Artificial Intelligence: Logic agents

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Artificial Intelligence Logic agents
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AI in the News

• April 6, 2005 GIDEON - Global Infectious Disease
  and Epidemiology Network. Media review by Vincent
  J. Felitti. JAMA, the Journal of the American
  Medical Association ( Vol. 293, No. 13, pages
  1674-1675 subscription req'd.). "GIDEON The
  Global Infectious Disease and Epidemiology
  Network is a superbly designed expert system
  created to help physicians diagnose any
  infectious disease (337 recognized) in any
  country of the world (224 included). The program
  was created and has been progressively refined
  over more than a decade by a talented group of
  Americans, Canadians, and Israelis
• December 13, 2004 WebMed dispenses advice to
  students. By Robyn Shelton. Orlando Sentinel.
  "The site -- 24/7 WebMed -- takes students
  through questions, judges the severity of their
  symptoms and offers guidance for what to do next.
  ... 'It's decision-support systems, or artificial
  intelligence in a way,' said Dr. Scott Gettings,
  DSHI medical director. 'The system learns about
  you as you flow through and answer questions and
  determines how ill you are.' It makes no attempt
  to go further and diagnose the patient's illness
  -- but gauges the seriousness of the symptoms.
  'This is not intended to take the place of human
  interaction, but to augment it,' said Dr. Michael
  Deichen, associate director of clinical services
  at the UCF Student Health Center. 'It really just
  helps the students know with what urgency they
  should be evaluated.'"

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Thinking Rationally

• Computational models of human thought processes
• Computational models of human behavior
• Computational systems that think rationally
• Computational systems that behave rationally

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Logical Agents

• Reflex agents find their way from Arad to
  Bucharest by dumb luck
• Chess program calculates legal moves of its king,
  but doesnt know that a piece cannot be on 2
  different squares at the same time
• Logic (Knowledge-Based) agents combine general
  knowledge with current percepts to infer hidden
  aspects of current state prior to selecting
  actions
• Crucial in partially observable environments

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Outline

• Knowledge-based agents
• Wumpus world
• Logic in general
• Propositional and first-order logic
• Inference, validity, equivalence and
  satifiability
• Reasoning patterns
• Resolution
• Forward/backward chaining

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Knowledge Base

• Knowledge Base set of sentences represented in
  a knowledge representation language and
  represents assertions about the world.
• Inference rule when one ASKs questions of the
  KB, the answer should follow from what has been
  TELLed to the KB previously.

tell
ask
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Generic KB-Based Agent
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Abilities KB agent

• Agent must be able to
• Represent states and actions,
• Incorporate new percepts
• Update internal representation of the world
• Deduce hidden properties of the world
• Deduce appropriate actions

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Description level

• The KB agent is similar to agents with internal
  state
• Agents can be described at different levels
• Knowledge level
• What they know, regardless of the actual
  implementation. (Declarative description)
• Implementation level
• Data structures in KB and algorithms that
  manipulate them e.g propositional logic and
  resolution.

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A Typical Wumpus World
Wumpus
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Wumpus World PEAS Description
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Wumpus World Characterization

• Observable?
• Deterministic?
• Episodic?
• Static?
• Discrete?
• Single-agent?

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Wumpus World Characterization

• Observable? No, only local perception
• Deterministic? Yes, outcome exactly specified
• Episodic? No, sequential at the level of actions
• Static? Yes, Wumpus and pits do not move
• Discrete? Yes
• Single-agent? Yes, Wumpus is essentially a
  natural feature.

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Exploring the Wumpus World

• 1,1 The KB initially contains the rules of the
  environment. The first percept is none,
  none,none,none,none, move to safe cell e.g. 2,1
• 2,1 breeze which indicates that there is a pit
  in 2,2 or 3,1, return to 1,1 to try next
  safe cell

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Exploring the Wumpus World

• 1,2 Stench in cell which means that wumpus is
  in 1,3 or 2,2
• YET not in 1,1
• YET not in 2,2 or stench would have been
  detected in 2,1
• THUS wumpus is in 1,3
• THUS 2,2 is safe because of lack of breeze in
  1,2
• THUS pit in 1,3
• move to next safe cell 2,2

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Exploring the Wumpus World

• 2,2 move to 2,3
• 2,3 detect glitter , smell, breeze
• THUS pick up gold
• THUS pit in 3,3 or 2,4

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What is a logic?

