Logical Agents - CSC 411: AI Fall 2013 - Nc State Universitystamant/411/lectures/07-logic/logic-1.pdf · Logical Agents CSC 411: AI Fall 2013 NC State University 1 / 111 Logical Agents

Logical AgentsCSC 411: AI

Fall 2013

NC State University 1 / 111Logical Agents

Outline

Knowledge-based agents

Wumpus world

Logic in general – models and entailment

Propositional (Boolean) logic

Equivalence, validity, satisfiability

Inference rules and theorem proving

• Forward chaining• Backward chaining• Resolution


Knowledge basesInference engine ← Domain-independent algorithmsKnowledge base ← Domain-specific content

A knowledge base is a set of sentences in a formallanguage.

The declarative approach to building an agent (or othersystem):

• TELL it what it needs to know.• Then it can ASK itself what to do.

Knowledge level versus implementation level.


A simple knowledge-based agent

KB-AGENT(percept)static t = 0TELL(KB, MAKE-PERCEPT-SENTENCE(percept, t))action = ASK(KB, MAKE-ACTION-QUERY(t))TELL(KB, MAKE-ACTION-SENTENCE(action, t))t = t + 1return action


A simple knowledge-based agent

The agent must be able to

• Represent states, actions, etc.• Incorporate new percepts• Update internal representations of the world• Deduce hidden properties of the world• Deduce appropriate actions


Wumpus world PEAS description

Performance measure:

Gold: +1000Death: −1000Taking a step: −1Using an arrow: −10

PIT

1 2 3 4

1

2

3

4

START

Stench

Stench

Breeze

Gold

PIT

PIT

Breeze

Breeze

Breeze

Breeze

Breeze

Stench


Wumpus world PEAS descriptionEnvironment:Smelly when adjacent to Wumpus.Breezy when adjacent to pit.Glitter iff gold is in same square.Shooting kills Wumpus if you arefacing it.Shooting uses up only arrow.Grabbing picks up gold if in samesquare.Releasing drops the gold in samesquare.

PIT

1 2 3 4

1

2

3

4

START

Stench

Stench

Breeze

Gold

PIT

PIT

Breeze

Breeze

Breeze

Breeze

Breeze

Stench


Wumpus world PEAS description

Sensors:Stench, Breeze, Glitter,Bump, ScreamActuators:Left turn, Right turn, For-ward, Grab, Release, Shoot

PIT

1 2 3 4

1

2

3

4

START

Stench

Stench

Breeze

Gold

PIT

PIT

Breeze

Breeze

Breeze

Breeze

Breeze

Stench


Wumpus world characterization

Fully Observable? No, only local perception.

Deterministic? Yes, outcomes are exactly specified.

Episodic? No, sequential at the level of actions.

Static? Yes, the Wumpus and pits do not move.

Discrete? Yes.

Single-agent? Yes, the Wumpus is essentially a naturalfeature.


Exploring a Wumpus world
















Logic in general

Logics are formal languages for representing informationsuch that conclusions can be drawn.

Syntax defines the sentences in the language; semanticsdefine the “meaning” (i.e., truth values) of sentences.

The language of arithmetic:

x + 2 ≥ y is a sentencex2 + y > () is not a sentencex + 2 ≥ y is true in a world where x = 7, y = 1x + 2 ≥ y is false in a world where x = 0, y = 6


Entailment

The knowledge base KB entails sentence α if and only ifα is true in all worlds where KB is true: KB |= α.

Said differently, a set of premises KB logically entails aconclusion α if and only if every interpretation thatsatisfies the premises also satisfies the conclusion.

{p} |= (p ∨ q)

{p, q} |= (p ∧ q)

Entailment is a relationship between sentences (i.e.syntax), based on semantics.


Models

Logicians typically think in terms of models, formallystructured worlds with respect to which truth can beevaluated.

We can think of a model as a possible world.

In the semantics of arithmetic, x + y = 4 is true in a worldwhere x is 2 and y is 2, but false in a world where x is 1and y is 10.


