CS G120 Artificial Intelligence

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CS G120 Artificial Intelligence

• Prof. C. Hafner
• Class Notes Jan 22, 2009

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Topics for today

• Probability of success in Wumpus World
• Finish discussion of propositional logic
• Horn clauses
• Example
• Forward chaining
• Backward chaining
• First order logic
• Unification
• CNF for first order logic (if time permits)

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Exploring a wumpus world
(1,4)
(4,4)
X
Y
Z
(1,1)
(4,1)
Change S in location (2,1) to B Calculate
Prob(pit at Z)
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Important formulas

• P(A v B) P(A) P(B) - P(A and B)
• P(A B) P(A B) / P(B)
• P(A B) P(A) P(B)
• if they are independent events

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Breeze in (1,2) B1 Breeze in (2, 1)
B2 Prob(B1) Prob (PX v PY) 0.2 0.2 0.04
0.36 Prob(PX B1) Prob (PX B1) / Prob(B1)
0.2 / 0.36 0.555 Prob ((PX PY)
B1) Prob(PX PY B1) / Prob(B1) 0.04 /
0.36 0.111
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Prob(B1 B2) Prob(PY v (PX and PZ)) Prob
(PY) Prob(PX PZ) Prob(PX PY PZ)
0.2 0.04 - 0.008 0.232 Prob(PZ (B1
B2)) Prob (PZ B1 B2) / Prob(B1 B2)
0.2 0.36 / 0.232 0.31 Therefore agent
should proceed to Z if Prob(winning) gt .31
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Reasoning with Propositional Logic (cont.)Horn
Clauses

• P1 v P2 v . . . v Pn v Q
• Same as P1 P2 . . . Pn ? Q

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Example

• KB a database of assertions (sometimes called
  facts and rules)
• fruit ? edible
• vegetable ? edible
• vegetable green ? healthy
• apple ? fruit
• banana ? fruit
• spinach ? vegetable
• spinach ? green
• edible healthy ? recommended
• ? apple

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Forward chaining data driven reasoning triggered
by a new percept (fact)

• Basis of forward chaining
• P ? Q is an assertion in the KB
• P is a new percept
• ----------
• Conclude Q
• Two views of KB
• All implications are made explicit vs.
• Reasoning on demand
• Pure forward chaining suggests the former

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Example

• KB fruit ? edible
• vegetable ? edible
• vegetable green ? healthy
• edible healthy ? recommended
• apple ? fruit
• banana ? fruit
• spinach ? vegetable
• spinach ? green
• apple
• Lets make all the implications explicit
• Now consider this KB without the fact ? apple,
  and consider a new percept ? spinach. What
  happens

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Forward chaining algorithm

• forwardChain(KB, percept) returns updated KB
• new-knowledge percept
• while new-knowledge is not empty
• p ? pop(new-knowledge)
• add p to KB
• for each rule r in KB with p contained in
  LHS(r)
• if all elements of LHS(r) are in KB
• push (RHS(r), new-knowledge)
• return KB
• --------------------------------------------------
  ----------------
• Requires an index of the rule set by LHS symbols

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Forward chaining (cont.)

• To make a KB explicit
• KB ? makeExplicit(KB)
• for each fact f in KB
• KB ? forwardChain(KB, f)
• Note possibility of infinite loop how to avoid
• for each rule r in KB with p contained in
  LHS(r)
• if all elements of LHS(r) are in KB and
  RHS(r) is not in KB
• push (RHS(r), new-knowledge)

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Backward chaining goal-driven reasoning
triggered by a question being asked

• KBfruit ? edible
• vegetable ? edible
• edible green ? healthy
• apple ? fruit
• banana ? fruit
• spinach ? vegetable
• spinach ? green
• edible healthy ? recommended
• apple
• Consider some queries
• ? apple ? fruit ? banana ? edible ?
  healthy

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Sketch of Backward Chaining algorithm

