Showing papers on "Completeness (order theory) published in 2010"
TL;DR: In this paper, it was shown that affine completeness is a decidable property in congruence permutable varieties with higher commutators, and that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruense modular variety, it is possible to check whether two given polynomial terms induce the same function.
Abstract: We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.
71 citations
11 Jul 2010
TL;DR: It is proved that a final locally finite (dimensional) coalgebra is, equivalently, an initial iterative algebra of the category of real vector spaces and makes the connection to existing work on the semantics of recursive specifications.
Abstract: Stream circuits are a convenient graphical way to represent streams (or stream functions) computed by finite dimensional linear systems. We present a sound and complete expression calculus that allows us to reason about the semantic equivalence of finite closed stream circuits. For our proof of the soundness and completeness we build on recent ideas of Bonsangue, Rutten and Silva. They have provided a "Kleene theorem'' and a sound and complete expression calculus for coalgebras for endofunctors of the category of sets. The key ingredient of the soundness and completeness proof is a syntactic characterization of the final locally finite coalgebra. In the present paper we extend this approach to the category of real vector spaces. We also prove that a final locally finite (dimensional) coalgebra is, equivalently, an initial iterative algebra. This makes the connection to existing work on the semantics of recursive specifications.
68 citations
TL;DR: It is shown that the two NP-complete problems of Dodgson Score and Young Score have differing computational complexities when the winner is close to being a Condorcet winner and it is proved that the corresponding problem for Young elections is W[2]-complete.
Abstract: We show that the two NP-complete problems of Dodgson Score and Young Score have differing computational complexities when the winner is close to being a Condorcet winner. On the one hand, we present an efficient fixed-parameter algorithm for determining a Condorcet winner in Dodgson elections by a minimum number of switches in the votes. On the other hand, we prove that the corresponding problem for Young elections, where one has to delete votes instead of performing switches, is W[2]-complete. In addition, we study Dodgson elections that allow ties between the candidates and give fixed-parameter tractability as well as W[2]-completeness results depending on the cost model for switching ties.
65 citations
TL;DR: This article develops tableau-based decision procedures for the logics of subinterval structures over dense linear orderings and proves PSPACE completeness for both procedures and implement them in the generic tableAU-based theorem prover Lotrec.
Abstract: In this article, we develop tableau-based decision procedures for the logics of subinterval structures over dense linear orderings. In particular, we consider the two difficult cases: the relation of strict subintervals (with both endpoints strictly inside the current interval) and the relation of proper subintervals (that can share one endpoint with the current interval). For each of these logics, we establish a small pseudo-model property and construct a sound, complete and terminating tableau that searches systematically for existence of such a pseudo-model satisfying the input formulas. Both constructions are non-trivial, but the latter is substantially more complicated because of the presence of beginning and ending subintervals which require special treatment. We prove PSPACE completeness for both procedures and implement them in the generic tableau-based theorem prover Lotrec.
62 citations
20 Sep 2010
TL;DR: A general approach to ranking with partial abstention is proposed as well as evaluation metrics for measuring the correctness and completeness of predictions, able to achieve a reasonable trade-off between these two criteria.
Abstract: The prediction of structured outputs in general and rankings in particular has attracted considerable attention in machine learning in recent years, and different types of ranking problems have already been studied. In this paper, we propose a generalization or, say, relaxation of the standard setting, allowing a model to make predictions in the form of partial instead of total orders. We interpret such kind of prediction as a ranking with partial abstention: If the model is not sufficiently certain regarding the relative order of two alternatives and, therefore, cannot reliably decide whether the former should precede the latter or the other way around, it may abstain from this decision and instead declare these alternatives as being incomparable. We propose a general approach to ranking with partial abstention as well as evaluation metrics for measuring the correctness and completeness of predictions. For two types of ranking problems, we show experimentally that this approach is able to achieve a reasonable trade-off between these two criteria.
57 citations
06 Jun 2010
TL;DR: This paper studies partially closed databases from which both tuples and values may be missing, and proposes three models to characterize whether a c-instance T is complete for a query Q relative to master data.
