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1

Negation as a Resource: a Novel View on Answer Set Semantics

Costantini S., Formisano A.

Fundamenta Informaticae

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2015

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Vol. 140, nr 3/4

279--305

EN

In recent work, we provided a formulation of ASP programs in terms of linear logic theories. Answer sets were characterized in terms of maximal tensor conjunctions provable from such theories. In this paper, we propose a full comparison between Answer Set Semantics and its variation obtained by interpreting literals (including negative literals) as resources, which leads to a different interpretation of negation. We argue that this novel view can be of both theoretical and practical interest, and we propose a modified Answer Set Semantics that we call Resource-based Answer Set Semantics. An advantage is that of avoiding inconsistencies, as every program has a (possibly empty) resource-based answer set. This implies however the introduction of a different way of representing constraints. We provide a characterization of the new semantics as a variation of the answer set semantics, and also in terms of Autoepistemic Logic. The latter characterization leads to a way of computing resource-based answer set via answer set solvers.

2

Notes on nonmonotonic autoepistemic propositional logic

Suchenek M. A.

Zeszyty Naukowe Warszawskiej Wyższej Szkoły Informatyki

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2011

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nr 6

74--92

EN

This paper comprises an in-depth study of semantics of autoepistemic logic that is based on author’s may years of research in the subject matter. It begins with a brief review of semantics of common patterns of nonmonotonic deduction arising from a lack of knowledge, including autoepistemic deduction, in terms of the fixed-point equation F (T, E) = E. Then it narrowly investigates minimal expansion semantics for autoepistemic propositional logic and its „only knowing” consequence operation CnAE. In particular, the following minimalknowledge assumption M K A : ϕ ∈M K A (T) iff ϕ does not add modally positive S5- consequences to T is used to syntactically characterize the operation CnAE by means of suitable completeness theorem. The paper also offers a proof that the consequence operation CnS5 of modal logic S5 is the maximal monotonic consequence operation satisfying CnS5(T) ⊆ M K A (T) for every modal theory T.

PL

Artykuł stanowi dogłębne studium semantyki logiki autoepistemicznej wykorzystujące wyniki wieloletnich badań autora w tym przedmiocie. Rozpoczyna się ono od skrótowego przeglądu semantyk powszechnie stosowanych wzorców niemonotonicznej dedukcji biorącej się z braku wiedzy, włączając w to dedukcję autoepistemiczną, w terminach równania stałopunktowego F (T, E) = E. Następnie bada ono szczegółowo semantykę minimalnych ekspansji dla zdaniowej logiki autoepistemicznej oraz jej operację CnAE odpowiadającą schematowi wnioskowania opartemu na założeniu „wiedząc tylko”. W szczególności następujące założenie M K A o minimalności wiedzy: ϕ ∈M K A (T) wtedy, i tylko wtedy, gdy ϕ nie dokłada modalnie pozytywnych S5-konsekwencji do T jest używane w celu syntaktycznego scharakteryzowania operacji CnAE przy pomocy stosownego twierdzenia o pełności. Artykuł przedstawia też dowód, że operacja konsekwencji CnS5 logiki modalnej S5 jest maksymalną monotoniczną operacją konsekwencji spełniającą CnS5(T) ⊆ M K A (T) dla każdej teorii modalnej T.

3

Predicate Introduction for Logics with Fixpoint Semantics. Part II: Autoepistemic Logic

Vennekens J., Wittocx J., Mariën M., Denecker M.

Fundamenta Informaticae

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2007

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Vol. 79, nr 1-2

209-227

EN

We study the transformation of "predicate introduction" in non-monotonic logics. By this, we mean the act of replacing a complex formula by a newly defined predicate. From a knowledge representation perspective, such transformations can be used to eliminate redundancy or to simplify a theory. From a more practical point of view, they can also be used to transform a theory into a normal form imposed by certain inference programs or theorems. In a companion paper, we developed an algebraic theory that considers predicate introduction within the framework of "approximation theory," a fixpoint theory for non-monotone operators that generalizes all main semantics of various non-monotonic logics, including logic programming, default logic and autoepistemic logic. We then used these results to show that certain logic programming transformations are equivalence preserving under, among others, both the stable and well-founded semantics. In this paper, we now apply the same algebraic results to autoepistemic logic and prove that a transformation to reduce the nesting depth of modal operators is equivalence preserving under a family of semantics for this logic. This not only provides useful theorems for autoepistemic logic, but also demonstrates that our algebraic theory does indeed capture the essence of predicate introduction in a generally applicable way.

4

On Undecidability of Non-monotonic Logic

Suchenek M. A.

Studia Informatica : systems and information technology

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2006

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Vol. 1(7)

127--132

EN

The degree of undecidability of nonmonotonic logic is investigated. A proof is provided that arithmetical but not recursively enumerable sets of sentences definable by nonmonotonic default logic are elements of ∆n+1 but not Σ n nor Π n for some n ≥1 in Kleene- Mostowski hierarchy of arithmetical sets.