Many-one Research Articles

Complexity Theory for Operators in Analysis

We propose an extension of the framework for discussing the computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An advantage of using string functions is that we can define their size in a way inspired by higher-type complexity theory. This enables us to talk about computation on string functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.

Propositional proofs and reductions between [formula omitted] search problems

Total NP search problems (TFNP problems) typically have their totality guaranteed by some combinatorial property. This paper proves that, if there is a polynomial time Turing reduction between two TFNP problems, then there are quasipolynomial size, polylogarithmic height, constant depth, free-cut free propositional (Frege) proofs of the combinatorial property associated with the first TFNP problem from the property associated with the second problem. In addition, they have Nullstellensatz derivations of polylogarithmic degree. These results extend the previous work of Buresh-Oppenheim and Morioka first by applying to Turing reductions in place of many-one reductions and second by giving tight bounds on the height of the Frege proofs. As a corollary, PLS-complete problems are not polynomial time Turing reducible to PPA problems relative to a generic oracle. We establish a converse construction as well, by showing that a polynomial time Turing reduction can be obtained from a family of quasipolynomial size, polylogarithmic depth, propositional proofs which are based on “decision tree” substitutions. This establishes the optimality of our constructions, and partially resolves an open question posed by Buresh-Oppenheim and Morioka.We observe that the classes PPA, PPAD, PPADS, and PLS are closed under Turing reductions, and give an example of a TFNP class that is not closed under Turing reductions.

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.

MAKING POLONIA — FROM MANY ONE? OR MANY? OR NONE? PROBLEMATIZING POLISH-AMERICAN IDENTITY

Research Article| October 01 2011 MAKING POLONIA — FROM MANY ONE? OR MANY? OR NONE? PROBLEMATIZING POLISH-AMERICAN IDENTITY JOHN J. BUKOWCZYK JOHN J. BUKOWCZYK Search for other works by this author on: This Site Google The Polish Review (2011) 56 (3): 187–206. https://doi.org/10.2307/41550432 Cite Icon Cite Share Icon Share Facebook Twitter LinkedIn MailTo Permissions Search Site Citation JOHN J. BUKOWCZYK; MAKING POLONIA — FROM MANY ONE? OR MANY? OR NONE? PROBLEMATIZING POLISH-AMERICAN IDENTITY. The Polish Review 1 October 2011; 56 (3): 187–206. doi: https://doi.org/10.2307/41550432 Download citation file: Zotero Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All Scholarly Publishing CollectiveUniversity of Illinois PressThe Polish Review Search Advanced Search The text of this article is only available as a PDF. © Copyright 2011 Polish Institute of Arts and Sciences of America, Inc.2011 Article PDF first page preview Close Modal You do not currently have access to this content.

Low sets without subsets of higher many-one degree

Given a reducibility ⩽r, we say that an infinite set A is r-introimmune if A is not r-reducible to any of its subsets B with |A\B| = ∞. We consider the many-one reducibility ⩽m and we prove the existence of a low1 m-introimmune set in Π01 and the existence of a low1 bi-m-introimmune set.

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.

Space-efficient informational redundancy

We study the relation of autoreducibility and mitoticity for polylog-space many-one reductions and log-space many-one reductions. For polylog-space these notions coincide, while proving the same for log-space is out of reach. More precisely, we show the following results with respect to nontrivial sets and many-one reductions: 1. polylog-space autoreducible ⇔ polylog-space mitotic, 2. log-space mitotic ⇒ log-space autoreducible ⇒ ( log n ⋅ log log n ) -space mitotic, 3. relative to an oracle, log-space autoreducible ⇏ log-space mitotic. The oracle is an infinite family of graphs whose construction combines arguments from Ramsey theory and Kolmogorov complexity.

The orbit problem is in the GapL hierarchy

The Orbit problem is defined as follows: Given a matrix A?? n×n and vectors x,y?? n , does there exist a non-negative integer i such that A i x=y. This problem was shown to be in deterministic polynomial time by Kannan and Lipton (J. ACM 33(4):808---821, 1986). In this paper we place the problem in the logspace counting hierarchy GapLH. We also show that the problem is hard for C=L with respect to logspace many-one reductions.

Fine hierarchies and m-reducibilities in theoretical computer science

This is a survey of results about versions of fine hierarchies and many-one reducibilities that appear in different parts of theoretical computer science. These notions and related techniques play a crucial role in understanding complexity of finite and infinite computations. We try not only to present the corresponding notions and facts from the particular fields but also to identify the unifying notions, techniques and ideas.

