3 research outputs found
Semantical Characterizations and Complexity of Equivalences in Answer Set Programming
In recent research on non-monotonic logic programming, repeatedly strong
equivalence of logic programs P and Q has been considered, which holds if the
programs P union R and Q union R have the same answer sets for any other
program R. This property strengthens equivalence of P and Q with respect to
answer sets (which is the particular case for R is the empty set), and has its
applications in program optimization, verification, and modular logic
programming. In this paper, we consider more liberal notions of strong
equivalence, in which the actual form of R may be syntactically restricted. On
the one hand, we consider uniform equivalence, where R is a set of facts rather
than a set of rules. This notion, which is well known in the area of deductive
databases, is particularly useful for assessing whether programs P and Q are
equivalent as components of a logic program which is modularly structured. On
the other hand, we consider relativized notions of equivalence, where R ranges
over rules over a fixed alphabet, and thus generalize our results to
relativized notions of strong and uniform equivalence. For all these notions,
we consider disjunctive logic programs in the propositional (ground) case, as
well as some restricted classes, provide semantical characterizations and
analyze the computational complexity. Our results, which naturally extend to
answer set semantics for programs with strong negation, complement the results
on strong equivalence of logic programs and pave the way for optimizations in
answer set solvers as a tool for input-based problem solving.Comment: 58 pages, 6 tables. The contents were partially published in:
Proceedings 19th International Conference on Logic Programming (ICLP 2003),
pp. 224-238, LNCS 2916, Springer, 2003; and Proceedings 9th European
Conference on Logics in Artificial Intelligence (JELIA 2004), pp. 161-173,
LNCS 3229, Springer, 200
Characterising equilibrium logic and nested logic programs: Reductions and complexity
Equilibrium logic is an approach to nonmonotonic reasoning that extends the
stable-model and answer-set semantics for logic programs. In particular, it
includes the general case of nested logic programs, where arbitrary Boolean
combinations are permitted in heads and bodies of rules, as special kinds of
theories. In this paper, we present polynomial reductions of the main reasoning
tasks associated with equilibrium logic and nested logic programs into
quantified propositional logic, an extension of classical propositional logic
where quantifications over atomic formulas are permitted. We provide reductions
not only for decision problems, but also for the central semantical concepts of
equilibrium logic and nested logic programs. In particular, our encodings map a
given decision problem into some formula such that the latter is valid
precisely in case the former holds. The basic tasks we deal with here are the
consistency problem, brave reasoning, and skeptical reasoning. Additionally, we
also provide encodings for testing equivalence of theories or programs under
different notions of equivalence, viz. ordinary, strong, and uniform
equivalence. For all considered reasoning tasks, we analyse their computational
complexity and give strict complexity bounds
Omission-based Abstraction for Answer Set Programs
Abstraction is a well-known approach to simplify a complex problem by
over-approximating it with a deliberate loss of information. It was not
considered so far in Answer Set Programming (ASP), a convenient tool for
problem solving. We introduce a method to automatically abstract ASP programs
that preserves their structure by reducing the vocabulary while ensuring an
over-approximation (i.e., each original answer set maps to some abstract answer
set). This allows for generating partial answer set candidates that can help
with approximation of reasoning. Computing the abstract answer sets is
intuitively easier due to a smaller search space, at the cost of encountering
spurious answer sets. Faithful (non-spurious) abstractions may be used to
represent projected answer sets and to guide solvers in answer set
construction. For dealing with spurious answer sets, we employ an ASP debugging
approach to help with abstraction refinement, which determines atoms as badly
omitted and adds them back in the abstraction. As a show case, we apply
abstraction to explain unsatisfiability of ASP programs in terms of blocker
sets, which are the sets of atoms such that abstraction to them preserves
unsatisfiability. Their usefulness is demonstrated by experimental results.Comment: Under consideration in Theory and Practice of Logic Programming
(TPLP