Showing papers on "Completeness (order theory) published in 1994"

Categories for Types

Abstract: This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory. which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specialising in category theory.

Completeness results for basic narrowing

Abstract: In this paper we analyze completeness results for basic narrowing. We show that basic narrowing is not complete with respect to normalizable solutions for equational theories defined by confluent term rewriting systems, contrary to what has been conjectured. By imposing syntactic restrictions on the rewrite rules we recover completeness. We refute a result of Holldobler which states the completeness of basic conditional narrowing for complete (i.e. confluent and terminating) conditional term rewriting systems without extra variables in the conditions of the rewrite rules. In the last part of the paper we extend the completeness results of Giovannetti and Moiso for level-confluent and terminating conditional systems with extra variables in the conditions to systems that may also have extra variables in the right-hand sides of the rules.

What Is the Search Space of the Regular Inference

Abstract: This paper revisits the theory of regular inference, in particular by extending the definition of structural completeness of a positive sample and by demonstrating two basic theorems. This framework enables to state the regular inference problem as a search through a boolean lattice built from the positive sample. Several properties of the search space are studied and generalization criteria are discussed. In this framework, the concept of border set is introduced, that is the set of the most general solutions excluding a negative sample. Finally, the complexity of regular language identification from both a theoritical and a practical point of view is discussed.

Linguistics, logic, and finite trees

Abstract: A modal logic is developed to deal with finite ordered binary trees as they are used in (computational) linguistics A modal language is introduced with operators for the `mother of', `first daughter of' and `second daughter of' relations together with their transitive reflexive closures The relevant class of tree models is defined and three linguistic applications of this language are discussed: context free grammars, command relations, and trees decorated with feature structures An axiomatic proof system is given for which completeness is shown with respect to the class of finite ordered binary trees A number of decidability results followArticle

On completeness of certain families of semi-Riemannian manifolds

Abstract: Semi-Riemannian manifolds with a suitable set of conformal symmetries are shown to be complete. Locally warped products are studied and warped-completeness is introduced. In the case of definite and complete basis, several assumptions on the growth of the warping function yield some of the three kinds of completeness. The case of 1-dimensional basis (including a known family of relativistic space-times) is specially studied. Null warped-completeness is related to the completeness of a certain conformal metric on the basis. Several examples and counter-examples explaining the main results are also given.

Modeling Object-Oriented Program Execution

Abstract: This paper describes a way of organizing information about an object-oriented program's execution. The context is our system for visualizing that execution. The critical constraints are completeness, compactness, and efficient retrieval. We describe our design and how it meets these constraints.

Instance complexity

Abstract: We introduce a measure for the computational complexity of mdiwdual instances of a decision problem and study some of Its properties. The instance complexity of a string ~ with respect to a set A and time bound t, ict(x : A). is defined as the size of the smallest special-case program for A that run> m time t,decides x correctly, and makes no mistakes on other strings (“don’t know” answers are permitted). We prove that a set A is m P if and only if there exist a polynomial t and a constant c such that ic’(x : A) < c for all X; on the other hand, If A ]s NP-hard and P # NP, then for all polynomials t and constants c. lc’(~ : A) > c log I ~ I for ]nfimtely many x. Obserwng that Kf(x), the t-bounded Kolmogorov complexity of x, N roughly an upper bound on ]Ct(.t : A), we proceed to investigate the existence of mdiwdually hard problem Instances. ].e , strings whose instance complexity E close to their Kolmogorov complexity. We prove that if t(n)z n is a time-constructible function and A 1s a recurswe set not in DTIME(t), there then exist a constant c and mfimtely many I such that ic’(x : ,4) z K’ (x) – c. for some Prehmmary versions of parts of this work have appeared under the titles “What 1sa hard instance of a computational problem?” m Proceedings of tize Conference on Structare m Cornplexm Theory (Berkeley, Calif., June i 986), and “On the instance complexity of NP-hard problems” in Procecduzgs of the 5tk .4nrrual Conference on StntctLwe m Cowrpkwty Theory (Barcelona, Spain, July 1990). These Proceedings have been published by Springer-Verlag, Berlin, and IEEE, New York, respectively. The research of P. Orponen was supported by the Academy of Finland, and the research of K. Ko in part by National Science Foundation (NSF) grant CCR 8S-01575. Authors’ current addresses: P. Orponen, Department of Computer Science, Unnerslty of Helsinkl, FIN-0001 4 Helsinki, Finland; K. Ko, Department of Computer Science, State Unwersity of New York at Stony Brook, Stony Brook, NY 11794; U. Schomng, Abteiltrng Theoretische Informatik, Umversltat Ulm, D-89069 Ulm, Germany; O. Watanabe, Department of Computer Science, Tohyo Institute of Technology, Tokyo 152, Japan. Permission to copy without fee all or part of this material IS granted provided that the copies are not made or distributed for duect commercial advantage, the ACM copyright notice and the title of the pubhcdtion and Its date appear, and notice K given that copying 1s by permission of the Association for Computing Machinery. To copy otherwse, or to repubhsh, requmes a fee and/or specific permission. 01994 ACM 0004-5411/94/’0100-0096 $03.50 Journal of the AwocI.tIon for Compuh.g Md.hlncry, Vii 41 No 1, January 1YY4 pp Y6-121 Instance Complexity 97 time bound t‘(n)dependent on the complexity of recognizing A. Under the stronger assumptions that the set A is NP-hard and DEXT # NEXT, we prove that for any polynomia~ t there exist a polynomial f‘ and a constant c such that for infinitely many x, ict(x : A) z K“(x) – c. If A is DEXT-hard, then the same result holds unconditionally. We also prove that there is a set A E DEXT such that for some constant c and all x, ic’xp(x : A) s K’xp (x) – 2 log ZCexPr(x)– C, where exp(n) = 2“ and exp’(n) = cn2zn + c.

