Logic Programming as Constructivism

The features of logic programming that seem unconventional from the viewpoint of classical logic can be explained in terms of constructivistic logic. We motivate and propose a constructivistic proof theory of non-Horn logic programming. Then, we apply this formalization for establishing results of practical interest. First, we show that 'stratification can be motivated in a simple and intuitive way. Relying on similar motivations, we introduce the larger classes of 'loosely stratified' and 'constructively consistent' programs. Second, we give a formal basis for introducing quantifiers into queries and logic programs by defining 'constructively domain independent* formulas. Third, we extend the Generalized Magic Sets procedure to loosely stratified and constructively consistent programs, by relying on a 'conditional fixpoini procedure

Classical propositional logic and decidability of variables in intuitionistic propositional logic

We improve the answer to the question: what set of excluded middles for propositional variables in a formula suffices to prove the formula in intuitionistic propositional logic whenever it is provable in classical propositional logic

A Hypercomputation in Brouwer's Constructivism

In contrast to other constructivist schools, for Brouwer, the notion of "constructive object" is not restricted to be presented as `words' in some finite alphabet of symbols, and choice sequences which are non-predetermined and unfinished objects are legitimate constructive objects. In this way, Brouwer's constructivism goes beyond Turing computability. Further, in 1999, the term hypercomputation was introduced by J. Copeland. Hypercomputation refers to models of computation which go beyond Church-Turing thesis. In this paper, we propose a hypercomputation called persistently evolutionary Turing machines based on Brouwer's notion of being constructive.Comment: This paper has been withdrawn by the author due to crucial errors in theorems 4.6 and 5.2 and definition 4.

Five Observations Concerning the Intended Meaning of the Intuitionistic Logical Constants

This paper contains five observations concerning the intended meaning of the intuitionistic logical constants: (1) if the explanations of this meaning are to be based on a non-decidable concept, that concept should not be that of 'proof'; (2) Kreisel's explanations using extra clauses can be significantly simplified; (3) the impredicativity of the definition of → can be easily and safely ameliorated; (4) the definition of → in terms of 'proofs from premises' results in a loss of the inductive character of the definitions of ∨ and ∃; and (5) the same occurs with the definition of ∀ in terms of 'proofs with free variables

Undeciding the decidable

Heinz von Foerster’s influential distinction between decidable and undecidable decisions may be taken to imply an ethics that is personal and pluralistic, summed up in invocations to decide the undecidable and to act in ways that increase the number of choices. While this approach is helpful as a critique of moralism and objectivity, it is of limited assistance in situations characterised by conflict, inequality, or the need for collective action. In this paper, I return to Foerster’s discussion to suggest a different way of thinking about ethics in terms of undecidability. I argue that it is not enough to decide upon (take responsibility for) undecidable questions. To confront the injustices that are embedded in the present world, decidable decisions—those that Foerster characterised as decided already by the frameworks in which they are asked—also need to be challenged. Whereas Foerster traces undecidability back to foundational metaphysical questions, positioning the ethical within the context of a choice between distinct worldviews, I situate decidability and undecidability as frames to move between within the context of practical situations. To complement the need to decide the undecidable, I explore the value of undeciding the decidable. By undeciding, what I mean to suggest is a process of reconceiving the framework in which a decidable decision is asked such that the framework is itself undecidable, thus requiring a decision to be made as to the decidability of the decision that is at stake. A consequence of putting decidability in question is that it is not sufficient to discharge one’s responsibilities as they arise. One must become responsible not just for one’s responsibilities but also for what these are and how their boundaries and scope are conceived. From this perspective, I offer an alternate reading of Foerster’s call to increase the number of choices, understanding this in the sense of acting to increase the number of decisions that are to be made rather than increasing the number of possibilities to be chosen between

The strength of countable saturation

We determine the proof-theoretic strength of the principle of countable saturation in the context of the systems for nonstandard arithmetic introduced in our earlier work.Comment: Corrected typos in Lemma 3.4 and the final paragraph of the conclusio

Church's thesis and related axioms in Coq's type theory

"Church's thesis" ($\mathsf{CT}$) as an axiom in constructive logic states that every total function of type $\mathbb{N} \to \mathbb{N}$ is computable, i.e. definable in a model of computation. $\mathsf{CT}$ is inconsistent in both classical mathematics and in Brouwer's intuitionism since it contradicts Weak K\"onig's Lemma and the fan theorem, respectively. Recently, $\mathsf{CT}$ was proved consistent for (univalent) constructive type theory. Since neither Weak K\"onig's Lemma nor the fan theorem are a consequence of just logical axioms or just choice-like axioms assumed in constructive logic, it seems likely that $\mathsf{CT}$ is inconsistent only with a combination of classical logic and choice axioms. We study consequences of $\mathsf{CT}$ and its relation to several classes of axioms in Coq's type theory, a constructive type theory with a universe of propositions which does neither prove classical logical axioms nor strong choice axioms. We thereby provide a partial answer to the question which axioms may preserve computational intuitions inherent to type theory, and which certainly do not. The paper can also be read as a broad survey of axioms in type theory, with all results mechanised in the Coq proof assistant