Models

We say m is a model of a sentence α if α is true in m.

M(α) is the set of all models of α.

KB |= α iff M(KB) ⊆ M(α).

{p, q} |= p because the worlds in which p is true is asuperset of the worlds in which p is true and q is true.


Entailment in the Wumpus world

You detect nothing in [1,1].

You move right.

You detect a breeze in [2,1].

Consider possible models forKB assuming only pits:

3 (Boolean) possibilities→8 possible models.


Wumpus models


Wumpus models

KB = Wumpus-world rules + observations


Wumpus models


α1 = “[1,2] is safe.” Does KB |= α1?


Wumpus models



Wumpus models


α2 = “[2,2] is safe.” Does KB |= α2?


Inference

KB ì α means that sentence α can be derived from KBby procedure i.

Soundness: i is sound if whenever KB ì α, it is also truethat KB |= α.

Completeness: i is complete if whenever KB |= α, it isalso true that KB ì α.


Propositional logic: Syntax

The proposition symbols P1, P2, etc. are sentences.

If S is a sentence, ¬S is a sentence (negation).

If S1 and S2 are sentences. . .

S1 ∧ S2 is a sentence (conjunction).

S1 ∨ S2 is a sentence (disjunction).

S1 ⇒ S2 is a sentence (implication).

S1 ⇔ S2 is a sentence (biconditional).


Propositional logic: Semantics

Each model specifies true or false for each propositionsymbol.

P1,2 P2,2 P3,1False True False

With these symbols, 8 possible models can be enumeratedautomatically.



Rules for evaluating truth with respect to a model m:¬S is true iff S is false.

S1 ∧ S2 is true iff S1 is true and S2 is true.S1 ∨ S2 is true iff S1 is true or S2 is true.

S1 ⇒ S2 is true iff S1 is false or S2 is true.S1 ⇔ S2 is true iff S1 ⇒ S2 is true and S2 ⇒ S1 is true.



A simple recursive process can evaluate an arbitrarysentence:¬P1,2 ∧ (P2,2 ∨ P3,1) = not false ∧ ( true ∨ false )

= true ∧ ( true ∨ false )= true ∧ true= true


Truth tables for connectives

P Q ¬P P ∧ Q P ∨ Q P⇒ Q P⇔ QFalse False True False False True TrueFalse True True False True True FalseTrue False False False True False FalseTrue True False True True True True


Wumpus world sentences

Let Pi,j be true if there is a pit in [i, j].

Let Bi,j be true if there is a breeze in [i, j].

Suppose we know

P1,1 and B1,2 and B2,1.

We also know that pits cause breezes in adjacent squares.Then

B1,2 ⇔ (P1,1 ∨ P2,2 ∨ P1,3).

B2,1 ⇔ (P1,1 ∨ P2,2 ∨ P3,1).


Truth tables for inference

B1,1 B2,1 P1,1 P1,2 P2,1 P2,2 P3,1 KB α1

F F F F F F F F TF F F F F F T F T

. . .F T F F F F F F T

. . .F T F F F F F T TF T F F F T F T T

. . .F T F F T F F F T

. . .T T T T T T T F F


Inference by enumerationTT-ENTAILS?(KB, α)

symbols = a list of symbols in KB and αreturn TT-CHECK-ALL(KB, α, symbols, {})

TT-CHECK-ALL(KB, α, symbols,model)if EMPTY?(symbols)

if PL-TRUE?(KB, model)return PL-TRUE?(α,model)

else return Trueelse p = First(symbols)

rest = Rest(symbols)return TT-CHECK-ALL(KB, α, rest, EXTEND(p,True,model))

and TT-CHECK-ALL(KB, α, rest, EXTEND(p,False,model))


Inference by enumeration

Depth-first enumeration of all models is sound andcomplete.

For n symbols, time complexity is O(2n); spacecomplexity is O(n).


Logical equivalence

Two sentences are logically equivalent iff they are true insame models:

α ≡ β iff α |= β and β |= α.