Lambek calculus and its relational semantics: Completeness and incompleteness

Abstract: The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girard's Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models.

On completeness of historical relational query languages

Abstract: Numerous proposals for extending the relational data model to incorporate the temporal dimension of data have appeared in the past several years. These proposals have differed considerably in the way that the temporal dimension has been incorporated both into the structure of the extended relations of these temporal models and into the extended relational algebra or calculus that they define. Because of these differences, it has been difficult to compare the proposed models and to make judgments as to which of them might in some sense be equivalent or even better. In this paper we define temporally grouped and temporally ungrouped historical data models and propose two notions of historical relational completeness, analogous to Codd's notion of relational completeness, one for each type of model. We show that the temporally ungrouped models are less expressive than the grouped models, but demonstrate a technique for extending the ungrouped models with a grouping mechanism to capture the additional semantic power of temporal grouping. For the ungrouped models, we define three different languages, a logic with explicit reference to time, a temporal logic, and a temporal algebra, and motivate our choice for the first of these as the basis for completeness for these models. For the grouped models, we define a many-sorted logic with variables over ordinary values, historical values, and times. Finally, we demonstrate the equivalence of this grouped calculus and the ungrouped calculus extended with a grouping mechanism. We believe the classification of historical data models into grouped and ungrouped models provides a useful framework for the comparison of models in the literature, and furthermore, the exposition of equivalent languages for each type provides reasonable standards for common, and minimal, notions of historical relational completeness.

Beyond NP-completeness for problems of bounded width (extended abstract): hardness for the W hierarchy

Abstract: The parameterized computational complexity of a collection of well-known problems including: BANDWIDTH, PRECEDENCE CONSTRAINED MULTIPROCESSOR SCHEDULING, LONGEST COMMON SUBSEQUENCE, DNA PHYSICAL MAPPING (or INTERNALIZING COLORED GRAPHS), PERFECT PHYLOGENY (or TRIANGULATING COLORED GRAPHS), COLORED CUTWIDTH, and FEASIBLE REGISTER ASSIGNMENT is explored. It is shown that these problems are hard for various levels of the W hierarchy. In the case of PRECEDENCE CONSTRAINED MULTIPROCESSOR SCHEDULING the results can be interpreted as providing substantial new complexity lower bounds on the outcome of [OPEN 8] of the Garey and Johnson list.

On the structure of complete sets

Abstract: The many types of resource-bounded reductions that are both an object of study and a research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The study of its structure can reveal properties that are general in that the complexity class, and the study of the structure of complete sets in different classes, can reveal secrets about the relation between these classes. Research into all sorts of aspects and properties of complete sets has been and will be a major topic in structural complexity theory. In this expository paper, we review the progress that has been made in recent years on selected topics in the study of complete sets. >

Reducibility and completeness in multi-party private computations

Abstract: We define the notions of reducibility and completeness in multi-party private computations. Let g be an n-argument function. We say that a function f is reducible to g if n honest-but-curious players can compute the function f n-privately, given a black-box for g (for which they secretly give inputs and get the result of operating g on these inputs). We say that g is complete (for multi-party private computations) if every function f is reducible to g. In this paper, we characterize the complete Boolean functions: we show that a Boolean function g is complete if and only if g itself cannot be computed n-privately (when there is no black-box available). Namely, for Boolean functions, the notions of completeness and n-privacy are complementary. This characterization gives a huge collection of complete functions (any non-private Boolean function!) compared to very few examples given (implicitly) in previous work. On the other hand, for non-Boolean functions, we show that these two notions are not complementary. Our results can be viewed as a generalization (for multi-party protocols and for (n/spl ges/2)-argument functions) of the two-party case, where it was known that Oblivious Transfer protocol (and its variants) are complete. >

Strong p-completeness of stochastic differential equations and the existence of smooth flows on noncompact manifolds

Abstract: Here we discuss the regularity of solutions of SDE's and obtain conditions under which a SDE on a complete Riemannian manifoldM has a global smooth solution flow, in particular improving the usual global Lipschitz hypothesis whenM=R n . There are also results on non-explosion of diffusions.

