Modal Logic CS 621 Seminar Group no.: 10 Kiran Sawant ( ) Joe Cheri Ross ( ) Sudha Bhingardive ( )

Modes of Truth Propositional logic is decidable but too restrictive. FOL and HOL have high expressivity but are not decidable. Modal logic extends PL to add expressivity without losing decidability. Consider the following: Either it rains or it does not rain. It may rain today. Dr. Manmohan Singh is Prime Minister of India. I believe that Ram believes that I know that he did it. The truth value of some of these sentences depends on the place, time and judgement of the person who uttered it.

What is Modal Logic? Study of modal propositions and logic relationships Modal propositions are propositions about what is necessarily the case and what is possibly the case Ex: It is possible for humans to travel to Mars It is necessary that either it is raining or it is not raining

Modal Operators: □ and ◊ □ is read as “necessarily” ◊ is read as “possibly” p : It will rain tomorrow □ p: It is necessary that it will rain tomorrow ◊ p: It is possible that it will rain tomorrow □ p ↔ ¬ ◊ ¬ p

Syntax The formulas of basic modal logic φ are defined by the following Backus Naur form (BNF): φ := p | ⊥ | ¬ φ | φ ∧ ψ | φ ∨ ψ | φ → ψ | φ ↔ ψ | □ φ | ◊ φ where "p" is any atomic formula Example: □p →□ □ p p ∧ ◊ (p → □ ¬r) □ (( ◊ q ∧ ¬r) → □ p)

Meanings of the Modal Operators □ ◊ Alethic logic p is necessarily true p is possibly true Deontic logicp is obligatory p is permitted Temporal logicp will always be true p will become true Sometime in future Epistemic logic p knows that P believes that

Semantics Kripke structures (possible worlds structures) are models of basic modal logic. A Kripke structure is a tuple M = (W,R,L) where  W is a non-empty set (possible Worlds)  R ⊆ WΧW is an accessibility relation (wRv) L : W →P, {true, false} is a labelling function

Example of Kripke Structure

Truth of Modal Formulas

Example of Kripke Structure

Inference Rules lUS – Rule of Uniform Substitution: The result of uniformly replacing any variables p1, …, pn in a theorem by any WFF φ1, …, φn respectively, is itself a theorem lMP – Modus Ponens lNR – Rule of Necessitation: If φ is a theorem, so is □ φ

Axioms and their Corresponding Properties on Accessibility Relations AxiomFormula SchemeProperty on R K□ (φ → ψ) → (□φ → □ψ ) T□φ → φReflexive Bφ → □◊φSymmetric D□φ → ◊φSerial 4□φ → □□φTransitive 5◊φ → □◊φEuclidean □φ ↔ ◊φFunctional Some modal logic systems take only a subset of this set All general, problem independent theorems can be derived from only these axioms and some additional, problem specific axioms describing the research problem

Which Formula Schemes Should Hold for these Readings of □? □φKTD45 It is necessarily true that φYYYYY It will always be true that φYY It ought to be that φYY Agent Q believes that φYYYY Agent Q knows that φYYYYY

Axiomatic Systems Systems: K := K + N T := K + T S4 := T + 4 S5 := S4 + 5 D := K + D

Example of Inference in Modal Logic Given: □ (p → q) and □ p Infer: □ q where, p: It rained.q: Grass is wet. 1. □ (p → q)[Given] 2. □ p[Given] 3. □ p → □ q[K, 1] 4. □ q[MP, 3 and 2]

Muddy Children Problem Statement Two children a and b coming to mother after playing Mother says “Atleast one of you has dirty forehead” She asks each one “Do you know whether your forehead is dirty ? “ If b says “yes”: a's forehead is not muddy If b says “no”: both foreheads are muddy

Muddy Children Kripke Structure (0 0) W1 W2 W3 (1 0) (0 1) W4 (1 1) (A,B) b b a a

Muddy Children Formalization A: a's forehead is dirty B: b's forehead is dirty Ki : Child i knows Initial: K a K b (A ∨ B) After first query: K a ¬K b B Final: K a A

Muddy Children Proof 1. K a K b (¬A → B) Premise (Mother said) 2. K a (K b ¬A → K b B) K- Axiom 3. K a ¬K b B → K a ¬K b ¬A (p→q)  (¬q → ¬p), K- Axiom 4. K a ¬K b B After 1st query 1. K a ¬ K b ¬ A 3,4- MP 6. K a (¬K b ¬A → K b A) Premise(Init) 7. K a K b A 5,6- Axiom K and MP 8. K a A 7- Axiom T

Conclusion Modal logic forms the basis for other kinds of logic. Modal logic extends the expressivity propositional logic. Modal logic is a non-numeric alternative to different logics like fuzzy logic, probabilistic logic, multiple-valued logic. Fuzzy logic operations on uncertainties derive uncertainties (better or worse), whereas in modal logic one can derive certainties from uncertainties. Relevant in various fields such as knowledge representation[6], linguistics[5], verification.

