Logic Programming

Chapter 15

Part 2

Breadth-first v Depth-first Search

• Suppose a query has compound goals (several propositions must be satisfied)

• Depth-first searches prove the first goal before looking at the others.

• Breadth-first works on goals in parallel.

• Prolog uses the depth-first approach.

Backtracking

• When a compound goal is being proved, it may be that a subgoal cannot be shown true.

• In that case, Prolog will back up and try to find another solution to a previous subgoal.

A Partial Family TreeFigure 15.3

A Small Family “Tree”Figure 15.4

Processing Queries

• ?- father(X, sue)Satisfied with the first comparison. X is instantiated with john.

• ?- mother(sue, X)Satisfied with X = nancy, X = jeff

• ?- mother(alice, ron)Fails

?- grandparent(Who, ron).

Instantiating grandparent rule from query:Grandparent(Who,ron):-

parent(Who,X),parent(X,ron).

First, find a fact that satisfies parent (Who,X)

This entails finding a fact to satisfy either mother (Who, X) or father(Who, X)

First try: mother(mary, sue) (“mother” rule is first)

Next, find a fact that satisfies parent(sue, ron)

By satisfying either mother(sue, ron) or father(sue, ron)

Prolog Lists

• The list is Prolog’s basic data structure

• Lists are a series of Prolog terms, separated by commas

• Each list element can be a(n)– atom – variable – sublist – etc.

Examples of Lists

• The empty list: [ ]• List with embedded list: [girls, [like, boys]]• List with variables: [x, V1, y, V2, [A, B]]

– V1, V2, A, and B are variables that may be instantiated with data at a later time.

• Multi-type lists: [boy, [1, 2, 3], ran]• [A, _, Z]

– The _ means “don’t care” – sometimes referred to as an unbound variable.

Working with Lists

• [Head|Tail] notation simplifies processing:– Head represents the first list element, Tail

represents everything else.– Head can be any Prolog term (list, variable,

atom, predicate, etc.)– If L = [a, b, c] then Head = a and Tail = [b,c]– Tail is always another list.

• What is the head of [a]? The tail?• Compare to car and cdr in Lisp, Scheme

The append Function• append is a built-in Prolog function that

concatenates two lists.• append(A, B, L)

concatenates the lists A and B and returns them as L.

• append([my, cat], [is, fat], L). yieldsL = [my, cat, is, fat]

• Compare to Scheme function

The Append Function

• append(L1, L2, L3):append([], X, X). %base case

append([Head|Tail], Y, [Head|Z]) :- append(Tail, Y, Z).

• This definition says:– The empty list concatenated with any list (X)

returns an unchanged list (X again).– If Tail is concatenated with Y to get Z, then a

list one element larger [Head | Tail] can be concatenated with Y to get [Head | Z].

?- Append([english, russian], [spanish], L).

H=english, T=[russian], Y=[spanish], L=[english,Z] 1 and Z = [russian, spanish]

Append([russian],[spanish], [Z]).

H = russian, T=[ ], Y=[spanish], Z=[russian|Z1] 2Append([ ], [spanish], [Z1]). So Z1= [spanish]

X=[spanish], Z1=[spanish] 3Append([ ], [spanish], [spanish]).

Using append

prefix(X, Z) :- append(X, Y, Z).

(finds all prefixes of a list Z)

suffix(Y, Z) :- append(X, Y, Z).

(finds all suffixes of Z)

Recursion/ member

• The function returns ‘yes’ or ‘true’ if X is a member of a given list.

member(X, [X | _ ]).member(X, [ _ | Y]) :- member(X, Y).

Member(X,Y)

• The test for membership succeeds if either:– X is the head of the list [X |_]– X is not the head of the list [_| Y] , but X is

a member of the list Y.

• Notes: pattern matching governs tests for equality.

• Don’t care entries (_) mark parts of a list that aren’t important to the rule.

Naming Lists

• Defining a set of lists:a([single]).a([a, b, c]).a([cat, dog, sheep]).

• When a query such as a(L), prefix(X, L). Is posed, all three lists will be processed.

• Other lists, such as b([red, yellow, green]), would be ignored.

a([single]).a([a, b, c]).a([cat, dog, sheep]).

prefix(X, Z) :- append(X, _, Z).suffix(Y, Z) :- append(_, Y, Z).

% To make queries about lists in the database:% suffix(X, [the, cat, is, fat]).% a(L), prefix(X, L).

A Sample List Program

?- a(L), prefix(X, L).

L = [single]X = [] ;

L = [single]X = [single] ;

L = [a, b, c]X = [] ;

L = [a, b, c]X = [a] ;

L = [a, b, c]X = [a, b] ;

L = [a, b, c]X = [a, b, c] ;

L = [cat, dog, sheep]X = []

Sample Output

Based on the program on the previous slide.

35 ?- a(L), append([cat], L, M).

L = [single]M = [cat, single] ;

L = [a, b, c]M = [cat, a, b, c] ;

L = [cat, dog, sheep]M = [cat, cat, dog, sheep] ;

Sample Output

Recursive Factorial Program

To see the dynamics of a function call, use the trace function. For example,given the following function:

factorial(0, 1).factorial(N, Result):- N > 0,M is N-1,factorial(M, SubRes), Result is N * SubRes. %is ~ assignment

Logic Programming

15.2.2: Practical Aspects

15.3: Example Applications

Using the Trace Function

• At the prompt, type “trace.”

• Then type the query.

• Prolog will show the rules it uses and the instantiation of unbound constants.

• Useful for understanding what is happening in a search process, or in a recursive function.

Tracing Output?- trace(factorial/2).