Logical equivalence(α ∧ β) ≡ (β ∧ α) Commutativity of ∧(α ∨ β) ≡ (β ∨ α) Commutativity of ∨

((α ∧ β) ∧ γ) ≡ (α ∧ (β ∧ γ)) Associativity of ∧((α ∨ β) ∨ γ) ≡ (α ∨ (β ∨ γ)) Associativity of ∨

¬(¬α) ≡ α Double negation(α⇒ β) ≡ (¬β ⇒ ¬α) Contraposition¬(α ∧ β) ≡ (¬α ∨ ¬β) de Morgan¬(α ∨ β) ≡ (¬α ∧ ¬β) de Morgan

(α ∧ (β ∨ γ)) ≡ ((α ∧ β) ∨ (α ∧ γ)) Distributivity, ∧/∨(α ∧ (β ∨ γ)) ≡ ((α ∨ β) ∧ (α ∨ γ)) Distributivity, ∨/∧


Validity

A sentence is valid if it is true in all models, e.g.,

TrueA ∨ ¬AA⇒ A

(A ∧ (A⇒ B))⇒ B

Validity is connected to inference via the DeductionTheorem:

KB |= α iff (KB⇒ α) is valid.


Satisfiability

A sentence is satisfiable if it is true in some model.

A sentence is unsatisfiable if it is true in no model.

Satisfiability is also connected to inference:

KB |= α iff (KB ∧ ¬α) is unsatisfiable.


Proof methods: Inference rules

One kind of proof method involves the application ofinference rules.

• Rules give sound generation of new sentences fromold.

• A proof is a sequence of inference rule applications.(We can use rules as operators in search.)

• This typically requires transformation of sentencesinto some normal form.


Proof methods: Model checking

Another approach is model checking, e.g.,

• Truth table enumeration (as we’ve seen).• Smarter backtracking (potentially more efficient).• Heuristic search or optimization in model space

(sound but incomplete).


Exercise: Bike riding

“You often ride your bike, but only on days when theweather is nice. Whenever you ride, you wear a helmet.Wearing your helmet always messes up your hair beyondeasy repair, but you like to be presentable: you neverarrive at work with messy hair, and you never go out ondinner dates with messy hair.”

What propositions should we define for this story?



“You often ride your bike, but only on days when theweather is nice. Whenever you ride, you wear a helmet.Wearing your helmet always messes up your hair beyondeasy repair, but you like to be presentable: you neverarrive at work with messy hair, and you never go out ondinner dates with messy hair.”

One possible set: NiceOut, BikeRide, Helmet, MessyHair,Work, DinnerDate.

Develop sentences to represent this story.



“You call me on your cell phone and tell me you’re out ona dinner date. If I wonder whether you rode your bike,should I ask you or figure it out on my own?”

How can I express this as a sentence in logic?


Resolution

We start with Conjunctive Normal Form (CNF).

Call a logical variable or its complement a literal.

A, B, ¬A, ¬B

Call a disjunction of literals (including a literal standingalone) a clause.

A, (B ∨ ¬A)

A sentence consisting of a conjunction of clauses is inCNF.


Resolution

The resolution rule for CNF:

a1 ∨ . . . ∨ ak b1 ∨ . . . ∨ bn

a1 ∨ . . . ∨ aj−1 ∨ aj+1 ∨ . . . ∨ ak ∨ b1 ∨ . . . ∨ bm−1 ∨ bm+1 ∨ . . . ∨ bn

where aj and bm are complementary literals.

Resolution is sound and complete for propositional logic.


Resolution in Wumpus world

P1,3 ∨ P2,2 ¬P2,2

P1,3


Resolution in Wumpus world

P1,3 ∨ P2,2 ¬P2,2

P1,3


Conversion to CNF

B1,1 ⇔ (P1,2 ∨ P2,1)

Eliminate⇔, replacing α⇔ β with (α⇒ β) ∧ (β ⇒ α).

(B1,1 ⇒ (P1,2 ∨ P2,1)) ∧ ((P1,2 ∨ P2,1)⇒ B1,1)

Eliminate⇒, replacing α⇒ β with ¬α ∨ β.

(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬(P1,2 ∨ P2,1) ∨ B1,1)

. . .


Conversion to CNF

. . .

(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬(P1,2 ∨ P2,1) ∨ B1,1)

Move ¬ inwards, using de Morgan’s rule:

(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ ((¬P1,2 ∧ ¬P2,1) ∨ B1,1)

Apply distributivity law (∧ over ∨) and flatten:

(¬B1,1 ∨ P1,2 ∨ P2,1) ∧ (¬P1,2 ∨ B1,1) ∧ (¬P2,1 ∨ B1,1)



In CNF:

¬NiceOut⇒ ¬BikeRideBikeRide⇒ HelmetHelmet⇒ MessyHair¬(MessyHair ∧ DinnerDate)¬(MessyHair ∧Work)



In CNF:

¬NiceOut⇒ ¬BikeRide NiceOut ∨ ¬BikeRideBikeRide⇒ Helmet Helmet ∨ ¬BikeRideHelmet⇒ MessyHair ¬Helmet ∨MessyHair¬(MessyHair ∧ DinnerDate) ¬DinnerDate ∨ ¬MessyHair¬(MessyHair ∧Work) ¬Work ∨ ¬MessyHair



In CNF:




In CNF:




In CNF:




In CNF:




A truth table, for enumeration

NiceOut BikeRide ¬NiceOut⇒ ¬BikeRide NiceOut ∨ ¬BikeRideT T T TT T T TF F F FF T T T


Resolution algorithm

The strategy is proof by contradiction.

To prove α, we show that there exist no models in whichKB ∧ ¬α is true.

That is, we prove KB ∧ ¬α is unsatisfiable.


Resolution algorithmPL-RESOLUTION(KB, α)

clauses = clauses in CNF representation of KB ∧ ¬αempty = the empty clausenew = the empty setwhile True

for Ci,Cj in clausesresolvent = PL-RESOLVE(Ci,Cj)if resolvent is the empty set

return Truenew = new ∪ resolvent

if new ⊆ clausesreturn False

clauses = clauses ∪ new



“You call me on your cell phone and tell me you’re out ona dinner date. . . ”

NiceOut ∨ ¬BikeRideHelmet ∨ ¬BikeRide¬Helmet ∨MessyHair¬DinnerDate ∨ ¬MessyHair¬Work ∨ ¬MessyHairDinnerDate

I’ll guess you didn’t ride your bike (¬BikeRide). Can Iprove it?



Recall. . .

a1 ∨ . . . ∨ ak b1 ∨ . . . ∨ bn

a1 ∨ . . . ∨ aj−1 ∨ aj+1 ∨ . . . ∨ ak ∨ b1 ∨ . . . ∨ bm−1 ∨ bm+1 ∨ . . . ∨ bn

Prove ¬BikeRide by resolution.

To start: Negate ¬BikeRide. . .



BikeRide Helmet ∨ ¬BikeRideHelmet

Helmet ¬Helmet ∨MessyHairMessyHair

MessyHair ¬DinnerDate ∨ ¬MessyHair¬DinnerDate

¬DinnerDate DinnerDate


Forward and backward chaining

Horn Form: KB is a conjunction of Horn clauses.

A Horn clause is an implication of a conjunction to apositive literal:

C, (B⇒ A), (C ∧ D⇒ B)

Modus Ponens (for Horn Form) is complete for HornKBs.

α1 . . . αn α1 ∧ . . . ∧ αn ⇒ ββ


Forward chaining

AB

P ⇒ QL ∧M ⇒ PB ∧ L ⇒ MA ∧ P ⇒ LA ∧ B ⇒ L

Q

P

M

L

BA


Forward chaining

Fire any rule whose premises are satisfied in the KB.

Add its conclusion to the KB.

Continue until the query is found.


