Lambda Y-Calculus With Priorities

Abstract

The lambdaY-calculus with priorities is a variant of the simplytyped lambda calculus designed for higher-order model-checking. The higher-order model-checking problem asks if a given parity tree automaton accepts the Böhm tree of a given term of the simply-typed lambda calculus with recursion. We show that this problem can be reduced to the same question but for terms of lambdaY-calculus with priorities and visibly parity automata; a subclass of parity automata. The latter question can be answered by evaluating terms in a simple powerset model with least and greatest fixpoints. We prove that the recognizing power of powerset models and visibly parity automata are the same. So, up to conversion to the lambdaY-calculus with priorities, powerset models with least and greatest fixpoints are indeed the right semantic framework for the model-checking problem. The reduction to lambdaY-calculus with priorities is also efficient algorithmically: it gives an algorithm of the same complexity as direct approaches to the higher-order model-checking problem. This indicates that the task of calculating the value of a term in a powerset model is a central algorithmic problem for higher-order model-checking.

Profinite lambda-terms and parametricity

“…Aware of this correspondence with Scott semantics, Salvati and Walukiewicz actively promoted a semantic approach to higherorder model checking [10] which would complement the intersection type approach developed by Kobayashi and Ong [11], [12]. However, besides the fascinating connections to Krivine environment machines and collapsible pushdown automata [13], [14], [15], it took several years for developing a precise connection between Scott semantics and intersection type systems for higher-order model checking, with the emergence of a notion of higher-order parity automaton [6] founded on the discovery of an unexpected relationship with linear logic [16], [17], [4], [5] combined with a comonadic translation designed by Mellies of the simply-typed λY -calculus into a λY µνcalculus with inductive and coinductive fixpoints [6], or in a λY -calculus with priorities [18].…”

“…To finish this brief recap of Stone duality, we also recall [37, VI.3.2] that, conversely to (18), one may instead extend Proposition 24 by taking the inductive completion of the category FinSet to get Set, and the projective completion of the category FinBA to obtain the category CABA of complete and atomic Boolean algebras [37,VII.1.16]. We thus have a different dual equivalence, ℘ : Set op ⇆ CABA : At , often referred to as discrete Stone duality.…”

Profinite lambda-terms and parametricity

“…Aware of this correspondence with Scott semantics, Salvati and Walukiewicz actively promoted a semantic approach to higher-order model checking [29] which would complement the intersection type approach developed by Kobayashi and Ong [18,19]. However, besides the fascinating connections to Krivine environment machines and collapsible pushdown automata [6,15,30], it took several years to develop a precise connection between Scott semantics and intersection type systems for higher-order model checking, with the emergence of a notion of higher-order parity automaton [20] founded on the discovery of an unexpected relationship with linear logic [7,8,13,14] combined with a comonadic translation designed by Melliès of the simply typed λY -calculus into a λY µν -calculus with inductive and coinductive fixpoints [20], or into a λY -calculus with priorities [32].…”