Modal Logic A Friendly Introduction

1. Modal LogicA Friendly Introduction First-Order Modal Logic M. Fitting and Richard L. Mendelson A New Introduction to Modal Logic G.E. Hughes and M.J. Cresswell Presented By Eran Yahav yahave@post.tau.ac.il http://www.cs.tau.ac.il/~yahave

2. Why Study Modal Logic • Generalizes important logics • Temporal logics • Dynamic logic • Logic of knowledge • And more... • Applications in • Program semantics • Verification and Model Checking • Artificial intelligence • And more…

3. What is a Modal? Classical propositional logic P  Q  R Jonathan is happy It is true that Jonathan is happy

4. Alethic Temporal Deontic Epistemic What is a Modal? Qualification over the truth of claim Necessarily Possibly Will be Was Has been Will have been May Can Must Certainly Probably Perhaps Surely Jonathan is _____ happy It is _____ true that Jonathan is happy

5. Outline • Propositional Modal Logic • Possible world semantics (Kripke Semantics) • Temporal logic, epistemic logic • First-Order Modal Logic • What the quantifiers quantify over • Constant and Varying Domains • Applications in program semantics • Summary

6. Propositional Modal Logic • Syntax • Propositional logic • Modal operators •  - necessarily (box) •  - possibly (diamond) (PQ)(PQ) PQ

7. Informal Interpretations Jonathan is happy Jonathan is happy under interpretation i0  Jonathan is happy  Jonathan is happy i Jonathan is happy under interpretation i i Jonathan is happy under interpretation i PP PP

8. Possible World Semantics • Possible interpretation = possible world • Accessibility relation  Jonathan is happy w w0Rw Jonathan is happy at w PP (w w0Rw P at w)  P at w0

9. Possible World Semantics • Valid propositional formula • True for every possible assignment to propositions • “all lines of the truth table” • Quantify over possible worlds P Q (PQ)P FF FT F F T F T T T F T TF TT T T T

10. Frames and Models • Frame • G – non-empty set of possible worlds • R – binary accessibility relation • Model • A frame <G,R> with an assignment V • V - which propositions are true at which worlds • Also denoted 

11. Truth in a Model • w  X  w  X • w  (XY)  w  X and w  Y • w   X  for every w’G, if (w,w’)R then w’  X • w   X  for some w’G, if (w,w’)R then w’  X

12. Example a b c P Q b  PQ a  P c  PQ a  Q a   (P  Q) a  P  Q

13. Example a b c P a  P a  P

14. More General Examples • Given a model G,R,, aG • a  (PQ)  a  P  Q • Under what terms • P  P for each world aG • P  P • P  P • P  P transitive reflexive symmetric symmetric and transitive

15. Important Modal Logics

16. Important Modal Logics S5  B S4  T K4   D K

17. Logical Consequence • Set of formulae  formula • Classically • S  X when X must be true whenever members of S are true • Modal setting • X is true at each world in which members of S are true? (local) • X is valid in every model in which members of S are valid? (global)

18. Logical Consequence • Answer: both • S  U  X • S – set of formulae – global assumptions • U – set of formulae – local assumptions • X – single formula • X is consequence of S and U when • For every model in which all members of S are valid • and for every world w in which satisfies U • we have w  X

19. Example a   {PP}  (P P) a  P a  PP b  P a  P a  P P At world a local assumptions are true but P P is not b P

20. Example {PP}   P P assume a  P Let a world b, aRb Then b  P But PP is valid in the model b P And thus a  P And therefore P P follows from the global assumption

21. Temporal Logic Jonathan is happy at time t Jonathan is happy in March 1999 Jonathan is happy in June 2000 t Jonathan is happy at time t t Jonathan is happy at time t  Jonathan is happy t Jonathan is happy at time t  Jonathan is happy t Jonathan is happy at time t

22. Temporal Logic • F P – will sometime be the case that P • P P – was sometime be the case that P • G P – will always be the case that P • H P – has always been P • In common temporal logic •  P = P G P •  P = P  F P

23. Epistemic Logic • Logic of knowledge • Ka P – a knows that P • Pa P – it is possible, for all that a knows, that P • Logic of belief • Ba P – a believes that P • Ca P – it is compatible with everything a believes in that P

24. Epistemic Logic Ka (P  Q)  (Ka P  Ka Q) Ka P  P Ka P  Ka Ka P Ka P  Ka Ka P Ka Kb P  Ka P Ka P  Pa P

25. Epistemic Logic Example Ka (P  Q)  (Ka P  Ka Q) E = I see my hand S = I’m dreaming (1) Ka(E  S) (2) KaS Ka (E  S)  (Ka E  Ka S) Ka (E  S)  (KaS  Ka E) By (1) and (2) Ka E

26. First-Order Modal Logic • , – quantify over possible worlds • , – quantify over individuals • Complications • What quantifiers quantify over? • Mixing modal operators and quantifiers

