Propositional Logic Russell and Norvig Chapter 7.

Propositional LogicPropositional Logic

Russell and NorvigChapter 7

Knowledge-Based AgentKnowledge-Based Agent

environmentagent

?

sensors

actuators

Knowledge base

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

Types of KnowledgeTypes of Knowledge

Procedural, e.g.: functions Such knowledge can only be used in one way -- by executing it

Declarative, e.g.: constraints It can be used to perform many different sorts of inferences

LogicLogic

Logic is a declarative language to:

Assert sentences representing facts that hold in a world W (these sentences are given the value true)

Deduce the true/false values to sentences representing other aspects of W

Wumpus World PEAS descriptionPerformance measure

gold +1000, death -1000 -1 per step, -10 for using the arrow

Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Releasing drops the gold in same square

Sensors: Stench, Breeze, Glitter, Bump, ScreamActuators: Left turn, Right turn, Forward, Grab, Release, Shoot

Wumpus world characterization

Fully Observable No – only local perceptionDeterministic Yes – outcomes exactly specifiedEpisodic No – sequential at the level of actionsStatic Yes – Wumpus and Pits do not moveDiscrete YesSingle-agent? Yes – Wumpus is essentially a natural feature

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Exploring a wumpus world

Logic in general

Logics are formal languages for representing information such that conclusions can be drawnSyntax defines the sentences in the languageSemantics define the "meaning" of sentences; i.e., define truth of a sentence in a

world

Connection World-Connection World-

RepresentationRepresentation

World W

Conceptualization

Facts about W

hold

hold

Sentences

represent

Facts about W

represent

Sentencesentail

Examples of LogicsExamples of Logics

Propositional calculus A B C First-order predicate calculus ( x)( y) Mother(y,x) Logic of Belief B(John,Father(Zeus,Cronus))

ModelModel

A model of a sentence is an assignment of a truth value – true or false – to every atomic sentence such that the sentence evaluates to true.

Model of a KBModel of a KB

Let KB be a set of sentences

A model m is a model of KB iff it is a model of all sentences in KB, that is, all sentences in KB are true in m.

Satisfiability of a KBSatisfiability of a KB

A KB is satisfiable iff it admits at least one model; otherwise it is unsatisfiable

KB1 = {P, QR} is satisfiable

KB2 = {PP} is satisfiable

KB3 = {P, P} is unsatisfiable

valid sentenceor tautology

Logical EntailmentLogical Entailment

KB : set of sentences : arbitrary sentence KB entails – written KB – iff every model of KB is also a model of Alternatively, KB iff {KB,} is unsatisfiable KB is valid

Inference RuleInference Rule

An inference rule {, } consists of 2 sentence patterns and called the conditions and one sentence pattern called the conclusion If and match two sentences of KB then the corresponding can be inferred according to the rule

InferenceInference

I: Set of inference rules KB: Set of sentences Inference is the process of applying successive inference rules from I to KB, each rule adding its conclusion to KB

Example: Modus PonensExample: Modus Ponens

From Battery-OK Bulbs-OK Headlights-Work Battery-OK Bulbs-OKInfer Headlights-Work

{ , }

{, }

Connective symbol (implication)

Logical entailment

Inference

KB iff KB is valid

SoundnessSoundness

An inference rule is sound if it generates only entailed sentences All inference rules previously given are sound, e.g.:modus ponens: { , } The following rule: { , } is unsound, which does not mean it is useless (an inference rule for abduction, outside scope of this course)

Is each of the following a sound inference rule?

{ , }

{ , }

CompletenessCompleteness

A set of inference rules is complete if every entailed sentences can be obtained by applying some finite succession of these rulesModus ponens alone is not complete, e.g.:from A B and B, we cannot get A

ProofProof

The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

ProofProof

The proof of a sentence from a set of sentences KB is the derivation of by applying a series of sound inference rules

1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Battery-OK Starter-OK by 5,610. Battery-OK Starter-OK Empty-Gas-Tank by 9,711. Engine-Starts by 2,1012. Engine-Starts Flat-Tire by 3,813. Flat-Tire by 11,12

Inference ProblemInference Problem

Given: KB: a set of sentence : a sentence

Answer: KB ?

