Notion Of Strong Equivalence Research Articles

A Lower Bound of the Number of Rewrite Rules Obtained by Homological Methods

It is well-known that some equational theories such as groups or boolean algebras can be defined by fewer equational axioms than the original axioms. However, it is not easy to determine if a given set of axioms is the smallest or not. Malbos and Mimram investigated a general method to find a lower bound of the cardinality of the set of equational axioms (or rewrite rules) that is equivalent to a given equational theory (or term rewriting systems), using homological algebra. Their method is an analog of Squier's homology theory on string rewriting systems. In this paper, we develop the homology theory for term rewriting systems more and provide a better lower bound under a stronger notion of equivalence than their equivalence. The author also implemented a program to compute the lower bounds, and experimented with 64 complete TRSs.

On the different types of collective attacks in abstract argumentation: equivalence results for SETAFs

Abstract Argumentation frameworks with collective attacks are a prominent extension of Dung’s abstract argumentation frameworks, where an attack can be drawn from a set of arguments to another argument. These frameworks are often abbreviated as SETAFs. Although SETAFs have received increasing interest recently, a thorough study on the actual behaviour of collective attacks has not been carried out yet. In particular, the richer attack structure SETAFs provide can lead to different forms of redundant attacks, i.e. attacks that are subsumed by attacks involving less arguments. Also the notion of strong equivalence, which is fundamental in nonmonotonic formalisms to characterize equivalent replacements, has not been investigated for SETAFs so far. In this paper, we first provide a classification of different types of collective attacks and analyse for which semantics they can be proven redundant. We do so for eleven well-established abstract argumentation semantics. We then study how strong equivalence between SETAFs can be decided with respect to the considered semantics and also consider variants of strong equivalence. Our results show that removing redundant attacks in a suitable way provides direct means to characterize strong equivalence by syntactical equivalence of so-called kernels, thus generalizing well-known results on strong equivalence between Dung AFs.

K-theory of derivators revisited

We define a $K$-theory for pointed right derivators and show that it agrees with Waldhausen $K$-theory in the case where the derivator arises from a good Waldhausen category. This $K$-theory is not invariant under general equivalences of derivators, but only under a stronger notion of equivalence that is defined by considering a simplicial enrichment of the category of derivators. We show that derivator $K$-theory, as originally defined, is the best approximation to Waldhausen $K$-theory by a functor that is invariant under equivalences of derivators.

Knowledge Forgetting in Answer Set Programming

The ability of discarding or hiding irrelevant information has been recognized as an important feature for knowledge based systems, including answer set programming. The notion of strong equivalence in answer set programming plays an important role for different problems as it gives rise to a substitution principle and amounts to knowledge equivalence of logic programs. In this paper, we uniformly propose a semantic knowledge forgetting, called HT- and FLP-forgetting, for logic programs under stable model and FLP-stable model semantics, respectively. Our proposed knowledge forgetting discards exactly the knowledge of a logic program which is relevant to forgotten variables. Thus it preserves strong equivalence in the sense that strongly equivalent logic programs will remain strongly equivalent after forgetting the same variables. We show that this semantic forgetting result is always expressible; and we prove a representation theorem stating that the HT- and FLP-forgetting can be precisely characterized by Zhang-Zhou's four forgetting postulates under the HT- and FLP-model semantics, respectively. We also reveal underlying connections between the proposed forgetting and the forgetting of propositional logic, and provide complexity results for decision problems in relation to the forgetting. An application of the proposed forgetting is also considered in a conflict solving scenario.

The cf2 argumentation semantics revisited

Abstract argumentation frameworks nowadays provide the most popular formalization of argumentation on a conceptual level. Numerous semantics for this paradigm have been proposed, whereby the cf2 semantics has shown to solve particular problems concerned with odd-length cycles in such frameworks. Due to the complicated definition of this semantics it has somehow been neglected in the literature. In this article, we introduce an alternative characterization of the cf2 semantics which, roughly speaking, avoids the recursive computation of subframeworks. This facilitates further investigation steps, like a complete complexity analysis. Furthermore, we show how the notion of strong equivalence can be characterized in terms of the cf2 semantics. In contrast to other semantics, it turns out that for the cf2 semantics strong equivalence coincides with syntactical equivalence. We make this particular behaviour more explicit by defining a new property for argumentation semantics, called the succinctness property. If a semantics σ satisfies the succinctness property, then for every framework F, all its attacks contribute to the evaluation of at least one framework F′ containing F. We finally characterize strong equivalence also for the stage and the naive semantics. Together with known results these characterizations imply that none of the prominent semantics for abstract argumentation, except the cf2 semantics, satisfies the succinctness property.

General default logic

In this paper, we propose a new nonmonotonic logic called general default logic. On the one hand, it generalizes Reiter's default logic by adding to it rule-like operators used in logic programming. On the other hand, it extends logic programming by allowing arbitrary propositional formulas. We show that with this new logic, one can formalize naturally rule constraints, generalized closed world assumptions, and conditional defaults. We show that under a notion of strong equivalence, sentences of this new logic can be converted to a normal form. We also investigate the computational complexity of various reasoning tasks in the logic, and relate it to some other nonmonotonic formalisms such as Lin and Shoham's logic of GK and Moore's autoepistemic logic.

Strong equivalence of logic programs under the infinite-valued semantics

We consider the notion of strong equivalence [V. Lifschitz, D. Pearce, A. Valverde, Strongly equivalent logic programs, ACM Transactions on Computational Logic 2 (4) (2001) 526–541] of normal propositional logic programs under the infinite-valued semantics [P. Rondogiannis, W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 (2) (2005) 441–467] (which is a purely model-theoretic semantics that is compatible with the well-founded one). We demonstrate that two such programs are strongly equivalent under the infinite-valued semantics if and only if they are logically equivalent in the corresponding infinite-valued logic. In particular, we show that strong equivalence of normal propositional logic programs is decidable, and more specifically coNP-complete. Our results have a direct implication for the well-founded semantics since, as we demonstrate, if two programs are strongly equivalent under the infinite-valued semantics, then they are also strongly equivalent under the well-founded semantics.

Sound and Complete Inference Rules for SE-Consequence

The notion of strong equivalence on logic programs with answer set semantics gives rise to a consequence relation on logic program rules, called SE-consequence. We present a sound and complete set of inference rules for SE-consequence on disjunctive logic programs.

Structural equivalence and ET0L grammars

For a given context-sensitive grammar G we construct ET0L grammars G 1 and G 2 that are structurally equivalent if and only if the language generated by G is empty, which implies that structural equivalence is undecidable for ET0L grammars. In contrast, structural equivalence is decidable for E0L grammars and for extended E0L grammars. In fact, we show that structural equivalence is undecidable for propagating ET0L grammars in which the number of tables is restricted to be at most two. A stronger notion of equivalence that requires the sets of syntax trees to be isomorphic is shown to be decidable for ET0L grammars.