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logic

This folder contains an object which implements a translator from first-order predicate logic propositions to conjunctive normal form and to clausal form. The translator code is partially based on code published in the book "Programming in Prolog" by W. F. Clocksin and C. S. Mellish.

The following operators are used for representing logic connectives:

negation:[126]

disjunction:[118]

conjunction:[38]

implication:[61, 62]

equivalence:[60, 61, 62]

Quantifiers are represented using the following notation:

universal:[97, 108, 108, 40, 88, 44, 32, 80, 41]

existential:[101, 120, 105, 115, 116, 115, 40, 88, 44, 32, 80, 41]

The two main object predicates are translate/2 and step_by_step/2. The first predicate, translate/2, translate a logic proposition to a list of clauses. The second predicate, step_by_step/2, performs the same translations as translate/2 but also prints the results of each conversion step.

Print Logtalk, Prolog backend, and kernel versions (if running as a notebook):

%versions

Start by loading the example:

logtalk_load(logic(loader)).

Translate a single logic proposition:

translator::translate((p v ~q) => (r & k), Cs).

<!-- r :- p. k :- p. q; r :- . q; k :- .

Cs = [cl([r],[p]),cl([k],[p]),cl([q,r],[]),cl([q,k],[])]. -->

Translate a single logic proposition printing each translation step:

translator::step_by_step((p v ~q) => (r & k), Cs).

<!-- Processing proposition: p v ~q=>r&k

1. Remove implications: ~ (p v ~q) v r&k
2. Distribute negation: ~p&q v r&k
3. Remove existential quantifiers: ~p&q v r&k
4. Convert to prenex normal form: ~p&q v r&k
5. Remove universal quantifiers: ~p&q v r&k
6. Convert to conjunctive normal form: (~p v r)&(~p v k)&((q v r)&(q v k))
7. Convert to clauses: [cl([r],[p]),cl([k],[p]),cl([q,r],[]),cl([q,k],[])]

Clauses in Prolog-like notation: r :- p. k :- p. q; r :- . q; k :- .

Cs = [cl([r],[p]),cl([k],[p]),cl([q,r],[]),cl([q,k],[])]. -->

Translate a single logic proposition printing each translation step:

translator::step_by_step(all(X, exists(Y, p(X) v ~q(X) => r(X, Y))), Cs).

<!-- Processing proposition: all(X, exists(Y, p(X)v~q(X)=>r(X, Y)))

1. Remove implications: all(X, exists(Y, ~ (p(X)v~q(X))v r(X, Y)))
2. Distribute negation: all(X, exists(Y, ~p(X)&q(X)v r(X, Y)))
3. Remove existential quantifiers: all(X, ~p(X)&q(X)v r(X, f1(X)))
4. Convert to prenex normal form: all(X, ~p(X)&q(X)v r(X, f1(X)))
5. Remove universal quantifiers: ~p(X)&q(X)v r(X, f1(X))
6. Convert to conjunctive normal form: (~p(X)v r(X, f1(X)))& (q(X)v r(X, f1(X)))
7. Convert to clauses: [cl([r(X, f1(X))], [p(X)]), cl([q(X), r(X, f1(X))], [])]

Clauses in Prolog-like notation: r(X, f1(X)) :- p(X). q(X); r(X, f1(X)) :- .

X = X, Y = f1(X), Cs = [cl([r(X, f1(X))], [p(X)]), cl([q(X), r(X, f1(X))], [])]. -->

Translate a single logic proposition printing each translation step:

translator::step_by_step(all(X, men(X) => mortal(X)), Cs).

<!-- Processing proposition: all(X, men(X)=>mortal(X))

1. Remove implications: all(X, ~men(X)v mortal(X))
2. Distribute negation: all(X, ~men(X)v mortal(X))
3. Remove existential quantifiers: all(X, ~men(X)v mortal(X))
4. Convert to prenex normal form: all(X, ~men(X)v mortal(X))
5. Remove universal quantifiers: ~men(X)v mortal(X)
6. Convert to conjunctive normal form: ~men(X)v mortal(X)
7. Convert to clauses: [cl([mortal(X)], [men(X)])]

Clauses in Prolog-like notation: mortal(X) :- men(X).

X = X, Cs = [cl([mortal(X)], [men(X)])]. -->