35 - Logical

35 - Logical#

The Origins of Prolog#

  • University of Aix-Marseille (Calmerauer & Roussel)

    • Natural language processing

  • University of Edinburgh (Kowalski)

    • Automated theorem proving

Terms#

  • Term: a constant, variable, or structure

  • Constant: an atom or an integer

  • Atom: symbolic value of Prolog, which consists of either:

    • a string of letters, digits, and underscores beginning with a lowercase letter

    • a string of printable ASCII characters delimited by apostraphes

  • 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)

Statement Types#

Fact Statements#

Headless Horn clauses:

female(shelley).
male(bill).
father(bill, jake).

Rule Statements#

Headed Horn clauses:

  • Right side: antecedent (if part)

    • May be a single term or a conjunction

  • Left side: consequent (then part)

    • Must be a single term

A conjunction is multiple terms separated by logical AND operators (implied).

For example: 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).

Goal Statements#

For theorem proving, the theorem is in the form of proposition that we want the 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)

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, you must find a chain of inference rules and/or facts. For goal Q:

P2 :- P1.
P3 :- P2.
...
Q :- Pn.

Process of proving a subgoal called matching, satisfying, or resolution.

Approaches#

  • Matching is the process of proving a proposition.

  • Proving a subgoal is called satisfying the subgoal

  • Bottom-up resolution, forward chaining

    • Begin with the facts and rules of database and attempt to find the sequence that leads to the goal

    • Works well with a large set of possibly correct answers

  • Top-down resolution, backward chaining

    • Begin with a goal and attempt to find a sequence that leads to some set of facts in the database

    • Works well with a small set of possibly correct answers

  • Prolog implementations use backward chaining

Subgoal Strategies#

  • When a 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

Backtracking#

With a goal with multiple subgoals, if you fail to show the truth of one of the subgoals, reconsider the previous subgoal to find an alternative solution: backtracking.

Begin search where previous search left off.

Can take lots of time and space because it may find all possible proofs to every subgoal.

Simple Arithmetic#

Prolog supports integer variables and integer arithmetic.

The is operator takes an arithmetic expression as the right operand and a variable as the left operand.

A is B / 17 + C

This is not the same as an assignment statement!

  • The following is illegal:

Sum is Sum + Number

For 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.

A query: distance(chevy, Chevy_Distance).

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)

For example:

likes(jake,chocolate).
likes(jake, apricots).
likes(darcie, licorice).
likes(darcie, apricots).

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
notrace.
../../_images/prolog-trace.png

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

Append example:

append([], List, List).
append([Head | List_1], List_2, [Head | List_3]) :-
    append(List_1, List_2, List_3).

More examples:

reverse([], []).
reverse([Head | Tail], List) :-
    reverse(Tail, Result),
        append(Result, [Head], List).

member(Element, [Element | _]).
member(Element, [_ | List]) :- member(Element, List).

The underscore character _ means an anonymous variable. It means we do not care what instantiation it might get from unification.

GNU Prolog#

ssh [email protected]

gprolog
| ?- X is 3 + 5.
X = 8
| ?- append([a,b], [c,d], X).
X = [a, b, c, d]
| ?- halt.

You can also type end_of_file. or CTRL-D to quit.

Steps to Using GNU Prolog#

  1. Use a text editor to create a Prolog database, say, first.pl to include facts and rules.

  2. Start Prolog by typing in gprolog.

  3. Consult a database by typing in consult('first.pl'). or simply [first]..

  4. Run queries/goals

Examples:

  • first.pl

  • cars.pl

  • food.pl

  • append.pl

  • reverse.pl

  • member.pl

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