• A formal language
• Syntax what expressions are legal (well-formed)
• Semantics what legal expressions mean
• in logic the truth of each sentence with respect
  to each possible world.
• E.g the language of arithmetic
• X2 gt y is a sentence, x2y is not a sentence
• X2 gt y is true in a world where x7 and y 1
• X2 gt y is false in a world where x0 and y 6

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Entailment

• One thing follows from another
• KB ?
• KB entails sentence ? if and only if ? is true
  in worlds where KB is true.
• E.g. xy4 entails 4xy
• Entailment is a relationship between sentences
  that is based on semantics.

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Models

• Logicians typically think in terms of models,
  which are formally structured worlds with respect
  to which truth can be evaluated.
• m is a model of a sentence ? if ? is true in m
• M(?) is the set of all models of ?

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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Logical inference

• The notion of entailment can be used for logic
  inference.
• Model checking (see wumpus example) enumerate
  all possible models and check whether ? is true.
• If an algorithm only derives entailed sentences
  it is called sound or truth preserving.
• Otherwise it just makes things up.
• i is sound if whenever KB -i ? it is also true
  that KB ?
• Completeness the algorithm can derive any
  sentence that is entailed.
• i is complete if whenever KB ? it is also
  true that KB-i ?

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Schematic perspective
If KB is true in the real world, then any
sentence ? derived From KB by a sound inference
procedure is also true in the real world.
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Standard Logical Equivalences
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Terminology

• A sentence is valid iff its truth value is t in
  all interpretations ( f)
• Valid sentences true, Øfalse, P V ØP
• A sentence is satisfiable iff its truth value is
  t in at least one interpretation
• Satisfiable sentences P, true, ØP
• A sentence is unsatisfiable iff its truth value
  is f in all interpretations
• Unsatisfiable sentences P ØP, false, Øtrue

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Examples
Sentence
Valid?
wealthy gt wealthy
valid
Øwealthy V wealthy
satisfiable, not valid
wealthy gt happy
w t, h f
inverse
satisfiable, not valid
(w gt h)gt(Øw gtØh)
wf, ht w gth t, Øw gtØh f
contrapositive
valid
(w gt h) gt (h gtØw)
w V h V (w gt h)
valid
w V h V Øw V h
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Examples
Sentence
Valid?
wealthy gtwealthy
valid
Øwealthy V wealthy
satisfiable, not valid
wealthy gt happy
w t, h f
inverse
satisfiable, not valid
(w gth) gt(Øw gtØh)
wf, ht w gtht, Øw gtØh f
contrapositive
valid
(w gth)gt(Øh gt Øw)
w V h V (w gt h)
valid
w V h V Øw V h
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Inference

• KB -i a
• Soundness Inference procedure i is sound if
  whenever KB -i a, it is also true that KB a
• Completeness Inference procedure i is complete
  if whenever KB a, it is also true that KB -i a

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Validity and Inference
((P V H) ØH) gt P
Ø


P

H

P
H

(P
H)

H
((P
H)

ØH) gt
P

V
V

V


T

T

T

F

T

T

F

T

T

T

F

T

T

F

T

F

F

F

F

T


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Rules of Inference

• a - b
• a b
• Valid Rules of Inference
• Modus Ponens
• And-Elimination
• And-Introduction
• Or-Introduction
• Double Negation
• Unit Resolution
• Resolution

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Examples in Wumpus World

• Modus Ponens a gt b, a - b(WumpusAhead
  WumpusAlive) gt Shoot, (WumpusAhead
  WumpusAlive) - Shoot
• And-Elimination a b - a(WumpusAhead
  WumpusAlive) - WumpusAlive
• Resolution a V b, Øb V g - a V g(WumpusDead V
  WumpusAhead), (ØWumpusAhead V Shoot)
  (WumpusDead V Shoot)

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Proof Using Rules of Inference

• Prove A gt B, (A B) gt C, Therefore A gt C
• A gt B - ØA V B
• A B gt C - Ø(A B) V C - ØA V ØB V C
• So ØA V B resolves with ØA V ØB V C deriving
• ØA V C
• This is equivalent to A gt C

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Rules of Inference (continued)

• And-Introduction a1, a2, , an a1 a2 an
• Or-Introduction ai a1 V a2 V ai
  V an
• Double NegationØØa a
• Unit Resolution (special case of resolution)a V
  b Alternatively Øa gt b Øb
  Øb a
  a