• backwardChain(KB, query) returns Boolean
• if query is in KB, return True
• for each rule r in KB such that RHS(r) query
• testing True
• for each element e of the LHS(r)
• if backwardChain(KB, e) False
• testing False
• break
• if testing True return True
• return False
• NOTE that backward chaining does not update the
  KB
• DISCUSS NEGATION BY FAILURE (example green is
  false if apple)

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Assignment 3

• Part I. Due Date January 29 (upload to
  Blackboard)
• Write a Python program to implement the forward
  chaining mechanism we defined for propositional
  logic. The following functions should be
  implemented
• KB ? readKB(filename) returns a knowledge
  base object
• KB ? makeExplicit(KB)
• KB ? forwardChain(KB, fact)
• You should define a class KB and objects of the
  class will be returned from these 3 functions.
  Other classes may be defined as needed.
• The makeExplicit and forwardChain functions
  should also produce printed output of the
  assertions added to the KB.
• The format of input for the readKB function is,
  one assertion per line
• Rule format p1 p2 q
• Fact format q

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Assignment 3 (cont.)

• Part 2. Due date Feb 5
• Create a model of the Boston subway (T) system
  using first order logic (FOL). Define some
  axioms that would support reasoning about how to
  get from one subway station to another. Be
  prepared to explain your model in class. You can
  find a map of the Boston subway system at the
  MBTA web site or
• http//www.tuftslife.com/images/map_mbta.gif
• More detailed information about this assignment
  will be posted on the Blackboard site in the next
  few days.

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First Order Logic (FOL)

• Why FOL?
• Syntax and semantics of FOL
• Using FOL
• Wumpus world in FOL
• Knowledge engineering in FOL

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Pros and cons of propositional logic

• ? Propositional logic is declarative
• ? Propositional logic allows partial/disjunctive/n
  egated information
• (unlike most data structures and databases)
• (Horn clauses an intermediate form)
• Propositional logic is compositional
• meaning of B1,1 ? P1,2 is derived from meaning of
  B1,1 and of P1,2
• ? Meaning in propositional logic is
  context-independent
• (unlike natural language, where meaning depends
  on context)
• ? Propositional logic has very limited expressive
  power
• (unlike natural language)
• E.g., cannot say "pits cause breezes in adjacent
  squares
• except by writing one sentence for each square

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First-order logic

• Whereas propositional logic assumes the world
  contains facts,
• first-order logic (like natural language) assumes
  the world contains
• Objects people, houses, numbers, colors,
  baseball games, wars,
• Relations red, round, prime, brother of, bigger
  than, part of, comes between,
• Functions father of, best friend, one more than,
  plus,
  

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Syntax of FOL Basic elements

• Constants KingJohn, 2, NU,...
• Predicates Brother, gt,...
• Functions Sqrt, LeftLegOf,...
• Variables x, y, a, b,...
• Connectives ?, ?, ?, ?, ?
• Equality
• Quantifiers ?, ?

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Atomic sentences

• Atomic sentence predicate (term1,...,termn)
  or term1 term2
• Term function (term1,...,termn)
  or constant or variable
• E.g., Brother(KingJohn,RichardTheLionheart)
• gt (Length(LeftLegOf(Richard)),
  Length(LeftLegOf(KingJohn)))
• Brother (Length(LeftLegOf(Richard)),
  Length(LeftLegOf(KingJohn)))

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Complex sentences

• Complex sentences are made from atomic sentences
  using connectives
  
• ?S, S1 ? S2, S1 ? S2, S1 ? S2, S1 ? S2,
• E.g. Sibling(KingJohn,Richard) ?
  Sibling(Richard,KingJohn)
• gt(1,2) ? (1,2)
• gt(1,2) ? ? gt(1,2)

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Truth in first-order logic

• Sentences are true with respect to a model and an
  interpretation
• Model contains objects (domain elements) and
  relations among them
• Interpretation specifies referents for
• constant symbols ? objects
• predicate symbols ? relations
• function symbols ? functional relations
• An atomic sentence predicate(term1,...,termn) is
  true
• iff the objects referred to by term1,...,termn
• are in the relation referred to by predicate

