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

ISBN-10: 9491216589

ISBN-13: 9789491216589

Solution 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, prognosis and database upkeep. besides the fact that, ASP isn't without delay appropriate for modeling issues of non-stop domain names. Such difficulties happen certainly in varied fields resembling the layout of fuel and electrical energy networks, machine imaginative and prescient and funding portfolios. to beat this challenge we research FASP, a mixture of ASP with fuzzy good judgment -- a category of manyvalued logics which may deal with continuity. We particularly specialise in the subsequent concerns: 1. an incredible query while modeling non-stop optimization difficulties is how we should always deal with overconstrained difficulties, i.e. difficulties that experience no ideas. in lots of instances we will choose to settle for a less than perfect resolution, i.e. an answer that doesn't fulfill the entire said principles (constraints). even though, this results in the query: what imperfect recommendations should still we elect? We examine this query and increase upon the cutting-edge via featuring an process according to aggregation features. 2. clients of a programming language frequently need a wealthy language that's effortless to version in. besides the fact that, implementers and theoreticians want a small language that's effortless to enforce and cause approximately. We create a bridge among those wishes through featuring a small middle language for FASP and by way of displaying that this language is able to expressing a lot of its universal extensions reminiscent of constraints, monotonically reducing capabilities, aggregators, S-implicators and classical negation. three. a widely known approach for fixing ASP includes translating a software P to a propositional conception whose versions precisely correspond to the reply units of P. We convey how this method could be generalized to FASP, paving tips to enforce effective fuzzy solution set solvers which may make the most of present fuzzy reasoners.

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

**Sample text**

Bn ; c1 , . . , cm ) as follows. Suppose I is an interpretation of a FASP program P then (1) I(l) = l if l ∈ L (2) I( f (b1 , . . , bn ; c1 , . . , cm )) = f (I(b1 ), . . , I(bn ); I(c1 ), . . , I(cm )) For a rule (r : a ← α) ∈ P we say that it is satisﬁed by an interpretation I of P iff I(a) I(α). Intuitively we can regard rules as residual implicators. 10 r is then satisﬁed by I iff I (I(α), I(a)) 1, where I is the residual implicator corresponding to r. 3. Let P be a FASP program. An interpretation I of P is a model of P iff every rule r ∈ P is satisﬁed by I.

Cm ) is called the body of the rule, denoted rb . In the remainder of this book we implicitly assume all lattices to be complete. 1) using r : a ← α. The Herbrand base of a rule r, denoted Br , is deﬁned as the set of atoms occurring in r. Similar to ASP, FASP rules can be divided in certain classes, depending on the conditions satisﬁed by their head and body. (1) A rule r : a ← f (b1 , . . , bn ; c1 , . . , cm ) on a lattice L is called a constraint if a ∈ L . (2) A rule r : a ← f (b1 , . .

Pn , the connectives ∧ and → and the truth constant 0 for 0. Formulas are deﬁned as usual: each propositional variable is a formula; 0 is a formula; if ϕ, ψ, are formulas, then ϕ ∧ ψ and ϕ → ψ are formulas. Further connectives are deﬁned as follows: (1) ϕ ψ = ϕ ∧ (ϕ → ψ) (2) ϕ ψ = ((ϕ → ψ) → ψ) ∧ ((ψ → ϕ) → ϕ) (3) ¬ϕ = ϕ → 0 (4) ϕ ↔ ψ = (ϕ → ψ) ∧ (ψ → ϕ) An evaluation of propositional variables is a mapping e assigning to each propositional variable p a truth value e(p) ∈ [0, 1]. This evaluation is extended to formulas as follows: (1) e(0) = 0 (2) e(ϕ → ψ) = IT (e(ϕ), e(ψ)) (3) e(ϕ ∧ ψ) = T (e(ϕ), e(ψ)) A formula ϕ is a 1-tautology of PC(T ) iff e(ϕ) = 1 for each evaluation e.

### Answer Set Programming for Continuous Domains: A Fuzzy Logic Approach by Jeroen Janssen, Steven Schockaert, Dirk Vermeir, Martine De Cock

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