Knowledge representation and reasoning
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Master of Science in Artificial Intelligence, 2009-2011. Knowledge Representation and Reasoning. University "Politehnica" of Bucharest Department of Computer Science Fall 2009 Adina Magda Florea http://turing.cs.pub.ro/krr_09 curs.cs.pub.ro. Lecture 4. Modal Logic Lecture outline

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Knowledge representation and reasoning

Master of Science in Artificial Intelligence, 2009-2011

Knowledge Representation and Reasoning

University "Politehnica" of Bucharest

Department of Computer Science

Fall 2009

Adina Magda Florea

http://turing.cs.pub.ro/krr_09

curs.cs.pub.ro


Lecture 4

Lecture 4

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

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

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

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

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

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 reasoning1

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

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

    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 logic1

    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

    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

    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 


    Semantics of modal logic1

    Semantics of modal logic

    • Using the model, we can define the semantics of formulae in modal logic and can compute the truth value of formulae.

    • M |=W  iff M |=/W  (or M |=W ¬)

    • M |=W   iff M |=W  and M |=W 

    • M |=W    iff M |=W  or M |=W 

    • M |=W  →  iff M |=W ¬ or M |=W 

      (¬   is true in W)

    • M |=W   iff w': R(w,w')  M |=W' 

    • M |=W   iff w': R(w,w')  M |=W' 


    Knowledge representation and reasoning

    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) = ?


    Knowledge representation and reasoning

    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

    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 systems1

    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 systems2

    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 systems3

    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 systems4

    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 systems5

    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 systems6

    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)


    Different modal logic systems7

    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

    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

    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 belief1

    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

    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 knowledge1

    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

    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

    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

    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

    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, )


    Knowledge representation and reasoning

    8. KB(WB)  WAcontrapositive 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

    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

    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 ctl1

    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


    Knowledge representation and reasoning

    LB- set of moment formula

    LS - set of path-formula

    Semantics

    M = <W, T, <, | |, R> - every tT has associated a world wtW

    M |=t  iff t||

     is true in the set of moments for which  holds

    M |=t pq iff M |=t p and M |=t q

    M |=t p iff M |=/t p

    M |=s,t pUq iff (t': tt' and M |=s,t' q and

    (t": t  t" t'  M |=s,t" p))

    p holds on a path s starting in the current moment t until q comes true

    Fp  true Up

    Gp  F p

    M |=tA p iff (s: sSt  M |=s,t p)Ep  A p

    s is a path, St - all paths starting at the present moment

    M |=s,tX p iff M |=s,t+1 p)

    38


    Knowledge representation and reasoning

    • 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 Franceq - It is spring time

    39


    Knowledge representation and reasoning

    • 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


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