Sequent and Hypersequent Calculi for Abelian and Lukasiewicz Logics

We present two embeddings of infinite-valued Lukasiewicz logic L into Meyer and Slaney's abelian logic A, the logic of lattice-ordered abelian groups. We give new analytic proof systems for A and use the embeddings to derive corresponding systems for L. These include: hypersequent calculi for A and L and terminating versions of these calculi; labelled single sequent calculi for A and L of complexity co-NP; unlabelled single sequent calculi for A and L.Comment: 35 pages, 1 figur

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution

A Spectrum of Applications of Automated Reasoning

The likelihood of an automated reasoning program being of substantial assistance for a wide spectrum of applications rests with the nature of the options and parameters it offers on which to base needed strategies and methodologies. This article focuses on such a spectrum, featuring W. McCune's program OTTER, discussing widely varied successes in answering open questions, and touching on some of the strategies and methodologies that played a key role. The applications include finding a first proof, discovering single axioms, locating improved axiom systems, and simplifying existing proofs. The last application is directly pertinent to the recently found (by R. Thiele) Hilbert's twenty-fourth problem--which is extremely amenable to attack with the appropriate automated reasoning program--a problem concerned with proof simplification. The methodologies include those for seeking shorter proofs and for finding proofs that avoid unwanted lemmas or classes of term, a specific option for seeking proofs with smaller equational or formula complexity, and a different option to address the variable richness of a proof. The type of proof one obtains with the use of OTTER is Hilbert-style axiomatic, including details that permit one sometimes to gain new insights. We include questions still open and challenges that merit consideration.Comment: 13 page

Fuzzy Maximum Satisfiability

In this paper, we extend the Maximum Satisfiability (MaxSAT) problem to {\L}ukasiewicz logic. The MaxSAT problem for a set of formulae {\Phi} is the problem of finding an assignment to the variables in {\Phi} that satisfies the maximum number of formulae. Three possible solutions (encodings) are proposed to the new problem: (1) Disjunctive Linear Relations (DLRs), (2) Mixed Integer Linear Programming (MILP) and (3) Weighted Constraint Satisfaction Problem (WCSP). Like its Boolean counterpart, the extended fuzzy MaxSAT will have numerous applications in optimization problems that involve vagueness.Comment: 10 page