Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set ℐ of the sentences that get limit probability 1 is a Π3—theory, in fact ℐ can be a Π3—complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Π3—sentences that get limit probability less than 1/2k. No set (...) ℐ as above can contain the set of all true Π3—sentences, although we exhibit some ℐ containing all the true Σ2—sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent. (shrink)

We investigate the variety corresponding to a logic, which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, and prove that (...) every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield, and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields. (shrink)

The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...) as well. We prove that the equational theory of \ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work. (shrink)

We describe a variety of sets internal to models of intuitionistic set theory that (1) manifest some of the crucial behaviors of potentially infinite sets as described in the foundational literature going back to Aristotle, and (2) provide models for systems of predicative arithmetic. We close with a brief discussion of Church’s Thesis for predicative arithmetic.

We develop a variant of Least Fixed Point logic based on First Orderlogic with a relaxed version of guarded quantification. We develop aGame Theoretic Semantics of this logic, and find that under reasonableconditions, guarding quantification does not reduce the expressibilityof Least Fixed Point logic. But we also find that the guarded version ofa least fixed point algorithm may have a greater time complexity thanthe unguarded version, by a linear factor.

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤be, which is a natural extension of s-reducibility ≤s. We show that ≤s, ≤be, and enumeration reducibility do not coincide on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} –sets, and the structure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\mathcal{D}_{\rm be}}}$$\end{document} of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory (...) of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\mathcal{D}_{\rm be}}}$$\end{document} is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987). (shrink)

A case is made for supposing that the total probability accounted for in a decision analysis is less than unity. This is done by constructing a measure on the set of all codes for computable functions in such a way that the measure of every effectively accountable subset is bounded by a number <1. The consistency of these measures with the Savage axioms for rational preference is established. Implications for applied decision theory are outlined.

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

We give an algorithm for deciding whether an embedding of a finite partial order [Formula: see text] into the enumeration degrees of the [Formula: see text]-sets can always be extended to an embedding of a finite partial order [Formula: see text].

In [2], Jockusch and Shore have introduced a new hierarchy of sets and operators called the REA hierarchy. In this note we prove analogues of the Friedberg Jump Theorem and the Sacks Jump Theorem for many REA operators. MSC: 03D25, 03D55.

For automatic and recursive graphs, we investigate the following problems: (A) existence of a Hamiltonian path and existence of an infinite path in a tree (B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree (C) existence of an infinite clique and an infinite version of set cover The complexity of these problems is determined for automatic graphs and, supplementing results from the literature, for recursive graphs. Our results show (...) that these problems (A) are equally complex for automatic and for recursive graphs ( $\Sigma _{1}^{1}\text{-complete}$ ), (B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy), (C) are much simpler for automatic than for recursive graphs (decidable and $\Sigma _{1}^{1}\text{-complete}$ , resp.). (shrink)

We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic.