Knowledge Representation and Reasoning

Master of Science in Artificial Intelligence, 2010-2012 Knowledge Representation and Reasoning University "Politehnica" of Bucharest Department of Computer Science Fall 2009 Adina Magda Florea http://turing.cs.pub.ro/krr_10 curs.cs.pub.ro

Lecture 6 Modal Logic Lecture outline • Introduction • Modal logic in CS • Syntax of modal logic • Semantics of modal logic • Logics of knowledge and belief • Temporal logics

1. Introduction • In first order logic a formula is either true or false in any model • In natural language, we distinguish between various “modes of truth”, e.g, “known to be true”, “believed to be true”, “necessarily true”, “true in the future” • “Barack Obama is the president of the US” is currently true but it will not be true at some point in the future. • “After program P is executed, A hold” is possibly true if the program performs what is intended to perform.

History • Classical logic is truth-functional = truth value of a formula is determined by the truth value(s) of its subformula(e) via truth tables for ,, ¬, and →. • Lewis tried to capture a non-truth-functional notion of “A Necessarily Implies B” (A → B) • We can take A → B to mean “it is impossible for A to be true and B to be false” • He chose a symbol, P, and wrote PA for “A is possible”; then: • ¬PA is “A is impossible” • ¬P¬A is “not-A is impossible” • Then he used the symbol N to stand for ¬P and expressed • NA := ¬P¬A “A is necessary” • Because → is logical implication, we can transform it like this: • A → B := N(A → B) = ¬P¬(A → B) = ¬P¬(¬A  B) = ¬P(A  ¬B)

Modal operators • P - “possibly true” • N - “necessarily true” • Modal logics - modes of truth:  • Basic modal logic:  - box, and  - diamond • The necessity / possibility - necessary, and  - possible • Logics about knowledge - what an agent knows / believes • Deontic logic -  - it is obligatory that, and  - it is permissible that

2. Modal logic in CS • Temporal logic • Dynamic logic • Logic of knowledge and belief • Model problems and complex reasoning The Lady and the Tiger Puzzle • There are two rooms, A and B, with the following signs on them: • A: In this room there is a lady, and in the other room there is a tiger” • B: “In one of these rooms there is a lady and in one of them there is a tiger” • One of the two signs is true and the other one is false. Q: Behind which door is the lady?

Modeling modal reasoning The King's Wise Men Puzzle • The King called the three wisest men in the country. • He painted a spot on each of their foreheads and told them that at least one of them has a white spot on his forehead. • The first wise man said: “I do not know whether I have a white spot” • The second man then says “I also do not know whether I have a white spot”. • The third man says then “I know I have a white spot on my forehead”. Q: How did the third wise man reason?

Modeling modal reasoning Mr. S. and Mr. P Puzzle • Two numbers m and n are chosen such that 2  m  n  99. • Mr. S is told their sum and Mr. P is told their product. • Mr. P: "I don't know the numbers. " • Mr. S: "I knew you didn't know. I don't know either." • Mr. P: "Now I know the numbers." • Mr S: "Now I know them too." Q: In view of the above dialogue, what are the numbers?

3. Modal logic - Syntax • Atomic formulae: p ::= p0 | p1 | p2 | q …. where pi , q are atoms in PL • Formulae: ::= p | ¬ |   |   |    |    |  →  where  and  are a wffs in PL Examples: • p → q • p → q •  (p1 → p2) → ((p1) → (p2)) • Schema: •  →  •  →   • ( →  ) → ( →  ) • Schema Instances: Uniformly replace the formula variables with formulae (inference) Examples: • p → p is an instance of  →  but • p → q is not

Deduction in modal logic • Axioms The 3 axioms of PL • A1.  () • A2. ( ( ))  (( )  ( )) • A3. ((¬)  (¬))  ( ) The axiom to specify distribution of necessity • A4. ( )  (    ) Distribution of modality

Deduction in modal logic • Inference rules • Substitution (uniform)   ’ • Modus Ponens , (  )   • The modal rule of necessity |-    • « for any formula , if  was proved then we can infer  »

4. Semantics of modal logic • Nonlinear model • The semantics of modal logic is known as the Kripke Semantics, also called the Possible World approach Directed graph (V, E) • Vertices V = {v, v1, v2, …} • Directed edges {(s1,t1), (s2,t2),…} from source vertex si V to the target vertex tiV for i = 1,2,… Cross product of a set V, V x V • {(v,w) | vV and wV} the set of all ordered pairs (v,w), where v and w are from V. Directed graph - a pair (V,E), where V = {v, v1, v2, …} and E  V x V is a binary relation over V.

