Csci360 12 Logic Programming

8/3/2019 Csci360 12 Logic Programming

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CSCI 360

Survey Of ProgrammingLanguages

12 Logic Programming

Spring, 2008Doug L Hoffman, PhD


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CSCI 360 Survey Of Programming Languages

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

Logicprogramming language or declarativeprogramming language

Express programs in a form of symbolic logic Use a logical inferencing process to produce

results

Declarativerather that procedural:

Only specification of resultsare stated (not detailedproceduresfor producing them)


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Predicate Calculus



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Proposition

A logical statement that may or may not be true

Consists of objects and relationships of objects toeach other


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Symbolic Logic

Logic which can be used for the basic needs offormal 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 predicatecalculus


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Object Representation

Objects in propositions are represented by simpleterms: 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 propositionsconsist 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 b

a b

a implies b

b 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 thatP 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 theoremsthat can be inferred from known axioms andtheorems

Resolution: an inference principle that allowsinferred propositions to be computed from givenpropositions


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Resolution

Unification: finding values for variables inpropositions that allows matching process tosucceed

Instantiation: assigning temporary values tovariables to allow unification to succeed

After instantiating a variable with a value, ifmatching fails, may need to backtrackandinstantiate 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, onlyrestricted 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 eachstatement

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 theprocess 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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Terms

Edinburgh Syntax

Term: a constant, variable, or structure

Constant: an atom or an integer

Atom: symbolic value of Prolog

Atom consists of either:

a string of letters, digits, and underscores beginning witha lowercase letter

a string of printable ASCII characters delimited byapostrophes


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Terms: Variables and Structures

Variable: any string of letters, digits, andunderscores beginning with an uppercase letter

Instantiation: binding of a variable to a value

Lasts only as long as it takes to satisfy one completegoal

Structure: represents atomic proposition

functor(parameter list)


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Fact Statements

Used for the hypotheses

Headless Horn clauses

female(shelley).

male(bill).

father(bill, jake).


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Rule Statements

Used for the hypotheses

Headed Horn clause

Right side: antecedent(ifpart) May be single term or conjunction

Left side: consequent(thenpart)

Must be single term

Conjunction: multiple terms separated by logicalAND operations (implied)


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Example Rules

ancestor(mary,shelley):- mother(mary,shelley).

Can use variables (universalobjects) to generalizemeaning:

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 ofproposition that we want system to prove ordisprovegoal statement

Same format as headless Horn

man(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 asubgoal

To prove a goal is true, must find a chain of inference rulesand/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 sequencethat 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 offacts 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 useeither

Depth-first search: find a complete proof for the firstsubgoal 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 showtruth of one of subgoals, reconsider previoussubgoal to find an alternative solution: backtracking

Begin search where previous search left off

Can take lots of time and space because may findall possible proofs to every subgoal


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Simple Arithmetic

Prolog supports integer variables and integerarithmetic

isoperator: takes an arithmetic expression as

right operand and variable as left operand

A is B / 17 + C

Not the same as an assignment statement!


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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 isSpeed * Time.


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Practical Matters

Loading a file?- consult('lists.pl').

?- reconsult('lists.pro').

?- [file1,file2,file3].

Exiting gprolog?- halt. or ^D

Manually enter code

?- [user].

^D

To see your code

?- listing.


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Trace

Built-in structure that displays instantiations ateach step

Tracing modelof execution - four events:

Call(beginning of attempt to satisfy goal)

Exit(when a goal has been satisfied)

Redo(when backtrack occurs)

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: Factorial Definition

factorial(0,1).

factorial(N,F) :- N>0,

N1 is N-1,

factorial(N1,F1),F is N * F1.

The Prolog goal to calculate the factorial of the number 3

responds with a value for W, the goal variable:

?- factorial(3,W).

W=6


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Example: Factorial Definition with Trace

?- trace.% The debugger will first creep -- showing everythingyes[trace]?- factorial(3,X).(1) 0 Call: factorial(3,_8140) ?

(1) 1 Head [2]: factorial(3,_8140) ?(2) 1 Call (built-in): 3>0 ?(2) 1 Done (built-in): 3>0 ?(3) 1 Call (built-in): _8256 is 3-1 ?(3) 1 Done (built-in): 2 is 3-1 ?

(4) 1 Call: factorial(2, _8270) ?...(1) 0 Exit: factorial(3,6) ?

X=6[trace]?- notrace.

% The debugger is switched off


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Negation as Failure

The negation-as-failure 'not' predicate could be defined inprolog as follows:

not(P) :- call(P), !, fail.

not(P).

The goal ?-call(P) forces an attempt to satisfy the goal P.

Gprolog uses \+ operator instead of not.

Another way one can write the 'not' definition is using the

Prolog implication operator '->' :

not(P) :- (call(P) -> fail ; true).

The body can be read "if call(P) succeeds (i.e., if P succeeds

as a goal) then fail, otherwise succeed". The semicolon ';'

appearing here has the meaning of exclusive logical-or.


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Example

It is important to realize that in a goal ?-not(g), if g hasvariables, and some instance of these variables lead tosatisfaction of g, then ?-not(g) fails.

For example, consider the following 'bachelor program:

bachelor(P) :- male(P), not(married(P)).

male(henry).male(tom).

married(tom).


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Example

Then?- bachelor(henry).yes?- bachelor(tom).no?- bachelor(Who).

Who= henry ;no?- not(married(Who)).no.

The first three responses are correct and as expected but the answer

to the fourth query might have been unexpected.The goal ?-not(married(Who)) fails because for the variable bindingWho=tom, married(Who) succeeds, and so the negative goal fails.

Thus, negative goals ?-not(g) with variables cannot be expected toproduce bindings of the variables for which the goal g fails.

CSC S O


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List Structures


CSCI 360 S Of P i L


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List Structures

Other basic data structure (besides atomicpropositions we have already seen): list

Listis a sequence of any number of elements

Elements can be atoms, atomic propositions, orother 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

CSCI 360 S Of P i L


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Applications of Logic Programming

Relational database managementsystems

Expert systems

Natural language processing

CSCI 360 S r e Of Programming Lang ages


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Summary




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Summary

Symbolic logic provides basis for logicprogramming

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 thecurrent state of logic programming it has been usedin a number of areas



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Csci360 12 Logic Programming