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Models for FOL Example
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Universal quantification

• ?ltvariablesgt ltsentencegt
• Everyone at NU is smart
• ?x At(x,NU) ? Smart(x)
• ?x P is true in a model m iff P is true with x
  being each possible object in the model
• Roughly speaking, equivalent to the conjunction
  of instantiations of P
• At(KingJohn,NU) ? Smart(KingJohn)
• ? At(Richard,NU) ? Smart(Richard)
• ? At(NUS,NU) ? Smart(NU)

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A common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x,NU) ? Smart(x)
• means Everyone is at NU and everyone is smart

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Existential quantification

• ?ltvariablesgt ltsentencegt
• Someone at NU is smart
• ?x At(x,NU) ? Smart(x)
• ?x P is true in a model m iff P is true with x
  being some possible object in the model
• Roughly speaking, equivalent to the disjunction
  of instantiations of P
• At(KingJohn,NU) ? Smart(KingJohn)
• ? At(Richard,NU) ? Smart(Richard)
• ? At(NU,NU) ? Smart(NU)
• ? ...

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Another common mistake to avoid

• Typically, ? is the main connective with ?
• Common mistake using ? as the main connective
  with ?
• ?x At(x,NU) ? Smart(x)
• is true if there is anyone who is not at NUS!

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Properties of quantifiers

• ?x ?y is the same as ?y ?x
• ?x ?y is the same as ?y ?x
• ?x ?y is not the same as ?y ?x
• ?x ?y Loves(x,y)
• There is a person who loves everyone in the
  world
• ?y ?x Loves(x,y)
• Everyone in the world is loved by at least one
  person
• Quantifier duality each can be expressed using
  the other
• ?x Likes(x,IceCream) ??x ?Likes(x,IceCream)
• ?x Likes(x,Broccoli) ??x ?Likes(x,Broccoli)

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Equality

• term1 term2 is true under a given
  interpretation if and only if term1 and term2
  refer to the same object
• E.g., definition of Sibling in terms of Parent
• ?x,y Sibling(x,y) ? ?(x y) ? ?m,f ? (m f) ?
  Parent(m,x) ? Parent(f,x) ? Parent(m,y) ?
  Parent(f,y)

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Using FOL

• The kinship domain
• Brothers are siblings
• ?x,y Brother(x,y) ? Sibling(x,y)
• One's mother is one's female parent
• ?m,c Mother(c) m ? (Female(m) ? Parent(m,c))
• Sibling is symmetric
• ?x,y Sibling(x,y) ? Sibling(y,x)

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A model M for the kinship domain

• Individuals J K L M N O P Q R
• Functions mom1 mom(N) ? M
• Relationsarity
• fem1 M, Q
• par2 M, N, N, R . .
• sib2 M, O, P, J, J, P
• --------------------Interpretation I
  -----------------------
• Constants John, Mary, Sue, Tom .. ..
• I(Mary) M, I(Sue) Q, . . .
• Function symbol Mother, I(Mother) mom
• Relation symbols Female, Parent, Sibling
• I(Female) fem, I(Parent) par, I(Sibling)
  sib

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Wumpus world in FOL

• First step define constants, function symbols,
  predicate symbols to express the facts
• Percept(data, t) means at step t, the agent
  perceived the data where data is a 5 element
  vector
• Stench, Breeze, Glitter, Bump, Scream
• Ex Percept(None, Breeze, None, None, None),2
• At(Agent, s, t) means agent is at square s at
  step t
• Ex At(Agent, 2,1, 2

