737 research outputs found

    Exponential Lower Bounds and Separation for Query Rewriting

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    We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of first-order and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use non-signature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive existential rewritings; while first-order rewritings can be superpolynomially more succinct than positive existential rewritings

    Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

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    It is shown that for any fixed i>0i>0, the Σi+1\Sigma_{i+1}-fragment of Presburger arithmetic, i.e., its restriction to i+1i+1 quantifier alternations beginning with an existential quantifier, is complete for ΣiEXP\mathsf{\Sigma}^{\mathsf{EXP}}_{i}, the ii-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between NEXP\mathsf{NEXP} and EXPSPACE\mathsf{EXPSPACE}. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the Σ1\Sigma_1-fragment of Presburger arithmetic: given a Σ1\Sigma_1-formula Φ(x)\Phi(x), it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure

    Complexity of ITL model checking: some well-behaved fragments of the interval logic HS

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    Model checking has been successfully used in many computer science fields, including artificial intelligence, theoretical computer science, and databases. Most of the proposed solutions make use of classical, point-based temporal logics, while little work has been done in the interval temporal logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS over finite Kripke structures (under the homogeneity assumption) and an EXPSPACE model checking procedure for two meaningful fragments of it have been proposed. In this paper, we show that more efficient model checking procedures can be developed for some expressive enough fragments of HS

    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

    Invariant Generation through Strategy Iteration in Succinctly Represented Control Flow Graphs

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    We consider the problem of computing numerical invariants of programs, for instance bounds on the values of numerical program variables. More specifically, we study the problem of performing static analysis by abstract interpretation using template linear constraint domains. Such invariants can be obtained by Kleene iterations that are, in order to guarantee termination, accelerated by widening operators. In many cases, however, applying this form of extrapolation leads to invariants that are weaker than the strongest inductive invariant that can be expressed within the abstract domain in use. Another well-known source of imprecision of traditional abstract interpretation techniques stems from their use of join operators at merge nodes in the control flow graph. The mentioned weaknesses may prevent these methods from proving safety properties. The technique we develop in this article addresses both of these issues: contrary to Kleene iterations accelerated by widening operators, it is guaranteed to yield the strongest inductive invariant that can be expressed within the template linear constraint domain in use. It also eschews join operators by distinguishing all paths of loop-free code segments. Formally speaking, our technique computes the least fixpoint within a given template linear constraint domain of a transition relation that is succinctly expressed as an existentially quantified linear real arithmetic formula. In contrast to previously published techniques that rely on quantifier elimination, our algorithm is proved to have optimal complexity: we prove that the decision problem associated with our fixpoint problem is in the second level of the polynomial-time hierarchy.Comment: 35 pages, conference version published at ESOP 2011, this version is a CoRR version of our submission to Logical Methods in Computer Scienc

    Existentially Restricted Quantified Constraint Satisfaction

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    The quantified constraint satisfaction problem (QCSP) is a powerful framework for modelling computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

    Path Checking for MTL and TPTL over Data Words

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    Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL

    On the Succinctness of Query Rewriting over OWL 2 QL Ontologies with Shallow Chases

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    We investigate the size of first-order rewritings of conjunctive queries over OWL 2 QL ontologies of depth 1 and 2 by means of hypergraph programs computing Boolean functions. Both positive and negative results are obtained. Conjunctive queries over ontologies of depth 1 have polynomial-size nonrecursive datalog rewritings; tree-shaped queries have polynomial positive existential rewritings; however, in the worst case, positive existential rewritings can only be of superpolynomial size. Positive existential and nonrecursive datalog rewritings of queries over ontologies of depth 2 suffer an exponential blowup in the worst case, while first-order rewritings are superpolynomial unless NPP/poly\text{NP} \subseteq \text{P}/\text{poly}. We also analyse rewritings of tree-shaped queries over arbitrary ontologies and observe that the query entailment problem for such queries is fixed-parameter tractable

    Ontology-based data access with databases: a short course

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    Ontology-based data access (OBDA) is regarded as a key ingredient of the new generation of information systems. In the OBDA paradigm, an ontology defines a high-level global schema of (already existing) data sources and provides a vocabulary for user queries. An OBDA system rewrites such queries and ontologies into the vocabulary of the data sources and then delegates the actual query evaluation to a suitable query answering system such as a relational database management system or a datalog engine. In this chapter, we mainly focus on OBDA with the ontology language OWL 2QL, one of the three profiles of the W3C standard Web Ontology Language OWL 2, and relational databases, although other possible languages will also be discussed. We consider different types of conjunctive query rewriting and their succinctness, different architectures of OBDA systems, and give an overview of the OBDA system Ontop

    Revisiting Synthesis for One-Counter Automata

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    We study the (parameter) synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some omega-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called AERPADPLUS, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of AERPADPLUS, (ii) we prove that the synthesis problem is decidable in general and in N2EXP for several fixed omega-regular properties. Finally, (iii) we give a polynomial-space algorithm for the special case of the problem where parameters can only be used in tests, and not updates, of the counter
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