CSE 5731 Lecture 18 Putting First-Order Logic to Work CSE 573 Artificial Intelligence I Henry Kautz Fall 2001

CSE 5732 Applications 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?

CSE 5733 Theorem 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

CSE 5734 Theorem 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.

CSE 5735 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

CSE 5736 Progress 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

CSE 5737 Programming 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 5738 Prolog: 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]

CSE 5739 Deductive 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).

CSE Ontologies 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.

CSE Example Ontology: Categories anything physical abstract machineanimalanimate robothuman position emotion happiness

CSE Example Ontology: Subtypes anything physical abstract machineanimalanimate robothuman position emotion happiness  x. (anything(x)  (physical(x)  abstract(x)))  x. (robot(x)  machine(x))

CSE Example Ontology: Relations anything physical abstract machineanimalanimate robothuman position emotion happiness  x. (physical(x)   y. (position(y)  location(x,y)) location experience

CSE Example Ontology: Instances anything physical abstract machineanimalanimate robothuman position emotion happiness robot(R2D2) location(R2D2, X24Y99Z33)  position(X24Y99Z33) location experience R2D2

CSE 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

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

CSE Representing Change