Abstract: Databases in real life are often neither entirely closed-world nor entirely open-world. Indeed, databases in an enterprise are typically partially closed, in which a part of the data is constrained by master data that contains complete information about the enterprise in certain aspects [21]. It has been shown that despite missing tuples, such a database may turn out to have complete information for answering a query [9].This paper studies partially closed databases from which both tuples and values may be missing. We specify such a database in terms of conditional tables constrained by master data, referred to as c-instances. We first propose three models to characterize whether a c-instance T is complete for a query Q relative to master data. That is, depending on how missing values in T are instantiated, the answer to Q in T remains unchanged when new tuples are added. We then investigate four problems, to determine (a) whether a given c-instance is complete for a query Q, (b) whether there exists a c-instance that is complete for Q relative to master data available, (c) whether a c-instance is a minimal-size database that is complete for Q, and (d) whether there exists a c-instance of a bounded size that is complete for Q. We establish matching lower and upper bounds on these problems for queries expressed in a variety of languages, in each of the three models for specifying relative completeness.
57 citations
TL;DR: This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels, which have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations.
Abstract: This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/order-theoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres.
56 citations
Journal Article•
TL;DR: The origin of these operations, their formal deflnition and a series of results concerning language properties, decidability and computational completeness of families of languages generated by insertion-deletion systems and their extensions with the graphcontrol.
Abstract: This article gives an overview of the recent developments in the study of the operations of insertion and deletion. It presents the origin of these operations, their formal deflnition and a series of results concerning language properties, decidability and computational completeness of families of languages generated by insertion-deletion systems and their extensions with the graphcontrol. The basic proof methods are presented and the proofs for the most important results are sketched.
43 citations
09 Feb 2010
TL;DR: This work begins a rigorous study of completeness for primitives that allow fair computation, and shows that for fair secure computation, no primitive of size O(logk) is complete, where k is a security parameter.
Abstract: For secure two-party and multi-party computation with abort, classification of which primitives are complete has been extensively studied in the literature. However, for fair secure computation, where (roughly speaking) either all parties learn the output or none do, the question of complete primitives has remained largely unstudied. In this work, we initiate a rigorous study of completeness for primitives that allow fair computation. We show the following results:
No “short” primitive is complete for fairness. In surprising contrast to other notions of security for secure two-party computation, we show that for fair secure computation, no primitive of size O(logk) is complete, where k is a security parameter. This is the case even if we can enforce parallelism in calls to the primitives (i.e., the adversary does not get output from any primitive in a parallel call until it sends input to all of them). This negative result holds regardless of any computational assumptions.
A fairness hierarchy. We clarify the fairness landscape further by exhibiting the existence of a “fairness hierarchy”. We show that for every “short” l=O(logk), no protocol making (serial) access to any l-bit primitive can be used to construct even a (l+1)-bit simultaneous broadcast.
Positive results. To complement the negative results, we exhibit a k-bit primitive that is complete for two-party fair secure computation. We show how to generalize this result to the multi-party setting.
Fairness combiners. We also introduce the question of constructing a protocol for fair secure computation from primitives that may be faulty. We show that this is possible when a majority of the instances are honest. On the flip side, we show that this result is tight: no functionality is complete for fairness if half (or more) of the instances can be malicious.
43 citations
TL;DR: This article shows that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent, and shows that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ.
Abstract: Continuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result? The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies an approximated form of strong completeness, whereby Σ⊧ φ (if and) only if Σ⊢ φ ∸2 −n for all n . This approximated form of strong completeness asserts that if Σ⊧ φ , then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth of φ . Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theory T is decidable if for every sentence φ , the value φ T is a recursive real, and moreover, uniformly computable from φ . If T is incomplete, we say it is decidable if for every sentence φ the real number φ T o is uniformly recursive from φ , where φ T o is the maximal value of φ consistent with T . As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.
40 citations
TL;DR: An algorithm that checks whether a given test suite is complete is given and it is demonstrated that the algorithm can be used for relatively large FSMs and test suites.
Abstract: In testing from a Finite State Machine (FSM), the generation of test suites which guarantee full fault detection, known as complete test suites, has been a long-standing research topic. In this paper, we present conditions that are sufficient for a test suite to be complete. We demonstrate that the existing conditions are special cases of the proposed ones. An algorithm that checks whether a given test suite is complete is given. The experimental results show that the algorithm can be used for relatively large FSMs and test suites.
