Showing papers on "Completeness (order theory) published in 2015"
14 Jan 2015
TL;DR: A framework to prove almost sure termination for probabilistic programs with real valued variables, based on ranking supermartingales, which is proven sound and complete for a meaningful class of programs involving randomization and bounded nondeterminism.
Abstract: We propose a framework to prove almost sure termination for probabilistic programs with real valued variables. It is based on ranking supermartingales, a notion analogous to ranking functions on non-probabilistic programs. The framework is proven sound and complete for a meaningful class of programs involving randomization and bounded nondeterminism. We complement this foundational insigh by a practical proof methodology, based on sound conditions that enable compositional reasoning and are amenable to a direct implementation using modern theorem provers. This is integrated in a small dependent type system, to overcome the problem that lexicographic ranking functions fail when combined with randomization. Among others, this compositional methodology enables the verification of probabilistic programs outside the complete class that admits ranking supermartingales.
141 citations
TL;DR: This work states that there is no efficient solution for this version of the motion planning problem, and the addition of differential constraints on robot motion or more general goal specifications makes motion planning even harder.
Abstract: Motion planning is a key problem in robotics that is concerned with finding a path that satisfies a goal specification subject to constraints. In its simplest form, the solution to this problem consists of finding a path connecting two states, and the only constraint is to avoid collisions. Even for this version of the motion planning problem, there is no efficient solution for the general case [1]. The addition of differential constraints on robot motion or more general goal specifications makes motion planning even harder. Given its complexity, most planning algorithms forego completeness and optimality for slightly weaker notions such as resolution completeness, probabilistic completeness [2], and asymptotic optimality.
104 citations
TL;DR: It is argued that, in some cases, natural parameterized problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem can be better understood in terms of their parameterized space or parameterized circuit complexity.
Abstract: The parameterized complexity of a problem is generally considered "settled" once it has been shown to be fixed-parameter tractable or to be complete for a class in a parameterized hierarchy such as the weft hierarchy. Several natural parameterized problems have, however, resisted such a classification. In the present paper we argue that, in some cases, this is due to the fact that the parameterized complexity of these problems can be better understood in terms of their parameterized space or parameterized circuit complexity. This includes well-studied, natural problems like the feedback vertex set problem, the associative generability problem, or the longest common subsequence problem. We show that these problems lie in and may even be complete for different parameterized space classes, leading to new insights into the problems' complexity. The classes we study are defined in terms of different forms of bounded nondeterminism and simultaneous time---space bounds.
56 citations
TL;DR: This paper provides a formal synchronization framework with bidirectional update propagation operations generated from a given TGG, which specifies the language of all consistently integrated source and target models.
Abstract: Triple graph grammars (TGGs) have been used successfully to analyze correctness and completeness of bidirectional model transformations, but a corresponding formal approach to model synchronization has been missing This paper closes this gap by providing a formal synchronization framework with bidirectional update propagation operations They are generated from a given TGG, which specifies the language of all consistently integrated source and target models As our main result, we show that the generated synchronization framework is correct and complete, provided that forward and backward propagation operations are deterministic Correctness essentially means that the propagation operations preserve and establish consistency while completeness ensures that the operations are defined for all possible inputs Moreover, we analyze the conditions under which the operations are inverse to each other All constructions and results are motivated and explained by a running example, which leads to a case study, using concrete visual syntax and abstract syntax notation based on typed attributed graphs
51 citations
TL;DR: In this paper, the authors investigated completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder.
49 citations
06 Jul 2015
TL;DR: It is shown that checking whether a 3-player NE (3-Nash) instance has an equilibrium in a ball of radius half in \(l_{\infty }\)-norm is ETR-complete, where ETR is the class Existential Theory of Reals.
Abstract: As a result of some important works [4, 5, 6, 10, 15], the complexity of 2-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, when equilibria with special properties are desired, the only result known is due to Schaefer and Stefankovic [18]: that checking whether a 3-player NE (3-Nash) instance has an equilibrium in a ball of radius half in \(l_{\infty }\)-norm is ETR-complete, where ETR is the class Existential Theory of Reals.
