CSE 573 1

Lecture 18Putting First-Order Logic to Work

Lecture 18Putting First-Order Logic to Work

CSE 573

Artificial Intelligence IHenry Kautz

Fall 2001

CSE 573 2

Applications of FOLApplications of FOL

• Proving Theorems in Mathematics

• Proving Programs Correct

• Programming

• Ontologies – what kids of things makes up the world?

• Planning – how does the world change over time?

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Theorem ProvingTheorem Proving

Mathematics• Small number of axioms• Deep results

In 1933, E. V. Huntington presented the following basis for Boolean algebra: • x + y = y + x. [commutativity] • (x + y) + z = x + (y + z). [associativity] • n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation]

Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one:

• n(n(x + y) + n(x + n(y))) = x. [Robbins equation] Algebras satisfying commutativity, associativity, and the Robbins equation

became known as Robbins algebras. Question: is every Robbins algebra Boolean?

Solved October 10, 1996, by the theorem prover EQP• Open question for 63 years! • William McCune, Argonne National Laboratory • 8 days on an RS/6000 processor

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Theorem ProvingTheorem Proving

Program Verification• Huge number of axioms

• Tedious (but vital!) results

J. Strother Moore (Boyer-Moore Theorem Prover)

Correctness of the AMD5k86 Floating-Point Division: If p and d are double extended precision floating-point numbers (d /= 0) and mode is a rounding mode specifying a rounding style and target format of precision n not exceeding 64, then the result delivered by the K5 microcode is p/d rounded according to mode.

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The TPTP (Thousands of Problems for Theorem Provers) Problem Library

The TPTP (Thousands of Problems for Theorem Provers) Problem Library

Logic Combinatory logic Logic calculi Henkin models Mathematics Set theory Graph theory Algebra Boolean algebra Robbins algebra Left distributive Lattices Groups Rings General algebra Number theory Topology Analysis

Geometry Field theory Category theory Computer science Computing theory Knowledge representation Natural Language Processing Planning Software creation Software verification Engineering Hardware creation Hardware verification Social sciences Management Syntactic Puzzles Miscellaneous

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ProgressProgress

Automated theorem proving “stalled” in 1980’s

Recent resurgence

• Massive memory, speed

• Code sharing via web– you can download a program to do your homework!

• Integration of propositional reasoning techniques into FOL theorem proving– clever heuristics for grounding formulas

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Programming in LogicProgramming in Logic

Prolog – FOL as a programming logic• FO Horn clauses

– still Turing complete!– restricted form of resolution theorem proving– Idea: Predicate = Program– Function symbols = way to build data structures

[ 1, 2, 3 ] = cons(1,cons(2,cons(3,nil)))

X, T . member(X, cons(X, T))

X, Y, T . (member(X, T) member(X, cons(Y,T)))

member(X, [X|T]). member(X, [Y|T]) :- member(X, T). query: member(3, [1, 2, 3, 4]) returns true

CSE 573 8

Prolog: Computing ValuesProlog: Computing Values

append([], L, L).

append([H|L1], L2, [H|L3]) :- append(L1,L2,L3).

queries:

• append([1,2],[3,4],[1,2,3,4]) returns true

• append([1,2],[3,4],X) returns X = [1,2,3,4]

• append([1,2],Y,[1,2,3,4]) returns Y=[3,4]

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Deductive DatabasesDeductive Databases

Datalog: • Facts = DB relations (tables)

• Rules = Prolog without function symbols

• Decidable, but PTIME-complete

salary_by_name(X,Y) :- ssn(X,N) & salary_by_ssn(N,Y).

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OntologiesOntologies

on·tol·o·gy n. The branch of metaphysics that deals with the nature of being.

AI definition: a set of axioms that describe some aspect of the world in terms of types of objects and relationships between objects.

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Example Ontology: CategoriesExample Ontology: Categories

anything

physicalabstract

machine animalanimate

robot human

position emotion

happiness

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Example Ontology: SubtypesExample Ontology: Subtypes

anything

physicalabstract

machine animalanimate

robot human

position emotion

happiness

x . (anything(x) (physical(x) abstract(x)))

x . (robot(x) machine(x))

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Example Ontology: RelationsExample Ontology: Relations

anything

physicalabstract

machine animalanimate

robot human

position emotion

happiness

x . (physical(x) y . (position(y) location(x,y))

location

experience

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Example Ontology: InstancesExample Ontology: Instances

anything

physicalabstract

machine animalanimate

robot human

position emotion

happiness

robot(R2D2)location(R2D2, X24Y99Z33) position(X24Y99Z33)

location

experience

R2D2

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Why Formalize Ontologies?Why Formalize Ontologies?

Knowledge exchange and reuse

• Common syntax not enough

• How do the your meanings relate to my meanings?– Is Bill Gate’s meaning of “expensive” the same as

mine?

• What to do when we have different ways of conceptualizing the world?– Eskimo’s words for snow

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Categories in Dyirbal, an aboriginal language of Australia

Categories in Dyirbal, an aboriginal language of Australia

• Bayi: men, kangaroos, possums, bats, most snakes, most fishes, some birds, most insects, the moon, storms, rainbows, boomerangs, some spears, etc.

• Balan: women, anything connected with water or fire, bandicoots, dogs, platypus, echidna, some snakes, some fishes, most birds, fireflies, scorpions, crickets, the stars, shields, some spears, some trees, etc.

• Balam: all edible fruit and the plants that bear them, tubers, ferns, honey, cigarettes, wine, cake.

• Bala: parts of the body, meat, bees, wind, yamsticks, some spears, most trees, grass, mud, stones, noises, language, etc.

CSE 573 17

Ontologies + XML = Semantic Web (Tim Berners-Lee)

Ontologies + XML = Semantic Web (Tim Berners-Lee)

CSE 573 18

Representing ChangeRepresenting Change