By Jeroen Janssen, Steven Schockaert, Dirk Vermeir, Martine De Cock

Resolution set programming (ASP) is a declarative language adapted in the direction of fixing combinatorial optimization difficulties. it's been effectively utilized to e.g. making plans difficulties, configuration and verification of software program, analysis and database upkeep. in spite of the fact that, ASP isn't really at once compatible for modeling issues of non-stop domain names. Such difficulties take place certainly in diversified fields equivalent to the layout of gasoline and electrical energy networks, laptop imaginative and prescient and funding portfolios. to beat this challenge we examine FASP, a mixture of ASP with fuzzy good judgment -- a category of manyvalued logics that may deal with continuity. We in particular specialize in the next matters: 1. a tremendous query while modeling non-stop optimization difficulties is how we must always deal with overconstrained difficulties, i.e. difficulties that experience no recommendations. in lots of situations we will be able to favor to settle for a less than excellent answer, i.e. an answer that doesn't fulfill the entire said principles (constraints). besides the fact that, this ends up in the query: what imperfect options should still we decide? We examine this question and increase upon the state of the art by way of providing an procedure in accordance with aggregation services. 2. clients of a programming language usually need a wealthy language that's effortless to version in. even if, implementers and theoreticians favor a small language that's effortless to enforce and cause approximately. We create a bridge among those wants through presenting a small middle language for FASP and by way of exhibiting that this language is in a position to expressing lots of its universal extensions equivalent to constraints, monotonically lowering services, aggregators, S-implicators and classical negation. three. a well known method for fixing ASP includes translating a application P to a propositional idea whose versions precisely correspond to the reply units of P. We express how this method will be generalized to FASP, paving the right way to enforce effective fuzzy resolution set solvers which can make the most of current fuzzy reasoners.

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**Extra resources for Answer Set Programming for Continuous Domains: A Fuzzy Logic Approach**

**Example text**

Second, we are not only interested in ﬁnding 49 50 Answer Set Programming for Continuous Domains:a Fuzzy Logic Approach any solution: if multiple solutions can be found, we are especially interested in the solution modeling the rules best. Hence, based on the degree to which the rules of the program are satisﬁed, it is of interest to deﬁne an ordering on the solutions, which cannot be meaningfully done using ﬁxed weights. In [Van Nieuwenborgh et al. (2007b)] the proposed solution is to attach an aggregator expression to a program.

Let P be a normal answer set program. Then any answer set A of P is a minimal model of the completion comp(P) of P. We can now answer the question why we need ASP. While the graph coloring example could be concisely encoded in SAT, this does not hold in general. For example, programs incorporating recursion require a more involved translation [Lin and Zhao (2004)]. 1. The following program illustrates the use of recursion on the problem of ﬁnding Hamilton cycles in a graph. 12. e. ﬁnding a path in the graph that visits every vertex exactly once.

Classical negation can be eliminated from the program by introducing for each literal ¬a a new atom a and adding the constraint ← a, a to the program. The constraint ensures that any model will be consistent, and thus ensures that the semantics are preserved. For more details see [Baral (2003)]. Unless stated otherwise, for all programs in the remainder of this chapter we assume that classical negation has been eliminated in this way. 3 Answer Set Programming for Continuous Domains:a Fuzzy Logic Approach Links to SAT There exist important links between ASP and the boolean satisﬁability problem (SAT), which we highlight in this section.