1. Modal, Dynamic and Temporal Logics
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2. 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”
<> read as possibility (<> X “possibly X”)
Also called “diamond X”
See
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3. 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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4. 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 <> X if X is satisfied in at least one world accessible from W1. W4 W1 W2 W3
Notation:
W1 |= <> X if and only if
W3 |= X or W4 |= X
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5. Proof Rules for Modal Logic
Modal Generalization A
[] A
Monotonicity of 
A  B
 A   B
Monotonicity of 
 [] A  []B
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6. An Axiom System for Prepositional Logic
(A  (B  C))  (A  B)  (A  C)
A  (B  A)
(( A  false )  false ) A
Modus Ponens
A, A -> B
  B
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7. An Axiom System for Predicate Logic
x (A(x)  B(x))  (xA(x)  xB(x))
x A(x)  A[t/x] provided t is free for x in A
A  x A(x) provided x is not free in A
Modus Ponens
A, A -> B B
Generalization A
x A(x)
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8. Some Facts About Modal Logic
A couple of Valid Modal Formulas:
 (A  B ) <-> ( A)  ( B)
[](A  B ) <-> ([] A)  ([] B)
 (false) (false)
( A)  ([]B)   (A  B )
Counter-examples to invalid modal formulas
( A)  ( [] A )
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9. Proving Modal Formulas
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10. A counter-example in Modal Logic
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11. 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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12. Dynamic Logic Issues: Two Levels
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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13. Prepositional Dynamic Logic Syntax
If A, B propositions and a, b programs,
Following are formulas
A /\ B, A  B,  A, A  B, [a]A, < a>A 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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14. Prepositional Dynamic Logic Semantics
A collection of states: S = {si : i >= 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 }
Ra;b = Ra ; Rb ={ (s1,s3) :  s2 (s1,s2) e Ra and (s2,s3) e Rb }
Ra* = U {Rai : i >=0 }. Where Rai is defined inductively as Ra(i+1) = Rai ; Ra and Ra0 = Identity.
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15. 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 |= [a]A, if for all states sj such that (si , sj) e Ra sj |= A
I.e. si |= <a>A, there is a state sj with (si , sj) e Ra and sj |= A
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16. PDL Axiom System Axioms of prepositional logic
[a] (A  B)  ([a]A [a]B)
[a] (A /\ B) <-> ([a]A /\ [a]B)
[a U b]A <-> ([a] A /\ [b] A)
[a ; b]A <-> [a] [b] A
[B?]A <-> (B /\ A)
B /\ [a] [a*] A <-> [a*] A
B /\ [a*]( A [a]A)  [a*] A
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17. PDL Axiom System: Rules
Modus Ponens
A, A -> B B
Modal Generalization A
[a] A
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18. Some Derived Rules for PDL
Monotonicity of <a>
A -> B
<a>A -> <a>B
Monotonicity of [a]
[a]A -> [a]B
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19. Some Provable Properties
[a] (A /\ B)  ([a]A /\[a]B)
<a> (A \/ B) <-> (<a>A \/ <a>B)
(<a>A /\ [a] B)  <a>(A /\ B)
[a ]A <-> ( <a>( A))
<a>false <-> false
<a><b>A <-> <a;b>A,
[a][b]A <-> [a;b] A
< a U b>A <-> (<a>A \/ <b>B)
[ a U b]A <-> ([a]A /\ [b]B)
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20. Translating Gires’s 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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21. 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 [a]A, <a>A 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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22. 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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23. Axiomatization Axioms All axioms for predicate logic
All axioms for PDL
A[t/x] <-> < x:= t>A(x)
A <-> A’, A’ is obtained by replacing any program a by z:=x; a’; x:=z, where a’ is a with all occurrences of x replaced by z, and z does not appear in a
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24. Axiomatization: Rules
modus ponens
A, A -> B B
Generalization
A A
[a] A  x A(x)
Infinitary convergence
A -> [an]B for all n
B -> [a*]B
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25. Some Example Reductions I
Reduce: X:=X+1; ((X:=a) U (X:=b))  A(X)
Step1:  X=X+1; (X=a)  (X=b)  A(X)
Step2:  X=X+1   (X=a)  A(X)  <X=X+1   (X=b)  A(X)
Step3:  X=X+1  A
Step4: A(a)  A(b)
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26. Some Example Reductions II
Reduce: [x:=x+1;(x:=a U x:=b)] B(X)
Step1: [x:=a+1 U x:=b+1]B(x)
Step 2: [x:=a+1]B(x) /\ [x:=b+1]B(x)
Step 3: B(a+1) /\ B(b+1)
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27. 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 (MacMillan’s Notation)
O = next time F
[] = always G
 = some times X
 = until U
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28. 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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29. Prepositional Semantics
A collection of Kripke Worlds including the current one.
Accessibility relation is evolution of time.
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30. 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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31. PTL Axioms and Rules I Axioms [](A ->B) ->([]A -> []B)
O(A ->B) -> (OA -> OB)
(O  A) <-> (OA)
[]A -> (A /\ O[]A)
[](A -> OA) -> (A -> []A)
A  B -> B
A  B <-> B \/ (A /\ O(A  B ))
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32. PTL Axioms and Rules II Rules modus ponens generalization A [] A O A
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