From the Publisher:
This mathematically oriented introduction to the theory of logic programming presents a systematic exposition of the resolution method for propositional, first-order, and Horn- clause logics, together with an analysis of the semantic aspects of the method. It is through the inference rule of resolution that both proofs and computations can be manipulated on computers, and this book contains elegant versions and proofs of the fundamental theorems and lemmas in the proof theory of logic programming. Advanced topics such as recursive complexity and negation as failure and its semantics are covered, and streamlined setups for SLD- and SLDNF-resolution are described.
No other book treats this material in such detail and with such sophistication. Doets provides a novel approach to resolution that is applied to the first-order case and the case of (positive) logic programs. In contrast to the usual approach, the concept of a resolvent is defined nonconstructively, without recourse to the concept of unification, allowing the soundness and completeness proofs to be carried out in a more economic way. Other new material includes computability results dealing with analytical hierarchy, results on infinite derivations and an exposition on general logic programs using 3-valued logic.

From Logic to Logic Programming

The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces L
 p(Rd) (in the case p > 1), but (in the case when 1/p(·) is log-Holder continuous and p- = inf{p(x): x ∈ Rd > 1) on the variable Lebesgue spaces L
 p(·)(Rd), too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type (1, 1). In the present note we generalize Besicovitch’s covering theorem for the so-called γ-rectangles. We introduce a general maximal operator Ms
 γδ, and with the help of generalized Φ-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function 1/p(·) is log-Holder continuous and p- ≥ s, where 1 ≤ s ≤ ∞ is arbitrary (or p- ≥ s), then the maximal operator Ms
 γδ is bounded on the space L
 p(·)(Rd) (or the maximal operator is of weak-type (p(·), p(·))).

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Weak- and strong-type inequality for the cone-like maximal operator in variable Lebesgue spaces

We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model \(\mathfrak {A}\) by some Horn theory, where \(\mathfrak {A}\) can be any model of a higher recursion theory, like primitive set recursion, \(\alpha \)-recursion, or \(\beta \)-recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set \(\mathfrak {A}\) is recursively enumerable in models of the above mentioned recursion theories.

Juan A. Nido Valencia

Papers

A representation of recursively enumerable sets through Horn formulas in higher recursion theory