Predicate Logic As Formal Language

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Abstract

Sentences in first-order predicate logic is usefully taken as programs during this paper the operational and fixpoint linguistics of predicate logic programs area unit outlined, and also the connections with the proof theory and model theory of logic area unit investigated it’s finished that operational linguistics could be a a part of proof theory which fixpoint linguistics could be a special case of model-theoretic linguistics key words and phrases predicate logic as a formal language, linguistics of programming languages, resolution theorem proving, operaUonal versus denotational linguistics, fixpoint characterization.

Introduction

Predicate logic plays a very important role in several formal models of pc programs. Here we tend to square measure involved with the interpretation of predicate logic as a artificial language. The programming language system based mostly upon the procedural interpretation, has been used for several formidable programming tasks. we tend to compare the ensuing linguistics with the classical linguistics studied by logicians. two sorts of linguistics, operational and fix point, are outlined for program languages. Operational linguistics defines the input-output relation computed by a program in terms of the operations induced by the program within a machine. The which means of a program to the input-output relation obtained by corporal punishment the program on the machine. As a machine freelance different to operational linguistics, fixpoint linguistics defines the mean of a program to be the input output relation that to the stripped fixpoint of a change assonated with the program. Fix purpose linguistics has been to justify existing ways for proving properties of programs and to inspire and justify new ways of proof relation, and truth.

A Syntax of Well-Formed Formulas

It is convention to limit attention to predicate logic programs written m grammatical construction type. Such programs have AN particularly easy syntax however retain all the communicative power of the full predicate logic. A sentence may be a finite set of clauses. A clause may be a disjunction Li V ‘ • • V Ln of literals L, that are atomic formulas P (tl, . . . , tn) or the negations of atomic formulas P(tl . . . . . tn) , wherever P may be a predicate symbol and t, are terms. Atomic formulas are positive literals. Negating of atomic formulas are negative literals.

The Procedural interpretation

It is best to interpret procedurally sets of clauses that contain at the most one posmve hteral per clause. Such sets of clauses square measure known as Horn sentences. we tend to distinguish 3 kinds of Horn clauses. The empty clause, containing no hterals and denoting the reality worth false, is interpreted as a halt statement.

B~ V ….. V B a clause consisting of no positive hterals and n -> one negative hterals ~s understood as a goal statement.

Model- Theoretic Semantics

There is general agreement among logicians regarding the linguistics of predicate logic. This linguistics provides an easy technique for determinative the denotation of a predicate symbol P during a set of clauses A. where X ~ Y implies that X logically implies Y. Dz(P) is that the denotation of P as determined by model-theoretic linguistics. The completeness of first-order logic implies that there exist illation systems such that durability coinodes with logical implication; i.e. for such reference systems X I- Y iff Sl = Y.

In order to create a comparison of the fixpoint and model-theoretic linguistics, we need a additional careful definition of D2. For this purpose, we tend to outline the notions of Herbrand interpretation and Herbrand model. An expression (term, literal, clause, set of clauses) is ground if it contains no variables. A Herbrand interpretation at the same time associates, with each n-ary predicate image in A, a novel n-ary relation over H. The relation is associated by I with the predicate image P during a.

A ground atomic formula A is true during a Herbrand interpretation I iff A E I.
A ground negative literal. A is true in I iff A ~ one.
A ground clause L~ V • • • V luminous flux unit is true in I iff a minimum of one literal L, is true in I.

Fixpoint Semantics

In the fixpoint linguistics, the denotation of a recursively outlined procedure is outlined to be the stripped-down fixpoint of a change related to the procedure definition.

Here we have a tendency to propose a samdar definition of semantics for predicate logic programs.

In order to justify our definition, we have a tendency to 1st disquiet the fixpoint semantics because it has been formulated for additional conventionally outlined recourse procedures. Our description follows the one given by American state Bakker.

Let P ~ B(P) be a procedure declaration in AN Algol-like. Language, wherever the primary occurrence of P because the procedure name, wherever B(P) is that the procedure body, and wherever the occurrence of P in B(P) all calls to P within the body of the procedure. Associated with B as a change T that maps sets I of input-output tuples into other such sets J = T(I). once the transformation T is monotonic T(I~) C T(I2) whenever I~ C lz) the denotetion of P as outlined as n, which is adenocele to the intersection of all fixpoints of T.

Model-Theoretic and Fixpoint Semantics:

We shall show that for sets of procedure declarations A, model-theoretic and fixpoint semantics coincide: D2 = Ds. It would be sufficient to show that NM(A) = AC(A), but it is easy to prove that even M(A) = C(A). In other words, a Herbrand interpretation I of A is a model of A iff I is closed under the transformation T assumed with A.

Conclusion

For impulsive sentences X and Y of first-order predicate logic, proof theory determines when X ~- Y and model theory determines once X ~ Y. we’ve got argued that m the procedural i n t e r p r e t a t i o n , operational semantics ~s proof theory and fixpoint linguistics is model theory. On the opposite hand, operational and fixpoint linguistics solely manage the case wherever Y could be a set of ground atomic formulas. Moreover, fixpoint linguistics solely deals with X, a group of procedure declarations. we tend to believe that the supplementary generality of proof theory and model theory has helpful consequences. The completeness theorem of first-order logic states that the relations b of derivabilty and ~ of logical lmplication area unit equivalent. For goal directed logical thinking systems this equivalence establishes that varied computation rules cipher the relation determined by the fixpoint linguistics. additional typically, this equivalence may be accustomed justify varied rules (such as Scott’s induction rule [6]) for proving properties of programs.

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