A Type-Theoretic Approach to Structural Resolution

Structural resolution (or S-resolution) is a newly proposed alternative to SLD-resolution that allows a systematic separation of derivations into term-matching and unification steps. Productive logic programs are those for which term-matching reduction on any query must terminate. For productive programs with coinductive meaning, finite term-rewriting reductions can be seen as measures of observation in an infinite derivation. Ability of handling corecursion in a productive way is an attractive computational feature of S-resolution. In this paper, we make first steps towards a better conceptual understanding of operational properties of S-resolution as compared to SLD-resolution. To this aim, we propose a type system for the analysis of both SLD-resolution and S-resolution. We formulate S-resolution and SLD-resolution as reduction systems, and show their soundness relative to the type system. One of the central methods of this paper is realizability transformation, which makes logic programs productive and non-overlapping. We show that S-resolution and SLD-resolution are only equivalent for programs with these two properties.Comment: LOPSTR 201

Operational Semantics of Resolution and Productivity in Horn Clause Logic

This paper presents a study of operational and type-theoretic properties of different resolution strategies in Horn clause logic. We distinguish four different kinds of resolution: resolution by unification (SLD-resolution), resolution by term-matching, the recently introduced structural resolution, and partial (or lazy) resolution. We express them all uniformly as abstract reduction systems, which allows us to undertake a thorough comparative analysis of their properties. To match this small-step semantics, we propose to take Howard's System H as a type-theoretic semantic counterpart. Using System H, we interpret Horn formulas as types, and a derivation for a given formula as the proof term inhabiting the type given by the formula. We prove soundness of these abstract reduction systems relative to System H, and we show completeness of SLD-resolution and structural resolution relative to System H. We identify conditions under which structural resolution is operationally equivalent to SLD-resolution. We show correspondence between term-matching resolution for Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201

Mixed Poisson approximation of node depth distributions in random binary search trees

We investigate the distribution of the depth of a node containing a specific key or, equivalently, the number of steps needed to retrieve an item stored in a randomly grown binary search tree. Using a representation in terms of mixed and compounded standard distributions, we derive approximations by Poisson and mixed Poisson distributions; these lead to asymptotic normality results. We are particularly interested in the influence of the key value on the distribution of the node depth. Methodologically our message is that the explicit representation may provide additional insight if compared to the standard approach that is based on the recursive structure of the trees. Further, in order to exhibit the influence of the key on the distributional asymptotics, a suitable choice of distance of probability distributions is important. Our results are also applicable in connection with the number of recursions needed in Hoare's [Comm. ACM 4 (1961) 321-322] selection algorithm Find.Comment: Published at http://dx.doi.org/10.1214/105051604000000611 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org