References 1.P. Blackburn, et. al., Modal Logic, Cambridge: Cambridge University Press, P. Blackburn, et. al., Handbook of Modal Logic, New York: Elsevier Science Inc, S. A. Kripke, "A Completeness Theorem in Modal Logic", The Journal of Symbolic Logic, vol. 24, no. 1, 1-14, Mar J. Doyle, "A Truth Maintenance System", Artificial Intelligence, vol. 12, no. 3, , L. S. Moss and H. Tiede, "Applications of Modal Logic in Linguistics", Elsevier Science. Linguistics, , R. Rosati, "Multi-modal Nonmonotonic Logics of Minimal Knowledge", Annals of Mathematics and Artificial Intelligence, vol. 48, no. 3-4, , Dec. 2006

BACKUP

Example 1:

Example 2:

Cards game: Kripke structure

Wise Men Puzzle Problem description 3 Wise men There are 3 Red hats and 2 white hats The King puts a hat on each of them and ask sequentially the color of their hat on their head 1 st man and 2 nd man say he doesn't know We have to prove whether 3 rd man now knows his hat is red

Wise Men Puzzle Solution Method Initially:- R R R R R W R W R R W W W R R W R W W W R WWW After 1 st man says he doesn't know:- R R R R R W R W R W R R W R W W W R R W W After 2 nd man says he doesn't know:- R R R R R W R W R W R R W R W W W R R W W Now 3 rd man knows that the hat he wears is Red

Wise Men Puzzle Initial Knowledge Pi means man i has red hat. ¬Pi means man i has white hat. Kj Pi means agent/man j knows that man i has a red hat. Let Γ be set of formulas:- {C(p1 ∨ p2 ∨ p3), C(p1 → K2 p1), C(¬p1 → K2 ¬p1), C(p1 → K3 p1), C(¬p1 → K3 ¬p1), C(p2 → K1 p2), C(¬p2 → K1 ¬p2), C(p2 → K3 p2), C(¬p2 → K3 ¬p2), C(p3 → K1 p3), C(¬p3 → K1 ¬p3), C(p3 → K2 p3), C(¬p3 → K2 ¬p3)}.

Wise Men Puzzle Formalisation Naive approach Γ, C(¬K1 p1 ∧ ¬K1 ¬p1), C(¬K2 p2 ∧ ¬K2 ¬p2) |− K3 p3 This doesn't capture time between events (2 nd man answers after 1 st ) To formalise correctly this has to be broken into 2 entailments, corresponding to each announcement

Wise Men Puzzle Correct Formalisation 1. Γ, C(¬K1 p1 ∧ ¬K1 ¬p1) |− C(p2 ∨ p3). 2. Γ, C(p2 ∨ p3), C(¬K2 p2, ∧ ¬K2 ¬p2) |− K3 p3.

Wise Men Puzzle Proof of Entailment 1 1 C(p1 ∨ p2 ∨ p3) premise 2 C(pi → Kj pi) premise, (i= j) 3 C(¬pi → Kj ¬pi) premise, (i = j) 4 C¬K1 p1 premise 5 C¬K1 ¬p1 premise 6 C 7 ¬p2 ∧ ¬p3 assumption 8 ¬p2 → K1 ¬p2 Ce 3 (i, j) = (2, 1) 9 ¬p3 → K1 ¬p3 Ce 3 (i, j) = (3, 1) 10 K1 ¬p2 ∧ K1 ¬p3 prop 7, 8, 9 11 K1 ¬p2 ∧ e K1 ¬p3 ∧ e2 10

13 K1 14 ¬p2 K1e ¬p3 K1e ¬p2 ∧ ¬p3 ∧ i 14, p1 ∨ p2 ∨ p3 Ce 1 18 p1 prop 16, K1 p1 K1i 13−18 20 ¬K1 p1 Ce 4 21 ⊥ ¬e 19, ¬(¬p2 ∧ ¬p3) ¬i 7−21 23 p2 ∨ p3 prop C(p2 ∨ p3) Ci 6−23

Wise Men Puzzle Proof of Entailment 2 1 C(p1 ∨ p2 ∨ p3) premise 2 C(pi → Kj pi) premise, (i = j) 3 C(¬pi → Kj ¬pi) premise, (i = j) 4 C¬K2 p2 premise 5 C¬K2 ¬p2 premise 6 C(p2 ∨ p3) premise 7 K3 8 ¬p3 assumption 9 ¬p3 → K2 ¬p3 CK 3 (i, j) = (3, 2) 10 K2 ¬p3 →e 9, 8

11 K2 12 ¬p3 K2e p2 ∨ p3 Ce 6 14 p2 prop 12, K2 p2 K2i 11−14 16 Ki ¬K2 p2 CK 4, for each i 17 ¬K2 p2 KT ⊥ ¬e 15, p3 PBC 8−18 20 K3 p3 K3i 7−19