in $\Sigma_{2}^{p}$ . In the case of linear matroids, which are represented over polynomially growing fields, we note that the problem is unlikely to be $\Sigma_{2}^{p}$ -complete and is co NP-hard. We show that when the rank of the matroid is bounded by a constant, linear matroid isomorphism, and matroid isomorphism are both polynomial time many-one equivalent to graph isomorphism. We give a polynomial time Turing reduction from graphic matroid isomorphism problem to the graph isomorphism problem. Using this, we are able to show that graphic matroid isomorphism testing for planar graphs can be done in deterministic polynomial time. We then give a polynomial time many-one reduction from bounded rank matroid isomorphism problem to graphic matroid isomorphism, thus showing that all the above problems are polynomial time equivalent. Further, for linear and graphic matroids, we prove that the automorphism problems are polynomial time equivalent to the corresponding isomorphism problems. In addition, we give a polynomial time membership test algorithm for the automorphism group of a graphic matroid.

Effective Strong Dimension in Algorithmic Information and Computational Complexity

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff [Math. Ann., 79 (1919), pp. 157-179], and packing dimension, developed independently by Tricot [Math. Proc. Cambridge Philos. Soc., 91 (1982), pp. 57-74] and Sullivan [Acta Math., 153 (1984), pp. 259-277]. Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz [Proceedings of the 15th IEEE Conference on Computational Complexity, Florence, Italy, 2000, IEEE Computer Society Press, Piscataway, NJ, 2000, pp. 158-169] has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual of—and every bit as simple as—the gale characterization of Hausdorff dimension. Effectivizing our gale characterization of packing dimension produces a variety of effective strong dimensions, which are exact duals of the effective dimensions mentioned above. In general (and in analogy with the classical fractal dimensions), the effective strong dimension of a set or sequence is at least as great as its effective dimension, with equality for sets or sequences that are sufficiently regular. We develop the basic properties of effective strong dimensions and prove a number of results relating them to fundamental aspects of randomness, Kolmogorov complexity, prediction, Boolean circuit-size complexity, polynomial-time degrees, and data compression. Aside from the above characterization of packing dimension, our two main theorems are the following. 1. If $\vec{\beta} = (\beta_0,\beta_1,\ldots)$ is a computable sequence of biases that are bounded away from 0 and $R$ is random with respect to $\vec{\beta}$, then the dimension and strong dimension of $R$ are the lower and upper average entropies, respectively, of $\vec{\beta}$. 2. For each pair of $\Delta^0_2$-computable real numbers $0 < \alpha \le \beta \le 1$, there exists $A \in {\rm E}$ such that the polynomial-time many-one degree of $A$ has dimension $\alpha$ in E and strong dimension $\beta$ in E. Our proofs of these theorems use a new large deviation theorem for self-information with respect to a bias sequence $\vec{\beta}$ that need not be convergent.

Almost complete sets

We show that there is a set that is almost complete but not complete under polynomial-time many-one (p-m) reductions for the class E of sets computable in deterministic time 2lin. Here a set in a complexity class C is almost complete for C under some given reducibility if the class of the problems in C that do not reduce to this set has measure 0 in C in the sense of Lutz's resource-bounded measure theory. We also show that the almost complete sets for E under polynomial time-bounded length-increasing one–one reductions and truth-table reductions of norm 1 coincide with the almost p-m-complete sets for E. Moreover, we obtain similar results for the class EXP of sets computable in deterministic time 2poly.

The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P � NP . As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT with a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT with a less redundant distribution unless P = NP . We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of DistNP, this relation is proper unless P = NP. (2) C contains DistP, but it is not contained in AveP unless DistNP \subseteq AveZPP. (3) C has a ≤ p m -complete set. (4) C has a ≤ p tt -complete set that is not ≤ p m -complete unless P = NP. This shows that under the assumption that P � NP, the two completeness notions differ on some nontrivial subclass of DistNP.

