How to Handle Negation in Prolog: Several Logical Approaches

tags

Logic Programming

Notes

the existence of negated sentences (in this case ¬(2 × 2 = 4)) presents a challenging question: how can they exist?

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How can we claim that we know 2 × 2 = 4? Do we need to know each instances x where 2 × 2 6= x

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the closed world assumption presupposed a challenging position: an S5 model for knowledge.

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the cost we have to pay is an increase in the complexity.

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Four Valued Logic for Prolog

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“insist-on-ground negation as failure” (GNAF)

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total order on the degree of computational priority of the given truth values.

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different truth val- ues when they appear as the first conjunct of a conjunction.

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equivalent formulae may be parsed by fol- lowing a different order.

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Extending Programs with Classical Negation

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include negative infor- mation explicitly.

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extended programs will not correspond to the positive Horn clauses.

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The answer set will be defined as the set of all ground literals which can be generated by using either the rules of the program or by utilizing classical logic.

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a consistent answer set S is the answer set of the program P if and only if S + is an answer set for the program P + .

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under some conditions (such as stability), one can reduce the extended programs to the general programs in such a way that not is well- handled.

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Extending the Models for Negation

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this logic has an intuitionistic sense as it does not adopt the law of excluded middle.

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computational restrictions that Prolog poses. One such restriction is the non-computability of point infinity.

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without the closed world assumption, one can suggest that the logic of prolog is rather intuitionistic

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from the perspective of programming languages, this seems inevitable considering the vast majority of practical implementations.

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what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.

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“sun raises from west” is not a knowledge. Because, we cannot verify that it will hold in each temporal instance of the cosmos. Therefore, in a sophisticated sense, we cannot know neither “sun raises from west” nor “sun does not raise from west”.

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handling of negation is not a current problem, but a problem which kept philosophers and logicians busy throughout the ages.

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most other approaches to the negation problem would be reduced one or another instance of either of these approaches.

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Restricted closed world assumption

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separate set of syntactical domains. One for positive formulae, one for the negative formulae

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spot where the program needs the closed world assumption

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search tree algorithms

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The mismatch between the formal logic and the programming paradigm is significant.

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