Showing papers on "Completeness (order theory) published in 1990"

d-Separation: From Theorems to Algorithms

Abstract: An efficient algorithm is developed that identifies all independencies implied by the topology of a Bayesian network. Its correctness and maximality stems from the soundness and completeness of d -separation with respect to probability theory. The algorithm runs in time 0 (|E|) where E is the number of edges in the network

No better ways to generate hard NP instances than picking uniformly at random

Abstract: Distributed NP (DNP) problems are ones supplied with probability distributions of instances. It is shown that every DNP problem complete for P-time computable distributions is also complete for all distributions that can be sampled. This result makes the concept of average-case NP completeness robust and the question of the average-case complexity of complete DNP problems a natural alternative to P=?NP. Similar techniques yield a connection between cryptography and learning theory.

Asymptotic completeness for N-body short-range quantum systems: A new proof

Abstract: We give an alternative geometrical proof of asymptotic completeness for an arbitrary number of quantum particles interacting through shortrange pair potentials. It relies on an estimate showing that the intercluster motion concentrates asymptotically on classical trajectories.

On Restrictions of Ordered Paramodulation with Simplification

Abstract: We consider a restricted version of ordered paramodulation, called strict superposition. We show that strict superposition (together with equality resolution) is refutationally complete for Horn clauses, but not for general first-order clauses. Two moderate enrichments of the strict superposition calculus are, however, sufficient to establish refutation completeness. This strictly improves previous results. We also propose a simple semantic notion of redundancy for clauses which covers most simplification and elimination techniques used in practice yet preserves completeness of the proposed calculi. The paper introduces a new and comparatively simple technique for completeness proofs based on the use of canonical rewrite systems to represent equality interpretations.

A completeness relation for the q-analogue coherent states by q-integration

Abstract: q-integration is defined for the q-oscillator realization of quantum groups. This is used to prove a completeness relation for the q-analogue of the usual coherent states. These states are overcomplete.

NP-completeness of some problems concerning voting games

Abstract: The problem of confirming lower bounds on the number of coalitions for which an individual is pivoting is NP-complete. Consequently, the problem of confirming non-zero values of power indices is NP-complete. The problem of computing the Absolute Banzhaf index is #P-complete. Related problems for power indices are discussed.

Some homological properties of complete modules

Abstract: In this paper A is a commutative noetherian ring, a an ideal of A and the A- modules are given the a-adic topology.It is a general feeling that completeness is a kind of finiteness condition. We make precise that feeling and, after a result concerning the homology of a complex of complete modules which can be used in place of Nakayama's Lemma, we establish analogies between complete modules and finitely generated ones, with respect to flat dimension, injective dimension, Bass numbers and the Koszul complex. This is particularly clear in the local case, where we have also some partial information on the support of a complete module. With respect to dimension however, the analogy fails, as shown by an example.

Algebra and query language for a historical data model

Abstract: We propose a «state» oriented view of historical databases. We propose an algebra for historical relations which contains classical as well as some new operators. The operators are simple to comprehend, unlike in other research proposals. Were are also able to formulate a completeness criteria for the proposed model. Finally, we extend the popular SQL query language for use with historical databases. Again, the extensions are consistent with the simple basis of standard SQL

# P-completeness via man-one reductions

Abstract: Valiant defined #P-completeness using Turing reductions. We propose a more restrictive definition, a variant of many-one reductions, and prove that the (0, 1)-permanent remains #P-complete under this definition.

The Structure of Complete Degrees

Abstract: The notion of NP-completeness has cut across many fields and has provided a means of identifying deep and unexpected commonalities. Problems from areas as diverse as combinatorics, logic, and operations research turn out to be NP-complete and thus computationally equivalent in the sense discussed in the next paragraph. PSPACE-completeness, NEXP-completeness, and completeness for other complexity classes have likewise been used to show commonalities in a variety of other problems. This paper surveys investigations into how strong these commonalities are.

Mechanizing inductive reasoning.

