Reasoning Algorithms in Propositional Logic

Reasoning Algorithmsin Propositional Logic

Examination will be a take-home exam;confirmation coming as soon as signed course

evaluation is received in registrar’s office

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Knowledge representationand reasoning

Propositions, general knowledge, facts, KB, model-> big truth table

Propositions KB: general knowledge & facts

t t f t f t f f f t t t t t t t t t t t t t t t t t t t t t t t t t t t t model

The reasoning problems:

1. Find t/f assignment(s) model(s) where KB is true

2. Answering questions “entailed” by KB

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Approaches to reasoning

N propositions to satisfy KB1. Search through 2N rows of truth table:

goal-based search, fitness is truth of KB (SAT)

2. Use inference: restrict attention to relevant propositions (assumes many models satisfy the KB and many propositions might be “don’t care”)

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Approaches to Reasoning:strategies

1. Searcha) Depth-first exhaustive search from start

state of ‘empty’ truth tableb) Hill-climbing from random start state of

true-false assignments

2. Inferencea) Forward chaining from KB to queryb) Backward chaining from query into KB

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Propositional satisfiability Problem (SAT)

Definition (Hoos & Stutzle, 2005)“Given a propositional formula F, the

problem is to decide whether or not F is satisfiable.”

F = KB (facts + general knowledge)

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Propositional satisfiability Problem (SAT)

State: a vector of truth values for the n propositions

State space: 2n nodesGoal state(s): KB is true (a model)e.g., n = 5, {P1,P2,P3,P4,P5}

t t f t f f t f t f

t t t t f

t - f

t - f

KB = (P1P2) (P2P1)

(P1 P2 P3)

(P2P1) (P4P3)

(P5P3) etc.

1 b)

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Propositional satisfiability Problem (SAT)

TT-ENTAILS is depth-first search, exhaustive, incremental Improvement in efficiency by pruning: DPLL –

p.221 early termination pure symbol heuristic unit clause heuristic

WALKSAT: complete state algorithm – reduce number of false clauses by flipping propositions

true<->false

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Propositional satisfiability Problem (SAT)

Answers a question: Is a sentence a true in the KB?i.e., is the sentence true in all models of the KB which

are true? OR is (KBa) true?

t t t t f

t t t

f

t

KB = (P1P2) (P2P1)

(P1 P2 P3)

(P2P1) (P4P3)

(P5P3) etc.

Question: (P1P5) ?

1 a)

at root – no truth values assigned

KB, α both true?

KB false

P both true and false

1 a)

t t t t f

t t t

f

t

KB = (P1P2) (P2P1)

(P1 P2 P3)

(P2P1) (P4P3)

(P5P3) etc.

Question: (P1P5) ?

Propositions KB

P1 P2 P3 P4 P5 (P1P2)(P2P1)(P1P2P3)(P2P1)(P4P3)(P5P3) (P1P5)

Question

t t t t t t tt f t t t

t t t t f t tt f t t t

f f f f f t ft t f t t

… … … …

KB Q

t

t

t

TT-ENTAILS returns true if KB Q is true for all cases;

i.e., there is no row with KB true and Q false

TT-ENTAILS1 a)

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Variations on TT-ENTAILS

For efficiency: (see p.221) Early termination (pruning) Pure symbol heuristic Unit clause heuristic

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Propositional satisfiability Problem (SAT)

WALKSAT, p.223 (complete state search)Checks satisfiabilityi.e., are there models of the KB which are true?

t t f t f f t f t f

t t t t f

t - f

t - f

KB = (P1P2) (P2P1)

(P1 P2 P3)

(P2P1) (P4P3)

(P5P3) etc.

Question: KB satisfiable?1 b)

1 b)

t t f t f f t f t f

t t t t f

t - f

t - f

KB = (P1P2) (P2P1)

(P1 P2 P3)

(P2P1) (P4P3)

(P5P3) etc.

Question: KB satisfiable?

f t f t frandom Satisfied? truey

flip t/f of random proposition in clause

flip t/f of proposition in clause that

minimizes number of false clauses

Probability p

pick random false clause

Give up? falsey

WALKSAT

1 b)

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WALKSAT performance

Not guaranteed to find solution(not exhaustive like TT-ENTAILS)

More effective in practice thanTT-ENTAILS, even with efficiencyheuristics (DPLL)

1 b)

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Approaches to reasoning

N propositions to satisfy KB1. Search through 2N rows of truth table:

goal-based search, fitness is truth of KB (SAT)

2. Use inference: restrict attention to relevant propositions (assumes many models satisfy the KB and many propositions might be “don’t care”)

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Inference rule: Resolution

elimination of complementary literals from sentences in CNF

(~W \/ ~Q \/ T) Λ (W \/ P) (~Q \/ T \/ P)

inference by resolution is Sound – only infers true statements Complete – anything entailed is derivable

Part of KB

New proposition

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Resolution: Example (P11 \/ P22 \/ P13) ~P11

~P22

resolve (P11 \/ P22 \/ P13), ~P11

(P22 \/ P13) resolve (P22 \/ P13), ~P22

P13

(from Wumpus world)

Part of KB

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Resolution algorithm

goal-directed proof by contradiction to prove P

assume ~P add ~P to KB resolve in KB till resulting sentence is

1. in KB (therefore P is false)2. empty (therefore ~P is contradictory so P is

true)

Figure 7.12 p.216

α leads to

contradition

therefore

α is true

α is consistent

with KB so

α is false

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Horn clause inference method compromise representation that is

human-readable basic of logic programming (Prolog) uses modus ponens, not resolution like CNF but restricted to only one

positive proposition(~W \/ ~Q \/ ~S \/ T)

=> ~(W Λ Q Λ S) \/ T => (W Λ Q Λ S) T

2 a)

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Forward chaining inference with Horn clauses

algorithm to determine if a particular proposition is true

O(n) in size of KB!! p. 219, Fig 7.14

2 a)

Figure 7.14 p.219

Reasoning by FORWARD chaining

•From the known data “forward” to unknown

•Doesn’t need goal – self-directed agent

2 a)

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Reasoning by BACKWARD chaining

Goal-directed reasoning – question answering agent

Backward (KB, Q) //answer query Q If Q true in KB, return true For each Horn clause (P=>Q) in KB,

If Backward (KB, P), return true Return false

2 b)