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Introduction

Computational Logic Lecture 1

Michael Genesereth Autumn 2011

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Computational Logic

q(b,c)p(a,b)

∀x.∀y.(p(x,y) ⇒ q(x,y))

∃x.p(x,d)

¬p(b,d)

p(c,b)∨ p(c,d)

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Mathematics

Group Axioms���

Theorem

Tasks:Proof CheckingProof Generation

(x × y) × z = x × (y × z)x × e = xe × x = xx × x −1 = e

x −1 × x = e

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Some Successes

Various Theorems4 color theorem (Appel and Haken)the limit of a sum is the sum of the limitsthe Bolzano-Weierstrass Theoremthe Fundamental Theorem of calculusEuler's identityGauss' law of quadratic reciprocitythe undecidability of the halting problemGodel's incompleteness theorem (Shankar)

Other Thousands of Problems for Theorem Provers (TPTP) CADE ATP Systems Competition (CASC)

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Software Engineering

Program

Partial Specification:

Tasks:Program VerificationProof of TerminationComplexity AnalysisPartial Evaluation

sorter L M

€

∀i.∀j.(i < j⇒ Mi < Mj )

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Hardware Engineering

Circuit: Premises:

Conclusion:

x ∧ y ⇒ ¬c

Applications:SimulationConfigurationDiagnosisTest Generation

o⇔ (x ∧ ¬y)∨ (¬x∧ y)a⇔ z ∧ ob⇔ x ∧ ys⇔ (o∧ ¬z)∨ (¬o∧ z)c⇔ a ∨b

xy

z

s

c

o

a

b

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Constraint Satisfaction Systems

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Database Tables

Queriesquery(X,Z) :- parent(X,Y) & parent(Y,Z)

Constraintsillegal :- parent(X,X)illegal :- parent(X,Y) & parent(Y,X)

Deductive Database Systems

parentart bobart beabea coe

parent(art,bob)parent(art,bea)parent(bob,coe)

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Data Integration

Side-by-sideComparison

Infomaster

Manufacturer 1Manufacturer 2

General Data

IntegratedSearch

Product analysis

SatisfactionRatings

Supplier 1Supplier 2

Supplier 3Supplier 4

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Logical Spreadsheets

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Examples of Non-Functional ConstraintsScheduling

–Start times must be before end times–Room 104 may not be scheduled after 5:00 pm–Only senior managers can reserve the third floorconference room

Travel Reservations–The number of lap infants in a group on a flightmust not exceed the number of adults.

Academic Programs–Students must take at least 2 math courses

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Regulations and Business Rules

Using the language of logic, it is possible to definenew relations.

Office mates are people who share an office.

officemate(X,Y) :- office(X,Z) ∧ office(Y,Z)

This includes the property of legality / illegality.

Managers and subordinates may not be office mates.

illegal :- manages(X,Y) ∧ officemate(X,Y)

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Michigan Lease Termination ClauseThe University may terminate this lease when the Lessee, havingmade application and executed this lease in advance ofenrollment, is not eligible to enroll or fails to enroll in theUniversity or leaves the University at any time prior to theexpiration of this lease, or for violation of any provisions of thislease, or for violation of any University regulation relative toresident Halls, or for health reasons, by providing the studentwith written notice of this termination 30 days prior to theeffective data of termination; unless life, limb, or property wouldbe jeopardized, the Lessee engages in the sales of purchase ofcontrolled substances in violation of federal, state or local law, orthe Lessee is no longer enrolled as a student, or the Lesseeengages in the use or possession of firearms, explosives,inflammable liquids, fireworks, or other dangerous weaponswithin the building, or turns in a false alarm, in which cases amaximum of 24 hours notice would be sufficient.

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Logical VersionA ⇐ A1 ∧ A2 ∧¬BA ⇐ A4 ∧¬BA ⇐ A5 ∧¬BA ⇐ A6 ∧¬BA ⇐ A7 ∧¬B

B ⇐ B1B ⇐ B2B ⇐ B3B ⇐ B4B ⇐ B5

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Elements

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Formal Mathematics

Algebra1. Formal language for encoding information2. Legal transformations3. Automation

Logic1. Formal language for encoding information2. Legal transformations3. Automation

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Algebra Problem

Xavier is three times as old as Yolanda. Xavier's ageand Yolanda's age add up to twelve. How old areXavier and Yolanda?

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Algebra Solution

Xavier is three times as old as Yolanda. Xavier's ageand Yolanda's age add up to twelve. How old areXavier and Yolanda?

Automation: Saint, Sin, Reduce, Macsyma,Mathematica

x − 3y = 0x + y = 12−4y = −12

y = 3x = 9

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Logic Problem

If Mary loves Pat, then Mary loves Quincy. If it isMonday, then Mary loves Pat or Quincy. If it isMonday, does Mary love Quincy?

If it is Monday, does Mary love Pat?

Mary loves only one person at a time. If it isMonday, does Mary love Pat?

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FormalizationSimple Sentences: Mary loves Pat. Mary loves Quincy. It is Monday.

Premises: If Mary loves pat, Mary loves Quincy. If it Monday, Mary loves Pat or Quincy. Mary loves one person at a time.

Questions: Does Mary love Pat? Does Mary love Qunicy?

p⇒ qm⇒ p∨ q

p∧ q⇒

⇒ p⇒ q

pqm

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Rule of Inference

Propositional Resolution

NB: If pi is the same as sj, it is okay to drop the twosymbols, with the proviso that only one such pairmay be dropped.

NB: If a constant is repeated on the left or the right,all but one of the occurrences can be deleted.

p1 ∧ ...∧ pk ⇒ q1 ∨ ...∨ qlr1 ∧ ...∧ rm ⇒ s1 ∨ ...∨ sn

p1 ∧ ...∧ pk ∧ r1 ∧ ...∧ rm ⇒ q1 ∨ ...∨ ql ∨ s1 ∨ ...∨ sn

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Examples

p ⇒ q⇒ p⇒ q

p ⇒ qq ⇒p ⇒

p ⇒ qq ⇒ rp ⇒ r

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Logic Problem Revisited

p ⇒ qm ⇒ p∨qm ⇒ q ∨qm ⇒ q

If Mary loves Pat, then Mary loves Quincy. If it isMonday, then Mary loves Pat or Quincy. If it isMonday, does Mary love Quincy?

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Logic Problem Concluded

m ⇒ qp∧ q ⇒m ∧ p ⇒

Mary loves only one person at a time. If it isMonday, does Mary love Pat?

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Automated Reasoning

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Logic TechnologyLanguages Knowledge Interchange Format (KIF) - ANSI Common Logic - W3C

Some Popular Automated Reasoning SystemsOtter / Snark / Vampire / …PTTP / Epilog

Knowledge Bases Definitions (Bachelor is an unmarried adult male.) Physical Laws (e.g. PV=nRT) Laws (e.g. 1040 necessary if earnings > $10,000.)

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Study Guide

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Multiple Logics

Propositional Logic

If it is raining, the ground is wet.

Relational LogicIf x is a parent of y, then y is a child of x.

Modal LogicJohn believes that all men are mortal.

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Common Topics

Common TopicsSyntax - expressions in the languageSemantics - meaning of expressionsComputational Matters

Contrasts Expressiveness - operators, variables, terms, ... Computational Hierarchy - polynomial? decidable? Tradeoffs - expressiveness versus computability

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Meta

We will frequently write sentences about sentences.

Sentence: When it rains, it pours.Metasentence: That sentence contains two verbs.

We will frequently prove things about proofs.

Proofs: formalMetaproofs: informal

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Mike took it twice!

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http://cs157.stanford.edu