Prover9 Manual Version June-2006

Glossary

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Terms, Clauses, Formulas, Interpretations

These definitions apply to first-order unsorted logic. See any book on first-order logic for more formal definitions of these concepts.
Term
A recursive definition of first-order unsorted terms.

Atomic Formula
An n-ary predicate symbol applied to n terms is an atomic formula.

Literal
A literal is an atomic formula or the negation of an atomic formula.

Clause
A clause is a disjunction of literals. All variables in a clause are assumed to be universally quantified.

Formula
This is a standard definition of formula. Prover9 has a more restricted notion of formula that excludes formulas with free variables.

Interpretation
An interpretation of a first-order language consists of
Given an interpretation, each term in the language evaluates to a member of the domain, and each formula in the language evaluates to TRUE or to FALSE.

Logic Transformations


Negation Normal Form (NNF)
A formula is in negation normal form if the only logic connectives are negation, conjunction, disjunction, quantification (universal or existential), and if all negation operations are applied directly to atomic formulas.

Conjunctive Normal Form (CNF)
This definition applies to quantifier-free formulas.

A formula is in conjunctive normal form if (1) the only logic connectives are negation, conjunction, and disjunction, (2) no negation is applied to a conjunction or a disjunction, and (3) no disjunction is applied to a conjunction.

Alternate definition: A formula is in CNF if it is a clause or a conjunction of clauses.


Skolemization
Skolemization is the process of replacing existentially quantified variables in a formula with new constants (called Skolem constants) or functions (called Skolem functions). If an existential quantifier is in the scope of some universal quantifiers, the new symbol is a function of the corresponding universally quantified variables. The result of Skolemization is not, strictly speaking, equivalent to the original formula, because new symbols may have been introduced, but the result is inconsistent iff the the original formula is inconsistent.

Clausification
Clausification is the process of translating a formula into a conjunction of clauses. A standard way is NNF conversion, Skolemization, moving universal quantifiers to the top (renaming bound variables if necessary), CNF conversion, and finally removing universal quantifiers. The variables in each resulting clause are implicitly universally quantified.

Each step produces an equivalent formula, except for Skolemization, which produces an eqconsistent formula, so the result of Clausification is inconsistent iff the original formula is inconsistent.



Term Ordering Terminology


Maximal Literal
A literal is maximal in a clause, with respect to some term ordering, if no literal in the clause is greater. The terms orderings used by Prover9 (LPO, KBO, RPO) are, in general, only partial, so clauses do not necessarily have maximum literals.


Inference and Simplification Rules


"From" and "Into" Clauses and Literals
A paramodulation inference consists of two parents and a child. The parent containing the equality used for the replacement is the from parent or the from clause, the equality is the from literal, and the side of the equality that unifies with the term being replaced is the from term.

The replaced term is the into term, the literal containing the replaced term is the into literal, and the parent containing the replaced term is the into parent or into clause.



Prover9 Terminology