On the density of honest subrecursive classes

The relation of honest subrecursive classes to the computational complexity of the functions they contain is briefly reviewed. It is shown that the honest subrecursive classes are dense under the partial ordering of set inclusion. In fact, any countable partial ordering can be embedded in the gap between an effective increasing sequence of honest subrecursive classes and an honest subrecursive class which is properly above the sequence (or in the gap between an eflective decreasing sequence and a class which is properly below the sequence). Information is obtained about the possible existence of least upper bounds (greater lower bounds) of increasing (decreasing) sequences of honest subrecursive classes. Finally it is shown that for any two honest subrecursive classes, one properly containing the other, there exists a pair of incomparable honest subrecursive classes such that the greatest lower bound of the pair is the smaller of the first two classes and the least upper bound of the pair is the larger of the first two classes

Minimal pairs of polynomial degrees with subexponential complexity

AbstractThe goal of extending work on relative polynomial time computability from computations relative to sets of natural numbers to computations relative to arbitrary functions of natural numbers is discussed. The principal techniques used to prove that the honest subrecursive classes are a lattice are then used to construct a minimal pair of polynomial degrees with subexponential complexity; that is two sets computable by Turing machines in subexponential time but not in polynomial time are constructed such that any set computable from both in polynomial time can be computed directly in polynomial time

Honest elementary degrees and degrees of relative provability without the cupping property

An element a of a lattice cups to an element b>ab>a if there is a c<bc<b such that a∪c=ba∪c=b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0<Ea<Eb0<Ea<Eb that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 25th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2022, which was held during April 4-6, 2022, in Munich, Germany, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2022. The 23 regular papers presented in this volume were carefully reviewed and selected from 77 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

1976/1977 UCI General Catalogue

General catalogue for the academic year 1976-1977