Download X - Al Akhawayn University

CSC 3315
Programming Paradigms
Prolog Language
Hamid Harroud
School of Science and Engineering, Akhawayn University
http://www.aui.ma/~H.Harroud/csc3315/
CSC3315 (Spring 2009)
1
Logic Programming
Languages








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
1-2
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)
1-3
Proposition

A logical statement that may or may not be
true

Consists of objects and relationships of objects to
each other
1-4
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
1-5
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
1-6
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
1-7
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(john)
like(nick, windows)
like(jim, linux)
1-8
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
1-9
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
1-10
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
1-11
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

1-12
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
1-13
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
1-14
Proof by Contradiction



Hypotheses: a set of pertinent propositions
Goal: negation of theorem stated as a
proposition
Theorem is proved by finding an
inconsistency
1-15
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
1-16
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
1-17
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)
1-18
The Origins of Prolog

University of Aix-Marseille


Natural language processing
University of Edinburgh

Automated theorem proving
1-19
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 with a lowercase letter
a string of printable ASCII characters delimited
by apostrophes
1-20
Terms: Variables and
Structures


Variable: any string of letters, digits, and
underscores beginning with an uppercase
letter
Instantiation: binding of a variable to a value


Lasts only as long as it takes to satisfy one
complete goal
Structure: represents atomic proposition
functor(parameter list)
1-21
Fact Statements


Used for the hypotheses
Headless Horn clauses
female(shelley).
male(bill).
father(bill, jake).
1-22
Rule Statements



Used for the hypotheses
Headed Horn clause
Right side: antecedent (if part)


Left side: consequent (then part)


May be single term or conjunction
Must be single term
Conjunction: multiple terms separated by
logical AND operations (implied)
1-23
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).
1-24
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 Horn
man(fred)

Conjunctive propositions and propositions
with variables also legal goals
father(X,mike)
1-25
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
1-26
Approaches

Bottom-up resolution, forward chaining



Top-down resolution, backward 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
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
1-27
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
1-28
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
1-29
Simple Arithmetic

Prolog supports integer variables and integer
arithmetic
is operator: takes an arithmetic expression
as right operand and variable as left operand
A is B / 17 + C

Not the same as an assignment statement!

1-30
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.
1-31
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, kiwi]
[]
(empty list)
[X | Y] (head X and tail Y)
1-32
Append Example
append([], List, List).
append([Head | List_1], List_2, [Head |
List_3]) :append (List_1, List_2, List_3).
1-33
Reverse Example
reverse([], []).
reverse([Head | Tail], List) :reverse (Tail, Result),
append (Result, [Head], List).
1-34
Example: mylength
mylength([],0).
mylength([_|Tail], Len) :mylength(Tail, TailLen),
Len is TailLen + 1.
?- mylength([a,b,c],X).
X = 3
Yes
?- mylength(X,3).
X = [_G266, _G269, _G272]
Yes
Counterexample: mylength
mylength([],0).
mylength([_|Tail], Len) :mylength(Tail, TailLen),
Len = TailLen + 1.
?- mylength([1,2,3,4,5],X).
X = 0+1+1+1+1+1
Yes
Example: sum
sum([],0).
sum([Head|Tail],X) :sum(Tail,TailSum),
X is Head + TailSum.
?- sum([1,2,3],X).
X = 6
Yes
?- sum([1,2.5,3],X).
X = 6.5
Yes
Example: gcd
gcd(X,Y,Z) :X =:= Y,
Z is X.
gcd(X,Y,Denom) :X < Y,
NewY is Y - X,
gcd(X,NewY,Denom).
gcd(X,Y,Denom) :X > Y,
NewX is X - Y,
gcd(NewX,Y,Denom).
Note: not just
gcd(X,X,X)
The gcd Predicate At Work
?- gcd(5,5,X).
X = 5
Yes
?- gcd(12,21,X).
X = 3
Yes
?- gcd(91,105,X).
X = 7
Yes
?- gcd(91,X,7).
ERROR: Arguments are not sufficiently instantiated
Example: factorial
factorial(X,1) :X =:= 1.
factorial(X,Fact) :X > 1,
NewX is X - 1,
factorial(NewX,NF),
Fact is X * NF.
?- factorial(5,X).
X = 120
Yes
?- factorial(20,X).
X = 2.4329e+018
Yes
?- factorial(-2,X).
No
Problem Space Search


