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Transcript of ISBN 0-321-49362-1 Chapter 16 Logic Programming Languages.
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ISBN 0-321-49362-1
Chapter 16
Logic Programming Languages
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Copyright © 2007 Addison-Wesley. All rights reserved. 1-2
Chapter 16 Topics
• Introduction• A Brief Introduction to Predicate Calculus• Predicate Calculus and Proving Theorems• An Overview of Logic Programming• The Origins of Prolog• The Basic Elements of Prolog• Deficiencies of Prolog• Applications of Logic Programming
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Introduction
• Logic programming language or declarative programming language
• Express programs in a form of symbolic logic
• Use a logical inferencing process to produce results
• Declarative rather that procedural:– Only specification of results are stated (not
detailed procedures for producing them)
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Proposition
• A logical statement that may or may not be true– Consists of objects and relationships of objects
to each other
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Symbolic Logic
• Logic which can be used for the basic needs of formal logic:– Express propositions– Express relationships between propositions– Describe how new propositions can be inferred
from other propositions
• Particular form of symbolic logic used for logic programming called predicate calculus
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Object Representation
• Objects in propositions are represented by simple terms: either constants or variables
• Constant: a symbol that represents an object
• Variable: a symbol that can represent different objects at different times– Different from variables in imperative
languages
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Compound Terms
• Atomic propositions consist of compound terms
• Compound term: one element of a mathematical relation, written like a mathematical function– Mathematical function is a mapping– Can be written as a table
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Parts of a Compound Term
• Compound term composed of two parts– Functor: function symbol that names the
relationship– Ordered list of parameters (tuple)
• Examples:student(jon)
like(seth, OSX)
like(nick, windows)
like(jim, linux)
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Forms of a Proposition
• Propositions can be stated in two forms:– Fact: proposition is assumed to be true– Query: truth of proposition is to be determined
• Compound proposition:– Have two or more atomic propositions– Propositions are connected by operators
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Logical Operators
Name Symbol Example Meaning
negation a not a
conjunction a b a and b
disjunction a b a or b
equivalence a b a is equivalent to b
implication
a ba b
a implies bb implies a
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Quantifiers
Name Example Meaning
universal X.P For all X, P is true
existential X.P There exists a value of X such that P is true
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Clausal Form
•Too many ways to state the same thing•Use a standard form for propositions•Clausal form:– B1 B2 … Bn A1 A2 … Am
– means if all the As are true, then at least one B is true
•Antecedent: right side•Consequent: left side
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Predicate Calculus and Proving Theorems
• A use of propositions is to discover new theorems that can be inferred from known axioms and theorems
• Resolution: an inference principle that allows inferred propositions to be computed from given propositions
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Resolution
• Unification: finding values for variables in propositions that allows matching process to succeed
• Instantiation: assigning temporary values to variables to allow unification to succeed
• After instantiating a variable with a value, if matching fails, may need to backtrack and instantiate with a different value
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Proof by Contradiction
• Hypotheses: a set of pertinent propositions
• Goal: negation of theorem stated as a proposition
• Theorem is proved by finding an inconsistency
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Theorem Proving
• Basis for logic programming• When propositions used for resolution,
only restricted form can be used• Horn clause - can have only two forms
– Headed: single atomic proposition on left side– Headless: empty left side (used to state facts)
• Most propositions can be stated as Horn clauses
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Overview of Logic Programming
• Declarative semantics– There is a simple way to determine the
meaning of each statement– Simpler than the semantics of imperative
languages
• Programming is nonprocedural– Programs do not state now a result is to be
computed, but rather the form of the result
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Example: Sorting a List
• Describe the characteristics of a sorted list, not the process of rearranging a list
sort(old_list, new_list) permute (old_list, new_list) sorted (new_list)
sorted (list) j such that 1 j < n, list(j) list (j+1)
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The Origins of Prolog
• University of Aix-Marseille– Natural language processing
• University of Edinburgh– Automated theorem proving
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Terminology - Edinburgh Syntax
• Atom: symbolic value which consists of either:
– a string of letters, digits, and underscores • beginning with a lowercase letter
– a string of printable ASCII characters • delimited by apostrophes
• Constant: an atom or an integer• Variable:
– a string of letters, digits, and underscores • beginning with an uppercase letter
• Structure: represents atomic propositionfunctor(parameter list)
• Term: a constant, variable, or structure• Instantiation: binding of a variable to a value
– Lasts only as long as it takes to satisfy one complete goal
• Example: sibling(X,Y):- mother(M,X), mother(M,Y),father(F,X), father(F,Y).