?- factorial(4, X).Call: ( 7) factorial(4, _G173)Call: ( 8) factorial(3, _L131)Call: ( 9) factorial(2, _L144)Call: ( 10) factorial(1, _L157)Call: ( 11) factorial(0, _L170)Exit: ( 11) factorial(0, 1)Exit: ( 10) factorial(1, 1)Exit: ( 9) factorial(2, 2)Exit: ( 8) factorial(3, 6)Exit: ( 7) factorial(4, 24)

X = 24

These are temporary variables

These are levels in the search tree

Tracing2 ?- trace(factorial/2).% factorial/2: [call, redo, exit, fail]true.

[debug] 3 ?- factorial(3, Result). T Call: (6) factorial(3, _G521) T Call: (7) factorial(2, _G599) T Call: (8) factorial(1, _G602) T Call: (9) factorial(0, _G605) T Exit: (9) factorial(0, 1) T Exit: (8) factorial(1, 1) T Exit: (7) factorial(2, 2) T Exit: (6) factorial(3, 6)Result = 6

User-entered commands are in red; other output is generated by the Prolog runtime system.

%remove() removes an element from a list.%To Call: remove(a, List, Remainder).% or remove(X, List, Remainder).% First parameter is the removed item, % 2nd parameter is the original list,% third is the final list

remove(X, [X|R], R). remove(X, [H|R], [H|S]):- remove(X, R, S).

18 ?- trace.

Yes18 ?- remove(a, [b, d, a, c], R). Call: (7) remove(a, [b, d, a, c], _G545) ? creep Call: (8) remove(a, [d, a, c], _G608) ? creep Call: (9) remove(a, [a, c], _G611) ? creep Exit: (9) remove(a, [a, c], [c]) ? creep Exit: (8) remove(a, [d, a, c], [d, c]) ? creep Exit: (7) remove(a, [b, d, a, c], [b, d, c]) ? creep

R = [b, d, c]

Revisiting The Factorial Function

Evaluation of clauses is from left to right.Note the use of is to temporarily assign values to M and Result

Trace of Factorial (4)

Simple Arithmetic

• Integer “variables” and integer operations are possible, but imperative language “assignment statements” don’t exist.

Sample Program

speed(fred, 60).speed(carol, 75).time(fred, 20).time(carol, 21).

distance(X, Y) :- speed(X, Speed), time(X, Time), Y is Speed * Time.

area_square(S, A) :- A is S * S.

Prolog Operators

• is can be used to cause a variable to be temporarily instantiated with a value.

• Compare to assignment statements in declarative languages, where variables are permanently assigned values.

• The not operator is used to indicate goal failure. For example not(P) is true when P is false.

Arithmetic

• Originally, used prefix notation +(7, X)• Modern versions have infix notation

X is Y * C + 3.• Qualification: Y and C must be instantiated, as in

the Speed program, but X cannot be (It’s not a traditional assignment statement).– X = X + Y is illegal.– X is X + Y is illegal.

“Arguments are not sufficiently instantiated”

More About Arithmetic

• Example of simple arithmetic, using something similar to Python’s calculator mode (not as part of a program).– ?- X is 3 + 7.– X = 10– Yes

• Arithmetic operators: +, -, *, /, ^ (exponentiation)

• Relational operators: <, >, =, =<, >=, \=

The cut & not Operators

• The cut (!) is used to control backtracking.• It tells Prolog not to retry the series of goals

that precede the cut symbol (if the goals have succeeded once).

• Reasons: Faster execution, saves memory• Not(P) will succeed when P fails.

– In some places it can replace the ! Operator.

Example: Revised Factorial

factorial(N, 1):- N < 1, !.factorial(N, Result):- M is N – 1,

factorial(M, P), Result is N * P.

factorial(N, 1):- N < 1.factorial(N, Result):- not(N < 1),

M is N–1, factorial(M, P),

Result is N * P.

When Cut Might Be Used(Clocksin & Mellish)

• To tell Prolog that it has found the right rule: – “if you get this far, you have picked the correct rule for

this goal.”

• To tell Prolog to fail a particular goal without trying other solutions:– “if you get to here, you should stop trying to satisfy the

goal.”

• “if you get to here, you have found the only solution to this problem and there is no point in ever looking for alternatives.”

Assert - Adding Facts• ?- assert(mother(jane, joe)).

adds another fact to the database.

• More sophisticated: assert can be embedded in a function definition so new facts and rules can be added to the database in real time.– Useful for learning programs, for example.

Symbolic Differentiation RulesFigure 15.9

Prolog Symbolic DifferentiatorFigure 15.10

Search Tree for the Query d(x, 2*x+1, Ans)Figure 15.11

Executing a Prolog Program

• Create a file containing facts and rules; e.g., familytree.pl

• Follow instructions in handout, which will be available Wednesday.

SWIplEdit “compile” error

• If SWI-Prolog finds an error in the .pl file it will give a message such as

ERROR: c:/temp/prologprogs/remove.pl:18:0: Syntax error: Illegal start of term

(18 is the line number)

Runtime Error Message

• The function samelength was called with one parameter when it needed 2:

21 ?- samelength(X).ERROR:Undefined procedure:

samelength/1ERROR:However, there are

definitions for: samelength/2

Runtime Errors

• Here, the error is an error of omission:

22 ?- samelength([a, b, c,],[a, b])

| Queries must end with a period. If you hit enter without typing a period SWIpl just thinks you aren’t through.

Using SWI Prolog

• If there is an error that you can’t figure out (for example you don’t get any answers, you don’t get a prompt, typing a semicolon doesn’t help) try “interrupt” under the Run button.

• If changes are made to the program, don’t forget to save the file and “consult” again.