[Paper Review] Two Puzzles About Computation

Samson Abramsky|arXiv (Cornell University)|Mar 19, 2014

Computability, Logic, AI Algorithms7 references3 citations

TL;DR

This paper identifies two foundational puzzles in computation: why we compute (information gain despite thermodynamic constraints) and what we compute (beyond functions to dynamic, interactive processes). It argues that a comprehensive theory of computation must incorporate agent-based information flow and a structural theory of processes, challenging classical views rooted in function computation and static computability.

ABSTRACT

The purpose of this note is to raise two different questions, which are rarely if ever considered, and to which, it seems, we lack convincing, systematic answers. These questions can be posed as: - Why do we compute? - What do we compute? The point is not so much that we have no answers to these puzzles, as that we have no established body of theory which gives satisfying, systematic answers, as part of a broader understanding. By raising these questions, we hope to stimulate some thinking in this direction.

Motivation & Objective

• To challenge the classical view of computation as merely function computation and highlight the need for a theory of dynamic, interactive processes.
• To address the paradox of information increase in computation despite thermodynamic and logical constraints.
• To investigate the nature of explicit vs. implicit knowledge in computation and its implications for epistemic logic and information theory.
• To question the adequacy of current models of computation for modern, interactive, and distributed systems.
• To advocate for a unified, fundamental theory of processes akin to the lambda calculus for functions.

Proposed method

• Analyzes the thermodynamic paradox of information increase by framing computation as occurring in open systems where subsystems can decrease entropy by exchanging energy and information with their environment.
• Applies the observer-relative notion of information increase, arguing that directionality of computation (e.g., multiplication vs. factorization) depends on the user’s perspective and intent.
• Uses the Curry-Howard isomorphism to model computation as proof normalization, distinguishing implicit knowledge (program/proof) from explicit knowledge (computed output).
• Introduces the concept of 'mechanical computation' as a process that transforms intensional descriptions into explicit, numerically representable outputs.
• Proposes that the true purpose of computation lies in extracting relevant information (the 'needle') from complex data (the 'haystack'), rather than in function evaluation.
• Draws analogies between computational dynamics and process calculi, arguing for a need for a unified, foundational calculus of processes, similar to the lambda calculus.

Experimental results

Research questions

• RQ1 How can information increase in a physical system without violating the second law of thermodynamics?
• RQ2 What accounts for the apparent informativeness of computation if the output is logically implied by the input?
• RQ3 Can information increase be objectively defined, or is it inherently observer-dependent?
• RQ4 What is the essential mathematical structure of a computational process beyond function computation?
• RQ5 Is there a universal model of interactive or concurrent computation analogous to the Church-Turing thesis for sequential computation?

Key findings

• Information increase in computation is possible because subsystems can decrease their entropy by exchanging energy and information with their environment, making open systems the fundamental unit of analysis.
• The direction of information increase is observer-relative: computing 3×5=15 is natural, but factoring 15 into 3×5 is a different kind of computation, depending on the user’s goal.
• Computation transforms implicit knowledge (a program or proof) into explicit knowledge (a numeral or normalized term), resolving the paradox of informative deduction.
• The distinction between intensional (program) and extensional (output) descriptions clarifies the epistemic value of computation, countering the problem of logical omniscience.
• The current proliferation of process calculi suggests a lack of a unified, foundational theory of processes, analogous to the lambda calculus for functions.
• The paper argues that modern computing—especially distributed, mobile, and interactive systems—demands a theory of computation centered on behavior and dynamics, not just function evaluation.