1-1 Introduction Logic programming languages, sometimes called declarative programming languages Express programs in a form of symbolic logic Use a logical.

1-1

Introduction

• Logic programming languages, sometimes called declarative programming languages

• 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)

1-2

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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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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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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The Origins of Prolog

• University of Aix-Marseille– Natural language processing

• University of Edinburgh– Automated theorem proving

1-9

Example Problem

We are given three colors – red, yellow and blue – with which to color all the countries on a map. The colors must be chosen so that no two countries that share a border have the same color. The shapes of the countries are arbitrary, but a country must consist of a single piece. Countries that meet a corner are not considered a border.

Is it possible?

1-10

Three Color Mapping Problem

Not always possible to solve

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Three Color Mapping Problem

• From the viewpoint of logic, it is easy to set down the parameters of the problem, that is, to assert what properties a solution, if it exist, must have.

• Take the map on the right, our solution should be a list of colors, A, B, C, D, E, F such that– A is different from B, C, D and F– B is different from C, E– C is different from D and E– D is different from E– E is different from F

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Three Color Mapping Problem

• To render the problem in Prolog, we must first state the fact that the colors red, yellow and blue are different, and assert the conditions just stated about this particular map.

different(yellow, red)

different(yellow, blue)

different(blue, yellow)

different(blue, red)

different(red, yellow)

different(red, blue)

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Three Color Mapping Problem

->map-coloring(A, B, C, D, E, F):- different(A, B), different(A, C), different(A, D), different(A, F), different(B, C), different(B, E), different(C, E), different(C, D), different(D, E), different(E, F)

yellow blue red blue yellow blueSatisfied

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Execution of Prolog Programs

• Prove that goal is satisfiable• Search of facts/rules is top-down• Execution of sub-goals is left to right• Closed-world assumption:

– Anything not in database is false

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Applications of Prolog

• Relational database queries• Expert systems• Parsing of context-free languages• Natural language processing• Teaching programming

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Other Examples

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

• Can you prove or disprove the following statements?

Fact:

father(fred,Mike) Query:parent(fred,Mike)

man(fred)

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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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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)

1-20

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).