Logic Programming

1
Logic Programming

• Chapter 15
• Part 2

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.

3
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.

4
A Partial Family Tree Figure 15.3
5
A Small Family Tree Figure 15.4
6

• 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

7

• ?- grandparent(Who, ron).
• Instantiating grandparent rule from query
• Grandparent(Who,ron)- parent(Who,X),pare
  nt(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)

8
Prolog Lists

• The list is Prologs basic data structure
• Lists are a series of Prolog terms, separated by
  commas
• Each list element can be a(n)
• atom
• variable
• sublist
• etc.

9
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 dont care sometimes referred to
  as an unbound variable.

10
Working with Lists

• HeadTail 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

11
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

12
The Append Function

• append(L1, L2, L3)
• append(, X, X). base case
• append(HeadTail, Y, HeadZ)
• - 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.

13
?- Append(english, russian, spanish, L).
Henglish, Trussian, Yspanish,
Lenglish,Z 1
and Z russian, spanish Append(rus
sian,spanish, Z). H russian, T ,
Yspanish, ZrussianZ1 2 Append( ,
spanish, Z1). So Z1
spanish Xspanish, Z1spanish
3 Append( , spanish, spanish).
14
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)

15
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).

16
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.
• Dont care entries (_) mark parts of a list that
  arent important to the rule.

17
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.

18
A Sample List Program
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).
19
Sample Output
?- 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
Based on the program on the previous slide.
20
Sample Output
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
21
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 gt 0,
• M is N-1,
• factorial(M, SubRes),
• Result is N SubRes. is assignment

22
Logic Programming

• 15.2.2 Practical Aspects
• 15.3 Example Applications

23
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.

24
Tracing Output
These are temporary variables

• ?- 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 levels in the search tree
25
Tracing

• 2 ?- 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.
26
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, XR,
R). remove(X, HR, HS)- remove(X,
R, S).
27
18 ?- trace. Yes 18 ?- 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
28
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
29
Trace of Factorial (4)
30
Simple Arithmetic

• Integer variables and integer operations are
  possible, but imperative language assignment
  statements dont exist.

31
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.

32
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.

33
Arithmetic

• Originally, used prefix notation (7, X)
• Modern versions have infix notationX is Y C
  3.
• Qualification Y and C must be instantiated, as
  in the Speed program, but X cannot be (Its not a
  traditional assignment statement).
• X X Y is illegal.
• X is X Y is illegal. Arguments are not
  sufficiently instantiated

34
More About Arithmetic

• Example of simple arithmetic, using something
  similar to Pythons calculator mode (not as part
  of a program).
• ?- X is 3 7.
• X 10
• Yes
• Arithmetic operators , -, , /,
  (exponentiation)
• Relational operators lt, gt, , lt, gt, \

35
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.

36
Example Revised Factorial
factorial(N, 1)- N lt 1, !. factorial(N,
Result)- M is N 1, factorial(M,
P), Result is N P.
factorial(N, 1)- N lt 1. factorial(N, Result)-
not(N lt 1), M is N1, factorial(M,
P), Result is N P.
37
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.

38
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.

39
Symbolic Differentiation Rules Figure 15.9
40
Prolog Symbolic Differentiator Figure 15.10
41
Search Tree for the Query d(x, 2x1, Ans) Figure
15.11
42
Executing a Prolog Program

• Create a file containing facts and rules e.g.,
  familytree.pl
• Follow instructions in handout, which will be
  available Wednesday.

43
SWIplEdit compile error

• If SWI-Prolog finds an error in the .pl file it
  will give a message such asERROR
  c/temp/prologprogs/remove.pl180 Syntax
  error Illegal start of term(18 is the line
  number)

44
Runtime Error Message

• The function samelength was called with one
  parameter when it needed 221 ?-
  samelength(X).ERRORUndefined procedure
  samelength/1ERRORHowever, there are
  definitions for samelength/2

45
Runtime Errors

• Here, the error is an error of omission22 ?-
  samelength(a, b, c,,a, b)
• Queries must end with a period. If you hit
  enter without typing a period SWIpl just thinks
  you arent through.

46
Using SWI Prolog

• If there is an error that you cant figure out
  (for example you dont get any answers, you dont
  get a prompt, typing a semicolon doesnt help)
  try interrupt under the Run button.
• If changes are made to the program, dont forget
  to save the file and consult again.