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Logical Agentsمنطقي عاملهاي

Chapter 7 (part 2)Modified by Vali Derhami

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Proof methods• Proof methods divide into (roughly) two kinds:

– Application of inference rules• Legitimate (sound) generation of new sentences from old• Proof = a sequence of inference rule applications

Can use inference rules as operators in a standard search algorithm

• Typically require transformation of sentences into a normal form

– Model checking• truth table enumeration (always exponential in n)• heuristic search in model space (sound but incomplete)

e.g., min-conflicts-like hill-climbing algorithms

•• improved backtracking, e.g., Davis--Putnam-

Logemann-Loveland (DPLL)

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Reasoning Patterns in Propositional Logic • Inference rules: The patterns of inference: 1-Modus Ponens ( استثنايي (قياس

2-And –Elimination

اساليد • در منطقي ارزهاي هم مي 28تمامي اول قسمت ازروند بكار استنتاج قاعده بعنوان توانند

اسنتاج : • قواعد كاربردهاي از اي دنباله اثباتتواند يكنواختي:• مي زماني فقط شده ايجاب جمالت مجموعه

. شود افزوده دانش پايگاه به اطالعاتي كه يابند افزايش

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Resolution تحليل: ساده: استنتاج روش يك تحليل

جستجوي الگوريتم هر با شدن همراه صورت دراستنتاج الگوريتم يك ميكند كاملكامل، ايجاد را

• In Wumpus world, consider the agent returns from [2,1] to [1,1] and then goes to [1,2], where it perceives a stench, but no breeze. We add the following facts to the

knowledge base:

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( ادامه ( تحليل

we can now derive the absence of pits in [2,2] and [1,3],([1,1] is already known(

:

Respect to R11 and R12:

Respect to R10 (R10 : P11), and R16:

Known from past:

Respect to R15 and Rr13:

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) تحليل ادامه(Conjunctive Normal Form (CNF)( نرمال (عطفيفرم

conjunction of disjunctions of literals (clauses)هم ANDتركيب با كه ليترال .ORاز اند شده

Literal: a atomic sentence. Ex; P , Q , or , AClause: a disjunction of literals.

E.g., (A B) (B C D)

• Unit resolution inference rule:

Where li and m are complementary literals

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) تحليل ادامه(• Full Resolution inference rule (for CNF):

l1 … lk, m1 … mn

l1 … li-1 li+1 … lk m1 … mj-1 mj+1 ... mn

where li and mj are complementary literals. E.g., P1,3 P2,2, P2,2

P1,3• Using any complete search, • Resolution is sound and complete

for propositional logic اثبات براي تواند مي هميشه تحليل توجه،

شود استفاده جمله يك ياتكذيب

••

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Conversion to CNFعطفي نرمال فرم به تبديل

B1,1 (P1,2 P2,1) 1. Eliminate , replacing α β with (α β)(β α). (B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)

2. Eliminate , replacing α β with α β.

3. Move inwards using de Morgan's rules and double-negation:(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

4. Apply distributivity law ( over ) and flatten:(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

–• (B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)

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Resolution algorithm

• Proof by contradiction, i.e., to Prove KB a, show KBα unsatisfiable

•

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Steps in Resolution algorithm• (KB a) is converted into CNF.• The resolution rule is applied to the resulting clauses.

Each pair that contains complementary literals is resolved to produce a new clause, which is added to the set if it is not already present.

• The process continues until one of two things happens: 1- two clauses resolve to yield the empty clause,

in which case KB entails a. 2- there are no new clauses that can be added, in which case KB does not entail a.

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Resolution example

• KB = (B1,1 (P1,2 P2,1)) B1,1 ,α = P1,2

•

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Forward and backward chaining• Horn Form (restricted): عطفي نرمال فرم از اي شده محدود فرم

است. ORتركيب • است مثبت آنها از يكي حداكثر كه ها ليترال از• : آن شرطي- 1مزاياي فرم به تبديل

P 1,1 B1,1 P1,2

(P 1,1 P1,2) B1,1

(P 1,1 P1,2) B1,1

KB = conjunction of Horn clauses– Horn clause =

• proposition symbol; or• (conjunction of symbols) symbol

– E.g., C (B A) (C D B)معين – بند را دارند مثبت ليترال يك دقيقا كه هورني مي )Definite clause(بندهاي

نامند

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Forward and backward chaining•2- : استثنايي قياس از استفاده با استناج اجازه

• Modus Ponens (for Horn Form): complete for Horn KBs

α1, … ,αn, α1 … αn ββ

• Can be used with forward chaining or backward chaining.

