Clase 13 - 11-07-00 Cap

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Clase 13 - 11-07-00Capítulo 7Lógica de Primer
Orden
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Clase 13 --04-07-00

• PARTE III - Conocimiento y RazonamientoCapítulo
  6. AGENTES QUE RAZONAN CON LÓGICA
  PROPOSITIVA-Capítulo 7. LÓGICA DE PRIMER ORDEN
  lt-Capítulo 8 . CONFORMACIÓN DE UNA BASE DE
  CONOCIMIENTOS-Capítulo 9 LA INFERENCIA EN LA
  LÓGICA DE PRIMER ORDEN-Capïtulo 10 SISTEMAS DE
  RAZONAMIENTO LÓGICO

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Lógica de Primer Orden

• Constantes
• representan objectos del mundo real
• juan, 0, 1, libro, etc. (notación a, b, c, )
• Funciones
• los nombres para objetos no representan
  relaciones en el mundo real (notación P, Q, R,
  )
• se_gustan(juan,maría), x gt y, valioso(oro)
• predicado especial para la igualdad ()
• Variables
• testaferros o guardasitios de objetos (notación
  x, y, z, )
• Conectivos o conectores y Cuantificadores
• ?,?, ?,?,?,?
• () se puede considerar que la igualdad es un
  signo de predicado con un significado previamente
  definido

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Lógica de Primer Orden

• Oraciones Atómicas (fórmulas atómicas)
• predicados (término1, término2, , términok)
• donde
• término funcción(término1, término2, ,
  términok) o constante, o variable
• Fórmulas Compuestas

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7.1

• 7.1 Syntax and Semantics
• - constants
• - predicates
• - functions (not evaluated)
• - terms refer to objects (constants or
  functions)
• - atomic sentences
• - complex sentences atomics plus connectives
• - quantifiers
• - universal
• - existential
• - equality

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7.2

• 7.2 Extensions and Notational Variations
• - higher-order logic
• - lambda operator
• - uniqueness quantifier
• - uniqueness operator
• - notational variations

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7.3

• 7.3 Using First-Order Logic
• - kinship domain
• - axioms, definitions and theorems
• - set domain
• - special notation for sets, lists and arithmetic
• - asking questions and getting answers (Prolog)

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7.4

• 7.4 Logical Agents for the Wumpus World Fig7.2
  Fig6.1
• - Percept (PerceptList,TimeStep)
• - Action Turn(Right), Turn(Left), Forward,
  Shoot, Grab, Release, Climb
• - Turn is a function, Release is new
• - reflex, model-based, goal-based

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7.5

• 7.5 A Simple Reflex Agent
• - percept(S,B,glitter,U,C,T) gt action(grab,T)
• - OR percept(S,B,glitter,U,C,T) gt atgold(T)
• atgold(T) gt action(grab,T)
• - no memory, does the same thing in the same
  situations
• - randomness may help (e.g., orientation)

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7.6-Representing Change

• The traditional ontology for reasoning about
  action is called the situation calculus. It
  involves three basic categories
• a "state" is a snapshot of the world,
• a "fluent" is a changing relationship between
  objects, and
• an "action" is an event that changes one state
  into another.
• States, fluents, and actions can be represented
  as logical terms
• let a and b be constants naming two blocks,
• stack(a,b) can name the action of placing a on b,
• do(s0, stack(a,b)) can name the state resulting
  from doing this action in the initial state s0.
• Axiomatic description of how the world changes
• Effect axioms describe the properties of
  situations resulting from actions
• e.g., "x"y"s Øblocked(x,s) Ù Øblocked(y,s) Þ
  above(x, y, do(s, stack(x,y)))
• Frame axioms represent how the world stays the
  same
• e.g., "x"y"z"s above(x,y,s) Ù a ¹ stack(x,z) Þ
  above(x, y, do(s,a))
• In general there will be too many frame axioms
  that can be stated