Forward chaining algorithmPL-FC-ENTAILS?(KB, q)

count = a table, where count[c] is # of symbols in c’s premiseinferred = a table, where inferred[s] is initially falseagenda = a list of symbols known to be true in KBwhile not Empty?(agenda)

p = Pop(agenda)if p is q return Trueif inferred[p] is False

inferred[p] = Truefor c such that p ∈ c.premise

count[c] = count[c]− 1if count[c] is 0

Push(c.conclusion, agenda)return False

Forward chaining is sound and complete for Horn KBs.NC State University 68 / 111Logical Agents

Forward chaining example

Agenda: A, B.

P⇒ Q : 1L ∧M ⇒ P : 2B ∧ L⇒ M : 2A ∧ P⇒ L : 2A ∧ B⇒ L : 2Pop A.



After processing A.

Agenda: B.

P⇒ Q : 1L ∧M ⇒ P : 2B ∧ L⇒ M : 2A ∧ P⇒ L : 1A ∧ B⇒ L : 1Pop B.



After processing B.

P⇒ Q : 1L ∧M ⇒ P : 2B ∧ L⇒ M : 1A ∧ P⇒ L : 1A ∧ B⇒ L : 0

Agenda: L.

Pop L.



After processing L.

P⇒ Q : 1L ∧M ⇒ P : 1B ∧ L⇒ M : 0A ∧ P⇒ L : 1A ∧ B⇒ L : 0

Agenda: M.

Pop M.



After processing M.

P⇒ Q : 1L ∧M ⇒ P : 0B ∧ L⇒ M : 0A ∧ P⇒ L : 1A ∧ B⇒ L : 0

Agenda: P.

Pop P.



After processing P.

P⇒ Q : 0L ∧M ⇒ P : 0B ∧ L⇒ M : 0A ∧ P⇒ L : 0A ∧ B⇒ L : 0

Agenda: Q, L.

Pop Q.


















Proof of completeness

FC derives every atomic sentence that is entailed by KB.

FC reaches a fixed point where no new atomic sentencesare derived.

Consider the final state to be a model m, assigningtrue/false to symbols.

Every clause in the original KB is true in m

a1 ∧ . . . ∧ ak ⇒ b

Then m is a model of KB.

If KB |= q, q is true in every model of KB, including m.


Backward chaining

Work backward from query q.

To prove q by BC,

• Check if q is known already, or• Prove by BC all premises of some rule concluding q.

Avoid loops by checking if a new subgoal is already onthe goal stack.

Avoid repeated work by checking if a new subgoal hasalready been proved true or has already failed.


Backward chaining example




















Forward vs backward chaining

FC is data-driven, automatic processing, e.g., objectrecognition, routine decisions.

It may do lots of work that is irrelevant to the goal.

BC is goal-driven, appropriate for problem solving, e.g.,Where are my keys? How do I get CS degree?

The complexity of BC can be much less than linear in sizeof KB.


Efficient propositional inference

Two families of efficient algorithms for propositionalinference are

• Complete backtracking search algorithms, such asDPLL, and

• Incomplete local search algorithms, such asWalkSAT.


The DPLL algorithm: Concepts

A clause is true if any literal is true.

A sentence is false if any clause is false.

A unit clause contains only one literal.

A pure symbol always appears in the same positive ornegative form in all clauses.

(A ∨ ¬B), (¬B ∨ ¬C), (C ∨ A)

A and B are pure; C is not.


The DPLL algorithm: Heuristics

Notice when a clause can be true if one of its literals istrue.

Notice that a sentence is false if one of its clauses is false.

Make the single literal in unit clauses true.

Make pure symbol literals true.


The DPLL algorithm

DPLL-SATISFIABLE?(s)clauses = the set of clauses in the CNF representation of ssymbols = propositional symbols in sreturn DPLL(clauses, symbols, {})


The DPLL algorithm

DPLL(clauses, symbols,model)if ALL-TRUE(clauses, model) return Trueif SOME-FALSE(clauses, model) return Falsep, value = FIND-PURE-SYMBOL(clauses, symbols, model)if p found

return DPLL(clauses, symbols \ p, [p = value | model])p, value = FIND-UNIT-CLAUSE(clauses, model)if p found

return DPLL(clauses, symbols \ p, [p = value | model])p = First(symbols)return DPLL(clauses, symbols \ p, [p = True | model]) or

DPLL(clauses, symbols \ p, [p = False | model])


The WalkSAT algorithm

WalkSat is an incomplete local search algorithm.

