Related papers: Introduction to clarithmetic I
We exhibit a sound and complete implicit-complexity formalism for functions feasibly computable by structural recursions over inductively defined data structures. Feasibly computable here means that the structural-recursive definition runs…
Computability logic (CoL) (see http://www.cis.upenn.edu/~giorgi/cl.html) is a recently introduced semantical platform and ambitious program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth…
Challenging the standard notion of totality in computable functions, one has that, given any sufficiently expressive formal axiomatic system, there are total functions that, although computable and "intuitively" understood as being total,…
The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see…
This is an exposition of facts about Arithmetic with an approach via mathematical logic. In Section 1 we present Peano Arithmetic, PA, and the complete theory of $\mathbb{N}$, and we show that $\mathbb{N}$ is a prime model of the theory of…
The basic notions of logic-predicate logic, Peano arithmetic, incompleteness theorems, etc.-have for long been an advanced topic. In the last decades, they became more widely taught, inphilosophy, mathematics, and computer science…
The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing…
The outcomes of this paper are twofold. Implicit complexity. We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class PTIME of languages computable in…
In a recently launched research program for developing logic as a formal theory of (interactive) computability, several very interesting logics have been introduced and axiomatized. These fragments of the larger Computability Logic aim not…
This article presents an overview of computability logic -- the game-semantically constructed logic of interactive computational tasks and resources. There is only one non-overview, technical section in it, devoted to a proof of the…
The present paper introduces a novel notion of `(effective) computability', called viability, of strategies in game semantics in an intrinsic (i.e., without recourse to the standard Church-Turing computability), non-inductive and…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
Computability logic (CL) (see http://www.cis.upenn.edu/~giorgi/cl.html ) is a research program for redeveloping logic as a formal theory of computability, as opposed to the formal theory of truth which it has more traditionally been.…
In this article, we investigate the arithmetical hierarchy from the perspective of realizability theory. An experimental observation in classical computability theory is that the notion of degrees of unsolvability for natural arithmetical…
Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and…
While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
Philosophy of science attempts to describe all parts of the scientific process in a general way in order to facilitate the description, execution and improvements of this process. So far, all proposed philosophies have only covered existing…
Logics with team semantics provide alternative means for logical characterization of complexity classes. Both dependence and independence logic are known to capture non-deterministic polynomial time, and the frontiers of tractability in…
We introduce Wave Arithmetic, a smooth analytical framework in which natural, integer, and rational numbers are represented not as discrete entities, but as integrals of smooth, compactly supported or periodic kernel functions. In this…