Title: 74.419 Artificial Intelligence Modal Logic
174.419 Artificial IntelligenceModal Logic
2Syntax 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 ? .
3Semantics 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
4Semantics 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.
- ...
5Semantics 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 ? - ?? ? ????
6Semantics 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 ?? ? ???? !
7Semantics of Modal Logic (? and ?)
- Other logical operators are interpreted as usual,
e.g. - ?w (???)1 iff ?w (??)0
8Semantics 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.
9Semantics 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.
- ...
10Semantics 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.)
11Semantics 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. - ? ?
12Semantics 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.
13Textbooks 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.
14Textbooks 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.
15Textbooks 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