6 logical agentsweb.ntnu.edu.tw/~tcchiang/ai/6_logical agents (s).pdf · 2010. 6. 9. · “logical...

Artificial Intelligence, Spring, 2010

Logical Agents (I)

Instructor: Tsung-Che [email protected]

Department of Computer Science and Information EngineeringNational Taiwan Normal University

2“Logical Agents,”Artificial Intelligence, Spring, 2010

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統計有誤結果有誤執行有誤實作有誤編譯有誤


Outline

Knowledge-based AgentsThe Wumpus WorldLogicPropositional LogicReasoning Patterns in Propositional LogicEffective Propositional InferenceAgents based on Propositional LogicSummary


Knowledge-based Agents

Humans know things and do reasoning.

Knowledge and reasoning play a crucial rolein dealing with partially observableenvironments. e.g. diagnosing a patient

Understanding natural language alsorequires reasoning.



Knowledge-based agents can benefit fromknowledge expressed in very general formsto suit many purposes.

They are able to accept new tasks explicitly described by goals, achieve competence by being told new

knowledge about the environment, and adapt to changes by updating the relevant

knowledge.



The central component of a knowledge-based agent is its knowledge base (KB).

A KB is a set of sentences.Each sentence is expressed in a knowledge

representation language and representssome assertion about the world.



TELL: add new sentences to the KBASK: query what is known

Both tasks may involve inference –derivingnew sentences from old.

When one ASKS a question of the KB, theanswer should “follow”what has beenTELLED (told) to the KB.



Artificial Intelligence: A Modern Approach, 2nd ed., Figure 7.1

The KB may initially contain some background knowledge.The details of the representation language are hidden inside three functions.The details of the inference mechanism are hidden inside TELL and ASK.



One can build a knowledge-based agent byTELLing it what it needs to know. But how?Declarative approach

It adds one by one the sentences that represent thedesigner’s knowledge.

Design of the representation language is important.

Procedural approach It encodes desired behaviors directly as program

code. Minimizing the role of explicit representation and

reasoning can result in a much more efficient system.



We will see both declarative andprocedural approaches later.

A successful agent must combine bothelements in its design.


The Wumpus World

PEAS description Environment:

It is a 44 grid of rooms. The agent always starts

from [1, 1], facing to theright.

The locations of the goldand the wumpus arechosen randomly.

Each square other thanthe start can be a pitwith probability 0.2.

The agent has only onearrow.



The Wumpus World

PEAS description Performance measure:

+1000 for picking up the gold -1000 for falling into a pit or being eaten by the

wumpus -1 for each action -10 for shooting the (only one) arrow


The Wumpus World

PEAS descriptionActuators:

Move forward Turn left/right by 90 Grab Shoot

Sensors: Stench Breeze Glitter Bump Scream



BreezeBreeze

StenchStench

GlitterGlitter

Breeze

Stench

Glitter

The Wumpus World


The Wumpus World

Fundamental property of reasoning

“In each case where the agent draws aconclusion from the available information, thatconclusion is guaranteed to be correct if theavailable information is correct.”


Logic

Sentences in the KB are expressedaccording to the syntax of therepresentation language. e.g. “x+y=4”is a well-formed sentence, whereas“x4y+=“is not.

A logic must also define the semantics ofthe language. It defines the truth of each sentence w.r.t.

each possible world.


Logic

ModelWe will use the term model in place of “possible

world.”We will say “m is a model of ”to mean that

sentence is true in model m.

EntailmentWe use ╞ to mean that the sentence

entails the sentence .╞ if and only if in every model in which is

true, is also true. The truth of is contained in the truth of .


Logic

ExampleThe agent has detected nothing in [1, 1] and a

breeze in [2, 1]. It is interested in whether the adjacent

squares [1, 2], [2, 2], and [3, 1] contain pits. There are 23 = 8 possible models.

The percepts and the rules of the wumpusworld constitute the KB.

The KB is true in models thatfollow what the agent knows.


Logic

8 possible models “nothing in [1, 1]”and“breeze in [2, 1]”


Logic


1 = “There is no pit in [1, 2]”KB╞ 1


Logic


2 = “There is no pit in [2, 2]”KB╞ 2


Logic

The previous example shows how aninference algorithm called “model checking”works.

It enumerates all possible models to checkthat is true in all models in which KB istrue.


Logic

If an inference algorithm i can derive from KB, we write

KB├i .

An inference algorithm is called sound or truth-preserving if it derives only entailed sentences. Model checking is a sound algorithm (when it is

applicable).

An inference algorithm is complete if it canderive any sentence that is entailed.


Logic

“If KB is true in the real world, then anysentence derived from KB by a soundinference procedure is also true in the realworld.”



Logic

GroundingHow do we know that KB is true in the real

world? The simple answer is that the agent’s sensors create

the connection.

What about the rest of the agent’s knowledge? Knowledge that is not a direct representation of a

single percept could be produced by a sentenceconstruction procedure called learning.

KB may not be true in the real world, but with goodlearning procedures there is reason for optimism.