هورن- 3• بندهاي در ايجاب مورد در گيري تصميم زمانخطي بصورت دانش پايگاه اندازه حسب بر تواند مي

باشد: • These algorithms are very natural and run in

linear time•

•

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Forward and backward chaining

A Horn clause with no positive literals can be written as an implication whose conclusion is the literal False.Ex: —wumpus cannot be in both [1,1] and [1,2]—is equivalent to . Such sentences are called integrity (کامل ( constraints in the database world,

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Forward chaining• Idea: 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 algorithm

• Forward chaining is sound and complete for Horn KB

•

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Forward chaining example

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Proof of completeness• FC derives every atomic sentence that is

entailed by KB1. FC reaches a fixed point where no new atomic

sentences are derived2. Consider the final state as a model m, assigning

true/false to symbols3. Every clause in the original KB is true in m

a1 … ak b4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB, including m

•

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Backward chainingIdea: work backwards from the query q:

to prove q by BC,check if q is known already, orprove by BC all premises of some rule concluding q

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal has already been proved true, or has already failed

•

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Backward chaining example

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Forward vs. backward chaining• FC is data-driven, automatic, unconscious processing

( هدف بي .appropritate for Design, Control ,(پروسس– e.g., object recognition, routine decisions

• May do lots of work that is irrelevant to the goal

• BC is goal-driven, appropriate for problem-solving, Diagnosis,– e.g., Where are my keys? How do I get into a PhD program?

• Complexity of BC can be much less than linear in size of KB

–

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Efficient propositional inferenceTwo families of efficient algorithms for propositional

inference on Model checking:

Complete backtracking search algorithms– DPLL algorithm (Davis, Putnam, Logemann, Loveland)

Incomplete local search algorithms– WalkSAT algorithm

–

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The DPLL algorithmDetermine if an input propositional logic sentence (in CNF) is satisfiable.

. مدل يافتن براي اول عمق روش از استفادهImprovements over truth table enumeration:

1. Early terminationA clause is true if any literal is true.A sentence is false if any clause is false.

2. Pure symbol heuristicPure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), (B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true.

كه شودند دهی مقدار اي گونه به تواند مي خالص هاي سمبل مقدار لذا باشد داشته مدلي جمله آگركند صحيح را اشان مربوطه .بند

3. Unit clause heuristicUnit clause: only one literal in the clauseThe only literal in a unit clause must be true.

درخت انشعاب از قبل واحد بندهای تمام دهی مقدار واقع در

•

•

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The DPLL algorithm

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The WalkSAT algorithm• Incomplete, local search algorithm

• . ارزيابي تابع كند ارضا را بندها تمام كه Wاست انتسابي يافتن هدف. آيد مي بر كار اين عهده از بشمارد را نشده ارضا بندهاي Wتعداد كه• On every iteration, the algorithm picks an unsatisfied

clause and picks a symbol in the clause to flip. It chooses randomly between two ways to pick which symbol to flip: A) a "min-conflicts" step that minimizes the number of unsatisfied clauses in the new state, and B) a random walk" step that picks the symbol randomly.

• Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses

• Balance between greediness and randomness

•

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The WalkSAT algorithm

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Hard satisfiability problems

• Consider random 3-CNF sentences. e.g.,(D B C) (B A C) (C B E) (E D B) (B E C)

m = number of clauses n = number of symbols

– Hard problems seem to cluster near m/n = 4.3 (critical point)

–

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Hard satisfiability problems

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Hard satisfiability problems

• Median runtime for 100 satisfiable random 3-CNF sentences, n = 50

•

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) 8، 5تکالیف : • - و ( د تا تست 9الف سواالت و