Polynomial-time Computable Functions Research Articles

Algorithms and Lower Bounds for Comparator Circuits from Shrinkage

In this paper, we initiate the study of average-case complexity and circuit analysis algorithms for comparator circuits. Departing from previous approaches, we exploit the technique of shrinkage under random restrictions to obtain a variety of new results for this model. Among them, we showAverage-case Lower Bounds For every k = k(n) with k geqslant log n, there exists a polynomial-time computable function f_k on n bits such that, for every comparator circuit C with at most n^{1.5}/O!left( kcdot sqrt{log n}right) gates, we have Prx∈0,1nC(x)=fk(x)⩽12+12Ω(k).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathop {{{\\,\\mathrm{\ extbf{Pr}}\\,}}}\\limits _{x\\in \\left\\{ 0,1\\right\\} ^n}\\left[ C(x)=f_k(x)\\right] \\leqslant \\frac{1}{2} + \\frac{1}{2^{\\Omega (k)}}. \\end{aligned}$$\\end{document} This average-case lower bound matches the worst-case lower bound of Gál and Robere by letting k=O!left( log nright) .#SAT Algorithms There is an algorithm that counts the number of satisfying assignments of a given comparator circuit with at most n^{1.5}/O!left( kcdot sqrt{log n}right) gates, in time 2^{n-k}cdot {{,textrm{poly},}}(n), for any kleqslant n/4. The running time is non-trivial (i.e., 2^n/n^{omega (1)}) when k=omega (log n).Pseudorandom Generators and textsf {MCSP} Lower Bounds There is a pseudorandom generator of seed length s^{2/3+o(1)} that fools comparator circuits with s gates. Also, using this PRG, we obtain an n^{1.5-o(1)} lower bound for textsf {MCSP} against comparator circuits.

On continuous one-way functions

The existence of one-way functions seems to depend, intuitively, on certain irregular properties of polynomial-time computable functions. Therefore, for functions with continuity properties, it suggests that all such functions are not one-way. It is shown here that in the formal complexity theory of real functions, this nonexistence of continuous one-way functions can be proved for one-to-one one-dimensional real functions, but fails for one-to-one two-dimensional real functions, if certain strong discrete one-way functions exist. Furthermore, for k-to-one functions, we can prove the existence of four-to-one one-dimensional one-way functions under the same assumption of the existence of strong discrete one-way functions. (A function f is k-to-one if for any y there exist at most k distinct values x such that f(x)=y.)

Bottleneck subset-type restricted matching problems

We introduce two problems related to finding in a weighted complete bipartite graph a special matching such that the maximum weight of its some subsets is minimal. We discuss their applications and show the strong NP-hardness of both problems. We show that one problem cannot be approximated in polynomial time within a factor of less than 2 and another problem cannot be approximated in polynomial time within a factor of $$\alpha (n)$$ , where $$\alpha (n)$$ is an arbitrary polynomial-time computable function, unless $$\hbox {P} = \hbox {NP}$$ .

A new algorithm design technique for hard problems

Heuristic algorithms have been used for a long time to tackle problems that are known or conjectured intractable. A heuristic algorithm is one that provides a correct decision for most inputs, but may fail on some. We focus on the case when failure means that the algorithm does not return any answer, rather than returning a wrong result. These algorithms are called errorless heuristics.A reasonable quality measure for heuristics is the failure rate over the set of n-bit instances. When no efficient exact algorithm is available for a problem, then, ideally, we would like one with vanishing failure rate. We show, however, that this is hard to achieve: unless a complexity theoretic hypothesis fails, some NP-complete problems cannot have a polynomial-time errorless heuristic algorithm with any vanishing failure rate.On the other hand, we prove that vanishing, even exponentially small, failure rate is achievable, if we use a somewhat different accounting scheme to count the failures. This is based on special sets, that we call α-spheres, which are the images of the n-bit strings under a bijective, polynomial-time computable and polynomial-time invertible encoding function α. Our main result is that polynomial-time errorless heuristic algorithms exist, with exponentially low failure rates on the α-spheres, for a large class of decision problems. This class includes, surprisingly, all known intuitively natural NP-complete problems. We also explore some connections to the theory of average case complexity.

