Alternating Turing machines and the analytical hierarchy

We use notions originating in Computational Complexity to provide insight into the analogies between computational complexity and Higher Recursion Theory. We consider alternating Turing machines, but with a modified, global, definition of acceptance. We show that a language is accepted by such a machine iff it is Pi-1-1. Moreover, total alternating machines, which either accept or reject each input, accept precisely the hyper-arithmetical (Delta-1-1) languages. Also, bounding the permissible number of alternations we obtain a characterization of the levels of the arithmetical hierarchy..The novelty of these characterizations lies primarily in the use of finite computing devices, with finitary, discrete, computation steps. We thereby elucidate the correspondence between the polynomial-time and the arithmetical hierarchies, as well as that between the computably-enumerable, the inductive (Pi-1-1), and the PSpace languages.

FINITARY REDUCIBILITY ON EQUIVALENCE RELATIONS

We introduce the notion of finitary computable reducibility on equivalence relations on the domainω. This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be${\rm{\Pi }}_{n + 2}^0$-complete under computable reducibility, we show that, for everyn, there does exist a natural equivalence relation which is${\rm{\Pi }}_{n + 2}^0$-complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy.

Computation over algebraic structures and a classification of undecidable problems

We consider a uniform model of computation over algebraic structures resulting from a generalization of the Turing machine and the BSS model of computation. This model allows us to gain more insight into the reasons for unsolvability of algorithmic decision problems from different perspectives. For example, classes of undecidable problems can be introduced in several ways by analogy with the classical arithmetical hierarchy and, for many structures, the different definitions lead to different hierarchies of undecidable problems. Here, we will investigate some classes of a hierarchy that is defined semantically by our deterministic oracle machines and that can be syntactically characterized by formulas whose quantifiers range only over an enumerable set. Starting from machines over algebraic structures endowed with some relations and containing an infinite recursively enumerable sequence of individuals, we will also consider this hierarchy for BSS RAM's over the reals and some undecidable problems defined by algebraic properties of the real numbers.

TRIAL AND ERROR MATHEMATICS I: DIALECTICAL AND QUASIDIALECTICAL SYSTEMS

We define and study quasidialectical systems, which are an extension of Magari’s dialectical systems, designed to make Magari’s formalization of trial and error mathematics more adherent to the real mathematical practice of revision: our proposed extension follows, and in several regards makes more precise, varieties of empiricist positions à la Lakatos. We prove several properties of quasidialectical systems and of the sets that they represent, called quasidialectical sets. In particular, we prove that the quasidialectical sets are ${\rm{\Delta }}_2^0$ sets in the arithmetical hierarchy. We distinguish between “loopless” quasidialectal systems, and quasidialectical systems “with loops”. The latter ones represent exactly those coinfinite c.e. sets, that are not simple. In a subsequent paper we will show that whereas the dialectical sets are ω-c.e., the quasidialectical sets spread out throughout all classes of the Ershov hierarchy of the ${\rm{\Delta }}_2^0$ sets.