A Theoretical Perspective of Coinductive Logic Programming

Abstract

In this paper we study the semantics of Coinductive Logic Programming and clarify its intrinsic computational limits, which prevent, in particular, the definition of a complete, computable, operational semantics. We propose a new operational semantics that allows a simple correctness result and the definition of a simple meta-interpreter. We compare, and prove the equivalence, with the operational semantics defined and used in other papers on this topic

Extending Coinductive Logic Programming with Co-Facts

“…Terms are (possibly infinite) trees where nodes are labeled with function and variable symbols and the number of children of a node corresponds to the symbol arity (for a more formal definition based on paths see, e.g., [4]). Atoms are (possibly infinite) trees where the root is labeled with a predicate symbol and other nodes are labeled with function and variable symbols, also accordingly with the arity.…”

“…This operational semantics is a generalization of SLD [10,7] and co-SLD [11,13,12] resolution. However, it is presented, rather than in the traditional small-step style, in big-step style, as introduced in [4]. This style turns out to be simpler since coinductive hypotheses (see below) can be kept local.…”

“…Moreover, it naturally leads to an interpreter, and the proof of soundness with respect to declarative semantics is more direct since we compare two inference systems. For a proof of equivalence of big-step and small-step co-SLD resolution see [4].…”

“…Coinductive logic programming [11,13,12,4] extends standard logic programming with the ability of reasoning about infinite objects and their properties. Whereas syntax of logic programs remains the same, semantics is different.…”

“…For what concerns operational semantics, in coinductive logic programming standard SLD resolution is replaced by co-SLD resolution [13,4], which, roughly speaking, keeps trace of the already encountered goals, called (dynamic) coinductive hypotheses, so that, when a goal is encountered the second time, it is considered successful. In this paper, we define an operational semantics of logic programs which is similar to co-SLD resolution, but takes co-facts into account, and prove its correctness with respect to the proposed declarative semantics.…”