Switching Cost-Aware Joint Frequency Assignment and Scheduling for Industrial Cognitive Radio Networks

The problem of inefficient and unevenly distributed spectrum usage in industrial wireless networks has led to the emergence of the concept of industrial cognitive radio (CR) networks, which have particularly important applications in automotive industry. Industrial CR networks are planned to function in a wide spectrum range; therefore, they have high energy consumption because of frequency switching while other wireless technologies do not have this problem. A distinctive feature of this switching cost is that it depends on the wideness between the two frequency bands. In this article, we formulate the joint frequency assignment and scheduling problem for multihop industrial CR networks with a single transceiver by considering varying amounts of energy consumption that occurs while CR devices switch to different frequency bands. Our optimization problem, which we formulate as an integer linear program, minimizes the energy cost related to frequency switching while making frequency and time slot allocation to the cognitive devices. We prove that even on star graphs, our formulated problem is inapproximable within any polynomial-time computable function f(n) in addition to being NP-Hard in the strong sense. Therefore, we propose a polynomial-time heuristic algorithm to solve the energy consumption problem due to channel switching. Simulation results demonstrate that the performance of our heuristic algorithm is very close to the results obtained from the integer linear programming implementation by CPLEX optimization software. We also compare our proposed method with the corresponding constant energy consumption for frequency switching case and two state-of-the-art algorithms and demonstrate that taking into account the different energy consumption while switching to different frequency bands is vital for joint frequency assignment and scheduling in multihop industrial CR networks with a single transceiver.

TRUTH AND FEASIBLE REDUCIBILITY

AbstractLet ${\cal T}$ be any of the three canonical truth theories CT− (compositional truth without extra induction), FS− (Friedman–Sheard truth without extra induction), or KF− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA (Peano arithmetic). We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA.Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every${\cal T}$-proof π of an arithmetical sentence ϕ, f (π) is a PA-proof of ϕ.

Polynomial time ultrapowers and the consistency of circuit lower bounds

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory $$\forall \mathsf {PV}$$ of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975), we show that every countable model of $$\forall \mathsf {PV}$$ is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of $$\forall \mathsf {PV}$$ we show that $$\forall \mathsf {PV}$$ is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajicek and Oliveira (Logical methods in computer science 13 (1:4), 2017).

A type-based complexity analysis of Object Oriented programs

A type system is introduced for a generic Object Oriented programming language in order to infer resource upper bounds. A sound and complete characterization of the set of polynomial time computable functions is obtained. As a consequence, the heap-space and the stack-space requirements of typed programs are also bounded polynomially. This type system is inspired by previous works on Implicit Computational Complexity, using tiering and non-interference techniques. The presented methodology has several advantages. First, it provides explicit big O polynomial upper bounds to the programmer, hence its use could allow the programmer to avoid memory errors. Second, type checking is decidable in polynomial time. Last, it has a good expressivity since it analyzes most object oriented features like inheritance, overload, override and recursion. Moreover it can deal with loops guarded by objects and can also be extended to statements that alter the control flow like break or return.

Game semantics approach to higher-order complexity

Game semantics was initially defined and used to characterize pcf functionals. We use this approach to propose a definition of complexity for such higher-order functions, as well as a class of polynomial time computable higher-order functions.

On the complexity of Wafer-to-Wafer Integration

In this paper we consider the Wafer-to-Wafer Integration problem. A wafer can be seen as a p-dimensional binary vector. The input of this problem is described by m multisets (called “lots”), where each multiset contains n wafers. The output of the problem is a set of n disjoint stacks, where a stack is a set of m wafers (one wafer from each lot). To each stack we associate a p-dimensional binary vector corresponding to the bit-wise AND operation of the wafers of the stack. The objective is to maximize the total number of “1” in the n stacks. We provide m1−ϵ and p1−ϵ non-approximability results even for n=2, f(n) non-approximability for any polynomial-time computable function f, as well as a pr-approximation algorithm for any constant r. Finally, we show that the problem is FPT when parameterized by p, and we use this FPT algorithm to improve the running time of the pr-approximation algorithm.

Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterministic Reductions

A circuit C compresses a function $${f : \{0,1\}^n\rightarrow \{0,1\}^m}$$f:{0,1}n?{0,1}m if given an input $${x\in \{0,1\}^n}$$x?{0,1}n, the circuit C can shrink x to a shorter l-bit string x? such that later, a computationally unbounded solver D will be able to compute f(x) based on x?. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size $${s=n^c}$$s=nc. Motivated by cryptographic applications, we focus on average-case $${(\ell,\epsilon)}$$(l,∈) incompressibility, which guarantees that on a random input $${x\in \{0,1\}^n}$$x?{0,1}n, for every size s circuit $${C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}$$C:{0,1}n?{0,1}l and any unbounded solver D, the success probability $${\Pr_x[D(C(x))=f(x)]}$$Prx[D(C(x))=f(x)] is upper-bounded by $${2^{-m}+\epsilon}$$2-m+∈. While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results: (1)Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$f:{0,1}n?{0,1} which is incompressible by size nc circuits with communication $${\ell=(1-o(1)) \cdot n}$$l=(1-o(1))·n and error $${\epsilon=n^{-c}}$$∈=n-c. Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).(2)We show that it is possible to achieve negligible error parameter $${\epsilon=n^{-\omega(1)}}$$∈=n-?(1) for nonboolean functions. Specifically, assuming that E is hard for exponential size $${\Sigma_3}$$Σ3-circuits, we construct a nonboolean function $${f:\{0,1\}^n\rightarrow \{0,1\}^m}$$f:{0,1}n?{0,1}m which is incompressible by size nc circuits with $${\ell=\Omega(n)}$$l=Ω(n) and extremely small $${\epsilon=n^{-c} \cdot 2^{-m}}$$∈=n-c·2-m. Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.(3)We show that the task of constructing an incompressible boolean function $${f:\{0,1\}^n\rightarrow \{0,1\}}$$f:{0,1}n?{0,1} with negligible error parameter $${\epsilon}$$∈ cannot be achieved by "existing proof techniques". Namely, nondeterministic reductions (or even $${\Sigma_i}$$Σi reductions) cannot get $${\epsilon=n^{-\omega(1)}}$$∈=n-?(1) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).

Cobham recursive set functions

This paper introduces the Cobham Recursive Set Functions (CRSF) as a version of polynomial time computable functions on general sets, based on a limited (bounded) form of ∈-recursion. This is inspired by Cobham's classic definition of polynomial time functions based on limited recursion on notation. We introduce a new set composition function, and a new smash function for sets which allows polynomial increases in the ranks and in the cardinalities of transitive closures. We bootstrap CRSF, prove closure under (unbounded) replacement, and prove that any CRSF function is embeddable into a smash term. When restricted to natural encodings of binary strings as hereditarily finite sets, the CRSF functions define precisely the polynomial time computable functions on binary strings. Prior work of Beckmann, Buss and Friedman and of Arai introduced set functions based on safe-normal recursion in the sense of Bellantoni-Cook. We prove an equivalence between our class CRSF and a variant of Arai's predicatively computable set functions.

Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey’s hierarchy

The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration ‘complete’ for the standard complexity classes NP and #P — even when restricted to smooth (=C∞) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n.The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k=k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to Λ(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k.It turns out that Maurice Gevrey’s 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-‘hardness’). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model.

Semigroups and one-way functions

We study the complexity classes 𝖯 and 𝖭𝖯 through a semigroup 𝖿𝖯 ("polynomial-time functions"), consisting of all polynomially balanced polynomial-time computable partial functions. The semigroup 𝖿𝖯 is non-regular if and only if 𝖯 ≠ 𝖭𝖯. The one-way functions considered here are based on worst-case complexity (they are not cryptographic); they are exactly the non-regular elements of 𝖿𝖯. We prove various properties of 𝖿𝖯, e.g. that it is finitely generated. We define reductions with respect to which certain universal one-way functions are 𝖿𝖯-complete.

Encoding Functions with Constant Online Rate, or How to Compress Garbled Circuit Keys

Randomized encodings of functions can be used to replace a “complex” function $f(x)$ by a “simpler” randomized mapping $\hat{f}(x;r)$ whose output distribution on an input $x$ encodes the value of $f(x)$ and hides any other information about $x$. One desirable feature of randomized encodings is low online complexity. That is, the goal is to obtain a randomized encoding $\hat{f}$ of $f$ in which most of the output can be precomputed and published before seeing the input $x$. When the input $x$ is available, it remains to publish only a short string $\hat{x}$, where the online complexity of computing $\hat{x}$ is independent of (and is typically much smaller than) the complexity of computing $f$. Yao's garbled circuit construction gives rise to such randomized encodings in which the online part $\hat{x}$ consists of $n$ encryption keys of length $\kappa$ each, where $n=|x|$ and $\kappa$ is a security parameter. Thus, the online rate $|\hat{x}|/|x|$ of this encoding is proportional to the security parameter $\kappa$. In this paper, we show that the online rate can be dramatically improved. Specifically, we show how to encode any polynomial-time computable function $f:\{0,1\}^n\to\{0,1\}^{m(n)}$ with online rate of $1+o(1)$ and with nearly linear online computation. More concretely, the online part $\hat{x}$ consists of an $n$-bit string and a single encryption key. These constructions can be based on the decisional Diffie--Hellman (DDH) assumption, the learning with errors (LWE) assumption, or the RSA assumption. We also present a variant of this result which applies to arithmetic formulas, where the encoding only makes use of arithmetic operations, as well as several negative results which complement our positive results. Our positive results can lead to efficiency improvements in most contexts where randomized encodings of functions are used. We demonstrate this by presenting several concrete applications. These include protocols for secure multiparty computation and for noninteractive verifiable computation in the preprocessing model which achieve, for the first time, an optimal online communication complexity, as well as noninteractive zero-knowledge proofs which simultaneously minimize the online communication and the prover's online computation.

