1,509 research outputs found

    Different Approaches to Proof Systems

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    The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper

    Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation

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    We present an efficient proof system for Multipoint Arithmetic Circuit Evaluation: for every arithmetic circuit C(x1,,xn)C(x_1,\ldots,x_n) of size ss and degree dd over a field F{\mathbb F}, and any inputs a1,,aKFna_1,\ldots,a_K \in {\mathbb F}^n, \bullet the Prover sends the Verifier the values C(a1),,C(aK)FC(a_1), \ldots, C(a_K) \in {\mathbb F} and a proof of O~(Kd)\tilde{O}(K \cdot d) length, and \bullet the Verifier tosses poly(log(dKF/ε))\textrm{poly}(\log(dK|{\mathbb F}|/\varepsilon)) coins and can check the proof in about O~(K(n+d)+s)\tilde{O}(K \cdot(n + d) + s) time, with probability of error less than ε\varepsilon. For small degree dd, this "Merlin-Arthur" proof system (a.k.a. MA-proof system) runs in nearly-linear time, and has many applications. For example, we obtain MA-proof systems that run in cnc^{n} time (for various c<2c < 2) for the Permanent, #\#Circuit-SAT for all sublinear-depth circuits, counting Hamiltonian cycles, and infeasibility of 00-11 linear programs. In general, the value of any polynomial in Valiant's class VP{\sf VP} can be certified faster than "exhaustive summation" over all possible assignments. These results strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed by Russell Impagliazzo and others. We also give a three-round (AMA) proof system for quantified Boolean formulas running in 22n/3+o(n)2^{2n/3+o(n)} time, nearly-linear time MA-proof systems for counting orthogonal vectors in a collection and finding Closest Pairs in the Hamming metric, and a MA-proof system running in nk/2+O(1)n^{k/2+O(1)}-time for counting kk-cliques in graphs. We point to some potential future directions for refuting the Nondeterministic Strong ETH.Comment: 17 page

    Epistemic virtues, metavirtues, and computational complexity

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    I argue that considerations about computational complexity show that all finite agents need characteristics like those that have been called epistemic virtues. The necessity of these virtues follows in part from the nonexistence of shortcuts, or efficient ways of finding shortcuts, to cognitively expensive routines. It follows that agents must possess the capacities – metavirtues –of developing in advance the cognitive virtues they will need when time and memory are at a premium

    Unary Pushdown Automata and Straight-Line Programs

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    We consider decision problems for deterministic pushdown automata over a unary alphabet (udpda, for short). Udpda are a simple computation model that accept exactly the unary regular languages, but can be exponentially more succinct than finite-state automata. We complete the complexity landscape for udpda by showing that emptiness (and thus universality) is P-hard, equivalence and compressed membership problems are P-complete, and inclusion is coNP-complete. Our upper bounds are based on a translation theorem between udpda and straight-line programs over the binary alphabet (SLPs). We show that the characteristic sequence of any udpda can be represented as a pair of SLPs---one for the prefix, one for the lasso---that have size linear in the size of the udpda and can be computed in polynomial time. Hence, decision problems on udpda are reduced to decision problems on SLPs. Conversely, any SLP can be converted in logarithmic space into a udpda, and this forms the basis for our lower bound proofs. We show coNP-hardness of the ordered matching problem for SLPs, from which we derive coNP-hardness for inclusion. In addition, we complete the complexity landscape for unary nondeterministic pushdown automata by showing that the universality problem is Π2P\Pi_2 \mathrm P-hard, using a new class of integer expressions. Our techniques have applications beyond udpda. We show that our results imply Π2P\Pi_2 \mathrm P-completeness for a natural fragment of Presburger arithmetic and coNP lower bounds for compressed matching problems with one-character wildcards

    A Casual Tour Around a Circuit Complexity Bound

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    I will discuss the recent proof that the complexity class NEXP (nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial size. The proof will be described from the perspective of someone trying to discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News, September 201

    Nondeterministic functions and the existence of optimal proof systems

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    We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system

    Toward accurate polynomial evaluation in rounded arithmetic

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    Given a multivariate real (or complex) polynomial pp and a domain D\cal D, we would like to decide whether an algorithm exists to evaluate p(x)p(x) accurately for all xDx \in {\cal D} using rounded real (or complex) arithmetic. Here ``accurately'' means with relative error less than 1, i.e., with some correct leading digits. The answer depends on the model of rounded arithmetic: We assume that for any arithmetic operator op(a,b)op(a,b), for example a+ba+b or aba \cdot b, its computed value is op(a,b)(1+δ)op(a,b) \cdot (1 + \delta), where δ| \delta | is bounded by some constant ϵ\epsilon where 0<ϵ10 < \epsilon \ll 1, but δ\delta is otherwise arbitrary. This model is the traditional one used to analyze the accuracy of floating point algorithms.Our ultimate goal is to establish a decision procedure that, for any pp and D\cal D, either exhibits an accurate algorithm or proves that none exists. In contrast to the case where numbers are stored and manipulated as finite bit strings (e.g., as floating point numbers or rational numbers) we show that some polynomials pp are impossible to evaluate accurately. The existence of an accurate algorithm will depend not just on pp and D\cal D, but on which arithmetic operators and which constants are are available and whether branching is permitted. Toward this goal, we present necessary conditions on pp for it to be accurately evaluable on open real or complex domains D{\cal D}. We also give sufficient conditions, and describe progress toward a complete decision procedure. We do present a complete decision procedure for homogeneous polynomials pp with integer coefficients, {\cal D} = \C^n, and using only the arithmetic operations ++, - and \cdot.Comment: 54 pages, 6 figures; refereed version; to appear in Foundations of Computational Mathematics: Santander 2005, Cambridge University Press, March 200
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