Modal, Dynamic and Temporal Logics

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Modal, Dynamic and Temporal Logics

• SWE 623

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Modal Logic

• Logic of Necessity and Possibility
• Has a philosophical background
• Syntax has two extra symbols
• read as necessity ( X is necessarily X)
• Also called box X
• ltgt read as possibility (ltgt X possibly X)
• Also called diamond X
• See http//turing.wins.uva.nl/mdr/AiML/background
  .html

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Kripke Semantics of Modal Logic

• The universe seen as a collection of worlds.
• Truth defined in each world.
• Say U is the universe.
• I.e. each w e U is a prepositional or predicate
  model.

W4
W1
W2
W3
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Kripke Semantics of Modal Logic

• W1 satisfies X if X is satisfied in each
  world accessible from W1.
• If W3 and W4 satisfy X.
• Notation
• W1 X if and only if
• W3 X and W4 X
• W1 W1 satisfies ltgt X if X is satisfied in at
  least one world accessible from W1.

W4
W1
W2
W3

• Notation
• W1 ltgt X if and only if
• W3 X or W4 X

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Proof Rules for Modal Logic

• Modal Generalization
• A
• A
• Monotonicity of ?
• A ? B
• ? A ? ? B
• Monotonicity of ?
• A ? B
• ? ? A ? ?B

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An Axiom System for Prepositional Logic

• (A ? (B ? C)) ? (A ? B) ? (A ? C)
• A ? (B ? A)
• (( A ? false ) ? false ) A
• Modus Ponens
• A, A -gt B
• ? ? B

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An Axiom System for Predicate Logic

• ?x (A(x) ? B(x)) ? (?xA(x) ? ?xB(x))
• ?x A(x) ? At/x provided t is free for x in A
• A ? ?x A(x) provided x is not free in A
• Modus Ponens
• A, A -gt B
• B
• Generalization
• A
• ?x A(x)

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Some Facts About Modal Logic

• A couple of Valid Modal Formulas
• ? (A ? B ) lt-gt (? A) ? (? B)
• (A ? B ) lt-gt ( A) ? ( B)
• ? (false) ?(false)
• (? A) ? (B) ? ? (A ? B )
• Counter-examples to invalid modal formulas
• (? A) ? ( A )

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Proving Modal Formulas
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A counter-example in Modal Logic
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Dynamic Logic

• A special kind of Modal Logic where each world is
  a system state.
• Definition of State
• The set of variables x1, xn.
• x1 a1, xn an. is a state, where each variable
  takes a value.
• Accessibility is state change perhaps due to
  executing code.
• x1 a1, xn an is changed to x1 b1, xn an
  by the program (x1 b1).

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Dynamic Logic

• Issues
• What kind of program constructs result in what
  type of state change
• What is the logic
• Two Levels
• Prepositional
• Only deals with state change at (abstract)
  symbolic level
• Predicate
• Details of variables, values and programming
  operators
• Deals well with non-determinism, concurrency etc.

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Prepositional Dynamic LogicSyntax

• If A, B propositions and a, b programs,
• Following are formulas
• A /\ B, A ? B, ? A, A ? B, aA, lt agtA are
  formulas.
• Following are programs
• U b non-deterministic choice
• a b sequential composition
• (A?) a test.
• a non-deterministic iteration

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Prepositional Dynamic LogicSemantics

• A collection of states S si i gt 0.
• For each state si a notion of satisfiability of
  atomic prepositions. I.e. si A for each A.
• For each each atomic program a, a relation Ra on
  SxS.
• Raub Ra u Rb
• R(A?) (s,s) s A
• Rab Ra Rb (s1,s3) ? s2 (s1,s2) e Ra and
  (s2,s3) e Rb
• Ra U Rai i gt0 . Where Rai is defined
  inductively as Ra(i1) Rai Ra and Ra0
  Identity.