Tea Time

Wumpus Worldhttp://www.youtube.com/watch?v=TgRXLA1EY4A

Wumpus World Gamehttp://www.inthe70s.com/games/wumpus/index.shtml#

http://www.funzac.com/play/Wumpus%20World.html


Propositional Logic

SyntaxA proposition symbol stands for a proposition

that can be true or false. special symbols: True and False

The atomic sentences are indivisible syntacticelements. They consist of a single propositionsymbol.

Complex sentences are constructed fromsimpler sentences using logical connectives.

A literal is either an atomic sentence or anegated atomic sentence.


Propositional Logic

Semantics In propositional logic, a model simply fixes the

truth value for every proposition symbol.The semantics must specify how to compute the

truth value of any sentence, given a model.


Truth table


Propositional Logic

“P Q”says that “If P is true, then I amclaiming that Q is true. Otherwise, I ammaking no claim.”

“P Q”shows that it is true wheneverboth “P Q”and “Q P”are true. e.g.

B1,1 (P1,2 P2,1)

B1,1 (P1,2 P2,1) is true but incomplete.


Propositional Logic

We often omit the parentheses by obeyingthe order of precedence (from highest to lowest):, , , , and .

PQR S is equivalent to ((P)(QR)) SWe allow ABC, ABC, and ABC.However, we do not allow ABC since it is

ambiguous. A(BC) and (AB)C havedifferent meaning.


Propositional Logic

A logical knowledge base is a conjunction ofsentences.

If we start with an empty KB and doTELL(KB, S1), …TELL(KB, Sn) then we haveKB = S1 …Sn.


Propositional Logic

A simple knowledge base for the wumpusworld (only considering the pits)There is no pit in [1, 1].

R1: P1, 1

A square is breezy if and only if there is a pit ina neighboring square. (True in all wumpus worlds)

R2: B1,1 (P1,2 P2,1) R3: B2,1 (P1,1 P2,2 P3,1)

Agent percepts R4: B1,1

R5: B2,1

P?

BOK


Inference

The aim of logical inference is to decidewhether KB╞for some sentence .

Our first algorithm will enumerate themodels and check that is true in everymodel in which KB is true. e.g. In the previous slide, we have seven

relevant proposition symbols. There are 27 =128 possible models, and in three of them KB istrue.


Inference

1 = P1, 2

BOK

R1: P1, 1

R2: B1,1 (P1,2 P2,1)R3: B2,1 (P1,1 P2,2 P3,1)R4: B1,1

R5: B2,1


128 models


Inference

ExerciseWrite down the related rules.Apply the truth table to do model checking.

Wumpus world: Is there a breeze in [2,2]?

Minesweeper: Where is the mine?

B

B?

1

1

1

1

1

A B

C

D


Inference



Inference

AnalysisTT-Entails is sound and complete. However, … If KB and contain n symbols, then there are 2n

possible models.The time complexity is O(2n) and the space

complexity is O(n).

We will see more efficient algorithms later. But every known inference algorithm for propositional

logic has a worst-case complexity that is exponentialin the size of the input. (Propositional entailment isco-NP-complete.)


Inference

Before we plunge into the details of logicalinference algorithms, we need someadditional concepts related to entailment.

equivalence validity satisfiability


Inference

Logical equivalenceTwo sentences and are logically equivalent if

they are true in the same set of models. Wewrite this as .

An alternative definition is

if and only if ╞and ╞.


Inference

Logical equivalence



Inference

Validity (tautology)A sentence is valid if it is true in all models.

The deduction theorem:

For any sentences and , ╞if and only ifthe sentence ( ) is valid.

We can think of the TT-ENTAILS algorithm aschecking the validity of (KB ). Conversely, every valid implication sentence describes

a legitimate inference.


Inference

SatisfiabilityA sentence is satisfiable if it is true in some

model. If a sentence is true in a model m, we say

that m satisfies or that m is a model of .

Determining the satisfiability of sentences inpropositional logic was the first problem provedto be NP-complete.


Inference

SatisfiabilityValidity and satisfiability are connected.

╞if and only if the sentence () isunsatisfiable.

Proving from by checking theunsatisfiability of () is called proof bycontradiction.

One assumes a sentence to be false and showsthat this leads to a contradiction with knownaxioms .


Reasoning Patterns in PL

Inference rulesModus Ponens

And-Elimination

And-Introduction

Or-Introduction

All of the logical equivalences in slide 41 can be used as inference rules.



Example

R1: P1, 1

R2: B1,1 (P1,2 P2,1)R3: B2,1 (P1,1 P2,2 P3,1)R4: B1,1

R5: B2,1

R6: (B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)R7: (P1,2 P2,1) B1,1

R8: B1,1 (P1,2 P2,1)R9: (P1,2 P2,1)R10: P1,2 P2,1

Biconditional elimination of R2

And elimination of R6

Contraposition of R7

Modus Ponens with R4 and R8

De Morgan’s Rule with R9

OK



The sequence of applications of inferencerules is called a proof.

Finding proofs is exactly like findingsolutions to search problems.The successor function can be defined to

generate all possible applications of inferencerules.