TL;DR: This work answers a question by proving that the Weihrauch-lattice is not a Brouwer algebra, and investigates the existence of infinite infima and suprema, as well as embeddings of the Medvedev-degrees into the Weil-degree.
Abstract: We answer a question by Vasco Brattka and Guido Gherardi by proving that the Weihrauch-lattice is not a Brouwer algebra. The computable Weihrauch-lattice is also not a Heyting algebra, but the continuous Weihrauch-lattice is. We further investigate the existence of infinite infima and suprema, as well as embeddings of the Medvedev-degrees into the Weihrauch-degrees.
Book Chapter•
01 Jan 2010
TL;DR: In this article, the authors trace the distinct historical roots of ordered algebras and logic, culminating with the theory of algebraizable logics, based on the pioneering work of Lindenbaum and Tarski and Blok and Pigozzi, that demonstrates the complementary nature of the two fields.
Abstract: Ordered algebras such as Boolean algebras, Heyting algebras, lattice-ordered groups, and MV-algebras have long played a decisive role in logic, although perhaps only in recent years has the significance of the relationship between the two fields begun to be fully recognized and exploited. The first aim of this survey article is to briefly trace the distinct historical roots of ordered algebras and logic, culminating with the theory of algebraizable logics, based on the pioneering work of Lindenbaum and Tarski and Blok and Pigozzi, that demonstrates the complementary nature of the two fields. The second aim is to explain and illustrate the usefulness of this theory, both from an ordered algebra and logic perspective, in the context of the relationship between residuated lattices and substructural logics. In particular, completions on the ordered algebra side, and Gentzen systems on the logic side, are used to address properties such as decidability, interpolation and amalgamation, and completeness.
TL;DR: The proposed model can also capture the consistency of data, in addition to its relative completeness, and provide characterizations for a database to be relatively complete, and for a query to allow a relatively complete database.
Abstract: This article investigates the question of whether a partially closed database has complete information to answer a query. In practice an enterprise often maintains master data Dm, a closed-world database. We say that a database D is partially closed if it satisfies a set V of containment constraints of the form q(D) ⊆ p(Dm), where q is a query in a language LC and p is a projection query. The part of D not constrained by (Dm, V) is open, from which some tuples may be missing. The database D is said to be complete for a query Q relative to (Dm, V) if for all partially closed extensions D' of D, Q(D') = Q(D), i.e., adding tuples to D either violates some constraints in V or does not change the answer to Q.We first show that the proposed model can also capture the consistency of data, in addition to its relative completeness. Indeed, integrity constraints studied for data consistency can be expressed as containment constraints. We then study two problems. One is to decide, given Dm, V, a query Q in a language LQ, and a partially closed database D, whether D is complete for Q relative to (Dm, V). The other is to determine, given Dm, V and Q, whether there exists a partially closed database that is complete for Q relative to (Dm, V). We establish matching lower and upper bounds on these problems for a variety of languages LQ and LC. We also provide characterizations for a database to be relatively complete, and for a query to allow a relatively complete database, when LQ and LC are conjunctive queries.
TL;DR: The work presented here investigates the combination of Kleene algebra with the synchrony model of concurrency from Milner’s SCCS calculus and isolates a class of pomsets which captures exactly synchronous Kleene algebras.
03 May 2010
TL;DR: The proof presented in this paper guarantees probabilistic completeness for a class of RRT-based algorithms given an appropriate projection operator for constraint manifolds of any fixed dimensionality.
Abstract: We present a proof for the probabilistic completeness of RRT-based algorithms when planning with constraints on end-effector pose Pose constraints can induce lower-dimensional constraint manifolds in the configuration space of the robot, making rejection sampling techniques infeasible RRT-based algorithms can overcome this problem by using the sample-project method: sampling coupled with a projection operator to move configuration space samples onto the constraint manifold Until now it was not known whether the sample-project method produces adequate coverage of the constraint manifold to guarantee probabilistic completeness The proof presented in this paper guarantees probabilistic completeness for a class of RRT-based algorithms given an appropriate projection operator This proof is valid for constraint manifolds of any fixed dimensionality
07 Aug 2010
TL;DR: In this article, the authors consider the operations of insertion and deletion working in a graph-controlled manner and show that computational completeness can be obtained by systems with insertion or deletion rules involving at most two symbols in a contextual or in a context-free manner and with the control graph having only four nodes.
