74.419 Artificial Intelligence Modal Logic

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74.419 Artificial IntelligenceModal Logic

• see reference last slide

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Syntax of Modal Logic (? and ?)

• Formulae in (propositional) Modal Logic ML
• The Language of ML contains the Language of
  Propositional Calculus, i.e. if P is a formula in
  Propositional Calculus, then P is a formula in
  ML.
• If ? and ? are formulae in ML, then
• ??, ???, ???, ???, ??, ??
• are also formulae in ML.
• Note The operator ? is often later introduced
  and defined through ? .

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Semantics of Modal Logic (? and ?)

• The semantics of a modal logic ML is defined
  through
• a set of worlds W w1, w2, ..., wn,
• an accessibility relation R?W?W, and
• an interpretation function ? ??0,1

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Semantics of Modal Logic (? and ?)

• The interpretation in ML of a formula P, Q, ...
  of the propositional language of ML corresponds
  to its truth value in the "current world"
• ?w (P)1 iff I(P) is true in w.
• ?w (P?Q)1 iff I(P?Q) is true in w.
• ...

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Semantics of Modal Logic (? and ?)

• We extend the semantics with an interpretation of
  the operators ? and ?, specified relative to a
  "current world" w.
• For all w?W
• ?w (??)1 iff ?w' (w,w')?R ? ?w' (?)1
• 0 otherwise.
• ?w (??)1 iff ?w' (w,w')?R ? ?w' (?)1
• 0 otherwise.
• Note Often, the operator ? is defined in terms
  of ?
• ?? ? ????

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Semantics of Modal Logic (? and ?)

• We can also prove the equivalence of ? and ? for
  our definitions above
• ?w (???)1 iff ?(?w (??)1) (or ?w (??)0)
• iff ??w' (w,w')?R ? ?w' (?)1
• iff ?w' (w,w')?R ? ?w' (?)0
• iff ?w' (w,w')?R ? ?w' (??)1
• iff ?w (???)1
• This means ??? ? ???
• Exercise Proof ?? ? ???? !

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Semantics of Modal Logic (? and ?)

• Other logical operators are interpreted as usual,
  e.g.
• ?w (???)1 iff ?w (??)0

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Semantics of ML - Complex Formulas

• The interpretation of a complex formula of ML is
  based on the interpretation of the atomic
  propositional symbols, and then composed using
  the interpretation function ? defined above, e.g.
  
• ?w (???)1 iff ?(?w' (w,w')?R ? ?w' (?)1)
• iff ?w' (w,w')?R ? ?w' (?)0
• Let's say ? ? (P?Q).
• ?w' (w,w')?R ? ?w' (P?Q)0
• ?w' (w,w')?R ? (?w' (P)0 ? ?w' (Q)0)
• "P or Q" is not necessarily true in world w, if
  there is a world w', accessible from w, in which
  P is false or Q is false.

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Semantics of Modal Logic - Grounding

• The interpretation in ML of a formula P, Q, ...
  of the propositional language of ML corresponds
  to its truth value in the "current world"
• ?w (P)1 iff I(P) is true in w.
• ?w (P?Q)1 iff I(P?Q) is true in w.
• ...

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Semantics of Modal Logic

• A formula ? is satisfied in a world w of a Model
  MltW,R,?gt, if it is true in this world w?W under
  the given interpretation ?, i.e. ?w (?)1.
• M, w ?
• A formula ? is true in a Model MltW,R,?gt, if it
  is satisfied in all worlds w?W of M.
• M ?
• A formula ? is valid, if it is true in all
  Models.
• ? ?
• A formula ? is satisfiable, if it is satisfied
  in at least one world w?W of one Model MltW,R,?gt.
  (or If its negation is not valid.)

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Semantics of Modal Logic

• A formula ? is satisfied in a world w of a Model
  MltW,R,?gt, if it is true in this world under the
  given interpretation ?, i.e. ?w (?)1.
• M, w ?
• A formula ? is true in a Model MltW,R,?gt, if it
  is satisfied in all worlds w?W of M.
• M ?
• A formula ? is valid, if it is true in all
  Models.
• ? ?
• A formula ? is satisfiable, if it is satisfied
  in at least one world w?W of one Model MltW,R,?gt.
  (or If its negation is not valid.)
• A formula ? is a consequence of a set of formulas
  ? in MltW,r,?gt, if in all worlds w?W, in which ?
  is satisfied, ? is also satisfied.
• ? ?

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Semantics of Modal Logic Terminology

• Sometimes the term "frame" is used to refer to
  worlds and their connection through the
  accessibility relation
• A frame ltW, Rgt is a pair consisting of a
  non-empty set W (of worlds) and a binary relation
  R on W.
• A model ltF, ?gt consists of a frame F, and a
  valuation ? that assigns truth values to each
  atomic sentence at each world in W.

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Textbooks on (Modal) Logic

• Richard A. Frost, Introduction to Knowledge-Base
  Systems, Collins, 1986 (out of print)
• Comments one of my favourite books contains
  (almost) everything you need w.r.t. foundations
  of classical and non-classical logic very
  compact, comprehensive and relatively easy to
  understand.
• Allan Ramsay, Formal Methods in Artificial
  Intelligence, Cambridge University Press, 1988
• Comments easy to read and to understand deals
  also with other formal methods in AI than logic
  unfortunately out of print a copy is on course
  reserve in the Science Library.

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Textbooks on (Modal) Logic

• Graham Priest, An Introduction to Non-Classical
  Logic, Cambridge University Press, 2001
• Comments the most poplar book (at least among
  philosophy students) on non-classical, in
  particular, (propositional) modal logic.
• Kenneth Konyndyk, Introductory Modal Logic,
  University of Notre-Dame Press, 1986 (with later
  re-prints)
• Comments relatively easy and nice to read
  contains propositional as well as first-order
  (quantified) modal logic, and nothing else.

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Textbooks on (Modal) Logic

• J.C. Beall Bas C. van Fraassen, Possibilities
  and Paradox, University of Notre-Dame Press, 1986
  (with later re-prints)
• Comments contains a lot of those weird things,
  you knew existed but you've never encountered in
  reality (during your university education).
• G.E. Hughes M.J. Creswell, A New Introduction
  to Modal Logic, Routledge, 1996
• Comments Location Elizabeth Dafoe Library, 2nd
  Floor, Call Number / Volume BC 199 M6 H85 1996