42 citations
31 Dec 2015
42 citations
14 Jan 2015
TL;DR: This work wants to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete or in the abstract, and introduces the completeness class of an abstraction as the set of all programs for which the abstraction is complete.
Abstract: We want to prove that a static analysis of a given program is complete, namely, no imprecision arises when asking some query on the program behavior in the concrete (ie, for its concrete semantics) or in the abstract (ie, for its abstract interpretation). Completeness proofs are therefore useful to assign confidence to alarms raised by static analyses. We introduce the completeness class of an abstraction as the set of all programs for which the abstraction is complete. Our first result shows that for any nontrivial abstraction, its completeness class is not recursively enumerable. We then introduce a stratified deductive system to prove the completeness of program analyses over an abstract domain A. We prove the soundness of the deductive system. We observe that the only sources of incompleteness are assignments and Boolean tests --- unlikely a common belief in static analysis, joins do not induce incompleteness. The first layer of this proof system is generic, abstraction-agnostic, and it deals with the standard constructs for program composition, that is, sequential composition, branching and guarded iteration. The second layer is instead abstraction-specific: the designer of an abstract domain A provides conditions for completeness in A of assignments and Boolean tests which have to be checked by a suitable static analysis or assumed in the completeness proof as hypotheses. We instantiate the second layer of this proof system first with a generic nonrelational abstraction in order to provide a sound rule for the completeness of assignments. Orthogonally, we instantiate it to the numerical abstract domains of Intervals and Octagons, providing necessary and sufficient conditions for the completeness of their Boolean tests and of assignments for Octagons.
36 citations
TL;DR: The uniqueness of Ricci flows on surfaces was proved in this paper, without any bound on the curvature or its growth at infinity, nor on the metric or the growth of the Ricci flow.
Abstract: We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, this completes the well-posedness theory for instantaneously complete Ricci flows on surfaces. 35K55, 53C44; 58J35
34 citations
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TL;DR: In this article, a survey of kernel analysis of Toeplitz operators is presented, starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.
Abstract: Toeplitz operators are met in different fields of mathematics such as stochastic processes, signal theory, completeness problems, operator theory, etc. In applications, spectral and mapping properties are of particular interest. In this survey we will focus on kernels of Toeplitz operators. This raises two questions. First, how can one decide whether such a kernel is non trivial? We will discuss in some details the results starting with Makarov and Poltoratski in 2005 and their succeeding authors concerning this topic. In connection with these results we will also mention some intimately related applications to completeness problems, spectral gap problems and P{\'o}lya sequences. Second, if the kernel is non-trivial, what can be said about the structure of the kernel, and what kind of information on the Toeplitz operator can be deduced from its kernel? In this connection we will review a certain number of results starting with work by Hayashi, Hitt and Sarason in the late 80's on the extremal function.
34 citations
TL;DR: In this article, generalized versions of the massless spin-boson model were considered and proved asymptotic completeness, which was later extended to generalized spinboson models.
11 Apr 2015
TL;DR: Optgen generates all local optimizations up to a given cost limit and verifies each rule using an SMT solver, guaranteeing correctness and completeness of the generated rule set.
Abstract: Every compiler comes with a set of local optimization rules, such as x + 0 → x and x & x → x, that do not require any global analysis. These rules reflect the wisdom of the compiler developers about mathematical identities that hold for the operations of their intermediate representation. Unfortunately, these sets of hand-crafted rules guarantee neither correctness nor completeness. Optgen solves this problem by generating all local optimizations up to a given cost limit. Since Optgen verifies each rule using an SMT solver, it guarantees correctness and completeness of the generated rule set. Using Optgen, we tested the latest versions of gcc, icc and llvm and identified more than 50 missing local optimizations that involve only two operations.
08 Sep 2015
TL;DR: The new situation in terms of an increasing semantic gap is discussed, and ISO 26262 should be amended with activities prescribing new refinements levels.
Abstract: The introduction of highly automated driving and autonomous road vehicles will imply new functional safety challenges. The higher complexity and the partly implicit definition of the tasks for the E/E systems will make it harder to argue completeness and correctness of the safety requirements in each phase of the ISO 26262 lifecycle. This paper discusses the new situation in terms of an increasing semantic gap, and recommends to perform more safety refinement steps. As a consequence, ISO 26262 should be amended with activities prescribing new refinements levels.
