An Infinitary Lambda Calculus with Global Trace Condition (Extended Abstract)
2026-06-22 • Logic in Computer Science
Logic in Computer Science
AI summaryⓘ
The authors study a version of the lambda calculus extended with numbers and conditions, along with a type system similar to Goedel's System T. They introduce a technical condition called the Global Trace Condition (GTC) that helps control infinite computations. They prove that if a term meets this condition, its infinite computations behave nicely and eventually produce a final result. Additionally, they show that all well-typed terms of number type reduce to actual numerals and connect their system to known results in cyclic proof theory and total function definitions.
lambda calculusinfinitary calculusGoedel's System Ttype systemGlobal Trace Condition (GTC)strong convergenceChurch-Rosser propertycyclic proofstotal functionsreduction
Authors
Stefano Berardi, Ugo de' Liguoro, Daisuke Kimura, Daniel Osorio-Valencia
Abstract
We consider an extension of the infinitary lambda calculus by Kennaway et al., with zero, successor, and conditional, and a type system akin to Goedel's system T. For terms that can be typed in this system, we define the Global Trace Condition (GTC), adapting the concept from Brotherston and Simpson's Cyclic Proofs, and show that any infinite reduction of a well-typed term satisfying the GTC is strongly convergent. As a corollary, we obtain the proof that any closed term of type Nat reduces to some numeral through any reduction by levels. We argue that the Church-Rosser in the limit holds for our calculus and, when restricted to regular terms, the calculus defines exactly the total functions defined in Das's Cyclic System T (an infinitary version of System T without $λ$), and hence in Goedel's System T.