Translating Datalog-Like Optimization Queries into ILOG Programs

Abstract

This paper presents a logic language, called \( \mathcal{N}\mathcal{P}{\mathbf{ }}Datalog \) suitable for expressing NP search and optimization problems. The ‘search’ language extends stratified Datalog with constraints and allows disjunction to define nondeterministically partitions of relations. It’s well known that \( \mathcal{N}\mathcal{P} \) search problems can be formulated as unstratified DATALOG queries under nondeterministic stable model semantics so that each stable model corresponds to a possible solution. \( \mathcal{N}\mathcal{P} \) optimization problems are then formulated by adding a max (or min) construct to select the stable model (thus, the solution) which maximizes (resp., minimizes) the result of a polynomial function applied to the answer relation. The problem in using DATALOG¬ to express search and optimization problems is that the unrestricted negation in programs is often neither simple nor intuitive and, besides, it does not allow to discipline the expressive power. Thus, we consider restricted forms of negation which force user to write programs in a more disciplined way without loosing of expressive power. More specifically, we consider the language \( \mathcal{N}\mathcal{P}{\mathbf{ }}Datalog \) which extends DATALOG¬s with two simple forms of unstratified negation embedded into built-in constructs: head disjunction and constraints. Thus the core of our language is stratified Datalog extended with two constructs allowing nondeterministic selections and with query goals enforcing conditions to be satisfied by stable models.