Things that can be made into themselves

One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $\alpha_0,\alpha_1,\ldots$ of all left-r.e. sets such that the index set $\{e: \alpha_e$ satisfies $P\}$ has the property $P$ as well. For example, the property of being Martin-L\"of random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of the present work is the investigation of the structure of left-r.e. sets under inclusion modulo a finite set. In contrast to the corresponding structure for r.e. sets, which has only maximal but no minimal members, both minimal and maximal left-r.e. sets exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly differs from Friedberg's classical construction of maximal r.e. sets. Finally, we investigate whether the properties of minimal and maximal left-r.e. sets can be made into themselves

Epistemic Circularity: Worry, Illusion, and Determination

Research on epistemic circularity has focused more on the problem of whether circular arguments are able to justify their conclusions rather than on the nature of circularity itself, and has kept a sharp delimitation from other types of circularities (logical, linguistic, and of definition). In this paper, I aim to move the focus toward the general concept of circularity, which I relate to the concept of ‘epistemic worry’. Through a brief overview on the known types of circularities, I show that circularity has multiple natures (not only that associated with its type), and the linguistic aspect is essential in qualifying a construct as circular, which in turn raises the problem of genuineness of a circularity. Within an extensional approach, I suggest generalizing the meaning of the key epistemological terms referred to in the classical definition of epistemic circularity to also cover additional associated concepts specific to other theoretical disciplines beyond epistemology. The proposed concepts and ideas are reflected in a structural unificatory account of circularity in terms of epistemic determination, which raises a challenge to the relational accounts of determination in regard to the transitivity of the determination relation

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

Characterizing Programming Systems Allowing Program Self-Reference ⋆

Abstract. The interest is in characterizing insightfully the power of program self-reference in effective programming systems (epses), the computability-theoretic analogs of programming languages. In an eps in which the constructive form of Kleene’s Recursion Theorem (KRT) holds, it is possible to construct, algorithmically, from an arbitrary algorithmic task, a self-referential program that, in a sense, creates a selfcopy and then performs that task on the self-copy. In an eps in which the not-necessarily-constructive form of Kleene’s Recursion Theorem (krt) holds, such self-referential programs exist, but cannot, in general, be found algorithmically. In an earlier effort, Royer proved that there is no collection of recursive denotational control structures whose implementability characterizes the epses in which KRT holds. One main result herein, proven by a finite injury priority argument, is that the epses in which krt holds are, similarly, not characterized by the implementability of some collection of recursive denotational control structures. On the positive side, however, a characterization of such epses of a rather different sort is shown herein. Though, perhaps not the insightful characterization sought after, this surprising result reveals that a hidden and inherent constructivity is always present in krt. Know thyself. – Greek prover

Computation and Logic in the Real World

The proceedings contain 78 papers. The topics discussed include: shifting and lifting of cellular automata; learning as data compression; a classification of viruses through recursion theorems; characterizing programming systems allowing program self-reference; thin maximal antichains in the turing degrees; effective computation for nonlinear systems; time-complexity semantics for feasible affine recursions; feasible depth; a continuous derivative for real-valued functions; refocusing generalised normalisation; parameterized complexity and logic; index sets of computable structures with decidable theories; operational semantics for positive relevant logics without distribution; unique existence and computability in constructive reverse mathematics; circuit complexity of regular language; definability in the homomorphic quasiorder of finite labeled forests; and membrane systems and their application to systems biology