First-order logic as a constraint satisfaction problem

Abstract

In this paper, we discourse an analysis of classical first-order predicate logic as a constraint satisfaction problem, CSP. First, we will offer our general framework for CSPs, and then apply it to first-order logic. We claim it would function as a new semantics, constraint semantics, for logic. Then, we prove the soundness and completeness theorems with respect to the constraint semantics. The latter theorem, which will be proven by a proof-search method, implies the cut-elimination theorem. Furthermore, using the constraint semantics, we make a new proof of the Craig interpolation theorem. Also, we will provide feasible algorithms to generate interpolants for some decidable fragments of first-order logic: the propositional logic and the monadic fragments. The algorithms, reflecting a ‘projection’ of an indexed relation, will show how to transform given formulas syntactically to obtain interpolants.

Appendices

Appendix I

Here, we give the sequent style proof system LK. Below, Greek capital letters ($\Gamma , \Delta , \ldots$) denote a finite sequence of formulas.

• Initial Sequent:        ${ A \Longrightarrow A}$       $\bot , \Gamma \Longrightarrow \Delta$

• Logical Inference rule:

The eigenvariable condition on $\forall :r$ and $\exists :l$:

there must not occur x freely in the lower sequent.

• Structural Inference rule:

Appendix II

Here, we present a counter-example when A is not in normal form and Theorem 6.1 fails. The following is such an example.

$$\begin{aligned}&A=\forall x(F_1(x) \vee F_2(x) \vee F_3(x)) \wedge (\exists x F_{3}(x) \vee \exists x F_{4}(x)) , \\&\quad ~\hbox { where} F_1=P_1\wedge P_2\wedge P_3;\\&F_2=P_1\wedge \lnot P_2\wedge \lnot P_3;\\&F_3=\lnot P_1\wedge \lnot P_2\wedge P_3;\\&F_4=P_1\wedge \lnot P_2\wedge P_3. \end{aligned}$$

Let $\pi _C=\pi _{J_{\forall xF_1(x)} \cap J_{\forall xF_3(x)}}$. Then, $(J_{\forall xF^{\pi }_1(x)}, R_{\forall xF^{\pi }_1(x)} )$ belongs to

$$\begin{aligned}&\pi _C(J_{\forall x(F_1(x) \vee F_2(x) \vee F_3(x))}, R_{\forall x(F_1(x) \vee F_2(x) \vee F_3(x))}) \\&\quad \cap \pi _C(J_{\exists x F_{3}(x) \vee \exists x F_{4}(x)}, R_{\exists x F_{3}(x) \vee \exists x F_{4}(x)}) , \end{aligned}$$

but does not belong to

$$\begin{aligned}&\pi _C[(J_{\forall x(F_1(x) \vee F_2(x) \vee F_3(x))}, R_{\forall x(F_1(x) \vee F_2(x) \vee F_3(x))}) \\&\quad \cap (J_{\exists x F_{3}(x) \vee \exists x F_{4}(x)}, R_{\exists x F_{3}(x) \vee \exists x F_{4}(x)})]. \end{aligned}$$