Here and There with Arithmetic

Abstract

Abstarct In the theory of answer set programming, two groups of rules are called strongly equivalent if, informally speaking, they have the same meaning in any context. The relationship between strong equivalence and the propositional logic of here-and-there allows us to establish strong equivalence by deriving rules of each group from rules of the other. In the process, rules are rewritten as propositional formulas. We extend this method of proving strong equivalence to an answer set programming language that includes operations on integers. The formula representing a rule in this language is a first-order formula that may contain comparison symbols among its predicate constants, and symbols for arithmetic operations among its function constants. The paper is under consideration for acceptance in TPLP.

Strong Equivalence of Logic Programs with Counting

“…Theorem 3 below shows that in some cases such a claim can be justified by operating with finite formulas—with first-order formulas of the signature . Instead of , we can use the logic of here-and-there with arithmetic (Lifschitz 2021) extended by the axiom schemas Defs :…”

“…We are interested in proving strong equivalence of programs in the input language of the grounder gringo (Gebser et al 2019) by deriving rules of each program from rules of the other. The possibility of such proofs has been demonstrated for the subset of that language called mini- gringo (Lifschitz et al 2019; Lifschitz 2021). Programs allowed in that subset may include comparisons, arithmetic operations, and simple choice rules, but not aggregates.…”

“…The process of proving strong equivalence uses the translation (Lifschitz et al 2019, Section 6) that transforms mini- gringo rules into first-order formulas with two sorts of variables – for numerals and for arbitrary precomputed terms. If two mini- gringo programs, rewritten as sets of sentences, are equivalent in the deductive system HTA (“here-and-there with arithmetic”), then they are strongly equivalent (Lifschitz 2021, Section 4).…”