Martingale families and dimension in P

We introduce a new measure notion on small complexity classes (called F -measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0 ’s and 1 ’s. On larger complexity classes ( E and above), F -measure is equivalent to Lutz resource-bounded measure. As applications to F -measure, we answer a question raised in [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818] by improving their result to: for almost every language A decidable in subexponential time, P A = BPP A . We show that almost all languages in PSPACE do not have small non-uniform complexity. We compare F -measure to previous notions and prove that martingale families are strictly stronger than Γ -measure [E. Allender, M. Strauss, Measure on small complexity classes, with application for BPP, in: Proc. of the 35th Ann. IEEE Symp. on Found. of Comp. Sci., 1994, pp. 807–818], we also discuss the limitations of martingale families concerning finite unions. We observe that all classes closed under polynomial many-one reductions have measure zero in EXP iff they have measure zero in SUBEXP . We use martingale families to introduce a natural generalization of Lutz resource-bounded dimension [J.H. Lutz, Dimension in complexity classes, in: Proceedings of the 15th Annual IEEE Conference on Computational Complexity, 2000, pp. 158–169] on P , which meets the intuition behind Lutz’s notion. We show that P -dimension lies between finite-state dimension and dimension on E . We prove an analogue of a Theorem of Eggleston in P , i.e. the class of languages whose characteristic sequence contains 1 ’s with frequency α , has dimension the Shannon entropy of α in P .

On the Computational Complexity of the Forcing Chromatic Number

We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$-chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F_{\chi}(G)$ of an $s$-chromatic graph $G$ is the smallest number of vertices which must be colored so that, with the restriction that $s$ colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of $\force G$ relating it to the complexity class US introduced by Blass and Gurevich [Inform. Control, 55 (1982), pp. 80-88]. We prove that recognizing whether $F_{\chi}(G)\le2$ is US-hard with respect to polynomial-time many-one reductions. Moreover, this problem is coNP-hard even under the promises that $F_{\chi}(G)\le3$ and $G$ is 3-chromatic. On the other hand, recognizing whether $F_{\chi}(G)\le k$, for each constant $k$, is reducible to a problem in US via a disjunctive truth-table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

Power from Random Strings

We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual many-one reductions. Let ${{R_{\rm C}}}, {{R_{\rm Kt}}}, {{R_{\rm KS}}}, {{R_{\rm KT}}}$ be the sets of strings x having complexity at least $|x|/2$, according to the usual Kolmogorov complexity measure ${\mbox{\rm C}}$, Levin's time-bounded Kolmogorov complexity ${\mbox{\rm Kt}}$ [L. Levin, Inform. and Control, 61 (1984), pp. 15-37], a space-bounded Kolmogorov measure ${\mbox{\rm KS}}$, and a new time-bounded Kolmogorov complexity measure ${\mbox{\rm KT}}$, respectively. Our main results are as follows: \begin{remunerate} \item ${{R_{\rm KS}}}$ and ${{R_{\rm Kt}}}$ are complete for ${{\rm{PSPACE}}}$ and {\mbox{\rm EXP}}, respectively, under ${\mbox{\rm P/poly}}$-truth-table reductions. Similar results hold for other classes with ${{\rm{PSPACE}}}$-robust Turing complete sets. \item ${\mbox{\rm EXP}} = {\mbox{\rm NP}}^{{{R_{\rm Kt}}}}.$ \item ${{\rm{PSPACE}}} = {\mbox{\rm ZPP}}^{{{R_{\rm KS}}}} \subseteq {\mbox{\rm P}}^{{{R_{\rm C}}}}$. \item The Discrete Log, Factoring, and several lattice problems are solvable in ${\mbox{\rm BPP}}^{{{R_{\rm KT}}}}$. \end{remunerate} Our hardness result for ${{\rm{PSPACE}}}$ gives rise to fairly natural problems that are complete for ${{\rm{PSPACE}}}$ under ${\mbox{$\leq^{\rm p}_{\rm T}$}}$ reductions, but not under ${\mbox{$\leq^{\rm log}_{\rm m}$}}$ reductions. Our techniques also allow us to show that all computably enumerable sets are reducible to ${{R_{\rm C}}}$ via ${\mbox{\rm P/poly}}$-truth-table reductions. This provides the first "efficient" reduction of the halting problem to ${{R_{\rm C}}}$.