Computing a Solution of Feigenbaum's Functional Equation in Polynomial Time

Lanford has shown that Feigenbaum's functional equation has an analytic solution. We show that this solution is a polynomial time computable function. This implies in particular that the so-called first Feigenbaum constant is a polynomial time computable real number.

Toward Wireless Security without Computational Assumptions—Oblivious Transfer Based on Wireless Channel Characteristics

Wireless security has been an active research area since the last decade. A lot of studies of wireless security use cryptographic tools, but traditional cryptographic tools are normally based on computational assumptions, which may turn out to be invalid in the future. Consequently, it is very desirable to build cryptographic tools that do not rely on computational assumptions. In this paper, we focus on a crucial cryptographic tool, namely 1-out-of-2 oblivious transfer. This tool plays a central role in cryptography because we can build a cryptographic protocol for any polynomial-time computable function using this tool. We present a novel 1-out-of-2 oblivious transfer protocol based on wireless channel characteristics, which does not rely on any computational assumption. We also illustrate the potential broad applications of this protocol by giving two applications, one on private communications and the other on privacy preserving password verification. We have fully implemented this protocol on wireless devices and conducted experiments in real environments to evaluate the protocol. Our experimental results demonstrate that it has reasonable efficiency.

On the complexity of constructing minimum changeover cost arborescences

The reload cost concept refers to the cost that occurs at a vertex along a path on an edge-colored graph when it traverses an internal vertex between two edges of different colors. This cost depends only on the colors of the traversed edges. Reload costs arise in various applications such as transportation networks, energy distribution networks, and telecommunications. Previous work on reload costs focuses on two problems of finding a spanning tree with minimum cost with respect to two different cost measures. In both problems the cost is associated with a set of paths from a given vertex r to all the leaves of the constructed tree. The first cost measure is the sum of the reload costs of all paths from r to the leaves. The second cost measure is the changeover cost, in which the cost of traversing a vertex by using two specific incident edges is paid only once regardless of the number of paths traversing it. The first problem is inapproximable within any polynomial time computable function of the input size [1], and the second problem is inapproximable within n1−ϵ for any ϵ>0[2]. In this paper we show that the first hardness result holds also for the second problem. Given this strong inapproximability result, we study the complexity and approximability properties of numerous special cases of this second problem. We mainly focus on bounded costs, and consider both directed and undirected graphs, bounded and unbounded number of colors, and both bounded and unbounded degree graphs. We also present polynomial time exact algorithms and an approximation algorithm for some special case. To the best of our knowledge, these are the first algorithms with a provable performance guarantee for the problem. Moreover, our approximation algorithm shows a tight bound on the approximability of the problem for a specific family of instances.

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic.The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

APPROXIMATING ASYMMETRIC TSP IN EXPONENTIAL TIME

Let G be a complete directed graph with n vertices and integer edge weights in range [0,M]. It is well known that an optimal Traveling Salesman Problem (TSP) in G can be solved in 2n time and space (all bounds are given within a polynomial factor of the input length, i.e., poly(n, log M)) and this is still the fastest known algorithm. If we allow a polynomial space only, then the best known algorithm has running time 4nnlog n. For TSP with bounded weights there is an algorithm with 1.657n · M running time. It is a big challenge to develop an algorithm with 2n time and polynomial space. Also, it is well-known that TSP cannot be approximated within any polynomial time computable function unless P=NP. In this short note we propose a very simple algorithm that, for any 0 < ε < 1, finds (1+ε)-approximation to asymmetric TSP in 2nε−1 time and ε−1 · poly(n, log M) space. Thereby, for any fixed ε, the algorithm needs 2n steps and polynomial space to compute (1 + ε)-approximation.