KB iff {KB,} is unsatisfiable

Deduction vs. Satisfiability Deduction vs. Satisfiability TestTest

Hence:• Deciding whether a set of

sentences entails another sentence, or not

• Testing whether a set of sentences is satisfiable, or not

are closely related problems

Complementary LiteralsComplementary Literals

A literal is a either an atomic sentence or the negated atomic sentence, e.g.: P, P

Two literals are complementary if one is the negation of the other, e.g.: P and P

Unit Resolution RuleUnit Resolution Rule

Given two sentences: L1 … Lp and M where Li,…, Lp and M are all literals, and M and Li are complementary literalsInfer: L1 … Li-1 Li+1 … Lp

ExamplesExamplesFrom:

Engine-Starts Car-OKEngine-Starts

Infer:Car-OK From:

Engine-Starts Car-OKCar-OK

Infer: Engine-Starts

Modus ponens

Modus tollens

Engine-Starts Car-OK

Shortcoming of Unit Shortcoming of Unit ResolutionResolution

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Empty-Gas-Tank

we can infer nothing!

Full Resolution RuleFull Resolution Rule

Given two clauses: L1 … Lp and M1 … Mq where Li and Mj are complementrary

Infer the clause: L1 … Li-1Li+1…LkM1 … Mj-1Mj+1…Mk

ExampleExample

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Empty-Gas-Tank

Infer:

Empty-Gas-Tank Flat-Tire Car-OK

ExampleExample

From:

P Q ( P Q)

Q R ( Q R)

Infer:

P R ( P R)

Not All Inferences are Not All Inferences are Useful! Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK

Not All Inferences are Not All Inferences are Useful!Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK

tautology

Not All Inferences are Not All Inferences are Useful!Useful!

From:

Engine-Starts Flat-Tire Car-OK

Engine-Starts Flat-Tire

Infer:

Flat-Tire Flat-Tire Car-OK True

tautology

ExampleExample1. Battery-OK Bulbs-OK Headlights-Work2. Battery-OK Starter-OK Empty-Gas-Tank Engine-Starts3. Engine-Starts Flat-Tire Car-OK4. Headlights-Work5. Battery-OK6. Starter-OK 7. Empty-Gas-Tank 8. Car-OK 9. Flat-Tire

We want to show Flat-Tire, given clauses 1-8. Using resolution, we can showthat clauses 1-8 along with clause 9 deduce an empty clause.

Can you trace the resolution steps?

Sentence Sentence Clause Form Clause FormExample:

(A B) (C D)

1. Eliminate (A B) (C D)2. Reduce scope of (A B) (C D)3. Distribute over

(A (C D)) (B (C D))(A C) (A D) (B C) (B D)

Set of clauses:{A C , A D , B C , B D}

Resolution Refutation Resolution Refutation AlgorithmAlgorithm

RESOLUTION-REFUTATION(KB)clauses set of clauses obtained from KB and new {}Repeat:

For each C, C’ in clauses dores RESOLVE(C,C’)If res contains the empty clause then

return yesnew new U res

If new clauses then return noclauses clauses U new

Efficient Propositional Inference

Two families of efficient algorithms for propositional inference:

Complete backtracking search algorithmsDPLL algorithm (Davis, Putnam, Logemann, Loveland)Incomplete local search algorithms

WalkSAT algorithm

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

Improvements over truth table enumeration:1. Early termination

A 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.

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

Horn ClausesHorn Clause

A clause with at most one positive literal. KB: A Horn clause with one positive literal which can be written as α1 … αn β

Query: A Horn clause without positive literal α1 … αn

I.e. ( α1 … αn )

Horn clause logic is the basis for Logic Programming

Forward chaining for Horn Clauses

Idea: fire any rule whose premises are satisfied in the KB,

add its conclusion to the KB, until query is found

Backward chaining for Horn Clasues

Idea: 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 subgoal1. has already been proved true, or2. has already failed

3.

SummarySummary

Propositional Logic Model of a KB Logical entailment Inference rules Resolution rule Clause form of a set of sentences Resolution refutation algorithm DPLL algorithm Horn clauses