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7.6.1

• 7.6.1 Representing Change in the World
• - retract(agent_location(X,Y)),
  assert(agent_location(X1,Y1))
• - but no recollection of route (okay for
  simulator, but not for agent)
• - situation calculus
• - at(agent,1,1,s0)
• - result(Action,Situation1) Situation2
• - frame axiom
• - (holding(X,S) and (A ! release)) gt
  holding(X,result(A,S))
• -

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7.6.2

• 7.6.2- successor-state axiom
• - holding(X,result(A,S)) ltgt (A grab) and
  present(X,S) and portable(X))
• V (holding(X,S)
  and (A ! release))
• - keeping track of location
• - at(Agent,Location,Situation)
• - orientation(Agent,Situation)
• - locationToward(Location,Orientation)
  NewLocation
• - locationAhead(Agent,Situation)
  AheadLocation
• -

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7.6.3

• 7.6.3 adjacent(Location1,Location2)
• - wall(Location)
• - agent location changes upon moving forward
  into a non-wall
• - agent orientation changes upon turning left
  or right
• - agent no longer has arrow after shoot
• - hasarrow(Agent,result(Action,Situation))
  ltgt hasarrow(Agent,Situation)
• 
  Action ! shoot
• - gold location, wumpus location and health

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7.6.4

• 7.6.4- wumpus dead if agent has arrow and shoots
  in direction of wumpus
• - dead(wumpus,result(Action,S)) ltgt Action
  shoot hasarrow(agent,S)
• 
  at(agent,X,Y,S)
• 
  orientation(agent,S) 0
• 
  at(wumpus,X1,Y1,S) Y Y1 X lt X1
• - three other disjuncts for other
  orientations
• - why would agent shoot?
• - need a goal
• -

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7.7

• 7.7 Deducing Hidden Properties of the World
• - causal rules (model-based)
• - at(wumpus,L1,S) adjacent(L1,L2) gt
  smelly(L2)
• - at(pit,L1,S) adjacent(L1,L2) gt breezy(L2)
• - at(wumpus,L,S) at(pit,L,S) ltgt ok(L)
• - diagnostic rules
• - at(agent,L,S) stench(S) gt smelly(L)
• - at(agent,L,S) breeze(S) gt breezy(L)
• - percept(none,none,G,U,C,T) at(agent,L1,S)
  adjacent(L1,L2) gt ok(L2)
• - compare to above causal rule
• - causal rules infer strongest conclusions

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7.8

• 7.8 Preferences Among Actions
• - action-value system
• - great, good, medium and risky actions
• - great(A,S) gt action(A,S)
• - good(A,S) E(A2) great(A2,S) gt
  action(A,S)
• - great(A,S) ltgt (A grab atgold(S)) V
• (A climb at(agent,1,1,S)
  holding(gold,S))

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7.9

• 7.9 Toward a Goal-Based Agent
• - above preferences will get the gold safely (if
  possible), but then what
• - holding(gold,S) gt goalLocation(1,1,S)
• - kill the wumpus
• - achieve via
• - inference (inefficient for long paths)
• - search
• - planning
• Logic and Knowledge Representation
• Semantics of First-Order Predicate Logic
• Proof Theory and the Notion of Derivation
• Resolution Mechanism
• Forward and Backward Chaining

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Transformation to FOPC3

• Mary got good grades in courses CS101 and passed
  CS102
• John passed CS102
• Student who gets good grades in a course passes
  that course
• Students who pass a course are happy
• A student who is not happy hasnt passed all
  his/her courses
• Only one student failed all the courses

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Transformation to FOPC4

• Usar habitualmente ? con ?
• e.g.,
• says, all humans are mortal
• but,
• say, everything is human and mortal
• Usually use Ù with
• e.g.,
• says, there is a bird that does not fly
• but,
• is also true for anything that is not a bird
• "xy is not the same as y"x
• e.g.,
• says, there is someone who loves everyone
• but,
• says, everyone is loved by at least one person

