Slide-1

Modal Logic and Its applications

Cheng-Chia Chen

Department of Computer Science, National Cheng-Chi University

Slide-2

Contents

• Classical propositional logic (CPL)

• Basic modal logic

• logic of knowledge and belief

• deontic logic

• logic of actions and programs(PDL)

Slide-3

Elements of a Logic

• Language • syntax (formal language)• semantics (model theory)• axiomatics (proof theory)• decidability & complexity (computation theory)• automated deduction (Theorem proving)

Slide-4

Classical Propositional Logic(CPL)

• The language L:– a set of proposition symbols (PV) :– p,q, r ... means it-is-raining, it-is-cloudy, ...

• logical connectives: /\ (and), ~ (negation)• (well-formed) formulas (abstract syntax):

P ::= p | P /\ Q | ~P • Definitions:

P \/ Q abbreviates ~(~P /\ ~Q)

P => Q abbreviates ~(P /\ ~Q)

Slide-5

The semantics for CPL

• Goals:– 1. define the contexts in which formulas can be

given truth values.– 2. define the truth conditions for formulas.

• interpretation (world, state): any assignment of truth value {1,0} to propositional symbols

• Truth conditions (or satisfaction relation) |= :• I |= p iff I(p)=T;• I |= P /\ Q iff I |= P and I |= Q• I |= ~P iff not I |= P

• If I |= A, then say I is a model of A.

Slide-6

Some logical notions

• A formula is satisfiable iff it is true in some world.• A formula is valid (a tautology) (|= A) if it is true in

all worlds.• A is a logical consequence of a set of formulas S

(S |= A) iff A is true in all models of S.

• Problems : How to characterize the set

{A | A is a tautology} ?

Slide-7

Calculus and provability

• A calculus C over a language L is a finite set of rules, each of the form:– (A1,A2, ..., An, B)– A1,A2,...,An : Premises – B: conclusion– if n = 0 => axioms

• Example: (A, B, A /\B), (A, A=>B, B),

(A=>B, B, A),...

Slide-8

Provability

• Given a calculus C,

• The set C = {A | A is C-provable(denoted |-C A)}

is defined recursively as follows:– Basis:If (A) is a rule, then A in C ---axioms– Ind: If (A1,..,An,B) is a rule &– A1,...,An in C, then B in C.

Slide-9

An axomatization for CPL

• Let CPL be the calculus:

(1) Axiom schema:– A => (B => A)– (A=>(B =>C)) => ((A=>B)=>(A=>C))– (~A => ~B) => (B => A)

(2) Inference rule:– from A and A => B infer B (MP)

• Theorem: A is valid in CPL iff A is CPL-provable

Slide-10

Basic Modal logic

• The logical study of necessity and possibility• The language:

– CPL augmented with two modal operators: [] (necessity) and ⃟ (possibility).

– P : any proposition , then []P (<>P) means “P is necessarily (possibly)

true”.– Meaning of []p:

• depends on the context it is used, not only determined by the truth value of p

• A family of logics instead of a single logic

Slide-11

Types of necessity

• logical necessity:– e.g, p \/ ~p is logically necessarily true.

• physical necessity:– F=ma

• Epistemic necessity:– e.g., It is believed(known) that ...

• Normal necessity:– e.g., It is obligated (permitted, forbidden) that ...

• time-related (always, eventual)• Others:

– After the programs terminates P must holds,...

Slide-12

Formal Definition

• The language:– Alphabet ():

• PV: a set of propositional variables.• logical connectives: ~ (not), /\ (and), [] (necessity)

– MF: a set of modal formulas defined inductively: • A ::= p | A /\ B | ~ A | []A

– Abbreviations (Macros)• (A \/ B) abbreviates ~(~A /\ ~B);• (A B) abbreviates ~(A /\ ~B)• ⃟ A abbreviates ~[]~A

Slide-13

Possible-world Semantics for modal logic

• Truth conditions for p /\ q, p \/ q, p q, and ~p .– Let p = “I win the game”,– q = “It is 5 p.m.”– Assume I win the game and – the present time is 3 p.m,– then p/\q: false, p\/q: true and pq: false.

• But how about the statement:

[]p =It must be the case that I win the game. “

Slide-14

Meaning of necessity and possibility:

• The game:– Two players A,B, each getting a card from four

cards labeled 1,2,3,4 randomly. • rule:

– The player who get a card larger than the other’s wins.

Slide-15

Scenario I: A gets “2”.

• Then consider the following sentences:– 1. “A may possibly win”

• = “It is possibly true that A win” = “ ⃟A_win”

– 2. “A may possibly not win”– 3. “A must win”– 4. “B must not get “2””

• Which is right ? why?

Slide-16

The answer:

• Statement 1 is right– since (2,1) may be the real world, in which A

wins.• Statement 2 is right

– since (2,3), (2,4) are possible, in which A does not win.

