Logic Programming in Tabular Allegories

We develop a new compilation scheme and categorical abstract machine for execution of logic programs based on allegories, the categorical version of the calculus of relations. In our case, both operational and denotational semantics are built upon the same theory, thus execution of a query is performed using algebraic reasoning, giving immediate proof of correctness properties. Our work serves two purposes: having an efficient and formal model of a logic programming compiler and machine and building the base for incorporating features typical of functional programming in a declarative way, while maintaining 100% compatibility with existing Prolog programs. The starting point is the construction of a Regular Lawvere Category from the program’s signature Σ. Then, the resulting Σ-allegory captures all the needed theory and meta-theory for Prolog. The categorical framework for reasoning and execution is based on relation composition, thus allowing the enrichment of the base category with types, functions and other constructions typical of functional programming languages. The single primitive of relation composition encompasses unification, garbage collection, parameter passing, environment creation and destruction. Composition happens between relations tabulated by a pair of maps, whose domain represents the global storage in use while its co-domain represents the number of temporary registers in use. This execution mechanism is efficient, given that shared state is faithfully represented by the notion of projections, corresponding to pointers in the implementation.