Modal and Temporal Logics An overview

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Modal and Temporal Logics- An overview

• Venkita Subramonian
• Research Seminar on Software Systems
• 05/03/2004

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What is Logic?

• Systematic study of how statements can be related
• Inference of knowledge based on just looking at
  such relations rather than the real world
• Reasoning
• Observation Reasoning Guesswork Scientific
  progress
• Applications Logic circuits, Boolean data
  types, Reasoning about programs, Automated
  reasoning, Artificial intelligence

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Basics

• Statement (Proposition) is a sentence which is
  either true or false
• Inference P1P2P3..Pn/C
• Inference schema, statement schema, schematic
  letters, interpretation
• Validity and soundness of an inference
• Validity based on logical correctness
• Soundness based on real world correctness

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Some Forms of Logic

• Propositional Logic
• Schematic letters represent whole statements
• Connectives ? ? ? ?
• Predicate Logic
• Two kinds of schematic letters
• a,b- to represent individual objects,entities
• A,B - to represent properties (or lack thereof)
  of individual objects
• Additional Operators - ? ?
• Modal Logic
• Temporal Logic

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Modal Logic

• Logic of possibility and necessity
• Modality
• the classification of logical propositions
  according to their asserting or denying the
  possibility, impossibility, contingency, or
  necessity of their content Merriam-Webster
  online

P is true in this state Q and R are both false
Each stage of the game, each of the following
propositions is either true or false P Noughts
wins Q Crosses wins R The game is drawn
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Modal Logic Operators

• Modal operators
• ? for possibility
• ? for necessity
• Given a proposition A,
• ?A Possibly, A is true
• ?A Necessarily, A is true
• The above two operators can be mixed with the
  propositional and predicate calculus operators

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Tic-Tac-Toe example (1/2)
W1
W3
W2
W4
W8
W5
W6
W7
W10
W9
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Tic-Tac-Toe example (2/2)
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Any visual representations?

• ?A Possibly, A is true
• There exists atleast one state accessible from
  the current state where A is true
• ?A Necessarily, A is true
• For all states accessible from current state, A
  is true

A
A
A
?A
A
?A
A
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Necessity what does it mean here?

• Note the meaning of necessity is different from
  the intuitive meaning of necessity
• E.g. In state W2, Q, ?P and ?R are true
• ?P and ?R are vacuously true, since there are no
  successor states for W2

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Complex Modalities

• Operators ? and ? can be prefixed to any
  statement that itself includes these operators
• E.g. ??Q is true in state W
• ?Q is true in a state accessible from W
• Q is true in a state accessible from a state
  accessible from W
• Consider this ?(??A ? ?A). This is true in every
  state

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Conditional validity
A

• Consider intransitive accessibility relationship
• ?(??A ? ?A) does not hold anymore

A
?A
? ?A
?A
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Syntax of PML

• Schematic letters
• P1 P2 P3
• Connectives
• ?? ? ?
• Modal operators
• ? ?
• Parentheses
• ()
• Rules of formation
• if Pi and Pj are schematic letters then
• Pi, Pi, (Pi ? Pj), (Pi ? Pj), (Pi ? Pj),(Pi ?
  Pj), ?Pi, ?Pi
• (2) If S is a schema, then so is any
  substitution-instance of S obtained by
  substituting schemas for the schematic letters in
  S
• (3) Nothing is a schema unless it can be
  generated by the rules 1 and 2 above

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Formal semantics

• A formal interpretation of PML is (W,R,I)
• W set of states
• R Accessibility relation on W
• I Interpretation function that maps schematic
  letters onto a subset of W, so that for the
  schematic letter P, I(P) is the set of states in
  which P is to count as true
• (W,R) called as a modal frame

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Extension of Interpretation function

• Interpretation function extended to cover complex
  schemas

• For a schematic letter A, I(A) I(A)
• I(A) W I(A)
• I(A?B) I(A) ? I(B)
• I(A?B) I(A) ? I(B)
• I(A?B) I(A ? B)
• I(A?B) I(A?B) ? I(B?A)
• I(?A) w?W for some w?I(A),wRw
• I(?A) w?W for every w?I(A),wRw

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Satisfaction and validity
For a schema A and a state w in W, W,R,IAw
means w is in I(A) A is satisfied by (W,R,I) at
state w W,R,IA means ? w ? W,
W,R,IAw (W,R,I) satisfies A For a modal frame
(W,R), If whichever I we choose, (W,R,I)
satisfies A Then W,RA and A is valid with
respect to (W,R,I) If A is valid in every modal
frame, A is universally valid A
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Universally valid modal schemes

• Examples
• ?A ? ?B ? ?(A ? B)
• ?(A?B) ? (?A??B)
• ?(A?B) ? (?A ? ?B)
• ?(A?B) ? (?A ? ?B)
• (?(A ? B) ? ?A ? ?B)

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Formal Proof System for PML
A
?-intro
?A
?(A?B) ?A
(K)
?B
? A
?-intro
?A
?A
?-elim
? A
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Temporal Logic

• Introduces both tenses past and future
• Operators for future
• FA It will be the case that A
• GA It will always be the case that A
• Operators for past
• PA It has been the case that A
• HA It has always been the case that A

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Modal vs Temporal

• Very similar, BUT some differences
• In modal logic, the relationship R in (W,I,R)
  could be symmetric or reflexive. But here it is
  difficult to imagine that happen.
• Think of earlier than relationship between two
  states.
• Instead of (W,R), here we write (T,lt) to denote a
  temporal frame

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Some important observations

• If in any universally valid modal schema, ? and ?
  are replaced by F and G respectively, then the
  result is a universally valid tense-logical
  schema
• If in any universally valid modal schema, ? and ?
  are replaced by P and H respectively, then the
  result is a universally valid tense-logical
  schema

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Examples
PML
Tense Logic

• ?(A?B) ? (?A??B)
• ?(A?B) ? (?A ? ?B)
• ?(A ? B) ? ?A ? ?B

• H(A?B) ? (HA?HB)
• G(A?B) ? (GA?GB)
• F(A?B) ? (FA ? FB)
• P(A?B) ? (PA ? PB)
• G(A ? B) ? (GA ? GB)
• H(A ? B) ? (HA ? HB)