On the Relation between Sized-Types Based Termination and Semantic Labelling

Abstract

We investigate the relationship between two independently developed termination techniques. On the one hand, sized-types based termination (SBT) uses types annotated with size expressions and Girard's reducibility candidates, and applies on systems using constructor matching only. On the other hand, semantic labelling transforms a rewrite system by annotating each function symbol with the semantics of its arguments, and applies to any rewrite system. First, we introduce a simplified version of SBT for the simply-typed lambda-calculus. Then, we give new proofs of the correctness of SBT using semantic labelling, both in the first and in the higher-order case. As a consequence, we show that SBT can be extended to systems using matching on defined symbols (e.g. associative functions).

Size-based termination of higher-order rewriting

“…Finally, Roux and the author proved in (Blanqui & Roux, 2009) that size annotations provide a quasi-model, and thus can be used in a semantic labeling. Terms whose type is annotated by ∞ (unknown size) are interpreted by using a technique introduced by Hirokawa and Middeldorp in (Hirokawa & Middeldorp, 2006).…”

“…It is also a tool for defining and 17:12 Size-based termination of higher-order rewriting 33 strengthening the higher-order recursive path ordering (Blanqui, 2006b;Jouannaud & Rubio, 2007;Blanqui et al, 2015). Finally, some relations between these notions have been formally established: size-change principle and dependency pairs (Thiemann & Giesl, 2005), semantic labeling and recursive path ordering (Kamin & Lévy, 1980), dependency pairs and recursive path ordering (Dershowitz, 2013), and size-based termination and semantic labeling (Blanqui & Roux, 2009).…”

Practical Subtyping for System F with Sized (Co-)Induction

“…Our type system also strongly relates to sized types [17] as our inductive and coinductive types carry ordinal numbers. Such a technique is widespread for handling induction [7,8,15,20,38] and even coinduction [3,4,37], in settings where termination is required.…”

Termination of rewrite relations on λ-terms based on Girard's notion of reducibility

“…Finally, Hamana developed a categorical semantics for terms with bound variables [Ham06] based on the work of Fiore, Plotkin and Turi [FPT99], that is complete for termination (which is not the case of van de Pol's interpretations), and extended to higher-order terms the technique of semantic labeling [Ham07] introduced for first-order terms by Zantema [Zan95]. However, Roux showed that its application to β-reduction itself is not immediate since the interpretation of β-reduction is not β-reduction [BR09,Rou11].…”

“…For instance, van de Pol proved that his interpretations on N can be obtained from a computability proof by adding information on the length of reductions [vdP96]. Conversely, the author and Roux proved that size-based termination [Gim98, Abe04, BFG + 04, Bla04], which is a refinement of computability, can to some extent be seen as an instance of Hamana's higher-order semantic labeling technique [BR09].…”