1. Logic Programming Languages
Chapter 16
Logic Programming Languages

2. 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

3. Introduction Definition: logic programming is programming that
uses a form of symbolic logic as a
programming language
Definition: logic programming languages are
languages based on symbolic logic
(also called declarative languages)
Use a logical inferencing process to produce results
Declarative rather that procedural:
only specification of results are stated
(not detailed procedures for producing them)

4. Advantages Programs are based on logic
likely to be logically organized and
written
Processing is naturally parallel
interpreters can take advantage of multi-processor machines
Programs are concise
development time is decreased
good for prototyping

5. A Brief Introduction to Predicate Calculus
Overview
Propositions
Clausal Form

6. Predicate Calculus Overview
Definition: a proposition is a logical statement that is true or not true
Consists of
1. objects, and
2. relationships of objects to each other
Definition: formal logic is a system of describing propositions and checking them for validity

7. Predicate Calculus Overview First Order Predicate Calculus
Definition: symbolic logic is a system meeting the three basic needs of formal logic:
(The system must be able to)
1. express propositions
2. express relationships between propositions
3. describe proposition inference
(used to create new propositions from other propositions)
The particular form of symbolic logic used for logic programming is called
First Order Predicate Calculus

8. Predicate Calculus Propositions - Objects
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)

9. Predicate Calculus Proposition Types
Atomic propositions
consist of compound terms
Compound term
one element of a mathematical relation
written like a mathematical function
A mathematical function is a mapping
Can be written as a table

10. Predicate Calculus Propositions – Compound Terms
A compound term is composed of two parts:
Functor = function symbol providing a name
Ordered list of parameters (tuple)
Examples:
student(jon)
like(seth, OSX)
like(nick, windows)
like(jim, linux)

11. Predicate Calculus Proposition Forms
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

12. Predicate Calculus 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

13. Predicate Calculus 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

14. Predicate Calculus Clausal Form
TOO MANY WAYS TO STATE THE SAME THING!
So…use a standard form for propositions, called
Clausal form:
B1  B2  ...  Bn  A1  A2  ...  Am
this means: if all the As are true,
then at least one B is true
Antecedent: right side
B1  B2  …  Bn  A1  A2  …  Am
Consequent: left side

15. Predicate Calculus and Proving Theorems
One use of propositions is to discover new theorems that can be inferred from known axioms and theorems
Definition: Resolution is
a method of inferencing
that allows propositions to be computed
from given propositions

16. Resolution Really a form of data reduction! Suppose P1  P2 Q1  Q2
If P1 and Q2 are identical, then
Q1  P2
But it’s more complex than this...

17. Resolution
Definition: Unification is finding values for variables in propositions that allows matching process to succeed
Definition: Instantiation is assigning temporary values to variables to allow unification to succeed
After instantiating a variable with a value,
if matching fails,
then it may be necessary to backtrack and instantiate with a different value

18. Proof by Contradiction
Theorem Proving
Proof by Contradiction
Definition: The hypotheses is a set of pertinent propositions
Definition: The goal is the negation of the theorem stated as a proposition
Theorem is proved by finding an inconsistency

19. Theorem Proving When propositions are used for resolution,
only a restricted “clausal form” can be used:
A 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

20. An Overview of Logic Programming Logic Programming is nonprocedural !
Declarative semantics:
There is a simple way to determine the meaning of each statement
Simpler than the semantics of imperative languages
Logic Programming is nonprocedural !
Logic programs do not state
how a result is to be computed,
but rather the form of the result.

21. An Overview of Logic Programming 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) A

22. The Basic Elements of Prolog
Edinburgh Syntax
Term a constant, variable, or structure
Constant an atom or an integer
Atom a symbolic value of Prolog
An 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

23. The Basic Elements of Prolog <functor>(<parameter list>)
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>)

24. Used for the hypotheses
Fact Statements
Used for the hypotheses
Headless Horn clauses
student(jonathan).
sophomore(ben).
brother(tyler, cj).

25. Rule Statements Used for the hypotheses Headed Horn clause
Right side: antecedent (if part)
May be single term or conjunction
Left side: consequent (then part)
Must be single term
Conjunction:
multiple terms separated
by logical AND operations
(usually implied)

26. 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
student(james)
Conjunctive propositions and propositions with variables also legal goals
father(X,joe)

27. Resolution
Definition: Unification is finding values for variables in propositions that allows matching process to succeed
Definition: Instantiation is assigning temporary values to variables to allow unification to succeed
After instantiating a variable with a value,
if matching fails,
then it may be necessary to backtrack and instantiate with a different value

28. Proof by Contradiction
Theorem Proving
Proof by Contradiction
Definition: The hypotheses is a set of pertinent propositions
Definition: The goal is the negation of the theorem stated as a proposition
Theorem is proved by finding an inconsistency

29. 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
student(james)
Conjunctive propositions and propositions with variables also legal goals
father(X,joe)

