Knowledge Representation and Reasoning

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Knowledge Representation and Reasoning
Master of Science in Artificial Intelligence,
2009-2011

• University "Politehnica" of Bucharest
• Department of Computer Science
• Fall 2009
• Adina Magda Florea
• http//turing.cs.pub.ro/krr_09
• curs.cs.pub.ro

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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

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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.

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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
• PA is not-A is impossible
• Then he used the symbol N to stand for P and
  expressed
• NA PA 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)

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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

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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?

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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?

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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?

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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

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Deduction in modal logic

• Axioms
• The 3 axioms of PL
• A1. ? ? (? ? ?)
• A2. (? ? (? ? ?)) ? ((? ? ?) ? (? ? ?))
• A3. ((?) ? (?)) ? (? ? ?)
• The axiom to specify distribution of necessity
• A4. ?(? ??) ? (? ? ? ? ?) Distribution of
  modality

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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 ?? 

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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.

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Semantics of modal logic

• A Kipke frame is a directed graph ltW, Rgt, 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 ltW, Rgt 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 ltW, R, Igt, where I is an
  interpretation of the formula on a Kripke frame
  ltW,Rgt which makes the formula true.
• This is denoted by M W ?

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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' ?

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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) ?
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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) ?
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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' ?
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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 serial iff
• (?w?W (?w'?W (w,w')?R))

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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 reflexive iff
• (?w?W (w,w)?R).

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Different modal logic systems

• The modal logic S4
• Add axiom
• ?X ? ??X
• An S4-model is a K-model whose accessibility
  relation is reflexive and transitive.
• A relation R on W is transitive iff
• (?w1,w2,w3 w?W
• (w1,w2)?R ? (w2, w3)?R ? (w1,w3)?R).

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Different modal logic systems

• The modal logic B
• Add axiom
• X ? ??X
• A B-model is a K-model whose accessibility
  relation is reflexive and symmetric.
• A relation R on W is symmetric iff
• (?w1,w2?W (w1,w2)?R ? (w2,w1)?R)

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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
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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)

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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
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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

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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 ltW, R, Igt
• with R ? A x W X W
• and I - interpretation of the formula on a Kripke
  frame ltW,Rgt which makes the formula true for
  agent a

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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 W K? 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 W B? iff ?w' R(w,w') ? Ma W' ?
• The difference between B and K is given by their
  properties

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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, ?) ? ?

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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

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Inference rules for knowledge

• (R1) Epistemic necessitation
• - ? ? K(a, ?)
• modal rule of necessity - ? ? ??
• (R2) Logical omniscience
• ? ? ? and K(a, ?) ? K(a, ?)
• problematic

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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

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Inference rules for belief

• (R1) Epistemic necessitation
• - ? ? B(a, ?) problematic
• modal rule of necessity - ? ? ??
• (R2) Logical omniscience
• ? ? ? and B(a, ?) ? B(a, ?)
• usually NO

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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, ?)

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• 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, A2 A2 K(a, ?) ? ? 5.
KB(?WA ? WB) 2, A2
6. KB(?WA) ? KB(WB) 5, A1 A1 K(a,? ??) ?
(K(a,?) ? K(a,?)) 7. ?WA ? KB(WB) 4, 6
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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, lt, 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
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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 lt
• 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

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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
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• LB - set of moment formula
• LS - set of path-formula
• Semantics
• M ltW, T, lt, , Rgt - 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 t A 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,t X p iff M s,t1 p)

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• 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
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• 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
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