159.302 LECTURE

159.302159.302 LECTURELECTUREPropositional Logic Propositional Logic

SyntaxSyntax

1

Source: MIT OpenCourseWare

Propositional Logic What is a logic?

A formal language


A formal language

• Syntax – what expressions are legal


A formal language

• Syntax – what expressions are legal• Semantics – what legal expressions mean


A formal language


• Proof System – a way of manipulating syntactic expressions to get other syntactic expressions (which will tell us something new)


A formal language


• Proof System – a way of manipulating syntactic expressions to get other syntactic expressions (which will tell us something new)

Why proofs?

• Two kinds of inferences an agent might want to make:


A formal language


• Proof System – a way of manipulating syntactic expressions to get other sytactic expressions (which will tell us something new)

Why proofs?

• Two kinds of inferences an agent might want to make:• Multiple percepts => conclusions about the world


A formal language


• Proof System – a way of manipulating syntactic expressions to get other sytactic expressions (which will tell us something new)

Why proofs?• Two kinds of inferences an agent might want to make:

• Multiple percepts => conclusions about the world

• Current state & operator => properties of next state

Propositional Logic

Syntax: what you’re allowed to write

• for( thing t=fizz; t == fuzz; t++ ){…}

Propositional Logic


• for(thing t=fizz; t == fuzz; t++){…}• Colorless green ideas sleep furiously.

Propositional Logic



Sentences (wwfs: well-formed formulas)

Propositional Logic




• true and false are sentences

Propositional Logic





• Propositional variables are sentences: P, Q, R, Z

Propositional Logic






• If φ and ψ are sentences, then so are ,,,),(

Propositional Logic






• If φ and ψ are sentences, then so are ,,,),(

• Nothing else is a sentence

Precedence


SemanticsSemantics

2


Propositional Logic Semantics

• Meaning of a sentence is truth value {t, f}


• Meaning of a sentence is truth value {t, f}• Interpretation is an assignment of truth values to the propositional variables

holds( , i) [Sentence is t in interpretation i]


• Meaning of a sentence is truth value {t, f}• Interpretation is an assignment of truth values to the propositional variables

holds( , i) [Sentece is t in interpretation i]

fails( , i) [Sentence is f in interpretation i]

Propositional Logic Semantic Rules

holds(true, i) for all i



fails(false, i) for all i




holds( , i) if and only if fails( , i) (negation)





holds( , i) iff holds( ,i) and holds( , i) (conjunction)






holds( , i) iff holds( ,i) or holds( , i) (disjunction)






holds( , i) iff holds( ,i) or holds( , i) (disjunction)

holds( P, i) iff i(P) = t

fails( P, i) iff i(P) = f

Propositional Logic Some important shorthand

(antecedent consequent)),( nimplicatiolconditiona



),( )()( eequivalencnalbiconditio




f f t f f t t tf t t f t t f ft f f f t f t ft t f t t t t t

P Q P QP QP QP PQ QP




f f t f f t t tf t t f t t f ft f f f t f t ft t f t t t t t

P Q P QP QP QP PQ QP

Note that implication is not “causality”, if P is f then is t QP

Implication doesn't mean "is related" in any kind of real-world way

Propositional Logic Terminology

A sentence is valid iff its truth value is t in all interpretations

Pfalse, Ptrue, :sentences Valid

A sentence is satisfiable iff its truth value is t in at least one interpretation

PP, true, :sentences eSatisfiabl

A sentence is unsatisfiable iff its truth value is f in all interpretations

trueP, false, P :sentences bleUnsatisfia

All are finitely decidable.

ExamplesSentence Valid? Interpretation that make

sentence’s truth value = f

valid

satisfiable, not validsatisfiable, not valid

valid

valid

smokesmoke smokesmoke

firesmoke ffiretsmoke ,

)()( fsfs tffs ,ffstfs )(,)(

)()( sffs contrapositive

)( dbdb dbdb

ExamplesExamplesSentence Valid? Interpretation that make

sentence’s truth value = f

valid

satisfiable, not validsatisfiable, not valid

valid

valid

smokesmoke smokesmoke

firesmoke ffiretsmoke ,

)()( fsfs tffs ,ffstfs )(,)(

)()( sffs contrapositive

)( dbdb dbdb

Satisfiability• Related to constraint satisfaction

• Given a sentence S, try to find an interpretation i such that holds(S,i)

• Analogous to finding an assignment of values to variables such that the constraints hold




• Brute force method: enumerate all interpretations and check




• Brute force method: enumerate all interpretations and check

• Better methods:

Heuristic searchConstraint propagationStochastic search

Satisfiability Problems• Scheduling nurses to work in a hospital

• propositional variables represent for example, that Pat is working on Tuesday at 2pm

• constraints on the schedule are represented using logical expressions over the variables

• Finding bugs in software

• propositional variables represent state of the program

• use logic to describe how the program works and to assert there is a bug

• if the sentence is satisfiable, you’ve found the bug!

