Fuzzy answer sets approximations

Abstract

Fuzzy answer set programming (FASP) is a recent formalism for knowledge representation that enriches the declarativity of answer set programming by allowing propositions to be graded. To now, no implementations of FASP solvers are available and all current proposals are based on compilations of logic programs into different paradigms, like mixed integer programs or bilevel programs. These approaches introduce many auxiliary variables which might affect the performance of a solver negatively. To limit this downside, operators for approximating fuzzy answer sets can be introduced: Given a FASP program, these operators compute lower and upper bounds for all atoms in the program such that all answer sets are between these bounds. This paper analyzes several operators of this kind which are based on linear programming, fuzzy unfounded sets and source pointers. Furthermore, the paper reports on a prototypical implementation, also describing strategies for avoiding computations of these operators when they are guaranteed to not improve current bounds. The operators and their implementation can be used to obtain more constrained mixed integer or bilevel programs, or even for providing a basis for implementing a native FASP solver. Interestingly, the semantics of relevant classes of programs with unique answer sets, like positive programs and programs with stratified negation, can be already computed by the prototype without the need for an external tool.

Stable Models of Fuzzy Propositional Formulas

“…Future work includes implementing this language using mixed integer programming solvers or bilevel programming solvers [15].…”

Fuzzy answer set computation via satisfiability modulo theories

“…The performance of fasp2smt was assessed on instances of a benchmark used to evaluate the FASP solver ffasp (Mushthofa et al 2014). The benchmark comprises two (synthetic) problems, the fuzzy versions of Graph Coloring and Hamiltonian Path, originally considered by Alviano and Peñaloza (2013). In Graph Coloring edges of an input graph are associated with truth degrees, and each vertex x is non-deterministically colored with a shadow of gray, i.e., truth degree 1 is distributed among the atoms black x and white x .…”

“…The performance of fasp2smt did not change, while ffasp improves considerably, especially regarding memory consumption. We also tested 180 instances (not reported in Table 1) of two simple problems called Stratified and Odd Cycle (Alviano and Peñaloza 2013;Mushthofa et al 2014), which both fasp2smt and ffasp solve in less than 1 second. The main picture resulting from the experimental analysis is that fasp2smt is slower than ffasp in Graph Coloring, but it is faster in Hamiltonian Path.…”

“…In practice, however, ASP offers many efficient solvers such as dlv (Alviano et al 2010), cmodels (Lierler and Maratea 2004), clasp (Gebser et al 2012), and wasp , which is not the case for FASP. A preliminary FASP solver for programs with atomic heads and Lukasiewicz conjunction, called fasp, was presented at ICLP'13 by (Alviano and Peñaloza 2013). It implements approximation operators and a translation into bilevel programming (Blondeel et al 2014).…”

Semantics for possibilistic answer set programs: Uncertain rules versus rules with uncertain conclusions

“…Current implementations of FASP translate programs to instances of other formalisms such as bilevel programming [49,50]. Other authors have looked at fixpoint operators for finding upper and lower bounds on the truth value of atoms [51,52], which can be useful as a preprocessing step for exact solvers, as an approximation of exact solvers, or as en exact method for restricted classes of FASP programs. While FASP is a framework to deal with multi-valuedness, it is not designed to model uncertainty.…”