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Some Wumpus axioms

• Axiom to interpreting perceptions in context
• ?x,t At(Agent, x, t) Breeze(t) ? Breezy(x)
• Definitional axiom
• ?a,b,c,d,t Percept(a, Breeze, b, c, d, t) ?
  Breeze(t)
• Diagnostic Axiom
• ?x Breezy(x) ? ?z Adjacent(z, x) Pit(z)
• Causal Axiom
• ?z Pit(z) ? (?x Adjacent(z, x) ? Breezy(x))
• World model axioms Adjacent(1,1,2,1) etc.
• ?x,y Adjacent(x, y) ?? Adjacent(y,x)
  

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Interacting with FOL KBs

• Suppose a wumpus-world agent is using an FOL KB
  and perceives a smell and a breeze (but no
  glitter) at t5
• Tell(KB,Percept(Smell,Breeze,None,5))
• Ask(KB,?a BestAction(a,5))
• I.e., does the KB entail some best action at t5?
• Answer Yes, a/Shoot ? substitution (binding
  list)
• Given a sentence S and a substitution s,
• Ss denotes the result of plugging s into S e.g.,
• S Smarter(x,y)
• s x/Hillary,y/Bill
• Ss Smarter(Hillary,Bill)
• Ask(KB,S) returns some/all s such that KB Ss

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Knowledge engineering in FOL

• Identify the taskAssemble the relevant knowledge
• Decide on a vocabulary of predicates, functions,
  and constants
• Encode general knowledge about the domain
• Encode a description of the specific problem
  instance
• Pose queries to the inference procedure and get
  answers
  Debug the knowledge base

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Inference in FOL

• Rules of Inference
• Unification
• Generalized Modus Ponens

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Universal instantiation (UI)

• Every instantiation of a universally quantified
  sentence is entailed by it
• ?v a
• Subst(v/g, a)
• for any variable v and ground term g
• E.g., ?x King(x) ? Greedy(x) ? Evil(x) yields
• King(John) ? Greedy(John) ? Evil(John)
• King(Richard) ? Greedy(Richard) ? Evil(Richard)
• King(Father(John)) ? Greedy(Father(John)) ?
  Evil(Father(John))
• .
• .
• .

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Existential instantiation (EI)

• For any sentence a, variable v, and constant
  symbol k that does not appear elsewhere in the
  knowledge base
  
• ?v a
• Subst(v/k, a)
• E.g., ?x Crown(x) ? OnHead(x,John) yields
• Crown(C1) ? OnHead(C1,John)
• provided C1 is a new constant symbol, called a
  Skolem constant
  

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Reduction to propositional inference

• Suppose the KB contains just the following
• ?x King(x) ? Greedy(x) ? Evil(x)
• King(John)
• Greedy(John)
• Brother(Richard,John)
• Instantiating the universal sentence in all
  possible ways, we have
• King(John) ? Greedy(John) ? Evil(John)
• King(Richard) ? Greedy(Richard) ? Evil(Richard)
• King(John)
• Greedy(John)
• Brother(Richard,John)
• The new KB is propositionalized proposition
  symbols are
• King(John), Greedy(John), Evil(John),
  King(Richard), etc.
  

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Unification

• We can get the inference immediately if we can
  find a substitution ? such that King(x) and
  Greedy(x) match King(John) and Greedy(y)
• ? x/John,y/John works
• Unify(a,ß) ? if a? ß?
• p q ?
• Knows(John,x) Knows(John,Jane)
• Knows(John,x) Knows(y,OJ)
• Knows(John,x) Knows(y,Mother(y))
• Knows(John,x) Knows(x,OJ)
• Standardizing apart eliminates overlap of
  variables, e.g., Knows(z17,OJ)

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Unification

• We can get the inference immediately if we can
  find a substitution ? such that King(x) and
  Greedy(x) match King(John) and Greedy(y)
• ? x/John,y/John works
• Unify(a,ß) ? if a? ß?
• p q ?
• Knows(John,x) Knows(John,Jane) x/Jane
• Knows(John,x) Knows(y,OJ)
• Knows(John,x) Knows(y,Mother(y))
• Knows(John,x) Knows(x,OJ)

