Non-deterministic Extension Research Articles

Token-passing Optimal Reduction with Embedded Read-back

We introduce a new interaction net implementation of optimal reduction for the pure untyped lambda calculus. Unlike others, our implementation allows to reach normal form regardless of the interaction net reduction strategy using the approach of so-called token-passing nets and a non-deterministic extension for interaction nets. Another new feature is the read-back mechanism implemented without leaving the formalism of interaction nets.

Combining relational algebra, SQL, constraint modelling, and local search

AbstractThe goal of this paper is to provide a strong integration between constraint modelling and relational DBMSs. To this end we propose extensions of standard query languages such as relational algebra and SQL, by adding constraint modelling capabilities to them. In particular, we propose non-deterministic extensions of both languages, which are specially suited for combinatorial problems. Non-determinism is introduced by means of a guessing operator, which declares a set of relations to have an arbitrary extension. This new operator results in languages with higher expressive power, able to express all problems in the complexity class NP. Some syntactical restrictions which make data complexity polynomial are shown. The effectiveness of both extensions is demonstrated by means of several examples. The current implementation, written in Java using local search techniques, is described.

Semantics and Expressive Power of Nondeterministic Constructs in Deductive Databases

Nondeterministic extensions are needed in logic-based languages, such as first-order relational languages and Datalog, to enhance their expressive power and support the efficient formulation of low-complexity problems and database queries. In this paper, we study the semantics and expressive power of the various nondeterministic constructs proposed in the past, including various versions of the choice operator and the witness operator. The paper develops a model-theoretic semantics, a fixpoint semantics, and an operational semantics for these constructs, and characterizes their power of expressing deterministic and nondeterministic queries. The paper presents various soundness and completeness results and establishes an expressiveness hierarchy that correlates the various operators with each other and with other constructs such as negation and fixpoint.

Datalog with non-deterministic choice computes NDB-PTIME

This paper addresses the issue of non-deterministic extensions of logic database languages. After providing a brief overview of the main proposals in the literature, we concentrate on the analysis of the dynamic choice construct from the point of view of the expressive power. We show how such construct is capable of expressing several interesting deterministic problems, such as computing the complement of a relation, and non-deterministic ones, such as computing an ordering of a relation. We then prove that Datalog augmented with the dynamic choice expresses exactly the non-deterministic time-polynomial queries. We thus obtain a complete characterization of the expressiveness of the dynamic choice, and conversely achieve a characterization of the class of non-deterministic time-polynomial queries (NDB-PTIME) by means of a simple, declarative, and efficiently implementable language.

A Nondeterministic Extension over Finitely Recursive ProcessModel

This paper extends the Finitely Recursive Process framework introduced by Inan and Varaiya for modelling Discrete Event Systems to encompass nondeterministic processes. Nondeterminism has been capt...

PROJECTIONS OF SEMICOMPUTABLE RELATIONS ON ABSTRACT DATA TYPES

We consider projections of semicomputable relations on abstract structures We show that they arise in several contexts, including nondeterministic extensions of while programs with arrays by means of arbitrary initializations and random assignments. They form a basis for an investigation of the concept of algorithmic specifications of relations with nondeterministic search. An important technique in this investigation is the study of computation trees for imperative programs, in order to prove characterization theorems for semicomputable sets, of a form first developed by E. Engeler.

Datalog extensions for database queries and updates

Deterministic and non-deterministic extensions of Datalog with fixpoint semantics are proposed, and their expressive power characterized. It is argued that fixpoint semantics provides an elegant way to overcome the limited expressive power available with purely declarative semantics. The Datalog extensions range from complete languages to languages capturing interesting complexity classes of queries and updates: PTIME and PSPACE in the non-deterministic case, and the fixpoint queries and while queries in the deterministic case. The connection between the Datalog extensions and explicitly procedural languages, as well as fixpoint extensions of first-order logic, is also investigated. In particular, a new family of non-deterministic fixpoint extensions of first-order logic is considered.

Non-determinism in logic-based languages

The use of non-determinism in logic-based languages is motivated using pragmatic and theoretical considerations. Non-deterministic database queries and updates occur naturally, and there exist non-deterministic implementations of various languages. It is shown that non-determinism resolves some difficulties concerning the expressive power of deterministic languages: there are non-deterministic languages expressing low complexity classes of queries/updates, whereas no such deterministic languages are known. Various mechanisms yielding non-determinism are reviewed. The focus is on two closely related families of non-deterministic languages. The first consists of extensions of Datalog with negations in bodies and/or heads of rules, with non-deterministic fixpoint semantics. The second consists of non-deterministic extensions of first-order logic and fixpoint logics, using thewitness operator. The expressive power of the languages is characterized. In particular, languages expressing exactly the (deterministic and non-deterministic) queries/updates computable in polynomial time are exhibited, whereas it is conjectured that no analogous deterministic language exists. The connection between non-deterministic languages and determinism is also explored. Several problems of practical interest are examined, such as checking (statically or dynamically) if a given program is deterministic, detecting coincidence of deterministic and non-deterministic semantics, and verifying termination for non-deterministic programs.

Fundamental properties of deterministic and nondeterministic extensions of Datalog

Fundamental properties of deterministic and nondeterministic extensions of Datalog from Abiteboul and Vianu (1988) are studied. The extensions involve the use of negative literals both in bodies and heads of rules. Negative literals in heads are interpreted as deletions. A deterministic semantics is obtained by firing in parallel all applicable rules. The nondeterministic semantics results from firing (nondeterministically) one rule at a time. In the nondeterministic case, programs do not describe functions but relations between database states. In both cases, the result is an increase in expressive power over Datalog. The price for it is that programs do not always terminate. We study when a program (i) is such that on a given input, all its successful computations reach a unique fixpoint, (ii) yields at least one output on every input and (iii) has only loop-free computations. We also show how to simulate programs containing loops by loop-free programs.

Statistical Analysis of Certain Binary Division Algorithms

Nondeterministic extensions of the nonrestoring method of binary division have been described by MacSorley [1]. One extension requires that the magnitudes of the divisor and partial remainders be normal, i. e., in the range [0.5, 1.0). This leads to a time improvement of more than two relative to conventional nonrestoring methods. Other extensions involve the use of several divisor multiples (or trial quotients). A Markov chain model is used here to analyze these methods. Steady-state distributions are determined for the division remainder and performance figures based on both this steady-state distribution and a random distribution are calculated. These are compared with the results of a computer simulation of 214 randmly-chosen division problems using two specific methods of division.