Semantics of modal logic • A Kipke frame is a directed graph <W, R>, where: • W is a non-empty set of worlds (points, vertices) and • R  W x W is a binary relation over W, called the accessibility relation. • An interpretation of a wff in modal logic on a Kripke frame <W, R> is a function I : W x L → {t,f} which tells the truth value of every atomic formula from the language L at every point (in every word) in W. • A Kripke model M of a formula  (an interpretation which makes the formula true) is • the triple <W, R, I>, where I is an interpretation of the formula on a Kripke frame <W,R> which makes the formula true. • This is denoted by M |=W 

W1 I(W1,p) = f I(W1,q) = f I(W1,r) = a W0 I(W0,p) = f I(W0,q) = f I(W0,r) = f W2 I(W2,p) = f I(W2,q) = f I(W2,r) = f Examples p – I am rich q – I am president of Romania r – I am holding a PhD in CS I(W0, p) = ? I(W0, p) = ? I(W0, r) = ? I(W0, r) = ?

w1 p, q, r w2 p, q, r w0 p, q, r w3 p, q, r Examples p -Alice visits Paris q - It is spring time r - Alice is in Italy I(W0, p) = ? I(W0, p) = ? I(W0, q) = ? I(W0, q) = ? I(W0, r) = ? I(W0, r) = ? I(W1, p) = ? I(W1, p) = ?

Different modal logic systems The modal logic K • A1.  () • A2. ( ( ))  (( )  ( )) • A3. ((¬)  (¬))  ( ) • A4. ( )  (    ) • X  X • Here is an invalidating model: R(w0,w1), I(w0,p)=f, I(w1,p)=t “it is impossible for A to be true and B to be false” M |=W   iff w': R(w,w')  M |=W' 

Different modal logic systems The modal logic D Add axiom • X X • In fact, D-models are K-models that meet an additional restriction: the accessibility relation must be serial. • A relation R on W is serialiff • (wW: (w'W: (w,w')R))

Different modal logic systems The modal logic T Add axiom • X  X • A T-model is a K-model whose accessibility relation is reflexive. • A relation R on W is reflexiveiff • (wW: (w,w)R).

Different modal logic systems The modal logic S4 Add axiom • X  X • An S4-model is a K-model whose accessibility relation is reflexiveand transitive. • A relation R on W is transitiveiff • (w1,w2,w3 wW: (w1,w2)R  (w2, w3)R  (w1,w3)R).

Different modal logic systems The modal logic B Add axiom • X  X • A B-model is a K-model whose accessibility relation isreflexive and symmetric. • A relation R on W is symmetriciff • (w1,w2W: (w1,w2)R  (w2,w1)R)

Different modal logic systems The modal logic S5 Add the axiom • X   X • An S5-model is a K-model whose accessibility relation is reflexive, symmetric, and transitive. • That is, it is an equivalence relation • Exercise: Find an S5-model in which X  X is false. S5 is the system obtained if every possible world is possible relative to every other world

Different modal logic systems The modal logic S5 • X   X • A relation is euclidian iff (w1,w2,w3W: (w1,w2)R  (w1, w3)R  (w2,w3)R)

reflexive Different modal logic systems D = K + D T = K + T S4 = T + 4 B = T + B S5 = S4 + B S5 symmetric transitive S4 B transitive symmetric T D reflexive serial K

5. Logics of knowledge and belief • Used to model "modes of truth" of cognitive agents • Distributed modalities • Cognitive agents  characterise an intelligent agent using symbolic representations and mentalistic notions: • knowledge - John knows humans are mortal • beliefs - John took his umbrella because he believed it was going to rain • desires, goals - John wants to possess a PhD • intentions - John intends to work hard in order to have a PhD • commitments - John will not stop working until getting his PhD

Logics of knowledge and belief • How to represent knowledge and beliefs of agents? • FOPL augmented with two modal operators K and B K(a,) - a knows  B(a,) - a believes  with LFOPL, aA, set of agents • Associate with each agent a set of possible worlds • Kripke model Ma of agent a for a formula  • Ma =<W, R, I> with R  A x W X W and I - interpretation of the formula on a Kripke frame <W,R> which makes the formula true for agent a

Logics of knowledge and belief • An agent knows a propositions in a given world if the proposition holds in all worlds accessible to the agent from the given world Ma |=WK iff w': R(w,w')  Ma |=W'  • An agent believes a propositions in a given world if the proposition holds in all worlds accessible to the agent from the given world Ma |=WB iff w': R(w,w')  Ma |=W'  • The difference between B and K is given by their properties

Properties of knowledge (A1) Distribution axiom: K(a, ) K(a,   )  K(a, ) "The agent ought to be able to reason with its knowledge" ( )  (  )(Axiom of distribution of modality) K(a, )  ( K(a,)  K(a,) ) (A2) Knowledge axiom: K(a, )   "The agent can not know something that is false"  (T) - satisfied if R is reflexive K(a, )  