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Cuantificadores

• " se reinterpreta como conjunción para todos
  los objetos del dominio
• p.ej.,
• se interpreta como
• cse reinterpreta como disyunction para todos
  los objetos del dominio
• p.ej.
• Se reinterpreta como
• Dualidad de los cuantificadores
• cada uno de ellos se puede expresar usando el
  otro
• lo cual es una aplicación de las leyes de
  deMorgan
• ejemplos

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Interpretations Models in FOPC6

• Definition An interpretation is a mapping which
  assigns
• objects in domain to constants in the language
• functional relationships in domain to function
  symbols
• relations to predicate symbols
• usual logical relationships to connectives and
  quantifiers Ø, Ù, Ú, Þ, Û, ",
• Definition Models
• An interpretation M is a model for a set of
  sentences S, if every sentence in S is true with
  respect to M (if S is a singleton s, then we
  say that M is a model for s).
• Notation S
• If there is a model M for S, then S is
  satisfiable
• If S is true in every interpretation M (every
  interpretation is a model for S), then S is valid

M
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Interpretations Models in FOPC7

• Example
• where N, L are predicate symbols, and f a
  function symbol
• interpretation 1
• domain positive integers
• N(x) x is a natural number
• L(x,y) x is less than y
• f(x) predecessor of x (i.e., x-1)
• then s says predecessor of any natural number
  is a natural number (of course this is false, so
  this interpretation is not a model for s)
• interpretation 2
• domain all people
• N(x) x is a person
• L(x,y) x likes y
• f(x) mother of x
• then s says everyone likes his/her mother

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Models as Sets of Atomic Formulas8

• If we assume the language has no quantifiers and
  variables, then models can be represented as sets
  of atomic formulas
• note that we can eliminate quantifiers and
  variables by completely expanding conjunctions of
  ground formulas (formulas without variables)
• let A be the set of all ground atomic formulas in
  the language, then a model M can be expressed as
  a subset of A (M Í A)
• for an atomic formula s, s Î M, means M is a
  model of s, otherwise s is false in M
• Example Consider KB consisting of
• if we assume that the named constants are the
  only objects in the domain, then A bird(sam),
  bird(tweety), flies(sam), flies(tweety)
• then, M bird(tweety), bird(sam), flies(sam)
  is a model for flies(sam), "x(bird(x)),
  x(bird(x) Ù flies(x)), but M is not a model for
  flies(tweety), "x(flies(x)), or x(Ø bird(x))
• Note that if there is a function symbol in the
  language, then A is infinite

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Semantics of FOPC Operators9

• Let F and G be FOPC Formulas, and M be any
  interpretation
• F Ù G is true in M iff both F and G are true in
  M
• F Ú G is true in M iff at least one of F or G
  is true in M
• ØF is true in M iff both F is false in M
• F Þ G is true in M iff either F is false in M
  or G is true in M
• F Û G is true in M iff both F and G are true in
  M or both are false in M
• So far this is the same as propositional how
  about quantifiers
• "x F is true in M iff for any object d in the
  domain, Fd is true in M, where Fd is the
  result of replacing every free occurrence of x in
  F with d
• x F is true in M iff for some object d in the
  domain, Fd is true in M, where Fd is the
  result of replacing every free occurrence of x in
  F with d
• Example Again consider KB
• x(bird(x) Ù flies(x)) is entailed by KB, since
  bird(tweety) Ù flies(tweety), is true in every
  model of KB (taking d tweety)

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Proof Theory of FOPC10

• The rules of inference for propositional logic
  still apply in the context of FOPC
• And-Introduction (AI)
• And-Elimination (AE)
• Or-Introduction (OI)
• Negation-Elimination(NE)
• Modes Ponens (MP)
• In addition we have inference rules
• for quantifiers
• Universal Instantiation (UI)
• where, t is a term replacing free occurrences
• of x in F (x must not occur in t)
• Existential Instantiation (EI)
• where, f is a new function symbol, and y
• is a free variable (not quantified in F)