Succinct circuit representations and leaf language classes are basically the same concept

This note connects two topics of complexity theory: The topic of succinct circuit representations initiated by Galperin and Wigderson (1983), and the topic of leaf language classes initiated by Bovet et al. (1992). It will be shown for any language that its succinct version is polynomial-time many-one complete for the leaf language class determined by it.

Structural Average Case Complexity

Levin introduced an average-case complexity measure, based on a notion of “polynomial on average,” and defined “average-case polynomial-time many-one reducibility” among randomized decision problems. We generalize his notions of average-case complexity classes, Random-NP and Average-P. Ben-Davidet al. use the notation of 〈C, F〉 to denote the set of randomized decision problems (L, μ) such thatLis a set in C andμis a probability density function in F. This paper introduces Aver〈C, F〉 as the class of randomized decision problems (L, μ) such thatLis computed by a type-C machine onμ-average andμis a density function in F. These notations capture all known average-case complexity classes as, for example, Random-NP= 〈NP, P-comp〉 and Average-P=Aver〈P, ∗〉, where P-comp denotes the set of density functions whose distributions are computable in polynomial time, and ∗ denotes the set of all density functions. Mainly studied are polynomial-time reductions between randomized decision problems: many–one, deterministic Turing and nondeterministic Turing reductions and the average-case versions of them. Based on these reducibilities, structural properties of average-case complexity classes are discussed. We give average-case analogues of concepts in worst-case complexity theory; in particular, the polynomial time hierarchy and Turing self-reducibility, and we show that all known complete sets for Random-NP are Turing self-reducible. A new notion of “real polynomial-time computations” is introduced based on average polynomial-time computations for arbitrary distributions from a fixed set, and it is used to characterize the worst-case complexity classesΔpkandΣpkof the polynomial-time hierarchy.

Simple characterizations of P(#P) and complete problems

In this paper, P(#P) and PF(#P) are characterized in terms of a largely different computation structure, where P(#P) (resp., PF(#P)) is the class of sets (resp., functions) that are polynomial-time Turing reducible to #P functions. Let MidP be the class of functions that give the medians in the outputs of metric Turing machines, where a metric Turing machine is a polynomial-time bounded nondeterministic Turing machine such that each branch writes a binary number on an output tape. Then it is shown that every function in PF(#P) is polynomial-time one-Turing reducible to a function in MidP and MidP ⊆ PF (#P); that is, PF(#P) = PF(MidP[1]). Furthermore, it is shown that for all sets L, L is in P(#P) if and only if there is a function F ∈ MidP, such that for every string x, x ∈ L, iff F(x) is odd. Thus the problem of computing medians in the outputs of metric Turing machines captures the computational complexity of P(#P) and PF(#P). As applications of the results, several natural polynomial-time many-one complete problems for P(#P) are shown, for example, given an undirected graph with integer edge weights, checking that the parity of the middle cost among all the simple circuits is complete for P(#P).

Padding, commitment and self-reducibility

A new property of NP sets called commitability is introduced in this paper. Roughly, a language L in NP is commitable if given any instance x and any string y, a string z can be found in polynomial time such that z is in L iff y is a prefix of a witness of x in the context of a PTIME relation that defines L. It is shown that all NP sets complete under polynomial-time many-one reductions are commitable. Commitability can be seen as a genralization of d-self-reducibility. Every d-self-reducible set is commited and yet, commitability retains an important aspect of d-self-reducibility: the tree search method of Berman is shown to be extendible to commitable sets. Therefore, certain known results about reductions to co-sparse sets and p-selective sets generalize from d-self-reductible sets to commitable sets. The connection with paddability is as follows: Every paddable and commitable set is shown to be d-self-reducible. It is also shown that paddability in NP implies kernel constructibility, which was introduced elsewhere to capture a notion of self-reducibility more general than d-self-reducibility.