• statement 3 is false – since there are cases (e.g., (2,3), (2,4)) in which

A does not win.• Statement 4 is true since in all possible cases B

does not get 2.

Slide-17

The Rule:

(2,1)A_win~B_2

(2,3)~A_win~B_2

(2,4)~A_win~B_2

Possible worlds

Impossible worlds

(2,?)

Real world

~[]A_win⃟ A_win⃟ ~A_win[] ~B_2

(3,4)

Slide-18

The Possible-world Semantics:

• Let W = the set of worlds

– e.g, {(x,y) | x = 1..4, y =1..4 & x y}

• Let V : W x PV -> {0,1} be a valuation function

s.t., V(w,p) =1 iff p is assigned true at world w.

– e.g, V((2,1), A-win) = 1

• R be a binary relation (I.e., subset of WxW) s.t.

wRw’ iff w’ is a possible world of w.

– e.g, (2,x)R(2,1), (2,x)R(2,3), (2,x)R(2,4).

• The triple M=<W,R,V> is called a (possible-world) structure.

Slide-19

Truth-conditions for modal formulas

M = <W,R,V>: a possible world structure; w: a world W, ∈

• The statement : “A is true at world w in structure M” is defined as follows:– M,w |= p iff V(w,p) = 1– M,w |= A /\ B iff M,w |= A and M,w |= B– M,w |= ~A iff not M,w |= A.– M,w |= ⃟ A iff – A is true at some possible world of w.– M,w |= [] A iff A is true at all possible worlds of w.

Slide-20

Some definitions

• A: modal formula, M: structure,• C: a class of structures• A is valid iff it is true in all worlds of all structures.• A is C-valid iff it is true at all worlds of all structur

es of C.• Problem: Given a class of structures C,

– {A | A is C-valid } = ?

Slide-21

Interesting classes of structures

• Class name Property of R • T reflexive: wRw.• D serial: for all w, there is w’ s.t. w R w’.• 4 transitive: wRw’ & w’Rw’’ ⇒ wRw’’.• 5 Eulidean: wRw’ & wRw’’ w’ R w’’.⇒• B symmetric: wRw’ w’Rw.⇒

• r: any string from {T,D,4,5,B} without repetition.• Kr = the class of the structures whose R satisfying all proper

ties mentioned in r.– (I.e., Every theorem of the logic Kr is valid in all Kr-strutur

e, and vice versa.)

Slide-22

Axiomatization of modal logics

• Axioms definitions • PC all truth-functional tautologies• K [](PQ) ([]P []Q)• T []P P• D []P ~[]~p• 4 []P [][]P• 5 ~[]P []~[]P• B ~P []~[]P.• Inference rule: MP: from P, P Q infer Q

Nec: from P infer []P

Slide-23

Axiomatizations of modal logic

• r: any subset {T,D,4,5,B}.• Kr = the axiom system (calculus) including axioms

K, PC and all of r and inference rules MP and Nec.• Kr-provable formulas are defined recursively as foll

ows: – 1. Every axioms of Kr is Kr-provable.– 2. If P, P Q are Kr-provable then so is Q (MP)– 3. If P is Kr-provable, then so is []P (Nec).

• Theorem[Chellas80]: – A is Kr-valid iff A is Kr-provable.

Slide-24

Some useful modal logics

• Logical system Property of R usage• S5 (KT45) equivalence logic of knowledge• KD serial deontic logic• KD45 almost equ. logic of belief• S4 (KT4) ref. tran. Intuitionistic logic• S4.3 linear(total) temporal logic

w

w

real world must be possiblereal world may and may not be possible

Worlds inside are fully connected

{w’ | w R w’}

Slide-25

Logic of Knowledge and Belief

• Modal logic of knowledge : KT45(S5)• Modal logic of belief: KD45( weak S5)• Epsitemic interpretation of knowledge&belief axioms

– KA means A is known; BA means A is believed.– T: []A A (knowledge axioms)– D: []A ~[]~A (belief axiom)– 4: []A [][] A (positive introspection)– 5:~[]A []~[]A (negative introspection)– K:[]A /\ [](A B) []B (distribution axiom)– Nec: From p infer []p -- agent knows the logic

Slide-26

Extensions to multimodal logics:

– S5 (KD45) can model only one single agent’s knowledge (believes)

– Multi-agent cases: n agents: 1,2,3,...,n;• 2n knowledge(and belief) operators K1,B1,...,Kn,Bn:

• KiA ( BiA ) means agent i knows(resp. believes) A.

– Resulting logic: S5nWS5n • N copies of S5, and N copies of KD45,

each for one agent.e.g., Tj: KjAA where j =1,..,n.

– semantics: Structure M=<W,{Ki,Bi}i=1..n, V> • Each Ki is an equivalence relation on W and Bi is a se

rial,trans. and euclidean relation.

