Page 124 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes In pointwise circumscription, each tuple of values is considered separately. For example, in the
formula P(a) ≡ P(b) one would consider the value of P(a) separately from P(b). A model is
minimal only it is not possible to turn any such value from true to false while still satisfying the
formula. As a result, the model in which P(a) = P(b) = true is selected by pointwise circumscription
because turning only P(a) into false does not satisfy the formula, and the same happens for P(b).

6.9.5 Domain and Formula Circumscription

An earlier formulation of circumscription by McCarthy is based on minimizing the domain of
first-order models, rather than the extension of predicates. Namely, a model is considered less
than another if it has a smaller domain, and the two models coincide on the evaluation of the
common tuples of values. This version of circumscription can be reduced to predicate
circumscription. Formula circumscription was a later formalism introduced by McCarthy. This
is a generalization of circumscription in which the extension of a formula is minimized, rather
than the extension of a predicate. In other words, a formula can be specified so that the set of
tuples of values of the domain that satisfy the formula is made as small as possible.

Self Assessment

State whether the following statements are true or false:

17. Circumscription was later used by Vladimir Lifschitz in an attempt to solve the frame
problem.
18. Pointwise circumscription is a variant of first-order circumscription that has been
introduced by McCarthy.

6.10 Modal Logic

Modal logic is a type of formal logic primarily developed in the 1960s that extends classical
propositional and predicate logic to include operators expressing modality. Modals—words
that express modalities—qualify a statement.

Example: The statement “John is happy” might be qualified by saying that John is
usually happy, in which case the term “usually” is functioning as a modal.

The traditional alethic modalities, or modalities of truth, include possibility (“Possibly, p”, “It
is possible that p”), necessity (“Necessarily, p”, “It is necessary that p”), and impossibility
(“Impossibly, p”, “It is impossible that p”) Other modalities that have been formalized in modal
logic include temporal modalities, or modalities of time (notably, “It was the case that p”, “It has
always been that p”, “It will be that p”, “It will always be that p”), deontic modalities (notably,
“It is obligatory that p”, and “It is permissible that p”), epistemic modalities, or modalities of
knowledge (“It is known that p”) and doxastic modalities, or modalities of belief (“It is believed
that p”). A formal modal logic represents modalities using modal operators.

Example: “It might rain today” and “It is possible that rain will fall today” both contain
the notion of possibility.
In a modal logic, this is represented as an operator, possibly, attached to the sentence “It will
rain today”. The basic unary (1-place) modal operators are usually written for necessarily and
◊ for possibly. In a classical modal logic, each can be expressed by the other with negation:

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