30. Inferencing Process Bottom-up resolution (forward chaining)
Begin with facts and rules, then try to find a sequence that leads to the goal
Good for a large set of possible answers
Top-down resolution (backward chaining)
Begin with the goal, then try to find a sequence that leads to a set of facts already known
Good for a small set of possible answers
Prolog implementations use backward chaining

31. When seeking a goal with multiple subgoals,
Inferencing Process
When seeking a goal with multiple subgoals, if
the truth of one of the subgoals cannot be shown,
then
reconsider the previous subgoal to find an alternative solution (called “backtracking”), beginning the search where the previous search left off
This can take lots of time and space because it may be necessary to find all possible proofs to every subgoal

32. integer variables and integer arithmetic
Simple Arithmetic
Prolog supports
integer variables and integer arithmetic
The is binary operator takes
right operand: an arithmetic expression
left operand: a variable
A is B / 10 + C
Not the same as an assignment statement!

33. List Structures Other basic data structure
(besides atomic propositions): 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)

34. Prolog append Function
Definition
append([],List,List).
append([Head|List_1],
List_2,
[Head|List_3]) :-
append(List_1, List_2, List_3).

35. Prolog reverse Function
Definition
reverse([],[]).
reverse([Head|Tail],List) :-
reverse(Tail,Result),
append(Result,[Head],List).

36. to generalize meaning:
Rule Statements
parent(kim,kathy):- mother(kim,kathy).
Can use variables
(universal objects)
to generalize meaning:
parent(X,Y) :- mother(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).

37. 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 is called
matching, satisfying, or resolution

38. Inferencing Process When goal has more than one subgoal,
it is possible to 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
Requires fewer computer resources

39. Trace Built-in structure that displays instantiations at each step
Tracing model of 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

40. Trace Model

41. Prolog Example 1 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.
distance(chevy, Chevy_Distance).

42. Prolog Example 1 trace. distance(chevy, Chevy_Distance).
(1) 1 Call: distance(chevy, _0)?
(2) 2 Call: speed(chevy, _5)?
(2) 2 Exit: speed(chevy, 105)
(3) 2 Call: time(chevy, _6)?
(3) 2 Exit: time(chevy, 21)?
(4) 2 Call: _0 is 105*21?
(4) 2 Exit: 2205 is 105*21
(1) 1 Exit: distance(chevy, 2205)
Chevy_Distance = 2205

43. Prolog Example 2 likes(jake,chocolate). likes(jake,apricots).
likes(darcie,licorice).
likes(darcie,apricots).
trace.
likes(jake,X),
likes(darcie,X).

44. Prolog Example 2 trace. likes(jake,X), likes(darcie,X).
(1) 1 Call: likes(jake, _0)?
(1) 1 Exit: likes(jake, chocolate)?
(2) 1 Call: likes(darcie, chocolate)?
(2) 1 Fail: likes(darcie, chocolate)?
(1) 1 Redo: likes(jake, _0)?
(1) 1 Exit: likes(jake, apricots)?
(3) 1 Call: likes(darcie, apricots)?
(3) 1 Exit: likes(darcie, apricots)?
X = apricots

45. Prolog append Function
Definition
append([],List,List).
append([Head|List_1],
List_2,
[Head|List_3]) :-
append(List_1, List_2, List_3).

46. Prolog reverse Function
Definition
reverse([],[]).
reverse([Head|Tail],List) :-
reverse(Tail,Result),
append(Result,[Head],List).

47. Prolog reverse Function
Definition
% Here is a more efficient (iterative/tail recursive) version.
reverse([],[]).
reverse(L,RL) :- reverse(L,[],RL).
reverse([],RL,RL).
reverse([X|L],PRL,RL) :- reverse(L,[X|PRL],RL).

48. Prolog member Function
Definition
member(Element, [Element | _]).
member(Element, [_ | List]) :- member(Element, List).

49. Logic Programming Advantages
Prolog programs are based on logic
they are likely to be more logically organized and written
Processing is naturally parallel
Prolog interpreters can take advantage of multi-processor machines
Programs are concise
development time is decreased
good for prototyping

50. Logic Programming Deficiencies
Resolution Order Control
Closed-World Assumption
The Negation Problem
Intrinsic Limitations

51. Resolution Order Control
Rule order (clause order) determines the order in which solutions are found
Goal order determines the search tree
Prolog uses Left-to-Right Resolution
Left recursion causes infinite loops
Same problems as Recursive Descent
Special explicit control of backtracking
“Cut” operator ( ! )
A goal that is always satisfied
Cannot be resatisfied after backtracking

52. Closed-World Assumption
Prolog cannot prove that a goal is false
True/Fail system, not a True/False system

53. The Negation Problem Negatives are hard to prove
The Horn clause is positive logic
B  A1  A2  ...  An
B can be shown to be true, but not false

54. Intrinsic Limitations
Prolog cannot sort well, unless the programmer provides the method
New inferencing techniques may correct this

55. Logic Programming Applications
Relational Database Management Systems
Expert Systems
Natural Language Processing