159.302159.302 LECTURELECTUREPropositional Logic ProofPropositional Logic Proof

3


A Good lecture?A Good lecture?• Imagine we knew that:

• If today is sunny, then Tomas will be happy

Should we conclude that the lecture will be good?

)( HS

• If Tomas is happy, the lecture will be good )( GH

)(S• Today is sunny

Checking InterpretationsChecking Interpretations

SS HH GG S HS H H GH G SS GGtt tt tt tt tt tt tttt tt ff tt ff tt fftt ff tt ff tt tt tttt ff ff ff tt tt ffff tt tt tt tt ff ttff tt ff tt ff ff ffff ff tt tt tt ff ttff ff ff tt tt ff ff

Good lecture!Good lecture!

Adding a variableAdding a variable

LL SS HH GG S HS H H GH G SS GGtt tt tt tt tt tt tt tttt tt tt ff tt ff tt fftt tt ff tt ff tt tt tttt tt ff ff ff tt tt fftt ff tt tt tt tt ff tttt ff tt ff tt ff ff fftt ff ff tt tt tt ff tttt ff ff ff tt tt ff ffff tt tt tt tt tt tt ttff tt tt ff tt ff tt ff…… …… …… ……

EntailmentEntailment•A knowledge base (KBKB) entailsentails a sentence SS iff every interpretation that makes KBKB true also makes SS true

Computing EntailmentComputing Entailment• enumerate all interpretations

• select those in which all elements of KBKB are true

• check to see if SS is true in all of those interpretations

Computing EntailmentComputing Entailment• enumerate all interpretations

• select those in which all elements of KBKB are true

• check to see if SS is true in all of those interpretations

• Way too many interpretations, in general!!

Entailment and ProofEntailment and Proof• A proofproof is a way to test whether a KB entails a sentence, without enumerating all possible interpretations

ProofProof• A proofproof is a sequence of sentences

• First ones are premises (KBKB)

• Then, you can write down on the next line the result of applying an inference ruleinference rule to previous lines

• When S is on a line, you have provedproved SS from KBKB

• If inference rules are soundsound, then any SS you can prove from KBKB is entailed by KBKB

• If inference rules are completecomplete, then any SS that is entailed by KBKB can be proven from KBKB

Natural DeductionNatural Deduction• Some inference rulesinference rules:

Modus ponens

Natural DeductionNatural Deduction• Some inference rules:

Modus ponens Modus tolens



And-Introduction



And-Introduction

And-Elimination

Natural Deduction ExampleNatural Deduction Example

StepStep FormulaFormula DerivationDerivation11 GivenGiven

22 GivenGiven33 GivenGiven

QP

RP

SRQ )(

Prove SProve S




44 PP 1 And-Elim1 And-Elim

QP

RP

SRQ )(

Prove SProve S




44 PP 1 And-Elim1 And-Elim55 RR 4,2 Modus-Ponens4,2 Modus-Ponens

QP

RP

SRQ )(

Prove SProve S




44 PP 1 And-Elim1 And-Elim55 RR 4,2 Modus-Ponens4,2 Modus-Ponens66 QQ 1 And-Elim1 And-Elim

QP

RP

SRQ )(

Prove SProve S




44 PP 1 And-Elim1 And-Elim55 RR 4,2 Modus-Ponens4,2 Modus-Ponens66 QQ 1 And-Elim1 And-Elim77 5,6 And-Intro5,6 And-Intro

QP

RP

SRQ )(

Prove SProve S

RQ




44 PP 1 And-Elim1 And-Elim55 RR 4,2 Modus-Ponens4,2 Modus-Ponens66 QQ 1 And-Elim1 And-Elim77 5,6 And-Intro5,6 And-Intro88 SS 7,3 Modus-Ponens7,3 Modus-Ponens

QP

RP

SRQ )(

Prove SProve S

RQ

Proof SystemsProof Systems• There are many natural deduction systems; they are typically “proof checkers”, sound but not complete


• Natural deductionNatural deduction uses lots of inference rules which introduces a large branching factor in the search for a proof.


• Natural deductionNatural deduction uses lots of inference rules which introduces a large branching factor in the search for a proof.

