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Concept
Autoepistemic logic
Summary
The autoepistemic logic is a formal logic for the representation and reasoning of knowledge about knowledge. While propositional logic can only express facts, autoepistemic logic can express knowledge and lack of knowledge about facts.
The stable model semantics, which is used to give a semantics to logic programming with negation as failure, can be seen as a simplified form of autoepistemic logic.
The syntax of autoepistemic logic extends that of propositional logic by a modal operator indicating knowledge: if is a formula, indicates that is known. As a result, indicates that is known and indicates that is not known.
This syntax is used for allowing reasoning based on knowledge of facts. For example, means that is assumed false if it is not known to be true. This is a form of negation as failure.
The semantics of autoepistemic logic is based on the expansions of a theory, which have a role similar to models in propositional logic. While a propositional model specifies which axioms are true or false, an expansion specifies which formulae are true and which ones are false. In particular, the expansions of an autoepistemic formula makes this distinction for every subformula contained in . This distinction allows to be treated as a propositional formula, as all its subformulae containing are either true or false. In particular, checking whether entails in this condition can be done using the rules of the propositional calculus. In order for an initial assumption to be an expansion, it must be that a subformula is entailed if and only if has been initially assumed true.
In terms of possible world semantics, an expansion of consists of an S5 model of in which the possible worlds consist only of worlds where is true. [The possible worlds need not contain all such consistent worlds; this corresponds to the fact that modal propositions are assigned truth values before checking derivability of the ordinary propositions.] Thus, autoepistemic logic extends S5; the extension is proper, since and are tautologies of autoepistemic logic, but not of S5.
Official source
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