Completeness of Quasi‐Uniform and Syntopological Spaces

Abstract: In this paper we begin to develop the filter approach to (completeness of) quasiuniform spaces, proposed in [8, Section V]. It will be seen that this permits a more powerful and elegant account of completion to be given than was feasible using sequences or nets [8]. Just as in the previous version [8], we find that received notions concerning convergence need to be revised and reformulated to deal adequately with the nonsymmetric (non-Hausdorff) situation. The detailed motivation for the revisions concerning filter convergence (Sections 1, 4, 5 below) is independent of that given previously [8] for revising the notions of convergence of sequences and nets; the fact that the two sets of revisions lead to results that are in good agreement with each other tends to confirm the soundness of the revisions. The filter approach is more general in scope than the sequence/net approach. In particular, we shall see below that sobrification is a special case of the filter completion construction. The connection with sober spaces and locales is certainly one that can be pursued further, and indeed we intend the work reported here as at least a step in the direction of a localic, point-free, or even information system (in the sense of Scott) view of quasi-uniformities. Although all the examples and applications we have in mind are quasi-uniform spaces, we have found it helpful to formulate much of the material in terms of a still more general concept, namely the syntopological spaces of Csaszar [2]. We have in fact used syntopological formulations in a piecemeal fashion in previous versions [8], but we shall be doing this more systematically here.

Reflexive graphs and parametric polymorphism

Abstract: The pioneering work on relational parametricity for the second order lambda calculus was done by Reynolds (1983) under the assumption of the existence of set-based models, and subsequently reformulated by him, in conjunction with his student Ma, using the technology of PL-categories. The aim of this paper is to use the different technology of internal category theory to re-examine Ma and Reynolds' definitions. Apart from clarifying some of their constructions, this view enables us to prove that if we start with a non-parametric model which is left exact and which satisfies a completeness condition corresponding to Ma and Reynolds "suitability for polymorphism", then we can recover a parametric model with the same category of closed types. This implies, for example, that any suitably complete model (such as the PER model) has a parametric counterpart. >

Linear logic, totality and full completeness

Abstract: I give a 'totality space' model for linear logic [4] derived by taking an abstract view of computations on a datatype. The model has similarities with both the coherence space model and game-theoretic models, but is based upon a notion of total object. Using this model, I prove a full completeness result. In other words, I show that the mapping of proofs to their interpretations (here collections of total objects uniform for a given functor) in the model is a surjection. >

Fischer's protocol in timed process algebra

Abstract: Timed algebraic process theories can be developed with quite different purposes in mind. One can aim for theoretical results about the theory itself (completeness, expressiveness, decidability), or one can aim for practical applicability to non-trivial protocols. Unfortunately, these aims do not go well together. In this paper we take two theories, which are probably of the first kind, and try to find out how well suited they are for practical verifications. We verify Fischer’s protocol for mutual exclusion in the settings of discrete-time process algebra (ACPdt) and real-time process algebra (ACPur). We do this by transforming the recursive specification into an equivalent linear specification, and then dividing out the maximal bisimulation relation. The required mutual exclusion result can then be found by reasoning about the obtained process graph. Finally, we consider the ease of the verification, and ways to adapt the theory to make it more practical. It will turn out that the theories investigated are quite unsatisfactory when verifying real-life protocols. 1991 Mathematics Subject Classification: 68Q10, 68Q22, 68Q60. 1991 CR Categories: D.1.3, D.2.4, D.3.1, F.1.2, F.3.1.

An Introduction to Analysis

Abstract: Contents: The rudiments of set theory.- Number systems.- Linear analysis.- Cardinal numbers.- Ordinal numbers.- Metric spaces.- Continuity and limits.- Completeness and compactness.- General topology.- Bibliography.- Index.

