A finite-valued solver for disjunctive fuzzy answer set programs

Fuzzy Answer Set Programming (FASP) is a declarative programming paradigm which extends the flexibility and expressiveness of classical Answer Set Programming (ASP), with the aim of modeling continuous application domains. In contrast to the availability of efficient ASP solvers, there have been few attempts at implementing FASP solvers. In this paper, we propose an implementation of FASP based on a reduction to classical ASP. We also develop a prototype implementation of this method. To the best of our knowledge, this is the first solver for disjunctive FASP programs. Moreover, we experimentally show that our solver performs well in comparison to an existing solver (under reasonable assumptions) for the more restrictive class of normal FASP programs

Non-clausal multi-ary alpha-generalized resolution calculus for a finite lattice-valued logic

Due to the need of the logical foundation for uncertain information processing, development of efficient automated reasoning system based on non-classical logics is always an active research area. The present paper focuses on the resolution-based automated reasoning theory in a many-valued logic with truth-values defined in a lattice-ordered many-valued algebraic structure - lattice implication algebras (LIA). Specifically, as a continuation and extension of the established work on binary resolution at a certain truth-value level α (called α-resolution), a non-clausal multi-ary α-generalized resolution calculus is introduced for a lattice-valued propositional logic LP(X) based on LIA, which is essentially a non-clausal generalized resolution avoiding reduction to normal clausal form. The new resolution calculus in LP(X) is then proved to be sound and complete. The concepts and theoretical results are further extended and established in the corresponding lattice-valued first-order logic LF(X) based on LIA

Local search and restart strategies for satisfiability solving in fuzzy logics

Satisfiability solving in fuzzy logics is a subject that has not been researched much, certainly compared to satisfiability in propositional logics. Yet, fuzzy logics are a powerful tool for modelling complex problems. Recently, we proposed an optimization approach to solving satisfiability in fuzzy logics and compared the standard Covariance Matrix Adaptation Evolution Strategy algorithm (CMA-ES) with an analytical solver on a set of benchmark problems. Especially on more finegrained problems did CMA-ES compare favourably to the analytical approach. In this paper, we evaluate two types of hillclimber in addition to CMA-ES, as well as restart strategies for these algorithms. Our results show that a population-based hillclimber outperforms CMA-ES on the harder problem class

Satisfiability checking in Łukasiewicz logic as finite constraint satisfaction

Although it is well-known that every satisfiable formula in Aukasiewicz' infinite-valued logic can be satisfied in some finite-valued logic, practical methods for finding an appropriate number of truth degrees do currently not exist. Extending upon earlier results by Aguzzoli et al., which only take the total number of variable occurrences into account, we present a detailed analysis of what type of formulas require a large number of truth degrees to be satisfied. In particular, we reveal important links between this number of truth degrees and the dimension, and structure, of the cycle space of an associated bipartite graph. We furthermore propose an efficient, polynomial-time algorithm for establishing a strong upper bound on the required number of truth degrees, allowing us to check the satisfiability of sets of formulas in , and more generally, sets of fuzzy clauses over Aukasiewicz logic formulas, by solving a small number of constraint satisfaction problems. In an experimental evaluation, we demonstrate the practical usefulness of this approach, comparing it with a state-of-the-art technique based on mixed integer programming

Foundations of fuzzy answer set programming

Answer set programming (ASP) is a declarative language that is tailored towards combinatorial search problems. Although ASP has been applied to many problems, such as planning, configuration and verification of software, and database repair, it is less suitable for describing continuous problems. In this thesis we therefore studied fuzzy answer set programming (FASP). FASP is a language that combines ASP with ideas from fuzzy logic -- a class of many-valued logics that are able to describe continuous problems. We study the following topics: 1. An important issue when modeling continuous optimization problems is how to cope with overconstrained problems. In many cases we can opt to allow imperfect solutions, i.e. solutions that do not satisfy all constraints, but are sufficiently acceptable. However, the question which one of these imperfect solutions is most suitable then arises. Current approaches to fuzzy answer set programming solve this problem by attaching weights to the rules of the program. However, it is often not clear how these weights should be chosen and moreover weights do not allow to order different solutions. We improve upon this technique by using aggregators, which eliminate the aforementioned problems. This allows a richer modeling language and bridges the gap between FASP and other techniques such as valued constraint satisfaction problems. 2. The wishes of users and implementers of a programming language are often in direct conflict with each other. Users prefer a rich language that is easy to model in, whereas implementers prefer a small language that is easy to implement. We reconcile these differences by identifying a core language for FASP, called core FASP (CFASP), that only consists of non-constraint rules with monotonically increasing functions and negators in the body. We show that CFASP is capable of simulating constraint rules, monotonically decreasing functions, aggregators, S-implicators and classical negation. Moreover we remark that the simulations of constraints and classical negation bear a great resemblance to their simulations in classical ASP, which provides further insight into the relationship between ASP and FASP. 3. As a first step towards the creation of an implementation method for FASP we research whether it is possible to translate a FASP program to a fuzzy SAT problem. We introduce the concept of the completion of a FASP program and show that for programs without loops the models of the completion coincide with the answer sets. Furthermore we show that if a program has loops, we can translate the program to a fuzzy SAT problem by generalizing the concept of loop formulas. We illustrate this on a continuous version of the k-center problem. Such a translation is important because it allows us to solve FASP programs by means of solvers for fuzzy SAT. Under the appropriate conditions it is for example possible to solve FASP programs by means of off-the-shelf solvers for mixed integer programming (MIP)