Small Spans in Scaled Dimension

Juedes and Lutz [SIAM J. Comput., 24 (1995), pp. 279--295] proved a small span theorem for polynomial-time many-one reductions in exponential time. This result says that for language A decidable in exponential time, either the class of languages reducible to A (the lower span) or the class of problems to which A can be reduced (the upper span) is small in the sense of resource-bounded measure and, in particular, that the degree of A is small. Small span theorems have been proved for increasingly stronger polynomial-time reductions, and a small span theorem for polynomial-time Turing reductions would imply $\BPP \not= \EXP$. In contrast to the progress in resource-bounded measure, Ambos-Spies et al. [{Proceedings of the 16th IEEE Conference on Computational Complexity, Philadelphia, PA, IEEE Computer Society, Los Alamitos, CA, 2001, pp. 210--217] showed that there is no small span theorem for the resource-bounded dimension of Lutz [SIAM J. Comput.}, 32 (2003), pp. 1236--1259], even for polynomial-time many-one reductions. Resource-bounded scaled dimension, recently introduced by Hitchcock, Lutz, and Mayordomo [J. Comput. System Sci., 69 (2004), pp. 97--122], provides rescalings of resource-bounded dimension. We use scaled dimension to further understand the contrast between measure and dimension regarding polynomial-time spans and degrees. We strengthen prior results by showing that the small span theorem holds for polynomial-time many-one reductions in the -3rd-order scaled dimension, but fails to hold in the -2nd-order scaled dimension. Our results also hold in exponential space. As an application, we show that determining the -2nd- or -1st-order scaled dimension in $\ESPACE$ of the many-one complete languages for $\E$ would yield a proof of $\mathrm{P} = \BPP$ or $\mathrm{P} \not= \PSPACE$. On the other hand, it is shown unconditionally that the complete languages for $\E$ have $-$3rd-order scaled dimension 0 in $\ESPACE$ and $-$2nd- and $-$1st-order scaled dimension 1 in $\E$.

On the Hardness of Graph Isomorphism

We show that the graph isomorphism problem is hard under DLOGTIME uniform AC{$^0$} many-one reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class {Mod}$_k$L and for the class DET of problems NC{$^1$} reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.

Completeness results for graph isomorphism

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC 2. In particular we show that tree isomorphism, when trees are encoded as strings, is NC 1-hard under AC 0-reductions. NC 1-completeness thus follows from Buss's NC 1 upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

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.

Reducing the complexity of reductions

We build on the recent progress regarding isomorphisms of complete sets that was reported in Agrawal et al. (1998). In that paper, it was shown that all sets that are complete under (non-uniform) AC0 reductions are isomorphic under isomorphisms computable and invertible via (non-uniform) depth-three AC0 circuits. One of the main tools in proving the isomorphism theorem in Agrawal et al. (1998) is a “Gap Theorem”, showing that all sets complete under AC0 reductions are in fact already complete under NC0 reductions. The following questions were left open in that paper:¶1. Does the “gap” between NC0 and AC0 extend further? In particular, is every set complete under polynomial-time reducibility already complete under NC0reductions?¶2. Does a uniform version of the isomorphism theorem hold?¶3. Is depth-three optimal, or are the complete sets isomorphic under isomorphisms computable by depth-two circuits?¶We answer all of these questions. In particular, we prove that the Berman—Hartmanis isomorphism conjecture is true for P-uniform AC0 reductions. More precisely, we show that for any class $ {\cal C} $ closed under uniform TC0-computable many-one reductions, the following three theorems hold:¶1. If $ {\cal C} $ contains sets that are complete under a notion of reduction at least as strong as Dlogtime-uniform AC0[mod 2] reductions, then there are such sets that are not complete under (even non-uniform) AC0 reductions.¶2. The sets complete for $ {\cal C} $ under P-uniform AC0 reductions are all isomorphic under isomorphisms computable and invertible by P-uniform AC0circuits of depth-three.¶3. There are sets complete for $ {\cal C} $ under Dlogtime-uniform AC0 reductions that are not isomorphic under any isomorphism computed by (even non-uniform) AC0 circuits of depth two.¶To prove the second theorem, we show how to derandomize a version of the switching lemma, which may be of independent interest. (We have recently learned that this result is originally due to Ajtai and Wigderson, but it has not been published.)

The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant

We show that approximating the shortest vector problem (in any $\ell_p$ norm) to within any constant factor less than $\sqrt[p]2$ is hard for NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai's constructive variant of Sauer's lemma that greatly simplifies Ajtai's original proof.

Resolution of Hartmanis’ conjecture for NL-hard sparse sets

We resolve a conjecture of Hartmanis from 1978 about sparse hard sets for nondeterministic logspace (NL). We show that there exists a sparse hard set S for NL under logspace many-one reductions if and only if NL= L (deterministic logspace).