It is desirable to answer queries' posed to deductive databases by computing fixpoints because such computations are directly amenable to set-oriented fact processing. However, the classical fixpoint procedures based on bottom-up processingthe naive and semi-naive methodsare rather primitive and often inefficient. In this article, we rely on bottom-up meta-interpretation for formalizing a new f...

It is desirable to answer queries posed to deductive databases by computing fix-points because such computations are directly amenable to set-oriented fact processing. However, the classical fixpoint procedures based on bottom-up reasoning-the naive and semi-naive methods-are rather primitive and often inefficient. In this article, we rely on bottom-up meta-interpretation for formalizing a new ...

We investigate the elimination of quantifiers in first-order formulas via Hilbert’s epsilon-operator (or -binder), following Bernays’ explicit definitions of the existential and the universal quantifier symbol by means of epsilon-terms. This elimination has its first explicit occurrence in the proof of the first epsilon-theorem in Hilbert–Bernays in 1939. We think that there is a lacuna in this...

Hilbert’s ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand’s Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon t...

Quantifier elimination (QE) over real closed fields has found numerous applications. Cylindrical algebraic decomposition (CAD) is one of the main tools for handling quantifier elimination of nonlinear input formulas. Despite of its worst case doubly exponential complexity, CAD-based quantifier elimination remains interesting for handling general quantified formulas and producing simple quantifi...

In this paper we investigate, for intuitionistic implicational logic, the relationship between normalization in natural deduction and cut-elimination in a standard sequent calculus. First we identify a subset of proofs in the sequent calculus that correspond to proofs in natural deduction. Then we define a reduction relation on those proofs that exactly corresponds to normalization in natural d...