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Fact Statements
• Used for the hypotheses• Headless Horn clausesfemale(shelley).
male(bill).
father(bill, jake).
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Rule Statements
• Used for the hypotheses• Headed Horn clause• Right side: antecedent (if part)
– May be single term or conjunction
• Left side: consequent (then part)– Must be single term
• Conjunction: multiple terms separated by logical AND operations (implied)
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Example Rules
ancestor(mary,shelley):- mother(mary,shelley).
• Can use variables (universal objects) to generalize meaning:parent(X,Y):- mother(X,Y).parent(X,Y):- father(X,Y).grandparent(X,Z):- parent(X,Y), parent(Y,Z).sibling(X,Y):- mother(M,X), mother(M,Y),
father(F,X), father(F,Y).
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Goal Statements
• For theorem proving, theorem is in form of proposition that we want system to prove or disprove – goal statement
• Same format as headless Hornman(fred)
• Conjunctive propositions and propositions with variables also legal goalsfather(X,mike)
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Inferencing Process of Prolog
• Queries are called goals• If a goal is a compound proposition, each of the
facts is a subgoal• To prove a goal is true, must find a chain of
inference rules and/or facts. For goal Q:B :- A
C :- B
…Q :- P
• Process of proving a subgoal called matching, satisfying, or resolution
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Approaches
• Bottom-up resolution, forward chaining– Begin with facts and rules of database and attempt to
find sequence that leads to goal– Works well with a large set of possibly correct answers
• Top-down resolution, backward chaining– Begin with goal and attempt to find sequence that leads
to set of facts in database– Works well with a small set of possibly correct answers
• Prolog implementations use backward chaining
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Subgoal Strategies
• When goal has more than one subgoal, can use either– Depth-first search: find a complete proof for
the first subgoal before working on others– Breadth-first search: work on all subgoals in
parallel
• Prolog uses depth-first search– Can be done with fewer computer resources
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Backtracking
• With a goal with multiple subgoals, if fail to show truth of one of subgoals, reconsider previous subgoal to find an alternative solution: backtracking
• Begin search where previous search left off
• Can take lots of time and space because may find all possible proofs to every subgoal
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Simple Arithmetic
• Prolog supports integer variables and integer arithmetic
• is operator: takes an arithmetic expression as right operand and variable as left operandA is B / 17 + C
• Not the same as an assignment statement!
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Review• Facts
female(shelley).
male(bill).
father(bill, jake).
• Rules ancestor(mary,shelley):- mother(mary,shelley).
• Can use variables to generalize meaning:
parent(X,Y):- mother(X,Y). parent(X,Y):- father(X,Y). grandparent(X,Z):- parent(X,Y), parent(Y,Z). sibling(X,Y):- mother(M,X), mother(M,Y),
father(F,X), father(F,Y).• Goal Statements
man(fred)
father(X,mike)
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Example
speed(ford,100).speed(chevy,105).speed(dodge,95).speed(volvo,80).time(ford,20).time(chevy,21).time(dodge,24).time(volvo,24).distance(X,Y) :- speed(X,Speed),
time(X,Time), Y is Speed * Time.
Distance(chevy, ChevyDistance).
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Goals
Query: man(bob).
Can be satisfied by the existance of a fact: man(bob)
Or a rule such as: father(bob). man(X) :- father(X).
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Goals
male(mike).male(david).parent(david, shelley).male(X), parent(X, shelley).