• In general, you need to do “proof by casesproof by cases” which introduces even more branching.

StepStep FormulaFormula112233

QP

RQ

RP

Prove R

Propositional ResolutionPropositional Resolution Resolution rule:

• Single inference rule is a sound and complete proof system

• Requires all sentences to be converted to conjunctive normal form


Conjunctive Normal Form (CNF)Conjunctive Normal Form (CNF)

4


Conjunctive Normal Conjunctive Normal FormForm

CNF Formulas:

)()()()( CBADBCBA


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA , which is a disjunction of literals, which is a disjunction of literals


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA

literals are ) ,, CandBA

, which is a disjunction of literals, which is a disjunction of literals


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA



, each of which is a variable or the negation of a variable., each of which is a variable or the negation of a variable.


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA




• Each clause is a requirement that must be satisfied and can be Each clause is a requirement that must be satisfied and can be satisfied in multiple wayssatisfied in multiple ways


CNF Formulas:

)()()()( CBADBCBA

clause a is )( CBA




• Each clause is a requirement that must be satisfied and can be Each clause is a requirement that must be satisfied and can be satisfied in multiple wayssatisfied in multiple ways

• Every sentence in propositional logic can be written in CNFEvery sentence in propositional logic can be written in CNF

Converting to CNFConverting to CNF1. Eliminate arrows using definitions


2. Drive in negations using De Morgan’s Laws

))(

))(



))(

))(

3. Distribute or over and

)()()(( CABACBA



))(

))(


)()()(( CABACBA

4. Every sentence can be converted to CNF, but it may grow exponentially in size

CNF Conversion ExampleCNF Conversion Example)()( DCBA

CNF Conversion ExampleCNF Conversion Example

1. Eliminate arrows using definitions

)()( DCBA

)()( DCBA




)()( DCBA

)()( DCBA

)()( DCBA





)()( DCBA

)()( DCBA

)()( DCBA

)()( DCBDCA

159.302159.302 LECTURELECTUREPropositional ResolutionPropositional Resolution

5


Propositional ResolutionPropositional Resolution Resolution rule:

• Convert all sentences to CNF

• Negate the desired conclusion (converted to CNF)

Resolution refutation:

• Apply resolution rule until either

• Derive false (a contradiction)

• Can’t apply anymore

Resolution refutation is soundsound and completecomplete• If we derive a contradiction, then the conclusion follows from the axioms

• If we can’t apply anymore, then the conclusion cannot be proven from the axioms

Propositional ResolutionPropositional Resolution


QP

RP

RQ

Prove R



44 Negated conclusionNegated conclusion55 1,21,266 2,42,477 3,43,488 5,75,799 ▪▪ 4,84,8

QP

RP RQ

R

RQ

P

Q

R



QP

RP

RQ

Prove R




QP

RP RQ

R

RQ

P

Q

RRfalse

falseR

falsefalse



QP

RP

RQ

Prove R




QP

RP RQ

R

RQ

P

Q

RRfalse

falseR

falsefalse

unnecessaryunnecessary

The Power of FalseThe Power of False


PP

Prove Z


22 GivenGiven33 Negated conclusionNegated conclusion44 ▪▪ 1,21,2

validis )( that Note ZPP

P

PZ

• Any conclusion follows from a contradiction – and so strict logic systems are very brittle.

Example ProblemExample Problem


33

QQP )(

Prove R

RPP )(

)()( QSSR

Convert to CNF

Example ProblemExample Problem

StepStep1122

33

44556677 NegNeg

Prove R

QP StepStep FormulaFormula1122

33

QQP )(

RPP )(

)()( QSSR

RP

RP SR

QR

QS

R

Example ProblemExample ProblemStepStep1122

33

44556677 NegNeg88 4,74,799 6,86,81010 1,91,91111 3,103,101212 ▪▪ 7,117,11

Prove R QP


33

QQP )(

RPP )(

)()( QSSR

RP

RP SR

QR

QS

R

Q

S

P

R

Proof StrategiesProof Strategies

• Produces a shorter clause – which is good since we are trying to produce a zero-length clause, that is, a contradiction.

Unit Preference: prefer a resolution step involving a unit clause (clause with one literal).

• We’re trying to produce a contradiction that follows from the negated goal, so these are “relevant” clauses

Set of support: choose a resolution involving the negated goal or any clause derived from the negated goal.

• If a contradiction exists, one can find one using the set-of-support strategy.

159.302 LECTURE

Documents

Transcript of 159.302 LECTURE