Searching for proofs is an alternative toenumerating models.Although inference in propositional logic is NP-

complete, finding a proof can be highlyefficient. e.g. The previous proof ignores B3,1, P1,1, P2,2, etc.

The simple truth-table algorithm, on the otherhand, would be overwhelmed by the exponentialexplosion of models.



MonotonicityThe set of entailed sentences can only increase

as information is added to the KB.

For any sentences and ,if KB╞, then (KB )╞.

It means that inference rules can be appliedwhenever suitable premises are found in theKBthe conclusion of the rule must followregardless of what else is in the KB.


Conjunctive Normal Form

A sentence expressed as a conjunction ofdisjunctions of literals is said to be inconjunctive normal form (CNF).A sentence in k-CNF has exactly k literals per

clause.

Every sentence can be transformed into aCNF sentence. exercise: B1,1 (P1,2 P2,1)


Conjunctive Normal Form

Example

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

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


Resolution

Unit resolution

Full resolution

li and m are complementary literals.

li and mj are complementary literals.


Resolution

The resulting clause should contain only onecopy of each literal. (The removal of multiplecopies of literals is called factoring.)

The resolution rule applied only todisjunctions of literals. (But, recall that everysentence can be transformed into a 3-CNF sentence.)

ABABA ,


Resolution

Any complete search algorithm, applyingonly the resolution rule, can derive anyconclusion entailed by any knowledge base.

Given that A is true, we cannot generate theconsequence AB. But we can answer whetherAB is true.

This is called refutation completeness, meaningthat resolution can always be used to eitherconfirm or refute a sentence.


A Resolution Algorithm

Resolution-based inference proceduresfollow the principle of proof bycontradiction.

To show that KB╞, we show that (KB)is unsatisfiable. First (KB) is converted into CNF.Then, each pair of clauses that contain

complementary literals is resolved to produce anew clause.



The process continues until one of twothings happen: there are no new clauses that can be added, in

which case KB does not entail ; or, two clauses (P and P) resolve to yield the

empty clause, in which case KB entails



ExampleR2: B1,1 (P1,2 P2,1) // (B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

R4: B1,1

We wish to prove = P1,2


KB

KB

CNF

Any clause containing two complementary literals can be discarded.



AnalysisThe resolution closure RC(S) of a set of clause S

is the set of all clauses derivable by repeatedapplication of the resolution rule to clauses in Sor their derivatives.

RC(S) must be finite because there are onlyfinitely many distinct clauses constructed bysymbols P1, …, Pk appearing in S. PL-RESOLUTION always terminates.

Note that the last sentence might not be true without the factoringstep that removes multiple copies of literals.



AnalysisThe Ground Resolution Theorem:

“If a set of clauses is unsatisfiable, then theresolution closure of those clauses contains theempty clause.”

We can prove this theorem by demonstratingits contrapositive: if the closure RC(S) does notcontain the empty clause, then S is satisfiable.



Analysis (contd.)We can construct a model for S with suitable

truth values for P1, …, Pk.

For i from 1 to k,

if there is a clause containing the literal Pisuch that all its other literals are false under theassignment of P1, …, Pi-1, then assign false to Pi.

otherwise, assign true to Pi.


Forward & Backward Chaining

The completeness of resolution makes it avery important inference method.

In many practical situations, however, thefull power of resolution is not needed.

Real-world KBs often contain only clausesof a restricted kind called Horn clauses.



Horn clausesA Horn clause is a disjunction of literals of

which at most one is positive. In the following algorithm, we assume that each clause

contains exactly one positive literal for simplicity. Exact one positive literal: definite clause Definite clause without negative literal: fact Without positive literal: integrity constraint

Every Horn clause can be written as animplication.



Horn clauses Inference with Horn clauses can be done

through the forward chaining and backwardchaining algorithms.

Deciding entailment with Horn clauses can bedone in time that is linear to the size of KB.



Forward chaining It begins from known facts (the clauses with

only a positive literal) in the KB. If all the premises of an implication are known,

then its conclusion is added to the set of knownfacts.

The process continues until the query is addedor until no further inferences can be made.




HEAD[c]:the positive literal of c



Analysis Forward chaining is sound: every inference is

essentially an application of Modus Ponens. Forward chaining is complete:

Consider the final state of the inferred table afterthe algorithm terminates.

The table contains true for each symbol inferredduring the process, and false for all other symbols.

We can view this table as a model of the KB. Any atomic sentence entailed by the KB must be true

in this model, and thus it is inferred by the algorithm.



Backward chainingAs its name suggest, it works backwards from

the query. If the query q is known to be true, no work is

needed.Otherwise, the algorithm finds those

implications in the KB that conclude q. If all the premises of one of those implications can be

proved true (by backward chaining), then q is true.



Forward chaining is an example of data-drivenreasoning. It can be used within an agent to derive conclusions from

incoming percepts.

Backward chaining is a form of goal-directedreasoning. It is useful for answering specific questions such as

“What should I do now.” Its cost is often lower since it touches only relevant

facts. An agent should share the work between forward and

backward reasoning.

6 logical agentsweb.ntnu.edu.tw/~tcchiang/ai/6_logical agents (s).pdf · 2010. 6. 9. · “logical...

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