Abstract: Automating proofs by induction is important in many computer science and artificial intelligence applications, in particular in program verification and specification systems. We present a new method to prove (and disprove) automatically inductive properties. Given a set of axioms, a well-suited induction scheme is constructed automatically. We call such a scheme a test-set. Then, for proving a property, we just instanciate it with terms from the test-set and apply pure algebraic simplification to the result. This method avoids completion and explicit induction. However it retains their positive features, namely the completeness of the former and the robustness of the latter.

Rigid E -unification: NP-completeness and applications to equational matings

Abstract: Rigid E -unification is a restricted kind of unification modulo equational theories, or E -unification, that arises naturally in extending Andrew's theorem proving method of matings to first-order languages with equality. This extension was first presented by J. H. Gallier, S. Raatz, and W. Snyder, who conjectured that rigid E -unification is decidable. In this paper, it is shown that rigid E -unification is NP-complete and that finite complete sets of rigid E -unifiers always exist. As a consequence, deciding whether a family of mated sets is an equational mating is an NP-complete problem. Some implications of this result regarding the complexity of theorem proving in first-order logic with equality are also discussed.

Tools for proving inductive equalities, relative completeness, and Ω-completeness

Abstract: Inductive theorems are properties valid in the initial algebra. A now popular tool for proving them in equational theories or abstract data types is based on proof by consistency. This method uses a completion procedure and requires two essential properties of the specification, namely relative completeness and ω-completeness. This paper investigates ways of proving them. For the first one, the complement algorithm is presented. It is based on unification and computation of coverings and complements. For the second one, a technique based on discrimination of pairs of normal forms is explained and illustrated through examples.

A new strategy for proving $ omega $ -completeness applied to process algebra

Abstract: A new technique for proving ω-completeness based on proof transformations is presented. This technique is applied to axiom systems for finite, concrete, sequential processes. It turns out that the number of actions is important for these sets to be ω-complete. For the axiom systems for bisimulation and completed trace semantics one action suffices and for traces 2 actions are enough. The ready, failure, ready trace and failure trace axioms are only ω-complete if an infinite number of actions is available. We also consider process algebra with parallelism and show several axiom sets containing the axioms of standard concurrency ω-complete.

Dynamic algebras as a well-behaved fragment of relation algebras

Abstract: The varieties RA of relation algebras and DA of dynamic algebras are similar with regard to definitional capacity, admitting essentially the same equational definitions of converse and star. They differ with regard to completeness and decidability. The RA definitions that are incomplete with respect to representable relation algebras, when expressed in their DA form are complete with respect to representable dynamic algebras. Moreover, whereas the theory of RA is undecidable, that of DA is decidable in exponential time. These results follow from representability of the free intensional dynamic algebras.

A New Strategy for Proving omega-Completeness applied to Process Algebra

Abstract: A new technique for proving ω-completeness based on proof transformations is presented. This technique is applied to axiom systems for finite, concrete, sequential processes. It turns out that the number of actions is important for these sets to be ω-complete. For the axiom systems for bisimulation and completed trace semantics one action suffices and for traces 2 actions are enough. The ready, failure, ready trace and failure trace axioms are only ω-complete if an infinite number of actions is available. We also consider process algebra with parallelism and show several axiom sets containing the axioms of standard concurrency ω-complete.

Strategies for modal resolution: results and problems

Abstract: This paper is concerned with the definition of strategies for resolution in modal logic. We propose the following strategies: deletion of subsumed clauses, extensions of classical strategies based on a static constraint, negative resolution, input and linear resolution. A class of strategies based on static constraints and the linear strategy are proved to be complete. For input and negative strategy we have completeness results provided some restrictions on the class of considered clauses. Some problems such as completeness of deletion of subsumed clauses are left open; we state and discuss them in the paper.

A Temporal Relational Algebra as Basis for Temporal Relational Completeness

Abstract: We define a tenlporal algebra that is applicable to any temporal relational data model supporting discrete linear bounded time. This algebra has the five basic relational algebra operators extended to the temporal domain and an operator of linear recursion. We show that this algebra has the expressive power of a safe temporal calculus based on the predicate temporal logic with the until and since temporal operators. In [CrC189], a historical calculus was proposed as a basis for historical relational completeness. We propose the temporal algebra defined in this paper and the equivalent temporal calculus as an alternative basis for temporal relational completeness.

