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Lecture 18
Putting First-Order Logic to Work
CSE 573
Artificial Intelligence I
Henry Kautz
Fall 2001
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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?
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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
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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.
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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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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
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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
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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]
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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).
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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.
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Example Ontology: Categories
anything
abstract
physical
position
machine
animate
animal
emotion
happiness
robot
human
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Example Ontology: Subtypes
anything
abstract
physical
position
machine
animate
animal
emotion
happiness
robot
human
 x . (anything(x) (physical(x)  abstract(x)))
 x . (robot(x)  machine(x))
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Example Ontology: Relations
anything
abstract
physical
position
machine
animate
emotion
animal
happiness
robot
human
 x . (physical(x)   y . (position(y)  location(x,y))
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Example Ontology: Instances
anything
abstract
physical
position
machine
animate
animal
emotion
happiness
robot
R2D2
human
robot(R2D2)
location(R2D2, X24Y99Z33)  position(X24Y99Z33)
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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
• 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.
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Ontologies + XML =
Semantic Web (Tim Berners-Lee)
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Representing Change
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