Properties of knowledge (A3) Positive introspection axiom K(a, )  K(a, K(a, )) X  X (S4) - satisfied if R is transitive K(a, )  K(a, K(a, )) (A4) Negative introspection axiom K(a, )  K(a, K(a, )) X   X (S5) - satisfied if R is euclidian

Inference rules for knowledge (R1) Epistemic necessitation |-  K(a, ) modal rule of necessity |-    (R2) Logical omniscience    and K(a, ) K(a, ) problematic

Properties of belief Distribution axiom: B(a, ) B(a,   )  B(a, ) YES Knowledge axiom: B(a, )   NO Positive introspection axiom B(a, )  B(a, B(a, )) YES Negative introspection axiom B(a, )  B(a, B(a, )) problematic

Inference rules for belief (R1) Epistemic necessitation |-  B(a, ) problematic modal rule of necessity |-    (R2) Logical omniscience    and B(a, ) B(a, ) usually NO

Some more axioms for beliefs Knowing what you believe B(a, ) K(a, B(a, )) Believing what you know K(a, ) B(a, ) Have confidence in the belief of another agent B(a1, B(a2,)) B(a1, )

8. KB(WB)  WA contrapositive of 7 9. KA(WA) 3, 8, R2 R2:    and K(a, ) inferK(a, ) Two-wise men problem - Genesereth, Nilsson (1) A and B know that each can see the other's forehead. Thus, for example: (1a) If A does not have a white spot, B will know that A does not have a white spot (1b) A knows (1a) (2) A and B each know that at least one of them have a white spot, and they each know that the other knows that. In particular (2a) A knows that B knows that either A or B has a white spot (3) B says that he does not know whether he has a white spot, and A thereby knows that B does not know 1. KA(WA  KB( WA) (1b) 2. KA(KB(WA WB)) (2a) 3. KA(KB(WB)) (3) Proof 4. WA  KB(WA) 1, A2A2: K(a, )   5. KB(WA  WB) 2, A2 6. KB(WA)  KB(WB) 5, A1A1: K(a, )  (K(a,)  K(a,)) 7. WA  KB(WB) 4, 6 34

6. Temporal logic • The time may be linear or branching; the branching can be in the past, in the future of both • Time is viewed as a set of moments with a strict partial order, <, which denotes temporal precedence. • Every moment is associated with a possible state of the world, identified by the propositions that hold at that moment Modal operators of temporal logic (linear) p U q - p is true until q becomes true - until Xp - p is true in the next moment - next Pp - p was true in a past moment - past Fp - p will eventually be true in the future - eventually Gp - p will always be true in the future – always Fp  true U p Gp  F p F – one time point G – each time point

Branching time logic - CTL • Temporal structure with a branching time future and a single past - time tree • CTL – Computational Tree Logic • In a branching logic of time, a path at a given moment is any maximal set of moments containing the given moment and all the moments in the future along some particular branch of < • Situation - a world w at a particular time point t, wt • State formulas - evaluated at a specific time point in a time tree • Path formulas - evaluated over a specific path in a time tree

Branching time logic - CTL CTL Modal operators over both state and path formulas From Temporal logic (linear) Fp - p will sometime be true in the future - eventually Gp - p will always be true in the future - always Xp - p is true in the next moment - next p U q - p is true until q becomes true - until (p holds on a path s starting in the current moment t until q comes true) Modal operators over path formulas(branching) Ap - at a particular time moment, p is true in all paths emanating from that point - inevitable p Ep - at a particular time moment, p is true in some path emanating from that point - optional p F – one time point G – each time point A – all path E – some path

s is true in each time point (G) and in all path (A) • r is true in each time point (G) in some path (E) • p will eventually (F) be true in some path (E) • q will eventually (F) be true in all path (A) s p s q F - eventually G - always A - inevitable E - optional AGs EGr EFp AFq r s r s r s q s q s r - Alice is in Italy p -Alice visits Paris s – Paris is the capital of France q - It is spring time 39

Each situation has associated a set of accessible words - the worlds the agent believes to be possible. Each such world is a time tree. • Within these worlds, the branching future represents the choices (options) available to the agent in selecting which action to perform • Similar to a decision tree in a game of chance Decision nodes Player 1 Dice • Each arc emanating from • a chance node corresponds • to a possible world Player 2 1/18 1/36 Chance nodes Dice • Each arc emanating from • a decision node corresponds • to a choice available in a • possible world Player 1 1/36 1/18 40

Knowledge Representation and Reasoning