Higher order E-unification

Abstract: In this extended abstract we report on an investigation of Higher-Order E-Unification, which consists of unifying typed lambda terms in the context of a first-order set of equations E. This problem subsumes both higher-order unification and general E-unification, and provides a theoretical background for reasoning systems which incorporate both algebraic and higher-order logic. The problem and its properties are discussed, a set of transformations (inference rules) extending those of Martelli-Montanari for standard unification is given, and then we prove the completeness of this non-deterministic algorithm. The completeness of restrictions of these rules for higher-order pre-E-unification and higher-order narrowing are corollaries of these results. Finally, we connect these results with previous work, and conclude with future directions and open problems. The major result is a set of inference rules for higher-order E-unification and a proof of its soundness and completeness (wrt complete sets of unifiers).

A Direct Proof for the Completeness of SLD-Resolution

Abstract: Lemma 2 Let A be an atom and P |= A. Then A has an implication tree wrt. P . Proof: We construct a model M of P . Let |M| be the set of all terms (with variables) and let fM(~t ) := f(~t ). Let rM(~t ) be true if and only if r(~t ) has an implication tree wrt. P . We claim that M is a model of P . If B is a fact of P then for every substitution (variable assignment) θ the atom Bθ is an implication tree wrt. P and M |= Bθ. If B ← C1 ∧ . . . ∧ Cn is a clause of P (1 ≤ n) and θ is a substitution and M |= C1θ ∧ . . . ∧ Cnθ, then C1θ, . . . , Cnθ have implication trees wrt. P and therefore Bθ and we have M |= Bθ. Since M is a model of P and P |= A, the atom A is true in M and has an implication tree wrt. P . 2

Foundation of compositional program refinement—safety properties

Abstract: The aim of this paper is twofold: first is to formulate a foundation for refinement of parallel programs that may synchronously communicate and/or share variables; programs rendered as 1st order transition systems. The second aim is to bring closer and to show the relevance of the algebraic theory of parallel processes to that of the refinement of such 1st order systems. We do this by first developing a notion of refinement and a complete verification criteria for it for algebraic, uninterpreted transition systems—basing ourselves on already existing theory. Then we show how 1st order transition systems can be translated—while preserving those aspects of their semantics that we are interested in—into uninterpreted transition systems. Since this translation is canonical, it is used to lift the algebraic refinement and verification criteria to the level of 1st order systems. Specifically, we show that they yield assertional methods for refinement of such systems that resemble the methods used in Z. Manna and A. Pnueli's temporal logic proof system.

Moduloïds and pseudomodules

Abstract: We determine here sufficient conditions for finite dimensional moduloids and pseudomodules to be lattices. In particular, we show that completeness of the dioid D of scalars is such a condition. The simplicity conditions for pseudomodules, which make the classification problem tractable, are also shown to be sufficient for the lattice structure. Since these conditions are clearly unrelated, both results show that neither one is necessary. A concluding example illustrates this remark.

Asymptotic completeness for N-body short-range quantum systems

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Empty types in polymorphic lambda-calculus

Abstract: The model theory of simply typed and polymorphic (second-order) lambda calculus changes when types are allowed to be empty. For example, the “polymorphic Boolean” type really has exactly two elements in a polymorphic model only if the “absurd” type ∀t.t is empty. The standard b-e axioms and equational inference rules which are complete when all types are nonempty are not complete for models with empty types. Without a little care about variable elimination, the standard rules are not even sound for empty types. We extend the standard system to obtain a complete proof system for models with empty types. The completeness proof is complicated by the fact that equational “term models” are not so easily obtained: in contrast to the nonempty case, not every theory with empty types is the theory of a single model.

The completeness problem in the algebra of partial functions of finite-valued logic

Abstract: All maximal subalgebras of the algebra of partial functions of k-valued logic (k≥4) are described using invariant relations. The problem of functional completeness in the system of partial functions of finite-valued logic is solved.