Logic as a distributive law (1610.02247v3)

Abstract: We present an algorithm for deriving a spatial-behavioral type system from a formal presentation of a computational calculus. Given a 2-monad Calc: Catv$\to$ Cat for the free calculus on a category of terms and rewrites and a 2-monad BoolAlg for the free Boolean algebra on a category, we get a 2-monad Form = BoolAlg + Calc for the free category of formulae and proofs. We also get the 2-monad BoolAlg $\circ$ Calc for subsets of terms. The interpretation of formulae is a natural transformation $\interp{-}$: Form $\Rightarrow$ BoolAlg $\circ$ Calc defined by the units and multiplications of the monads and a distributive law transformation $\delta$: Calc $\circ$ BoolAlg $\Rightarrow$ BoolAlg $\circ$ Calc. This interpretation is consistent both with the Curry-Howard isomorphism and with realizability. We give an implementation of the "possibly" modal operator parametrized by a two-hole term context and show that, surprisingly, the arrow type constructor in the $\lambda$-calculus is a specific case. We also exhibit nontrivial formulae encoding confinement and liveness properties for a reflective higher-order variant of the $\pi$-calculus.

• The paper introduces a novel categorical framework that derives spatial-behavioral type systems through a distributive law linking computational calculi with logical constructs.
• It employs 2-monads and Lawvere 2-theories to reinterpret calculi like the λ- and π-calculus, providing a bridge between algebraic structures and operational semantics.
• The approach has practical implications for developing robust type systems and automating program verification in modern programming languages.

An Analytical Overview of "Logic as a Distributive Law"

This paper explores a sophisticated algorithm for deriving spatial-behavioral type systems from formal presentations of computational calculi via the application of categorical logic tools, particularly through the notion of distributive laws. By leveraging a categorical framework that includes 2-monads and Lawvere 2-theories, the authors aim to advance the understanding and expressiveness of computational logic in the context of complex systems of computation and concurrent behavior.

The paper sets forth by defining a 2-monad Calc: Cat → Cat, denoting a free calculus over a certain category, and Form = BoolAlg + Calc as a 2-monad for the category of formulae and proofs. A key focus is the derivation of a distributive law δ: Calc ◦ BoolAlg ⇒ BoolAlg ◦ Calc, which underpins the semantics of formulae in terms of computational processes. This approach aligns with the Curry-Howard isomorphism, a fundamental perspective about the interplay between logic and computational constructs, and is consistent with realizability interpretations.

Several elements are crucial in the translation of theoretical frameworks into practical computational models:

• Lawvere Theories: The paper provides an intricate exposition on Lawvere theories, a foundational concept in categorical logic, and extends it to 2-categories. These theories are essential for understanding how algebraic and logical systems can be represented categorically.
• Interpretation of Calculi: An examination of computational calculi, such as the λ and π-calculus, is considered under this categorical lens. The λ-calculus illustrates functional computation, whereas the π-calculus exemplifies concurrent programming paradigms.
• Spatial-Behavioral Type System: The intricate relationship between a calculus's structural description and its dynamic properties is formalized through a natural transformation J−K: Form ⇒ BoolAlg ◦ Calc. This transformation elucidates the dynamic potential of programs defined over these calculi by interpreting term formulae and enforcing invariants through propositions.

The authors assert that logical formulae can be seen as collections of terms based on their structures. Since different collections manifest as monads, the distributive law, compounded with monad laws, enables the crafting of logical descriptions from computational constructs. Monads such as BoolAlg for Boolean algebra, and others such as list or tree, showcase the versatility and power of this approach.

In substantiating their framework, the authors offer concrete examples with detailed rigor, such as the derivation and manipulation of logic for monoids using both categorical and computational paradigms. Notably, they extend to a discussion about the SKI combinator calculus as a demonstration of the synthesis of structural and operational semantics through their framework. Furthermore, modal operators, a critical component of expressing temporal or eventual properties in logic, enrich the expressivity of the resultant logical systems.

The theoretical implications are profound, suggesting a pathway for enriching languages with spatial-behavioral types, thus providing new mathematical tools for the design and analysis of programming languages and computational models. Practically, this framework paves the way for developing robust type systems for languages, which can facilitate program specifications and automatic verification.

The algorithm's broad applicability includes creating logics for languages fronted by the K Framework, potentially impacting widely used programming environments like C++ or Java by lowering the barriers for these languages to be enhanced with advanced type systems. Additionally, the paper articulates a vision for utilizing these logical constructions in new language designs, exemplified through the development of languages for blockchain smart contracts.

The paper does not shy away from the open challenges in expanding these theoretical frameworks and encourages further exploration into the field of higher-order categorical logics and their impact on computing. This scholarly work anchors itself at the intersection of theoretical computer science and practical computational logic, offering fertile ground for future research in logical formalisms within the domain of distributed systems and beyond.