arXiv subjects

M. H. van Emden

Publications and source records attributed to M. H. van Emden.

26 records · Page 2Linked to original sources

arXiv2006

Compositional Semantics for the Procedural Interpretation of Logic

Semantics of logic programs has been given by proof theory, model theory and by fixpoint of the immediate-consequence operator. If clausal logic is a programming language, then it should also have a compositional semantics. Compositional semantics for programming languages follows the abstract syntax of programs, composing the meaning of a unit by a mathematical operation on the meanings of its constituent units. The procedural interpretation of logic has only yielded an incomplete abstract syntax for logic programs. We complete it and use the result as basis of a compositional semantics. We present for comparison Tarski's algebraization of first-order predicate logic, which is in substance the compositional semantics for his choice of syntax. We characterize our semantics by equivalence with the immediate-consequence operator.

cs.PL↗

arXiv2006

Towards Applicative Relational Programming

Functional programming comes in two flavours: one where ``functions are first-class citizens'' (we call this applicative) and one which is based on equations (we call this declarative). In relational programming clauses play the role of equations. Hence Prolog is declarative. The purpose of this paper is to provide in relational programming a mathematical basis for the relational analog of applicative functional programming. We use the cylindric semantics of first-order logic due to Tarski and provide a new notation for the required cylinders that we call tables. We define the Table/Relation Algebra with operators sufficient to translate Horn clauses into algebraic form. We establish basic mathematical properties of these operators. We show how relations can be first-class citizens, and devise mechanisms for modularity, for local scoping of predicates, and for exporting/importing relations between programs.

cs.PL↗

arXiv2004

Propagation by Selective Initialization and Its Application to Numerical Constraint Satisfaction Problems

Numerical analysis has no satisfactory method for the more realistic optimization models. However, with constraint programming one can compute a cover for the solution set to arbitrarily close approximation. Because the use of constraint propagation for composite arithmetic expressions is computationally expensive, consistency is computed with interval arithmetic. In this paper we present theorems that support, selective initialization, a simple modification of constraint propagation that allows composite arithmetic expressions to be handled efficiently.

math.NA↗

arXiv2003

Using Propagation for Solving Complex Arithmetic Constraints

Solving a system of nonlinear inequalities is an important problem for which conventional numerical analysis has no satisfactory method. With a box-consistency algorithm one can compute a cover for the solution set to arbitrarily close approximation. Because of difficulties in the use of propagation for complex arithmetic expressions, box consistency is computed with interval arithmetic. In this paper we present theorems that support a simple modification of propagation that allows complex arithmetic expressions to be handled efficiently. The version of box consistency that is obtained in this way is stronger than when interval arithmetic is used.

math.NA↗

arXiv2002

New Developments in Interval Arithmetic and Their Implications for Floating-Point Standardization

We consider the prospect of a processor that can perform interval arithmetic at the same speed as conventional floating-point arithmetic. This makes it possible for all arithmetic to be performed with the superior security of interval methods without any penalty in speed. In such a situation the IEEE floating-point standard needs to be compared with a version of floating-point arithmetic that is ideal for the purpose of interval arithmetic. Such a comparison requires a succinct and complete exposition of interval arithmetic according to its recent developments. We present such an exposition in this paper. We conclude that the directed roundings toward the infinities and the definition of division by the signed zeros are valuable features of the standard. Because the operations of interval arithmetic are always defined, exceptions do not arise. As a result neither Nans nor exceptions are needed. Of the status flags, only the inexact flag may be useful. Denormalized numbers seem to have no use for interval arithmetic; in the use of interval constraints, they are a handicap.

math.NA↗

arXiv2002

On the Significance of Digits in Interval Notation

To analyse the significance of the digits used for interval bounds, we clarify the philosophical presuppositions of various interval notations. We use information theory to determine the information content of the last digit of the numeral used to denote the interval's bounds. This leads to the notion of efficiency of a decimal digit: the actual value as percentage of the maximal value of its information content. By taking this efficiency into account, many presentations of intervals can be made more readable at the expense of negligible loss of information.

math.NA↗

arXiv2001

Computing Functional and Relational Box Consistency by Structured Propagation in Atomic Constraint Systems

Box consistency has been observed to yield exponentially better performance than chaotic constraint propagation in the interval constraint system obtained by decomposing the original expression into primitive constraints. The claim was made that the improvement is due to avoiding decomposition. In this paper we argue that the improvement is due to replacing chaotic iteration by a more structured alternative. To this end we distinguish the existing notion of box consistency from relational box consistency. We show that from a computational point of view it is important to maintain the functional structure in constraint systems that are associated with a system of equations. So far, it has only been considered computationally important that constraint propagation be fair. With the additional structure of functional constraint systems, one can define and implement computationally effective, structured, truncated constraint propagations. The existing algorithm for box consistency is one such. Our results suggest that there are others worth investigating.

cs.PL↗

arXiv2001

Factored Notation for Interval I/O

This note addresses the input and output of intervals in the sense of interval arithmetic and interval constraints. The most obvious, and so far most widely used notation, for intervals has drawbacks that we remedy with a new notation that we propose to call factored notation. It is more compact and allows one to find a good trade-off between interval width and ease of reading. We describe how such a trade-off can be based on the information yield (in the sense of information theory) of the last decimal shown.

math.NA↗