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Unification

• We can get the inference immediately if we can
  find a substitution ? such that King(x) and
  Greedy(x) match King(John) and Greedy(y)
• ? x/John,y/John works
• Unify(a,ß) ? if a? ß?
• p q ?
• Knows(John,x) Knows(John,Jane) x/Jane
• Knows(John,x) Knows(y,OJ) x/OJ,y/John
• Knows(John,x) Knows(y,Mother(y))
• Knows(John,x) Knows(x,OJ)

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Unification

• We can get the inference immediately if we can
  find a substitution ? such that King(x) and
  Greedy(x) match King(John) and Greedy(y)
• ? x/John,y/John works
• Unify(a,ß) ? if a? ß?
• p q ?
• Knows(John,x) Knows(John,Jane) x/Jane
• Knows(John,x) Knows(y,OJ) x/OJ,y/John
• Knows(John,x) Knows(y,Mother(y)) y/John,x/Mother
  (John)
• Knows(John,x) Knows(x,OJ)

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Unification

• We can get the inference immediately if we can
  find a substitution ? such that King(x) and
  Greedy(x) match King(John) and Greedy(y)
• ? x/John,y/John works
• Unify(a,ß) ? if a? ß?
• p q ?
• Knows(John,x) Knows(John,Jane) x/Jane
• Knows(John,x) Knows(y,OJ) x/OJ,y/John
• Knows(John,x) Knows(y,Mother(y)) y/John,x/Mother
  (John)
• Knows(John,x) Knows(x,OJ) fail
• Standardizing apart eliminates overlap of
  variables, e.g., Knows(z17,OJ)

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Unification

• To unify Knows(John,x) and Knows(y,z),
• ? y/John, x/z or ? y/John, x/John,
  z/John
  
• The first unifier is more general than the
  second.
• There is a single most general unifier (MGU) that
  is unique up to renaming of variables.
• MGU y/John, x/z

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The unification algorithm
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The unification algorithm
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Generalized Modus Ponens (GMP)

• p1', p2', , pn', ( p1 ? p2 ? ? pn ?q)
• q?
• p1' is King(John) p1 is King(x)
• p2' is Greedy(y) p2 is Greedy(x)
• ? is x/John,y/John q is Evil(x)
• q ? is Evil(John)
• GMP used with KB of definite clauses (exactly one
  positive literal)
• All variables assumed universally quantified

where pi'? pi ? for all i
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Soundness of GMP

• Need to show that
• p1', , pn', (p1 ? ? pn ? q) q?
• provided that pi'? pi? for all I
• Lemma For any sentence p, we have p p? by UI
• (p1 ? ? pn ? q) (p1 ? ? pn ? q)? (p1? ?
  ? pn? ? q?)
  
• p1', \ , \pn' p1' ? ? pn' p1'? ? ?
  pn'?
• From 1 and 2, q? follows by ordinary Modus Ponens

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Example knowledge base

• The law says that it is a crime for an American
  to sell weapons to hostile nations. The country
  Nono, an enemy of America, has some missiles, and
  all of its missiles were sold to it by Colonel
  West, who is American.
• Prove that Col. West is a criminal

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Example knowledge base contd.

• ... it is a crime for an American to sell weapons
  to hostile nations
• American(x) ? Weapon(y) ? Sells(x,y,z) ?
  Hostile(z) ? Criminal(x)
• Nono has some missiles, i.e., ?x Owns(Nono,x) ?
  Missile(x)
  
• Owns(Nono,M1) and Missile(M1)
• all of its missiles were sold to it by Colonel
  West
• Missile(x) ? Owns(Nono,x) ? Sells(West,x,Nono)
• Missiles are weapons
• Missile(x) ? Weapon(x)
• An enemy of America counts as "hostile
• Enemy(x,America) ? Hostile(x)
• West, who is American
• American(West)
• The country Nono, an enemy of America
• Enemy(Nono,America)