Relativizing relativized computations

This paper introduces a technique of relativizing already relativized computations and gives two interesting applications. The techniques developed here are simpler than the usual methods for constructing oracles that satisfy several requirements simultaneously. The first application shows that a result of Karp and Lipton (if sets in NP are decidable with polynomial-size circuits, then Σ P 2 = Π P 2) cannot be strengthened in the presence of certain oracles. This means that relativizable proof techniques cannot strengthen the conclusion to, say, P= NP. Such a stronger conclusion would be desirable as it would establish the equivalence of polynomial-time programs and polynomial-size circuits for solving NP-complete problems and would extend the known equivalence of polynomial-time programs and programs that are allowed a single query to a polynomial-size table. The second application gives an oracle C for which P C ≠ (NP C ∩ coNP C ) ≠ NP C and NP C ∩ coNP C has complete sets under polynomial-time many-one reductions. This complements a result of Sipser in which an oracle B is constructed for which NP B ∩ coNP B has no complete sets. These results suggest that current proof methods will not settle whether NP ∩ coNP has complete sets.

Collapsing degrees

An m-degree is a collection of sets equivalent under polynomial-time many-one (Karp) reductions; for example, the complete sets for NP or PSPACE are m-degrees. An m-degree is collapsing iff its members are p-isomorphic, i.e., equivalent under polynomial-time, 1-1, onto, polynomial-time-invertible reductions. L. Berman and J. Hartmanis showed that all the then known NP-complete sets are isomorphic, and conjectured that the m-degree of the NP-complete sets collapses, in essence claiming that there is only one NP-complete set. However, until now no nontrivial collapsing m-degree was known to exist. In this paper we provide the first examples of such degrees. In particular, we show that there is a collapsing degree which is btt-complete for EXP (the exponential-time-decidable sets) and that, for every set A, there is a collapsing degree which is hard for A. We also obtain analogous results for noncollapsing degrees.

An Inhomogeneity in the Structure of Karp Degrees

We show that there is a recursive nonzero Karp (polynomial time many-one) degree which is not supremum of a minimal pair, i.e. of an incomparable pair of degrees with infimum 0, the degree of polynomial time computable sets. By existence of minimal pairs, this implies that there are nonisomorphic initial segments of Karp degrees.

Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis

A set S ⊂ Σ ∗ is sparse if there is a polynomial p such that the number of strings in S of length at most n is at most p( n). All known NP-complete sets, such as SAT, are polynomial-time isomorphic and are not sparse. The main result of this paper is that if there is a sparse NP-complete set under polynomial-time many-one reductions, then P = NP. We also show that if there is a sparse NP-complete set under polynomial-time Turing reductions, then the polynomial-time hierarchy collapses to Δ 2 P .

The maximum value problem and NP real numbers

The concept of nondeterministic computation has been playing an important role in discrete complexity theory. In this paper the concept of nondeterminism is applied to a numerical problem—finding the maximum value of a polynomial time computable real function. The class of all these maximum values is characterized as the class of nondeterministic polynomial time ( NP) computable real numbers. The completeness of real numbers is then investigated. The result that NP real numbers cannot be polynomial time many-one complete in NP (unless P = NP) shows a basic difference between the maximum value problem and many natural NP combinatorial problems. It is also shown that real numbers are not complete in r.e. sets or PSPACE (unless P = PSPACE) and this seems to be a general phenomenon. Finally, the relationship between NP real numbers and NP sets over a single-letter alphabet and the existence of hardest NP real numbers are also discussed.

On γ-reducibility versus polynomial time many-one reducibility

We prove that each element of a class of functions (denoted by NPC t P), whose graphs can be accepted in nondeterministic polynomial time, can be evaluated in deterministic polynomial time if and only if γ-reducibility is equivalent to polynomial time many-one reducibility. We also modify the proof technique used to obtain part of this result to obtain the stronger result that if every γ-reduction can be replaced by a polynomial time Turing reduction, then every function in NPC t P can be evaluated in deterministic polynomial time.