Related papers: Computational Irreducibility and Computational Ana…

Our main models of computation (the Turing Machine and the RAM) make fundamental assumptions about which primitive operations are realizable. The consensus is that these include logical operations like conjunction, disjunction and negation,…

We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding…

Computation is commonly defined as the execution of abstract algorithms over symbolic representations, with physical systems treated as substrates that realise predefined operations. While effective for engineered machines, this separation…

In this paper we have investigated enumeration orders of elements of r.e. sets enumerated by means of Turing machines. We have defined a reducibility based on enumeration orders named "Enumeration Order Reducibility" on computable functions…

This article presents a formal model demonstrating that genuine autonomy, the ability of a system to self-regulate and pursue objectives, fundamentally implies computational unpredictability from an external perspective. we establish…

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy…

For readers already familiar with Partial Information Decomposition (PID), we show that PID's definition of synergy enables quantifying at least four different notions of irreducibility. First, we show four common notions of "parts" give…

The Church-Turing thesis states that any sufficiently powerful computational model which captures the notion of algorithm is computationally equivalent to the Turing machine. This equivalence usually holds both at a computability level and…

The study of computability has its origin in Hilbert's conference of 1900, where an adjacent question, to the ones he asked, is to give a precise description of the notion of algorithm. In the search for a good definition arose three…

The benchmark for computation is typically given as Turing computability; the ability for a computation to be performed by a Turing Machine. Many languages exploit (indirect) encodings of Turing Machines to demonstrate their ability to…

The notion of computable reducibility between equivalence relations on the natural numbers provides a natural computable analogue of Borel reducibility. We investigate the computable reducibility hierarchy, comparing and contrasting it with…

Regular functions from infinite words to infinite words can be equivalently specified by MSO-transducers, streaming $\omega$-string transducers as well as deterministic two-way transducers with look-ahead. In their one-way restriction, the…

This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…

We introduce new methods of equivalence checking and simulation based on Computing Range Reduction (CRR). Given a combinational circuit $N$, the CRR problem is to compute the set of outputs that disappear from the range of $N$ if a set of…

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect…

It is common practice to compare the computational power of different models of computation. For example, the recursive functions are strictly more powerful than the primitive recursive functions, because the latter are a proper subset of…

The concept of ``countable set'' is attributed to Georg Cantor, who set the boundary between countable and uncountable sets in 1874. The concept of ``computable set'' arose in the study of computing models in the 1930s by the founders of…

Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are…