27. It is a necessary truththat everything is F Each thing is such thatit has F necessarily xF(x) xF(x) de re de dicto Necessity de re & de dicto Everything is necessarily F

28. What the Quantifiers Quantify Over • Universal Instantiation (classical logic) • x (x)  (y) • First order modal logic • object need not exist in more than one world • Free variables take values from the domain of the model, not domain of world we are in • Validity depends on possible-world semantics • Holds in constant domain semantics • Does not hold in varying domain semantics

29. Constant Domains • Augmented Frame G,R,D • G,R as in PML • D – domain of the frame • Model G,R,D,L • L – interpretation assigning to each n-ary predicate P and each world w some n-ary relation on D • Valuation • Assign a member v(x)  D to each free variable x

30. Constant Domains • Quantification over objects of the model • Universal instantiation holds • x (x)  (y) • Possiblist quantification • Simpler to handle • yx(x=y) • If pigs could fly

31. Varying Domains • Augmented Frame G,R,D • G,R as in PML • D – domain function, mapping worlds to non-empty sets • Model G,R,D,L • L – interpretation assigning to each n-ary predicate P and each world w some n-ary relation on DF • Valuation • Assign a member v(x)  DF to each free variable x • Simulate constant domain model • For all w D(w) = DF

32. Varying Domains • Two ways for individual e to have  at w • e is  at w and e is in w • e is  at w but e not in w • Something does not exist in this world • But still has the property  • Quantification only over objects in world’s domain --- actualist quantification • Pigs cannot fly

33. Existence Relativization • Define a special unary predicate  • For a formula , we define  • If a is atomic A = A • (X) = (X) • (XY) = (X Y) • (X) = X • (x ) = x (x)   • (x ) = x (x) • A sentence  is valid in every varying domain model iff  is valid in every constant domain model

34. Barcan & Converse Barcan Formulae • Modal operators = FO quantifiers of different sort • In classical FO • xy   yx  • yx   xy • xy  yx  • Any of these translate to FOML?

35. Barcan & Converse Barcan Formulae x     x  x   x   Barcan • x  x   x    x  Converse Barcan Barcan is valid Anti-monotonic Converse Barcan is valid  Monotonic Barcan and Converse Barcan are valid  Locally constant

36. a b a Varying Domain Example P = { b } x  P(x)  x P(x)

37. Equality • What do we want? • keep equality across worlds • ( x = y )   ( x = y ) • Normal Model G,R,D,L • For each w  G, L(=,w) is the equality relation on the domain of the model DM • = is the same across worlds

38. Existence in Varying Domains • E(x) = (y y=x) • Fixing universal instantiation • x (x)  E(y)  (y) • When y exists, it has the property  • In classical logic E(y) is always true

39. x y x y x y x y x y x y x y x y x y x y x y x y Program Semantics as Kripke Structure y = x while (y != NULL && y->data != d) { y = y->n } … …

40. Finite State Programs • Propositional Kripke Structure • Possible worlds = global states • Accessibility relation = transition relation • May fold Kripke-Structure • Merge states that have same labeling • Result with abstraction of all computations • Check temporal properties over this model (rather than over infinite computations) • Results with what MC community calls Kripke Structure

41. Program Semantics in Modal Perspective • Propositional modal logic • propositions = properties of interest • Program global states = possible worlds • Transition relation = accessibility relation • Modal claims = properties of computations • First-Order modal logic • Global states are first-order logical structures • Closer to concrete semantic

42. Summary • Propositional modal logic • It is all about the accessibility relation • Generalizes other common logics • Wide range of applications in CS • First order modal logic • All about quantification domains • Constant/varying semantics models are equivalent • Should choose what’s more suitable for you • Closer to concrete program semantics • Currently no common applications in CS

43. References • M. Fitting, First Order Modal Logic • And his homepage http://comet.lehman.cuny.edu/fitting/ • G.E. Hughes and M.J. Cresswell, A New Introduction to Modal Logic • http://plato.stanford.edu/entries/logic-modal/ • Some theorem provers and tools • http://www.cs.man.ac.uk/~schmidt/tools/

44. The End http://www.cs.tau.ac.il/~yahave

45. A A A A A A A A A Traces • Linear paths of program execution • Program semantic = set of all program traces • A Kripke-structure is an abstraction of traces A A …

46. Common Wisdom • Assumption on program behavior = limit the set of traces considered • How? • Algorithmically --- e.g., Streett acceptance • Augmenting LTL property --- verify assumption goal • Disadvantages • assumption may be non-observable under abstraction • Pay more to express an assumed knowledge

47. Fairness Reduction Program • Advantages • Pay for verification of simplified properties • Simplified properties may be observable under abstraction even when original goal is not Simpler claims Modular Decomposition CFGDecomposition Progress Reduction GoalProperty Assumptions