Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

Revisiting the Complexity of Stability of Continuous and Hybrid Systems

We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using first-order formulas over the real numbers, and reduce stability problems to the delta-decision problems of these formulas. The framework allows us to obtain a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the stability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we give upper bounds on their degrees of unsolvability

A Galois connection between Turing jumps and limits

Limit computable functions can be characterized by Turing jumps on the input side or limits on the output side. As a monad of this pair of adjoint operations we obtain a problem that characterizes the low functions and dually to this another problem that characterizes the functions that are computable relative to the halting problem. Correspondingly, these two classes are the largest classes of functions that can be pre or post composed to limit computable functions without leaving the class of limit computable functions. We transfer these observations to the lattice of represented spaces where it leads to a formal Galois connection. We also formulate a version of this result for computable metric spaces. Limit computability and computability relative to the halting problem are notions that coincide for points and sequences, but even restricted to continuous functions the former class is strictly larger than the latter. On computable metric spaces we can characterize the functions that are computable relative to the halting problem as those functions that are limit computable with a modulus of continuity that is computable relative to the halting problem. As a consequence of this result we obtain, for instance, that Lipschitz continuous functions that are limit computable are automatically computable relative to the halting problem. We also discuss 1-generic points as the canonical points of continuity of limit computable functions, and we prove that restricted to these points limit computable functions are computable relative to the halting problem. Finally, we demonstrate how these results can be applied in computable analysis

Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page