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PDL Semantics - Satisfaction

• Prepositional connectives as usual
• I.e. si A /\ B if si A and si B
• I.e. si A ? B if si A or si B
• Modal Connectives as in Modal Logic
• I.e. si aA, if for all states sj such that
  (si , sj) e Ra sj A
• I.e. si ltagtA, there is a state sj with (si ,
  sj) e Ra and sj A

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PDL Axiom System

• Axioms of prepositional logic
• a (A ? B) ? (aA ?aB)
• a (A /\ B) lt-gt (aA /\ aB)
• a U bA lt-gt (a A /\ b A)
• a bA lt-gt a b A
• B?A lt-gt (B /\ A)
• B /\ a a A lt-gt a A
• B /\ a( A ?aA) ? a A

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PDL Axiom System Rules

• Modus Ponens
• A, A -gt B
• B
• Modal Generalization
• A
• a A

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Some Derived Rules for PDL

• Monotonicity of ltagt
• A -gt B
• ltagtA -gt ltagtB
• Monotonicity of a
• A -gt B
• aA -gt aB

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Some Provable Properties

• a (A /\ B) ? (aA /\aB)
• ltagt (A \/ B) lt-gt (ltagtA \/ ltagtB)
• (ltagtA /\ a B) ? ltagt(A /\ B)
• a A lt-gt (? ltagt(? A))
• ltagtfalse lt-gt false
• ltagtltbgtA lt-gt ltabgtA,
• abA lt-gt ab A
• lt a U bgtA lt-gt (ltagtA \/ ltbgtB)
• a U bA lt-gt (aA /\ bB)

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Translating Giress Style Pre/Post Conditions to
PDL

• Skip True?
• Fail false?
• If A then a else b (A?a) U (?A?b)
• While A do a (A?a) (?A?)

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First-Order Dynamic Logic

• Syntax
• The same definition as predicate logic except for
  the additions
• If A is a formula and a is a program, then aA,
  ltagtA are formulas.
• If A is a formula, then A? is a test. (I.e. a
  program)
• If A is quantifier free then its said to be a
  basic test, and otherwise a rich test.

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First-Order Dynamic Logic

• Semantics Transitions between states defined as
• R(X a) (S, S) if S(x) S(a) and
  S(y) S(y) for Y ! X
• R(A?) (S,S) S A
• Definitions of U, are same as in the
  prepositional case.

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Axiomatization

• Axioms
• All axioms for predicate logic
• All axioms for PDL
• At/x lt-gt lt x tgtA(x)
• A lt-gt A, A is obtained by replacing any program
  a by zx a xz, where a is a with all
  occurrences of x replaced by z, and z does not
  appear in a

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Axiomatization Rules

• modus ponens
• A, A -gt B
• B
• Generalization
• A A
• a A ? x A(x)
• Infinitary convergence
• A -gt anB for all n
• B -gt aB

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Some Example Reductions I

• Reduce ?XX1 ((Xa) U (Xb)) ? A(X)
• Step1 ? XX1 (Xa) ? (Xb) ? A(X)
• Step2 ? XX1 ? ? (Xa) ? A(X) ? ltXX1
  ? ? (Xb) ? A(X)
• Step3 ? XX1 ? A
• Step4 A(a) ? A(b)

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Some Example Reductions II

• Reduce xx1(xa U xb) B(X)
• Step1 xa1 U xb1B(x)
• Step 2 xa1B(x) /\ xb1B(x)
• Step 3 B(a1) /\ B(b1)

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Temporal Logic

• Special kind of modal logic to reason about time.
• There are many kinds of Temporal Logics
• Linear and Branching Time
• Future and Past times
• Discrete and Continuous time
• Operators in Temporal Logics (MacMillans
  Notation)
• O next time F
• always G
• ? some times X
• ? until U

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Prepositional Syntax

• Atomic Proposition letters p, q etc.
• If p, q are propositions then so are.
• Meaning Logical Notation Model Checking
• Next Time p Op Xp
• All ways p p Gp
• In the future p ?p Fp
• p until q p ? q pUq

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Prepositional Semantics

• A collection of Kripke Worlds including the
  current one.
• Accessibility relation is evolution of time.

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Prepositional Semantics II

• Op if some world accessible from the current
  satisfies p.
• p if every world accessible from the current
  satisfies p.
• ? p if some world in the future from the
  current satisfies p.

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PTL Axioms and Rules I

• Axioms
• (A -gtB) -gt(A -gt B)
• O(A -gtB) -gt (OA -gt OB)
• (O ? A) lt-gt (?OA)
• A -gt (A /\ OA)
• (A -gt OA) -gt (A -gt A)
• A ? B -gt ?B
• A ? B lt-gt B \/ (A /\ O(A ? B ))

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PTL Axioms and Rules II

• Rules
• modus ponens
• generalization
• A
• A
• A
• O A