Slide-27

Related Issues[Halpern85]

• Logical Omniscience Problem:• Agents with S5 (KD45) ability are perfect logical reasoners,

but human never be.

• Common knowledge, Distributed knowledge– [E]P = [1]P /\ [2]P.../\[n]P – [C]P = [E]P /\[E][E]P /\ [E][E][E]P /\ ...

= [E]P /\[E][C]P– [D]P = P can be known by an agent who knows all w

hat others known (the wisest man).– Needed and useful in many fields (Economics,distrib

uting sys,AI ...)

Slide-28

Deontic interpretation of modal logic

• Deontic logic (D or KD)– PA means A is permitted; OA means A is obligat

ed; FA means A is forbidden.– A is (strongly) forbidden =

• Doing A or bringing about A will result in punishment (dangerous, disastrous) worlds.

– A is obligated = not doing A or not bring about A will result in punishment. = ~A is forbidden.

– A is (weekly) permitted = A is not forbidden = doing A may not result in punishment.

– Another possible pairs:– weekly forbidden/strongly permitted

Slide-29

Semantic analysis of forbidden, obligation and permission

commit-crime or dead (undesired world)

~drive-carmurder~pay-tax~pay-tax~dead

drive-car~dead pay-tax~ murder

drive-carmurderpay-taxdead

~drive-carpay-tax~murder~dead

~drive-car~pay-tax~pay-taxdead~murder

Permitted worlds

current world

sets of worlds which may become the real world

F murder : since all murder-worlds are red.O pay-tax: since all ~pay-tax world are red.P drive-car: some drive-car-world is white.

Slide-30

Formalization of Deontic logic

• W: The set of all possible worlds• D: A set of undesired, punishment world• V: WXPV -> {0,1} with the constraint that

– V(w,v) = 1 iff w D. ∈• I.e., we use v to denote all sanction or punishment wor

lds.

• R: a binary relation on W, s.t.– wRw’ means w’ is a possible world that the agent

may choose to become the real world from w.

Slide-31

Truth conditions for PA,OA, &FA

– M,w |= FA iff M,w |= [] (Av)• ie., for all w’, if wRw’ & M,w|=A then M,w |= v.

– M,w |= OA iff M,w |= F~A iff M,w |= [](~A v)

– M,w |= PA iff M,w |=~FA iff M,w |= ⃟(A/\ ~v)• I.e., there is a world w’ s.t. wRw’ & M,w |= A /\ ~v.

Slide-32

Properties of the deontic logic:

• By definition:– FA = [] (A v) ;– OA = F~A = [](~A v); – PA = ~FA = ⃟ (A /\ ~v);

• All KD axioms(K, D)• Desirable property: OA => PA: not valid in K but

valid in KD (I.e., R must be serial)

Slide-33

Temporal interpretation of modal logic

real historynow

possible futurereal past

possible past

real future

Taxonomy of temporal structures: • linear v.s. branch-time,• past time v.s. future time v.s. past&future• continuous v.s. discrete

Slide-34

Linear discrete time temporal logic

• Temporal operators:– FA means A is eventually true– GA means A is always true– A U B means A is true until B becomes true– 0A: A is true at the next time.

Slide-35

Meaning of temporal formulas

0 1 2 3 ..... n n+1 m

Fp p Gq q q q q .... q..... q 0r rAUBA A A A B

•Linear discrete-time temporal structure:

initial world

Slide-36

Meaning of temporal formulas

• linear discrete temporal logic:• W = N = {0,1,2,3,...} :time point set• V:NXPV -> {0,1}• Truth conditions:

– M,n |= 0A iff M,n+1 |= A.– M,n |= FA iff there is m n s.t., M,m |= A– M,n |= GA iff for all m n, M,m |= A.– M,n |= A U B iff there is m n s.t., M,m|= B &

for all m > s n, M,s |= A.

Slide-37

Logic of programs and actions

• Modal logic of programs (Dynamic Logic)• PDL: propositional version of DL• The language:

– Primitive programs: a,b,c,...– Primitive propositions: p,q,r... – program constructs: “ ;”, “|”,”*”,”?”. – logic connectives: /\,~, [A] for each program A.

Slide-38

– (Compound) Programs A ::= • a | any primitive program is a program (x++ in C)

• A;B | doing A and then doing B• A+B | doing A or doing B nondeterministically• A* | iterate A a nondeterminstic number of times• A* = t + A + A;A + A;A;A + ...• P? | test if P is true.

Syntax of Programs

Slide-39

Syntax of Formulas

– Formulas(assertions): P ::=– p any primitive proposition is a formula– P /\ Q both P and Q are true– ~P P is not true– [A]P After A terminates, P will be true.– <A>P = ~[A]~P means P holds at some

execution of A.

