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NML3: a non-monotonic formalism with explicit defaults.
PhD Thesis. In series: Link√∂ping Studies in Science and Technology. Dissertations #258. Link√∂pings tekniska h√∂gskola. 290 pages. ISBN: 91-7870-816-8.
The thesis is a study of a particular approach to defeasible reasoning based on the notion of an information state consisting of a set of partial interpretations constrained by an information ordering. The formalism proposed, called NML3, is a non-monotonic logic with explicit defaults and is characterized by the following features: (1) The use of the strong Kleene three-valued logic as a basis. (2) The addition of an explicit default operator which enables distinguishing tentative conclusions from ordinary conclusions in the object language. (3) The use of the technique of preferential entailment to generate non-monotonic behavior. The central feature of the formalism, the use of an explicit default operator with a model theoretic semantics based on the notion of a partial interpretation, distinguishes NML3 from the existing formalisms. By capitalizing on the distinction between tentative and ordinary conclusions, NML3 provides increased expressibility in comparison to many of the standard non-monotonic formalisms and greater flexibility in the representation of subtle aspects of default reasoning.In addition to NML3, a novel extension of the tableau-based proof technique is presented where a signed formula is tagged with a set of truth values rather than a single truth value. This is useful if the tableau-based proof technique is to be generalized to apply to the class of multi-valued logics. A refutation proof procedure may then be used to check logical consequence for the base logic used in NML3 and to provide a decision procedure for the propositional case of NML3.A survey of a number of non-standard logics used in knowledge representation is also provided. Various formalisms are analyzed in terms of persistence properties of formulas and their use of information structures.
A three-valued approach to non-monotonic reasoning.
Licentiate Thesis. In series: Link√∂ping Studies in Science and Technology. Thesis #230. Link√∂ping University. ISBN: 91-7870-672-6.
Abstract: The subject of this thesis is the formalization of a type of non-monotonic reasoning using a three-valued logic based on the strong definitions of Kleene. Non-monotonic reasoning is the rule rather than the exception when agents, human or machine, must act where information about the environment is uncertain or incomplete. Information about the environment is subject to change due to external causes, or may simply become outdated. This implies that inferences previously made may no longer hold and in turn must be retracted along with the revision of other information dependent on the retractions. This is the variety of reasoning we would like to find formal models for.We start by extending Kleene-s three-valued logic with an \"external negation\" connective where ~ a is true when a is false or unknown. In addition, a default operator D is added where D a is interpreted as \"a is true by default. The addition of the default operator increases the expressivity of the language, where statements such as \"a is not a default\" are directly representable. The logic has an intuitive model theoretic semantics without any appeal to the use of a fixpoint semantics for the default operator. The semantics is based on the notion of preferential entailment, where a set of sentences G preferentially entails a sentence a, if and only if a preferred set of the models of G are models of a. We also show that one version of the logic belongs to the class of cumulative non-monotonic formalisms which are a subject of current interest.A decision procedure for the propositional case, based on the semantic tableaux proof method is described and serves as a basis for a QA-system where it can be determined if a sentence a is preferentially entailed by a set of premises G. The procedure is implemented.