Taylor introduced a variable binding scheme for logic variables in his PARMA system, that uses cycles of bindings rather than the linear chains of bindings used in the standard WAM representation. Both the HAL and dProlog languages make use of the PARMA representation in their Herbrand constraint solvers. Unfortunately, PARMA's trailing scheme is considerably more expensive in both time and space consumption. The aim of this paper is to present several techniques that lower the cost. First, we introduce a trailing analysis for HAL using the classic PARMA trailing scheme that detects and eliminates unnecessary trailings. The analysis, whose accuracy comes from HAL's determinism and mode declarations, has been integrated in the HAL compiler and is shown to produce space improvements as well as speed improvements. Second, we explain how to modify the classic PARMA trailing scheme to halve its trailing cost. This technique is illustrated and evaluated both in the context of dProlog and HAL. Finally, we explain the modifications needed by the trailing analysis in order to be combined with our modified PARMA trailing scheme. Empirical evidence shows that the combination is more effective than any of the techniques when used in isolation.

The unification problem in algebras capable of describing sets has been tackled, directly or indirectly, by many researchers and it finds important applications in various research areas, e.g. deductive databases, theorem proving, static analysis, rapid software prototyping. The various solutions proposed are spread across a large literature. In this paper we provide a uniform presentation of unification of sets, formalizing it at the level of set theory. We address the problem of deciding existence of solutions at an abstract level. This provides also the ability to classify different types of set unification problems. Unification algorithms are uniformly proposed to solve the unification problem in each of such classes. The algorithms presented are partly drawn from the literature – and properly revisited and analyzed – and partly novel proposals. In particular, we present a new goal-driven algorithm for general $ACI1$ unification and a new simpler algorithm for general $(Ab)(C\ell)$ unification.

Defeasible reasoning is a simple but efficient approach to nonmonotonic reasoning that has recently attracted considerable interest and that has found various applications. Defeasible logic and its variants are an important family of defeasible reasoning methods. So far no relationship has been established between defeasible logic and mainstream nonmonotonic reasoning approaches. In this paper we establish close links to known semantics of logic programs. In particular, we give a translation of a defeasible theory $D$ into a meta-program $P(D)$. We show that under a condition of decisiveness, the defeasible consequences of $D$ correspond exactly to the sceptical conclusions of $P(D)$ under the stable model semantics. Without decisiveness, the result holds only in one direction (all defeasible consequences of $D$ are included in all stable models of $P(D)$). If we wish a complete embedding for the general case, we need to use the Kunen semantics of $P(D)$, instead.

This paper looks at logic programming with three kinds of negation: default, weak and strict negations. A 3-valued logic model theory is discussed for logic programs with three kinds of negation. The procedure is constructed for negations so that a soundness of the procedure is guaranteed in terms of 3-valued logic model theory.