Slide-40

An Example:

• integer x,y,z– x := 3 ; – y := (1,4); – z := x+1 | y := x

• Problems: – Is it true that z > 0 or y x-2 after executing the

program, suppose initially the program state is (4,3,2) ?

Slide-41

Formalization of the problem:

• two primitive propositions:– p = “z > 0” ; q = “z x-2”

• four primitive programs:– a = “x := 3”, b = “y :=(1,4)”,– c = “z := x+1” , d = “y := x”.

• The program : A = a;b; (c | d)• The problem: is [A] (p \/ q) true ?

Slide-42

Analysis:

• A program state is triple (I,j,k) of integers,– which denote the possible simultaneous values

of variables (x,y,z).• Let W = {(i,j,k) | i,j,k are integers} be the set of all

possible program states.

Slide-43

a = “x := 3”, b = “y :=(1,4)”, c = “z := x+1” , d = “y := x”.

p = “z > 3” , q = “z >= x+1”

(3,1,4)

(3,3,2)

(3,4,2)

(3,1,2)(4,3,2)a b

b

(3,3,2)

(3,3,2)

(3,4,4)

c

c

d

d

p

~p

p

~p

q

~q

q

~q

p\/q

~(p\/q)

p\/q

~(p\/q)

a;b

c+d

a;b;(c+d)

initial programstate

Slide-44

(i,j,k)

(3,j,k)

a: x:=3

(i,1,k)

(i,j,i+1)

b: y:=(1,4)

(i,4,k)

(i,4,k)

b

d: y := x

c: z:= x+1

Slide-45

The Semantic rules

• 0. Let W = the set of all possible program states• 1. Each primitive proposition has a truth value in a pr

ogram state: – denoted by a function: V: W x PV {1,0} s.t.– V(w,p) = 1 iff p is true at state w.

• 2. Each primitive program a is a state transformer, denoted by a binary relation R(a): WxW s.t.,

• w R(a) w’ means the program state can become w’ from w by executing a.

• M=<W,R,V> is called a (program) structure.

Slide-46

Composition rule for programs:

• R(A;B) = R(A)R(B) = {(w,w’’) | there is w’ s.t., w R w’ and w R w’’.

• R(A+B) = R(A) U R(B);• R(A)* = I UR(A) UR(A)R(A) U ...

= R(A)* I.e., ref. and trans closure of R(A).• R(P?) = {(w,w) | P is true at w}.• Define classical program constructs:

– if P then A else Bif P then A else B P?;A + ~P?;B– while P do Awhile P do A (P?;A)* ; (~P?)– Repeat A until PRepeat A until P A;(~P?;A)*;P?

Slide-47

Truth conditions for Formulas

– M,w |= p iff V(w,p)=1– M,w |= P /\ Q iff M,w|=P and M,w|=Q.– M,w|=~P iff not M,w|=P.– M,w|= [A]P iff for all w’, w RA w’ then M,w’|=P.– M,w|=<A>P iff there is w s.t. wRAw’ & M,w’|=p.

• A formula is valid iff it is true at every world of every program structure.

• A formula is satisfiable if it is true at some world of some program structure.

• Subsume Hoare logic: P {A} Q (P [A] Q)

Slide-48

Variants of PDL [Harel84]

• DPDL – atomic programs are deterministic

• SPDL (structure PDL)– remove + and *– add “if then else” and “while do”.

• SDPDL (structure DPDL):– atomic programs are deterministic– replace + and * by “if then else” and “while do”.

Slide-49

PDL as a logic of actions

• Too strong part:– The *-operator may not be necessary– The +-operator is not very natural

• Too weak part:– need a notion of not doing something

• (I.e., A: an action => -A : an action (not doing A)

– need a notion of concurrent/parallel execution of actions. A,B: actions =>

• A&B means (doing A and B in parallel))• A \/ B means A;B + B;A + A&B

– Need internal free choice: A B

Slide-50

Axiomatize PDL

• The following formulas are valid in PDL

1. CPL: all tautologies of propositonal logic

2. K: [A](PQ) /\ [A]P [A]Q

3. cmp: [A;B]P <-> [A][B]P

4. union: [A+B]P <->([A]P /\ [B]P)

5. test: [P?]Q <-> (PQ)

6. mix: [A*]P -> (P /\[A]P /\ [A][A]P /\ …)

∴ [A*]P -> (P /\ [A][A*]P)

7. induction: (P /\ [A*](P [A]P)) [A*]P

Slide-51

PDL

• Valid inference rules in PDL:– MP: From P and P Q infer Q– Gen: From P infer [A]P

• TheoremTheorem::– 1. P is valid in PDL iff P can be proved from the

above calculus. (In symbols, |=PDLP|-PDLP)

– 2. The set {A | A is a valid in PDL} is EXPTIME-complete