TL;DR: In this article, the spectral properties of discrete Laplacians were investigated based on the Hardy inequality and the use of super-harmonic functions, and lower bounds for the bottom of the spectrum and of the essential spectrum were obtained for weakly spherically symmetric graphs.
Abstract: In this paper, we investigate spectral properties of discrete Laplacians. Our study is based on the Hardy inequality and the use of super-harmonic functions. We recover and improve lower bounds for the bottom of the spectrum and of the essential spectrum. In some situation, we obtain Weyl asymptotics for the eigenvalues. We also provide a probabilistic representation of super-harmonic functions. Using coupling arguments, we set comparison results for the bottom of the spectrum, the bottom of the essential spectrum and the stochastic completeness of different discrete Laplacians. The class of weakly spherically symmetric graphs is also studied in full detail.
13 Jun 2015
TL;DR: The procedure reduces the analysis of higher-order programs to checking satisfiability of a sequence of quantifier-free formulas over theories, thus enabling the use of efficient satisfiability modulo theory (SMT) solvers.
Abstract: We present a verification procedure for pure higher-order functional Scala programs with parametric types. We show that our procedure is sound for proofs, as well as sound and complete for counter-examples. The procedure reduces the analysis of higher-order programs to checking satisfiability of a sequence of quantifier-free formulas over theories such as algebraic data types, integer linear arithmetic, and uninterpreted function symbols, thus enabling the use of efficient satisfiability modulo theory (SMT) solvers. Our solution supports arbitrary function types and arbitrarily nested anonymous functions (which can be stored in data structures, passed as arguments, returned, and applied). Among the contributions of this work is supporting even those cases when anonymous functions cannot be statically traced back to their definition, ensuring completeness of the approach for finding counter-examples. We provide a proof of soundness and counter-example completeness for our system as well as initial evaluation in the Leon verifier.
TL;DR: To deduce an analogous basis for decision implications, the notion of decision premise is introduced and form the so-called decision implication canonical basis is shown, which is complete, non-redundant and minimal among all complete sets of decision implications.
TL;DR: In this article, the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n) were established for tensor products of fundamental representations.
Abstract: We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra gl(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.
20 Apr 2015
TL;DR: In this paper, the completeness property of root functions of general boundary value problems for first order systems of ordinary differential equations on a finite interval was studied, and it was shown that the system of root vectors of the general Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses.
Abstract: The paper is concerned with the completeness property of root functions of general boundary value problems for $n \times n$ first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be non-weakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval.
It is also shown that the system of root vectors of the general $n \times n$ Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in $L^2([0,1]; \mathbb{C}^n)$. Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.
TL;DR: In this article, the notion of geodesic completeness for semi-Riemannian metrics of low regularity was defined in the framework of the geometric theory of generalized functions, and completeness of a wide class of impulsive gravitational wave space-times was shown.
Abstract: We define the notion of geodesic completeness for semi-Riemannian metrics of low regularity in the framework of the geometric theory of generalized functions. We then show completeness of a wide class of impulsive gravitational wave space-times.
TL;DR: The notion of Borel completeness was introduced in this article for @0-stable theories, and it was shown that if T either has eniDOP or is eni-deep, then its class of countable models is Borel complete.
Abstract: We study @0-stable theories, and prove that if T either has eniDOP or is eni-deep, then its class of countable models is Borel complete. We introduce the notion of �-Borel completeness and prove that such theories are �-Borel complete. Using this, we conclude that
01 Dec 2015
TL;DR: In this article, the authors introduce the class of fuzzy real valued p-absolutely summable multiple sequences on probabilistic normed spaces and prove some inclusion results involving this sequence space.
Abstract: In this article we introduce the class of fuzzy real valued p-absolutely summable multiple sequences \({}_n\ell _F^p \) on probabilistic normed spaces. We study some properties of this sequence space like completeness solidness, symmetricity, convergence free, sequence algebra etc. We prove some inclusion results involving this sequence space.