Abstract: In this article, we consider the operations of insertion and deletion working in a graph-controlled manner. We show that like in the case of context-free productions, the computational power is strictly increased when using a control graph: computational completeness can be obtained by systems with insertion or deletion rules involving at most two symbols in a contextual or in a context-free manner and with the control graph having only four nodes.
09 Jun 2010
TL;DR: It is proved that even in a completely relationless world which assumes no commutativity nor associativity, permanent remains VNP-complete, and determinant can polynomially simulate any arithmetic formula, just as in the standard commutative, associative world of Valiant.
Abstract: This paper extends Valiant’s work on VP and VNP to the settings in which variables are not multiplicatively commutative and/or associative. Our main result is a theory of completeness for these algebraic worlds. We define analogs of Valiant’s classes VP and VNP, as well as of the polynomials permanent and determinant, in these worlds. We then prove that even in a completely relationless world which assumes no commutativity nor associativity, permanent remains VNP-complete, and determinant can polynomially simulate any arithmetic formula, just as in the standard commutative, associative world of Valiant. In the absence of associativity, the completeness proof gives rise to the following combinatorial problem: what is the smallest binary tree which contains as minors all binary trees with n leaves. We give an explicit construction of such a universal tree of polynomial size, a result of possibly independent interest. Given that such non-trivial reductions are possible even without commutativity and associativity, we turn to lower bounds. In the non-associative, commutative world we prove exponential circuit lower bounds on explicit polynomials, separating the non-associative commutative analogs of VP and VNP. Obtaining such lower bounds and a separation in the complementary associative, non-commutative world has been open for about 30 years.
01 Jan 2010
TL;DR: In this paper, the authors present the first examples of massless relativistic quantum field theories which are interacting and asymptotically complete, obtained by an application of a deformation procedure, introduced recently by Grosse and Lechner, to chiral conformal quantum field theory.
Abstract: This paper presents the first examples of massless relativistic quantum field theories which are interacting and asymptotically complete. These two-dimensional theories are obtained by an application of a deformation procedure, introduced recently by Grosse and Lechner, to chiral conformal quantum field theories. The resulting models may not be strictly local, but they contain observables localized in spacelike wedges. It is shown that the scattering theory for waves in two dimensions, due to Buchholz, is still valid under these weaker assumptions. The concepts of interaction and asymptotic completeness, provided by this theory, are adopted in the present investigation.
20 Mar 2010
TL;DR: To establish the completeness theorem for local state, it is necessary to reformulate the informal theory of Plotkin and Power as an enriched algebraic theory in the sense of Kelly and Power (JPAA, 89:163–179).
Abstract: Every algebraic theory gives rise to a monad, and monads allow a meta-language which is a basic programming language with side-effects. Equations in the algebraic theory give rise to equations between programs in the meta-language. An interesting question is this: to what extent can we put equational reasoning for programs into the algebraic theory for the monad?
In this paper I focus on local state, where programs can allocate, update and read the store. Plotkin and Power (FoSSaCS'02) have proposed an algebraic theory of local state, and they conjectured that the theory is complete, in the sense that every consistent equation is already derivable. The central contribution of this paper is to confirm this conjecture. To establish the completeness theorem, it is necessary to reformulate the informal theory of Plotkin and Power as an enriched algebraic theory in the sense of Kelly and Power (JPAA, 89:163–179). The new presentation can be read as 14 program assertions about three effects.
The completeness theorem for local state is dependent on certain conditions on the type of storable values. When the set of storable values is finite, there is a subtle additional axiom regarding quotient types.
TL;DR: In this paper, the notion of injectivity in Pos-S of S-posets for a pomonoid S is studied and a homological classification of pomonoids and pogroups is given.
Abstract: In this paper we study the notion of injectivity in the category Pos-S of S-posets for a pomonoid S. First we see that, although there is no non-trivial injective S-poset with respect to monomorphisms, Pos-S has enough (regular) injectives with respect to regular monomorphisms (sub S-posets). Then, recalling Banaschewski’s theorem which states that regular injectivity of posets with respect to order-embeddings and completeness are equivalent, we study regular injectivity for S-posets and get some homological classification of pomonoids and pogroups. Among other things, we also see that regular injective S-posets are exactly the retracts of cofree S-posets over complete posets.
