PECR: A Formal System Exploring Computability Logic

Core Concepts

PECR is a formal system exploring computability logic through programs on real-world computers.

Abstract

The content introduces PECR, a formal system designed to explore computability of programs on real-world computers. It discusses the main features of PECR, its practical applications, and its implementation in exploring the laws of nature through computer models. The structure covers topics like lists, programs, atomic programs, computable program extensions, IOT and substitution rules, disjunctions and conjunctions, applications, machine numbers, discrete boxes, and derivable rules of PECR. It delves into the background of mathematics and the challenges faced in expressing real-world phenomena using continuous mathematics. The article emphasizes a shift towards computational validity in theories and explores the interplay between formal systems and experimental computation. Introduction: Mathematics has explored abstract objects but faces challenges translating them into real-world phenomena. Theoretical models expressed in continuous mathematics pose translation difficulties for computer models. Real-world Machine Computation: PECR implements computability logic entirely by machine computation. Focuses on rigorous program construction based on CoL inference rule. Current Projects: PECR explores computability of real-world applications directly on machines. VPC software package implements interactive proof assistant based on CoL. Measure vs Patterns: Shift towards information-based descriptions over quantity-based measures. Computer models focus on structural patterns rather than arithmetic operations. Machine Parameters: Constraints imposed by finite resources define machine environment parameters. Encourages computations without excessive demands on resources. Lists: Discusses list elements, lists of lists, singleton lists, list operations like concatenation and intersection. Programs: Introduces atomic programs as building blocks for program lists. Defines value assignments for input/output elements in programs.

Stats

PECR is a formal system designed to explore properties of computability - G. Pantelis - March 22, 2024

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PECR

by G. Pantelis at arxiv.org 03-25-2024

https://arxiv.org/pdf/2403.14880.pdf

Deeper Inquiries

How does PECR address the limitations of traditional mathematical approaches

PECR addresses the limitations of traditional mathematical approaches by focusing on computability and real-world machine computation. Unlike theories expressed in continuous mathematics, which can be challenging to translate into computational languages, PECR operates directly within a machine environment with finite resources. By incorporating the properties of real-world machines and exploring computability through programs, PECR offers a more practical approach to understanding phenomena. This shift allows for rigorous validation based on computations rather than abstract mathematical formulations.

What implications does the shift towards computational validity have for scientific research

The shift towards computational validity has significant implications for scientific research. It emphasizes the importance of information as fundamental to observable objects and their behavior, aligning with digital physics concepts where discrete formulations are favored over continuous ones. By expressing laws of nature through programs instead of equations, researchers can create computer models that directly reflect real-world phenomena without relying on abstractions or approximations. This approach opens up new avenues for exploration and analysis in various scientific fields.

How can experimental mathematics complement or challenge the axiomatic method used in pure mathematics

Experimental mathematics can complement the axiomatic method used in pure mathematics by providing empirical evidence and insights that challenge established theoretical frameworks. While pure mathematicians rely on formal proofs derived from axioms, experimental mathematicians use computational methods to explore patterns, make conjectures, and test hypotheses through numerical experiments. This dynamic interplay between theory and experimentation can lead to new discoveries, alternative perspectives on existing problems, and potentially drive innovations in both fields by bridging the gap between abstract reasoning and practical applications.