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Wumpus World KB

• Proposition Symbols for each i,j
• Let Pi,j be true if there is a pit in square i,j
• Let Bi,j be true if there is a breeze in square
  i,j
• Sentences in KB
• There is no pit in square 1,1R1 ØP1,1
• A square is breezy iff pit in a neighboring
  squareR2 B1,1 ? (P1,2 V P2,1)R3 B1,2 ? (P1,1
  V P1,3 V P2,2)
• Square 1,1 has no breeze, Square 1,2 has a
  breezeR4 ØB1,1R5 B1,2

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Inference in Wumpus World

• Apply biconditional elimination to R2R6 (B1,1
  gt (P1,2 V P2,1)) ((P1,2 V P2,1) gt B1,1)
• Apply AE to R6R7 ((P1,2 V P2,1) gt B1,1)
• Contrapositive of R7R8 (ØB1,1 gt Ø(P1,2 V
  P2,1))
• Modus Ponens with R8 and R4 (ØB1,1)R9 Ø(P1,2 V
  P2,1)
• de MorganR10 ØP1,2 ØP2,1

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Searching for Proofs

• Finding proofs is exactly like finding solutions
  to search problems.
• Can search forward (forward chaining) to derive
  goal or search backward (backward chaining) from
  the goal.
• Searching for proofs is not more efficient than
  enumerating models, but in many practical cases,
  its more efficient because we can ignore
  irrelevant propositions

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Full Resolution Rule Revisited

• Start with Unit Resolution Inference Rule
• Full Resolution Rule is a generalization of this
  rule
• For clauses of length two

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Resolution Applied to Wumpus World

• At some point we determine the absence of a pit
  in square 2,2R13 ØP2,2
• Biconditional elimination applied to R3 followed
  by modus ponens with R5R15 P1,1 V P1,3 V P2,2
• Resolve R15 and R13R16 P1,1 V P1,3
• Resolve R16 and R1R17 P1,3

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Resolution Complete Inference Procedure

• Any complete search algorithm, applying only the
  resolution rule, can derive any conclusion
  entailed by any knowledge base in propositional
  logic.
• Refutation completeness Resolution can always be
  used to either confirm or refute a sentence, but
  it cannot be used to enumerate true sentences.

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Conjunctive Normal Form

• Conjunctive Normal Form is a disjunction of
  literals.
• Example(A V B V ØC) (B V D) (Ø A) (BVC)

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CNF Example

• Example (A V B) ? (C gt D)
• Eliminate ?
• ((A V B) gt (C gt D)) ((C gt D) gt (A V B)
• Eliminate gt
• (Ø (A V B) V (ØC V D)) (Ø(ØC V D) V (A V B) )
• Drive in negations((ØA ØB) V (ØC V D)) ((C
  ØD) V (A V B))
• Distribute(ØA V ØC V D) (ØB V ØC V D) (C V A
  V B) (ØD V A V B)

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Resolution Algorithm

• To show KB a, we show (KB Øa) is
  unsatisfiable.
• This is a proof by contradiction.
• First convert (KB Øa) into CNF.
• Then apply resolution rule to resulting clauses.
• The process continues until
• there are no new clauses that can be added (KB
  does not entail a)
• two clauses resolve to yield empty clause (KB
  entails a)

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Simple Inference in Wumpus World

• KB R2 R4 (B1,1 ? (P1,2 V P2,1)) ØB1,1
• Prove ØP1,2 by adding the negation P1,2
• Convert KB P1,2 to CNF

• PL-RESOLUTION algorithm

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Horn Clauses

• Real World KBs are often a conjunction of Horn
  clauses
• Horn clause
• proposition symbol or
• (conjunction of symbols) gt symbol
• ExamplesC (B gt A) (C D gt B)

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Forward Chaining

• Fire any rule whose premises are satisfied in the
  KB.
• Add its conclusion to the KB until query is
  found.

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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining

• Motivation Need goal-directed reasoning in order
  to keep from getting overwhelmed with irrelevant
  consequences
• Main idea
• Work backwards from query q
• To prove q
• Check if q is known already
• Prove by backward chaining all premises of some
  rule concluding q

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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Backward Chaining Example
P gt Q L M gt P B L gt M A P gt L A B gt
L A B
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Forward Chaining vs. Backward Chaining

• FC is data-drivenit may do lots of work
  irrelevant to the goal
• BC is goal-drivenappropriate for problem-solving