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Trace
• Built-in structure that displays instantiations at each step
• Tracing model of execution - four events:(1) Call (beginning of attempt to satisfy goal)(2) Exit (when a goal has been satisfied)(3) Redo (when backtrack causes an attempt to
resatisfy a goal)(4) Fail (when goal fails)
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Example
likes(jake,chocolate).
likes(jake,apricots).
likes(darcie,licorice).
likes(darcie,apricots).
trace.
likes(jake,X),
likes(darcie,X).
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Example
likes(jake,chocolate).
likes(jake,apricots).
likes(darcie,licorice).
likes(darcie,apricots).
trace.
likes(jake,X),likes(darcie,X).
(1) 1 Call: likes(jake, _0)?
(1) 1 Exit: likes (jake, chocolate)
(2) 1 Call: likes darcie, chocolate)?
(2) 1 Fail: likes(darcie, chocolate)
(1) 1 Redo: likes (jake, _0)?
(1) 1 Exit: likes (jake, apricots)
(3) 1 Call: likes(darcie, apricots)?
(3) 1 Exit: likes(darcie, apricots)
X = apricots
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Exercise
Write the following English conditional statements in Prolog:
a. If Fred is the father of Mike, then Fred is an ancestor of Mike.
b. If Mike is the father of Joe and Mike is the father of Mary, then Mary is the sister of Joe.
c. If Mike is the brother of Fred and Fred is the father of Mary, then Mike is the uncle of Mary.
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Exercise
a. If Fred is the father of Mike, then Fred is an ancestor of Mike. ancestor(fred, mike) :- father(fred, mike).
Generalized version. Facts: father(fred, mike). father(mike, james). Rules: ancestor(X, Y) :- father(X, Y). Query: ancestor(fred, mike). ancestor(mike, fred).
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Exercise
b. If Mike is the father of Joe and Mike is the father of Mary, then Mary is the sister of Joe.
sister(mary, joe) :- father(mike, joe), father(mike, mary).
Generalize.
Facts: father(mike, joe). father(mike, mary).
Rule: sister(X,Y) :- father(Z, X), father(Z, Y). Query: sister(mary, joe). sister(joe, mary). Is this a problem? How to solve?
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Exercise c. If Mike is the brother of Fred and Fred is the father of Mary, then Mike is the
uncle of Mary.
uncle(mike, mary) :- brother(mike, fred), father(fred, mary).
Generalize. Facts: brother(mike, fred). father(fred, mary).
Rule: uncle(X,Y) :- brother(X, Z), father(Z, Y). Query: uncle(mike, mary) . uncle(mary,mike).
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Exercise
Facts: cousin(david, brian). cousin(david, susan). cousin(carmen, yi). cousin(carmen, david).
Rules: cousin(X,Y) :- cousin(X, Z), cousin(Z, Y).
Goals: cousin(carmen, brian). cousin(david, yi). ? Problem with this
1-41
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List Structures
• Other basic data structure (besides atomic propositions we have already seen): list
• List is a sequence of any number of elements
• Elements can be atoms, atomic propositions, or other terms (including other lists)
[apple, prune, grape, kumquat]
[] (empty list)[X | Y] (head X and tail Y)
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Append Example
append([], List, List).
append([Head | List_1], List_2, [Head | List_3]) :-
append (List_1, List_2, List_3).
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Reverse Example
reverse([], []).
reverse([Head | Tail], List) :-
reverse (Tail, Result),
append (Result, [Head], List).
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Deficiencies of Prolog
• Resolution order control• The closed-world assumption• The negation problem• Intrinsic limitations
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Applications of Logic Programming
• Relational database management systems• Expert systems• Natural language processing
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Summary
• Symbolic logic provides basis for logic programming
• Logic programs should be nonprocedural• Prolog statements are facts, rules, or goals• Resolution is the primary activity of a Prolog
interpreter• Although there are a number of drawbacks
with the current state of logic programming it has been used in a number of areas
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Quantifiers
Name Example Meaning
universal X.P For all X, P is true
existential X.P There exists a value of X such that P is true
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Example: Sorting a List
• Describe the characteristics of a sorted list, not the process of rearranging a list
sort(old_list, new_list) permute (old_list, new_list) sorted (new_list)
sorted (list) j such that 1 j < n, list(j) list (j+1)