Its evaluation function is the min-conflict heuristic ofminimizing the number of unsatisfied clauses.

It balances between greediness and randomness.


The WalkSAT algorithm

WALKSAT(clauses, p,max-flips)model = a random assignment of True/False to symbols in clausesfor j = 1 to max-flips

if SATISFIES?(model, clauses)return model

clause = CHOOSE-RANDOM-FALSE-CLAUSE(model)if Sample(0, 1) < p

model = FLIP(Random-Element(clause), clause,model)else symbol = ARG-MAX(clause, MAXIMIZE-SATISFIED)

model = FLIP(symbol, clause, model)return Failure


Hard satisfiability problems

Consider random 3-CNF sentences.

(¬D ∨ ¬B ∨ C) ∧ (B ∨ ¬A ∨ ¬C) ∧ (¬C ∨ ¬B ∨ E)∨(E ∨ ¬D ∨ B) ∨ (B ∨ E ∨ ¬C)

Let

m = number of clauses andn = number of symbols.

Hard problems seem to cluster near a critical point,m/n = 4.3.






Inference in the Wumpus world¬P1,1¬W1,1Bx,y ⇔ (Px,y+1 ∨ Px,y−1 ∨ Px+1,y ∨ Px−1,y)Sx,y ⇔ (Wx,y+1 ∨Wx,y−1 ∨Wx+1,y ∨Wx−1,y)W1,1 ∨W1,2 ∨ . . . ∨W4,4¬W1,1 ∨W1,2¬W1,1 ∨W1,3. . .


A propositional Wumpus world agent

PL-WUMPUS-AGENT(percept)static: KB, x, y, orientation, visited, action, planUPDATE(x, y, orientation, visited, action)if percept.stench

TELL(KB, stenchx,y)else TELL(KB, ¬stenchx,y)if percept.breeze

TELL(KB, breezex,y)else TELL(KB, ¬breezex,y)if percept.glitter

return action = Grab. . .


A propositional Wumpus world agent

PL-WUMPUS-AGENT(percept). . .else if not Empty?(plan)

return action = Pop(plan)else for j, k on FRINGE(visited)

if ASK(KB,Pj,k ∧ ¬Wj,k) is True orASK(KB,Pj,k ∨Wj,k) is False

plan = PLAN-PATH(x, y, orientation, j, k,visited)return action = Pop(plan)

else return action = CHOOSE-RANDOM-MOVE()


Expressiveness limitation ofpropositional logic

KB contains “physics” sentences for every single square.

For every time t and every x, y location Lx,y,

Lx,y(t) ∧ FacingRight(t) ∧ Forward(t)⇒ Lx+1,y(t + 1),etc.

Rapid proliferation of clauses.


Summary

Logical agents apply inference to a knowledge base toderive new information and make decisions.

• Syntax: Formal structure of sentences.• Semantics: Truth of sentences with respect models.• Entailment: Necessary truth of one sentence given

another.• Inference: Deriving sentences from other sentences.• Soundness: Derivations produce only entailed

sentences.• Completeness: Derivations can produce all entailed

sentences.NC State University 110 / 111

Logical Agents

Last observations

An agent, as in Wumpus world, needs to represent partialand negated information, reason by cases, etc.

Resolution is complete for propositional logic.

Forward and backward chaining run in linear time; theyare complete for Horn clauses.

Propositional logic lacks expressive power.


Documents

Logical Agents - CSC 411: AI Fall 2013 - Nc State Universitystamant/411/lectures/07-logic/logic-1.pdf · Logical Agents CSC 411: AI Fall 2013 NC State University 1 / 111 Logical Agents