Primitive Recursive Dependent Type Theory: A Formal System for Capturing Primitive Recursive Functions

Core Concepts

Restricting the elimination principle of the natural numbers type in Martin-Löf Type Theory to a universe of types not containing Π-types ensures that all definable functions are primitive recursive, extending the concept of primitive recursiveness to general types.

Abstract

The paper presents a formal system called Primitive Recursive Dependent Type Theory (PRTT), which is a subtheory of Martin-Löf Type Theory (MLTT). The key idea is to restrict the elimination principle of the natural numbers type N to a universe U0 that does not contain Π-types. This ensures that all definable functions N → N are primitive recursive.
The authors first define the notion of primitive recursive functions and explain why one might expect MLTT with natural numbers but without Π-types to capture exactly primitive recursive functions. They then introduce the synthetic Tait computability framework, which is used to construct a model of PRTT in a topos of sheaves on a site of primitive recursive functions.
The main technical results are:

PRTT is sound and complete with respect to primitive recursion, in the sense that any primitive recursive function can be defined in it.
There is a sound interpretation of PRTT in a topos R, where the natural numbers object yN represents exactly the primitive recursive functions ℕ → ℕ.
By gluing the interpretation in R with the standard interpretation in Set, the authors show that all PRTT-definable functions N → N are primitive recursive.

The paper also discusses extensions of PRTT, such as adding a comonadic modality, an internal universe of codes for primitive recursive constructions, and connections to cubical type theory.

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Key Insights Distilled From

Primitive Recursive Dependent Type Theory

by Ulrik Buchho... at arxiv.org 04-02-2024

https://arxiv.org/pdf/2404.01011.pdf

Deeper Inquiries

What are the potential applications of PRTT beyond the results presented in the paper, such as in reverse mathematics or formal metatheory

Primitive Recursive Dependent Type Theory (PRTT) has various potential applications beyond the results presented in the paper. One significant application is in the field of reverse mathematics. PRTT can serve as a foundational system for reverse mathematics, allowing researchers to explore the relationships between different mathematical principles and the strength of various axioms. By providing a framework that captures primitive recursive functions and their definability, PRTT can aid in analyzing the computational complexity of mathematical theorems and systems.
Another application of PRTT could be in formal metatheory. PRTT's ability to define and reason about primitive recursive functions can be leveraged to develop formal systems for metatheoretical analysis. This could involve studying the properties of formal systems, proofs, and logical frameworks within a computationally bounded setting, leading to insights into the limits of formal reasoning and computation.
Furthermore, PRTT could find applications in the development of proof assistants and automated theorem provers. By ensuring that all definable functions are primitive recursive, PRTT provides a computationally tractable framework for formalizing and verifying mathematical proofs. This could enhance the efficiency and reliability of automated proof-checking systems, making them more robust and capable of handling complex mathematical reasoning tasks.

How could the techniques used to construct the model of PRTT be extended to capture other notions of computability, such as polynomial time computability

The techniques used to construct the model of PRTT could be extended to capture other notions of computability, such as polynomial time computability. By adapting the framework to incorporate the principles of polynomial time computability, researchers could develop a type theory that reflects the computational complexity of algorithms and functions that run in polynomial time.
One approach to extending the model to capture polynomial time computability could involve defining a new universe in the type theory that represents polynomial time computable functions. This universe would restrict the functions to those that can be computed in polynomial time, similar to how PRTT restricts functions to being primitive recursive. By incorporating the principles of polynomial time computability into the type theory, researchers could analyze and reason about the efficiency of algorithms and computational processes within a formalized setting.
Additionally, the techniques used in synthetic Tait computability could be adapted to characterize polynomial time computable functions and establish soundness results within the extended type theory. This would involve defining appropriate semantics and interpretations that capture the essence of polynomial time computability, enabling the formal verification of algorithms and computations that fall within this complexity class.

Are there any fundamental limitations or obstacles to transferring the authors' approach to higher topoi, and what would be the implications for type theory in that setting

Transferring the authors' approach to higher topoi may face fundamental limitations and obstacles due to the inherent complexity and structure of higher topoi. One potential limitation is the intricate nature of higher topoi, which may introduce additional complexities in defining and interpreting the semantics of type theories within these higher categorical settings. The higher categorical structures and properties of topoi could pose challenges in extending the techniques used in the paper to higher topoi, requiring a deeper understanding of higher category theory and topos theory.
Another obstacle could be the need to adapt the model of PRTT to accommodate the unique features and properties of higher topoi. Higher topoi often exhibit richer structures and more intricate categorical properties than ordinary topoi, necessitating a thorough reevaluation and modification of the model to align with the higher categorical context. This process may involve developing new techniques and methodologies to capture the essence of type theory in higher topoi accurately.
The implications for type theory in a higher topoi setting could be profound, offering a more sophisticated and nuanced framework for reasoning about types, functions, and computations within higher categorical structures. By overcoming the limitations and obstacles in transferring the approach to higher topoi, researchers could unlock new insights into the interplay between type theory and higher category theory, leading to advancements in computational and logical reasoning within these advanced mathematical contexts.