27 Sep 2010
TL;DR: The statistical analysis of the specification’s completeness indicates that use case descriptions lead to more complete requirements specifications, and an objective evaluation scheme for assessing the completeness of specification documents is recommended.
Abstract: Providing high-quality software within budget is a goal pursued by most software companies. Incomplete requirements specifications can have an adverse effect on this goal and thus on a company’s competitiveness. Several empirical studies have investigated the effects of requirements engineering methods on the completeness of a specification. In order to increase this body of knowledge, we suggest using an objective evaluation scheme for assessing the completeness of specification documents, as objectifying the term completeness facilitates the interpretation of evaluations and hence comparison among different studies. This paper reports experience from applying the scheme to a student experiment comparing a use case with a textual approach common in industry. The statistical analysis of the specification’s completeness indicates that use case descriptions lead to more complete requirements specifications. We further experienced that the scheme is applicable to experiments and delivers meaningful results.
TL;DR: The completeness theorem becomes available for the infinitary versions of many “first order” logical systems that appear in the area of logic or computer science.
Abstract: The completeness of the infinitary language Lω1,ω was proved by Carol Karp in 1964. We express and prove the completeness of infinitary firstorder logics in the institution-independent setting by using forcing, a powerful method for constructing models. As a consequence of this abstraction, the completeness theorem becomes available for the infinitary versions of many “first order” logical systems that appear in the area of logic or computer science.
TL;DR: In this paper, the authors examined the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity.
Abstract: We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity.
Journal Article•
TL;DR: Two new approximation algorithms for Unique Games based on new methods for partitioning graphs by cutting small fractions of edges when the graph can be embedded in a suitable metric space are presented.
Abstract: We present two new approximation algorithms for Unique Games. The first generalizes the results of [2, 15] who give polynomial time approximation algorithms for graphs with high conductance. We give a polynomial time algorithm assuming only good local conductance, i.e. high conductance for small subgraphs. The second algorithm runs in mildly exponential time, e, but makes no assumptions about the underlying constraint graph. As the completeness approaches 1 (completeness 1 − ), the constant α in the running time rapidly approaches 0 (α = exp(−Ω (1/ )).) The value of the solutions returned by these algorithms depend only on the completeness of the Unique Game and either the local conductance or the allowed running time respectively. In particular, the performance of these algorithms does not depend on the number of labels in the Unique Game. Both algorithms are based on new methods for partitioning graphs by cutting small fractions of edges when the graph can be embedded in a suitable metric space. ∗Work supported by NSF grant CCF-0832797 †Work supported by the Simonyi Fund, The Bell Company Fellowship and Fund for Math, and NSF grants DMS083573 and CCF-0832797. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. ‡This research is supported by NSF grant CCF-0947262 from the Division of Computing and Communication Foundations. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. §Work supported by NSF grant CCF-0832797
TL;DR: It is shown that, with respect to full reduction, it is not possible to define a sound and complete intersection and union type assignment system for λ ¯ μ μ .
TL;DR: In this paper, the authors present an exploratory study that gives account of some students' perceptions of the completeness property of the set of real numbers and their scripts are analysed trying to understand how operational is the notion of completeness for them and to what extent they take this notion for granted.
Abstract: This article presents an exploratory study that gives account of some students’ perceptions of the completeness property of the set of real numbers. Students taking three undergraduate correlative courses in Calculus and Analysis answered a written questionnaire; their scripts are analysed trying to understand how operational is the notion of completeness for them and to what extent they take this notion for granted.
04 Mar 2010
TL;DR: In this article, it was shown that the volume growth condition for Riemannian manifolds does not imply that the manifold is stochastically complete, and that the suggested implication is not true in general.
Abstract: It was suggested in 1999 that a certain volume growth condition for geodesically complete Riemannian manifolds might imply that the manifold is stochastically complete. This is motivated by a large class of examples and by a known analogous criterion for recurrence of Brownian motion. We show that the suggested implication is not true in general. We also give counterexamples to a converse implication.