Prolog’s strength is (obviously) not numeric
computation
The kinds of problems it does best on are
those that involve problem space search


You give a logical definition of the solution
Then let Prolog find it
The Knapsack Problem




You are packing for a camping trip
Your pantry contains these items:
Item
Weight in kilograms
Calories
bread
4
9200
pasta
2
4600
peanut butter
1
6700
baby food
3
6900
Your knapsack holds 4 kg.
What choice <= 4 kg. maximizes calories?
Greedy Methods Do Not Work



Item
Weight in kilograms
Calories
bread
4
9200
pasta
2
4600
peanut butter
1
6700
baby food
3
6900
Most calories first: bread only, 9200
Lightest first: peanut butter + pasta, 11300
(Best choice: peanut butter + baby food,
13600)
Search

No algorithm for this problem is known that




Always gives the best answer, and
Takes less than exponential time
So brute-force search is used here
That’s good, since search is something
Prolog does really well
Representation



We will represent each food item as a term
food(N,W,C)
Pantry in our example is
[food(bread,4,9200),
food(pasta,2,4500),
food(peanutButter,1,6700),
food(babyFood,3,6900)]
Same representation for knapsack contents
The Knapsack Problem
/*
weight(L,N) takes a list L of food terms, N is the sum of
all the Weights. */
weight([],0).
weight([food(_,W,_) | Rest], X) :weight(Rest,RestW),
X is W + RestW.
/*
calories(L,N) takes a list L of food terms, N is the sum of
all the Calories. */
calories([],0).
calories([food(_,_,C) | Rest], X) :calories(Rest,RestC),
X is C + RestC.
The Knapsack Problem
/*
subseq(X,Y) succeeds when list X is the same as
list Y, but with zero or more elements omitted.
*/
subseq([],[]).
subseq([Item | RestX], [Item | RestY]) :subseq(RestX,RestY).
subseq(X, [_ | RestY]) :subseq(X,RestY).


A subsequence of a list is a copy of the list
with any number of elements omitted
(Knapsacks are subsequences of the pantry)
The Knapsack Problem
?- subseq([1,3],[1,2,3,4]).
Yes
?- subseq(X,[1,2,3]).
X
X
X
X
X
X
X
X
=
=
=
=
=
=
=
=
No
[1, 2,
[1, 2]
[1, 3]
[1] ;
[2, 3]
[2] ;
[3] ;
[] ;
3] ;
;
;
;
Note that subseq can do more
than just test whether one list is a
subsequence of another; it can
generate subsequences, which is
how we will use it for the
knapsack problem.
The Knapsack Problem
/*
knapsackDec(Pantry,Capacity,Goal,Knapsack) takes
a list Pantry of food terms, a positive number
Capacity, and a positive number Goal. We unify
Knapsack with a subsequence of Pantry representing
a knapsack with total calories >= goal, subject to
the constraint that the total weight is =< Capacity.
*/
knapsackDec(Pantry,Capacity,Goal,Knapsack) :subseq(Knapsack,Pantry),
weight(Knapsack,Weight),
Weight =< Capacity,
calories(Knapsack,Calories),
Calories >= Goal.
The Knapsack Problem
?- knapsackDec(
|
[food(bread,4,9200),
|
food(pasta,2,4500),
|
food(peanutButter,1,6700),
|
food(babyFood,3,6900)],
|
4,
|
10000,
|
X).
X = [food(pasta, 2, 4500), food(peanutButter,
1, 6700)]
Yes

This decides whether there is a solution that meets the
given calorie goal
The 8-Queens Problem

Chess background:




Played on an 8-by-8 grid
Queen can move any number of spaces
vertically, horizontally or diagonally
Two queens are in check if they are in the same
row, column or diagonal, so that one could
move to the other’s square
The problem: place 8 queens on an empty
chess board so that no queen is in check
Representation