The formula F is derivable (provable) from KB,
if 1. F is already in KB (a fact or axiom) 2. F
is the result of applying a rule of inference
to sentences derivable from KB
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Universal / Existential Instantiation11

• Universal Instantiation (UI)
• where, t is a term replacing free occurrences
• of x in F (x must not occur in t)
• Example
• From "y(likes(jean,y)) we can infer
  likes(jean,joe), likes(joe, mother_of(joe)), etc

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Universal / Existential Instantiation11

• Existential Instantiation (EI)
• where, f is a new function symbol, and y
• is a free variable (not quantified in F)
• Example Consider y(likes(x,y)) we can infer
  likes(x,f(x)), where f is a new function symbol
  representing an object that satisfies
  y(likes(x,y)) (f is called a Skolem function)
• Note If there are no free variables in F, then
  we can use a new constant symbol (a function with
  no arguments)
• Consider y"x(likes(x,y)) we can infer
  "x(likes(x,a), where a is a new constant symbol
  (a is called a Skolem constant)

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Example of Derivation12

• Let KB parent(john,mary), parent(john,joe),
• This derivation shows that KB

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Soundness and Completeness of FOPC13

• Soundness of FOPC
• given a set of sentences KB and a sentence s,
  then
• KB s implies KB s
• note that if s is derived from KB, but KB does
  not entail s, then at least one of the inference
  rules used to derive s must have been unsound
• Completeness of FOPC
• given a set of sentences KB and a sentence s,
  then
• KB s implies KB s
• note that if s is entailed by KB, but we cannot
  derive s from KB, then out inference system (set
  of inference rules) must be incomplete
• However, note that entailment for FOPC is
  semi-decidable

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Forward and Backward Chaining

• Inference rules (e.g., resolution, generalized
  Modes Ponens) can be used in forward or backward
  chaining systems
• Forward Chaining
• Start with KB, infer new consequences using
  inference rule(s), add new consequences to KB,
  continue this process (possibly until a goal is
  reached)
• In a knowledge-based agent this amounts to
  repeated application of the TELL operation
• May generate many irrelevant conclusions, so not
  usually suitable for solving for a specific goal
• Useful for building a knowledge base
  incrementally as new facts come in
• Backward Chaining
• Start with goal to be proved, apply inference
  rules in a backward manner to obtain premises,
  then try to solve for premises until known facts
  (already in KB) are reached
• This is useful for solving for a particular goal
• In a knowledge-based agent this amounts to
  applications of the ASK operation
• The proofs can be viewed as a tree
• root is the goal to be proved
• for each node, its children are the conjuncts
  that make up the premises for a clause
• see example in next slide

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Proof Trees for Backward Chaining
KB
z john
z z2
z z1
x x1
x x2
fail
x1 bob
x2 sue
z1 john
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Logical Reasoning Agents

• Recall the general template for a knowledge-based
  agent

• Water-Jug Problem
• percepts may be in the form
• Precept(x, y, t), where x, y represent
• contents of 4 and 3 gallon jugs and t
• represents the current time instance
• actions may be of the form
• fill(4-gal), fill(3-gal), empty(4-gal),
• empty(3-gal), dump(4-gal, 3-gal), etc.
• e.g., agent tries to determine what is the best
  action at time 7, by ASKing if
• x Action(x,7), which might give an answer such
  as x fill(3-gal).

loop forever Input percepts time 0 KB
tell(KB, make-sentence(percept)) action
ask(KB, action-query) Output action KB
tell(KB, make-sentence(action)) time time
1 end
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Logical Reasoning Agents

• In the simple reflex agent, the KB might include
  rules that directly (or indirectly) connect
  percepts with actions
• e.g., Percept(x,y, t) Ù (xy ³ 4) Ù (y gt 0) Þ
  Action(dump(3-gal, 4-gal), t)
• However, for the agent to be able to reason about
  the results of its actions in a reasonable
  manner, it must be able to specify a model of the
  world and how it changes

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FINAL DE CLASE

• -