Normalization for Fitch-Style Modal Calculi
Journal article, 2022

Fitch-style modal lambda calculi enable programming with necessity modalities in a typed lambda calculus by extending the typing context with a delimiting operator that is denoted by a lock. The addition of locks simplifies the formulation of typing rules for calculi that incorporate different modal axioms, but each variant demands different, tedious and seemingly ad hoc syntactic lemmas to prove normalization. In this work, we take a semantic approach to normalization, called normalization by evaluation (NbE), by leveraging the possible-world semantics of Fitch-style calculi to yield a more modular approach to normalization. We show that NbE models can be constructed for calculi that incorporate the K, T and 4 axioms of modal logic, as suitable instantiations of the possible-world semantics. In addition to existing results that handle beta-equivalence, our normalization result also considers eta-equivalence for these calculi. Our key results have been mechanized in the proof assistant Agda. Finally, we showcase several consequences of normalization for proving meta-theoretic properties of Fitch-style calculi as well as programming-language applications based on different interpretations of the necessity modality.

Fitch-style lambda calculi

Possible-world semantics

Normalization by Evaluation


Nachiappan Valliappan

Chalmers, Computer Science and Engineering (Chalmers), Information Security

Fabian Ruch

University of Gothenburg

Carlos Tomé Cortiñas

Chalmers, Computer Science and Engineering (Chalmers), Information Security

Proceedings of the ACM on Programming Languages

24751421 (eISSN)

Vol. 6 ICFP 118

Areas of Advance

Information and Communication Technology

Subject Categories


Computer Science

Computer Systems



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4/2/2024 1