Logic program proportions

Abstract

The purpose of this paper is to present a fresh idea on how symbolic learning might be realized via analogical reasoning. For this, we introduce directed analogical proportions between logic programs of the form “P transforms into Q as R transforms into S” as a mechanism for deriving similar programs by analogy-making. The idea is to instantiate a fragment of a recently introduced abstract algebraic framework of analogical proportions in the domain of logic programming. Technically, we define proportions in terms of modularity where we derive abstract forms of concrete programs from a “known” source domain which can then be instantiated in an “unknown” target domain to obtain analogous programs. To this end, we introduce algebraic operations for syntactic logic program composition and concatenation. Interestingly, our work suggests a close relationship between modularity, generalization, and analogy which we believe should be explored further in the future. In a broader sense, this paper is a further step towards a mathematical theory of logic-based analogical reasoning and learning with potential applications to open AI-problems like commonsense reasoning and computational learning and creativity.

Sequential composition of propositional logic programs

“…where the sixth identity follows from the idempotency of union. This shows that Krom programs distribute from the left which implies that the space of all proper 4 Krom programs forms a semiring (cf. Corollary 9).…”

“…So the main difference between the above principle and the sequential composition operation of this paper is that the former operates on single programs and on single body atoms whereas the latter is defined as a binary operation between two programs and operates on all body atoms at once. Antić [4] uses the sequential composition and decomposition of (first-order) logic programs for logic program synthesis via analogical proportions [2], and [5] uses it for defining a qualitative notion of syntactic program similarity which allows one to answer queries across different domains. Antić [7] provides algebraic characterizations of least model and uniform equivalence of propositional Krom logic programs in terms of sequential composition.…”

“…This is partly motivated by recent work on algebraic logic program synthesis via analogical proportions in[4].…”

Boolean proportions

“…In the context of artificial intelligence it is particularly interesting to study analogical proportions between more sophisticated objects such as, for example, programs, neural networks, automata, et cetera. A recent paper in that direction is the study of logic program proportions for logic program synthesis via analogy-making in [Ant23c].…”