Finitely representable databases (extended abstract)

Abstract: We study classes of infinite but finitely representable databases based on constraints, motivated by new database applications such as geographical databases. The mathematical framework is based on classical decidable first-order theories. We investigate the theory of finitely representable models and prove that it differs strongly from both classical model theory and finite model theory. In particular, we show that most of the well known theorems of either one fail (compactness, completeness, locality, 0/1 laws, etc.). An immediate consequence is the lack of tools to consider the definability of queries in the relational calculus over finitely representable databases. We illustrate this very challenging problem through some classical examples.

Existence and nonexistence of complete refinement operators

Abstract: Inductive Logic Programming is a subfield of Machine Learning concerned with the induction of logic programs In Shapiro's Model Inference System — a system that infers theories from examples — the use of downward refinement operators was introduced to walk through an ordered search space of clauses Downward and upward refinement operators compute specializations and generalizations of clauses respectively In this article we present the results of our study of completeness and properness of refinement operators for an unrestricted search space of clauses ordered by θ-subsumption We prove that locally finite downward and upward refinement operators that are both complete and proper for unrestricted search spaces ordered by θ-subsumption do not exist We also present a complete but improper upward refinement operator This operator forms a counterpart to Laird's downward refinement operator with the same properties

A Complete and Recursive Feature Theory

Abstract: Various feature descriptions are being employed in logic programming languages and constrained-based grammar formalisms. The common notational primitive of these descriptions are functional attributes called features. The descriptions considered in this paper are the possibly quantified first-order formulae obtained from a signature of binary and unary predicates called features and sorts, respectively. We establish a first-order theory FT by means of three axiom schemes, show its completeness, and construct three elementarily equivalent models. One of the models consists of so-called feature graphs, a data structure common in computational linguistics. The other two models consist of so-called feature trees, a record-like data structure generalizing the trees corresponding to first-order terms. Our completeness proof exhibits a terminating simplification system deciding validity and satisfiability of possibly quantified feature descriptions.

Process algebra with guards: Combining Hoare logic with process algebra

Abstract: We extend process algebra with guards, comparable to the guards in guarded commands or conditions in common programming constructs such as 'if then - else - fi' and 'while - do - od'. The extended language is provided with an operational semantics based on transitions between pairs of a process and a (data-)state. The data-states are given by a data environment that also defines in which data-states guards hold and how atomic actions (non-deterministically) transform these states. The operational semantics is studied modulo strong bisimulation equivalence. For basic process algebra (without operators for parallelism) we present a small axiom system that is complete with respect to a general class of data environments. Given a particular data environment 5 P we add three axioms to this system, which is then again complete, provided weakest preconditions are expressible and 5 p is sufficiently deterministic. Then we study process algebra with parallelism and guards. A two phase- calculus is provided that makes it possible to prove identities between parallel processes. Also this calculus is complete. In the last section we show that partial correctness formulas can easily be expressed in this setting. We use process algebra with guards to prove the soundness of a Hoare logic for linear processes by translating proofs in Hoare logic into proofs in process algebra.

An extended completeness condition for exact cone-beam reconstruction and its application

Abstract: The completeness condition is an important concept that specifies the capability of exact image reconstruction from a set of cone-beam projections. Unfortunately, the existing completeness conditions associated with Tuy's (1983), Smith's (1985), and Grangeat's (1991) inversion formulae are too restrictive in various situations where some projections are partially measured. The authors derive an extended completeness condition that improves the existing results. The new condition is a direct consequence of a reconstruction algorithm that is in the form of space-variant filtering followed by cone-beam backprojection. The new condition is less restrictive compared with the existing ones and provides an effective means to incorporate partially measured projections into image reconstruction. >

A Completion-Based Method for Mixed Universal and Rigid E-Unification

Abstract: We present a completion-based method for handling a new version of E- unification, called “mixed” E-unification, that is a combination of the classical “universal” E-unification and “rigid” E-unification. Rigid E-unification is an important method for handling equality in Gentzen-type first-order calculi, such as free-variable semantic tableaux or matings. The performance of provers using E-unification can be increased considerably, if mixed E-unification is used instead of the purely rigid version. We state soundness and completeness results, and describe experiments with an implementation of our method.

Onm-separated projection spaces

Abstract: Closure operators in the category of projection spaces are investigated. It is shown that completeness, absolutes-closure ands-injectivity coincide in the subcategory of separated projection spaces and that there compactness with respect to projections implies completeness.

DPOCL: A Principled Approach to Discourse Planning

Abstract: Research in discourse processing has identified two representational requirements for discourse planning systems. First, discourse plans must adequately represent the intentional structure of the utterances they produce in order to enable a computational discourse agent to respond effectively to communicative failures \cite{MooreParisCL}. Second, discourse plans must represent the informational structure of utterances. In addition to these representational requirements, we argue that discourse planners should be formally characterizable in terms of soundness and completeness.