TL;DR: Two complete axiomatizations of the equational theories of the real numbers are given with respect to signatures of meadows with a single axiom scheme expressing formal realness.
TL;DR: In this paper, the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain were studied and an identity derived by Backer, Furstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), was proved both for Dirichlet and Neumann boundary conditions.
Abstract: In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Backer, Furstburger, Schubert, and Steiner, expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.
TL;DR: In this article, a quasi-metric generalization of Caristi's fixed point theorem for complete quasi-matric spaces is presented. But the generalization is restricted to a special case of complete quasimetric spaces.
Abstract: We obtain a quasi-metric generalization of Caristi’s fixed point theorem for a kind of complete quasi-metric spaces. With the help of a suitable modification of its proof, we deduce a characterization of Smyth complete quasi-metric spaces which provides a quasi-metric generalization of the well-known characterization of metric completeness due to Kirk. Some illustrative examples are also given. As an application, we deduce a procedure which allows to easily show the existence of solution for the recurrence equation of certain algorithms.
TL;DR: It is shown that the axioms proposed by Gabbay and Ciancia are not complete over the semantic interpretation they propose, and a slightly wider class of language models are identified over which they are sound and complete.
Abstract: Gabbay and Ciancia (2011) presented a nominal extension of Kleene algebra as a framework for trace semantics with statically scoped allocation of resources, along with a semantics consisting of nominal languages. They also provided an axiomatization that captures the behavior of the scoping operator and its interaction with the Kleene algebra operators and proved soundness over nominal languages. In this paper, we show that the axioms proposed by Gabbay and Ciancia are not complete over the semantic interpretation they propose. We then identify a slightly wider class of language models over which they are sound and complete.
TL;DR: In this article, the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra were established for the case of tensor products of fundamental representations.
Abstract: We establish the basics of the Bethe ansatz for the Gaudin model associated to the Lie superalgebra 𝔤𝔩(m|n). In particular, we prove the completeness of the Bethe ansatz in the case of tensor products of fundamental representations.
TL;DR: In this paper, normalized and strict NaP-preference on the same ground set are in a one-to-one correspondence, and the properties of injectivity and projectivity are a collectionwise extension of the antisymmetry and completeness of a single binary relation.
TL;DR: In this article, the authors investigated completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder.
Abstract: We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.
TL;DR: A theory of abstraction for the framework of parameterised Boolean equation systems, a first-order fixpoint logic, is presented and it is shown that for model checking the modal μ-calculus, the abstractions can be exponentially more succinct than GTSs.
Abstract: We present a theory of abstraction for the framework of parameterised Boolean equation systems, a first-order fixpoint logic. Parameterised Boolean equation systems can be used to solve a variety of problems in verification. We study the capabilities of the abstraction theory by comparing it to an abstraction theory for Generalised Kripke modal Transition Systems (GTSs). We show that for model checking the modal μ-calculus, our abstractions can be exponentially more succinct than GTSs and our theory is as complete as the GTS framework for abstraction. Furthermore, we investigate the completeness of our theory irrespective of the encoded decision problem. We illustrate the potential of our theory through case studies using the first-order modal μ-calculus and a real-time extension thereof, conducted using a prototype implementation of a new syntactic transformation for parameterised Boolean equation systems.
TL;DR: This paper tackles the problem of whether Åqvist's dyadic deontic systems E and F are complete with respect to their intended Hanssonian preference-based semantics, and establishes that, under either the maximality rule or the optimality rule, they are sound and complete.
Abstract: Abstract This paper tackles an open problem posed by Åqvist. It is the problem of whether his dyadic deontic systems E and F are complete with respect to their intended Hanssonian preference-based semantics. It is known that there are two different ways of interpreting what it means for a world to be best or top-ranked among alternatives. This can be understood as saying that it is optimal among them, or maximal among them. First, it is established that, under either the maximality rule or the optimality rule, E is sound and complete with respect to the class of all preference models, the class of those in which the betterness relation is reflexive, and the class of those in which it is total. Next, an analogous result is shown to hold for F. That is, it is established that, under either rule, F is sound and complete with respect to the class of preference models in which the betterness relation is limited, the class of those in which it is limited and reflexive, and the class of those in which it is limited and total.