We could represent a queen in column 2, row
5 with the term queen(2,5)
But it will be more readable if we use
something more compact
Since there will be no other pieces, we can
just use a term of the form X/Y
(We won’t evaluate it as a quotient)
Example
8
Q
7
Q
6
Q
5
Q
Q
4
Q
3
Q
2
1


Q
Our 8-queens solution
1
[8/4, 7/2, 6/7, 5/3,
4/6, 3/8, 2/5, 1/1]
2
3
4
5
6
7
8
The 8-Queens Problem
/*
nocheck(X/Y,L) takes a queen X/Y and a list
of queens. We succeed if and only if the X/Y
queen holds none of the others in check.
*/
nocheck(_, []).
nocheck(X/Y, [X1/Y1 | Rest]) :X =\= X1,
Y =\= Y1,
abs(Y1-Y) =\= abs(X1-X),
nocheck(X/Y, Rest).
The 8-Queens Problem
/*
legal(L) succeeds if L is a legal placement of
queens: all coordinates in range and no queen
in check.
*/
legal([]).
legal([X/Y | Rest]) :legal(Rest),
member(X,[1,2,3,4,5,6,7,8]),
member(Y,[1,2,3,4,5,6,7,8]),
nocheck(X/Y, Rest).
Adequate

This is already enough to solve the problem:
the query legal(X) will find all legal
configurations:
?- legal(X).
X = [] ;
X = [1/1] ;
X = [1/2] ;
X = [1/3]
8-Queens Solution


Of course that will take too long: it finds all
64 solutions with one queen, then starts on
those with two, and so on
To make it concentrate right away on
eight queens, we can give a different query:
?- X = [_,_,_,_,_,_,_,_], legal(X).
X = [8/4, 7/2, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1]
Yes
Room For Improvement


Slow
Finds trivial permutations after the first:
?- X = [_,_,_,_,_,_,_,_], legal(X).
X = [8/4, 7/2, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1] ;
X = [7/2, 8/4, 6/7, 5/3, 4/6, 3/8, 2/5, 1/1] ;
X = [8/4, 6/7, 7/2, 5/3, 4/6, 3/8, 2/5, 1/1] ;
X = [6/7, 8/4, 7/2, 5/3, 4/6, 3/8, 2/5, 1/1]
An Improvement


Clearly every solution has 1 queen in each
column
So every solution can be written in a fixed
order, like this:
X=[1/_,2/_,3/_,4/_,5/_,6/_,7/_,8/_]

Starting with a goal term of that form will
restrict the search (speeding it up) and avoid
those trivial permutations
The 8-Queens Problem
/*
eightqueens(X) succeeds if X is a legal
placement of eight queens, listed in order
of their X coordinates.
*/
eightqueens(X) :X = [1/_,2/_,3/_,4/_,5/_,6/_,7/_,8/_],
legal(X).
The 8-Queens Problem
nocheck(_, []).
nocheck(X/Y, [X1/Y1 | Rest]) :% X =\= X1, assume the X's are distinct
Y =\= Y1,
abs(Y1-Y) =\= abs(X1-X),
nocheck(X/Y, Rest).
legal([]).
legal([X/Y | Rest]) :legal(Rest),
% member(X,[1,2,3,4,5,6,7,8]), assume X in range
member(Y,[1,2,3,4,5,6,7,8]),
nocheck(X/Y, Rest).

Since all X-coordinates are already known to be in
range and distinct, these can be optimized a little
Improved 8-Queens Solution


Now much faster
Does not bother with permutations
?- eightqueens(X).
X = [1/4, 2/2, 3/7, 4/3, 5/6, 6/8, 7/5, 8/1] ;
X = [1/5, 2/2, 3/4, 4/7, 5/3, 6/8, 7/6, 8/1]
Cut & Negation


Cut (Minimum):

mini1(X, Y, X) :- X < Y.
mini1(X, Y, Y) :- X >= Y.

mini2(X, Y, X) :- X < Y, !.
mini2(_, Y, Y).

mini3(X, Y, Z) :- X < Y, !, Z = X.
mini3(X, Y, Y).
Negation:


not(P):- P, ! ,fail.
not(P). %sinon
Applications of Logic
Programming



Relational database management systems
Expert systems
Natural language processing
1-64
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
1-65