Reducing Propositional Theories in Equilibrium Logic to Logic Programs

Abstract

Abstract. The paper studies reductions of propositional theories in equilibrium logic to logic programs under answer set semantics. Specifically we are concerned with the question of how to transform an arbitrary set of propositional formulas into an equivalent logic program and what are the complexity constraints on this process. We want the transformed program to be equivalent in a strong sense so that theory parts can be transformed independent of the wider context in which they might be embedded. It was only recently established [2] that propositional theories are indeed equivalent (in a strong sense) to logic programs. Here this result is extended with the following contributions. (i) We show how to effectively obtain an equivalent program starting from an arbitrary theory. (ii) We show that in general there is no polynomial transformation if we require the resulting program to share precisely the vocabulary or signature of the initial theory. (iii) Extending previous work we show how polynomial transformations can be achieved if one allows the resulting program to contain new atoms. The program obtained is still in a strong sense equivalent to the original theory, and the answer sets of the theory can be retrieved from it.

A Revised Concept of Safety for General Answer Set Programs

“…While it is clear that many strong-equivalence preserving transformations (eg. from [4,5]) preserve safety under our definition, we do not have yet a full picture of precisely which syntactic classes have this property.…”

A Normal Form for Linear Temporal Equilibrium Logic

“…However, we can apply some simple transformations on implication, conjunction and disjunction that have been shown to be strongly equivalent at the (non-temporal) propositional level [15], and obtain a TLP without changing the signature V L . For each definition df (γ), we define the strongly equivalent set (understood as the conjunction) of temporal logic program rules df * (γ) as shown in Figure 1.…”

“…The reduction into normal form starts from the structure-preserving polynomial transformation presented in [15] for the non-temporal case. This transformation has as a main feature the introduction of an auxiliary atom per each subformula in the original theory.…”

System f2lp – Computing Answer Sets of First-Order Formulas

“…Given a set of formulas, f2lp first eliminates all quantifiers and then applies the transformation defined in [7] that turns the resulting quantifier-free formulas into logic program rules. In this section we describe how quantifier elimination is done in f2lp.…”

“…-Every quantifier-free formula (including propositional formula) is strongly equivalent to a logic program [6,7,4].…”