A short note on type-inhabitation: Formula-trees vs. game semantics

Abstract

Type-inhabitation is a topic of major importance, due to its close relationship to provability in logical systems and has been studied from di↵erent perspectives over the years. In 2000 a new proof method has been presented, evidencing the close relationship between the structure of types and their inhabitants. More recently, in 2011, another method has been given in the context of game semantics. In this paper we clarify the similarities between the two approaches.

Inhabitation in simply typed lambda-calculus through a lambda-calculus for proof search

“…Many approaches exist in the study of inhabitation problems in the simply typed λ-calculus, some adapting tools from automata and language theory (Dowek and Jiang 2009;Schubert et al 2015;Takahashi et al 1996) or game theory (Bourreau and Salvati 2011), others creating new representations like in the formula-tree method (Alves and Broda 2015;Broda and Damas 2005) and others through a direct combinatorial analysis of a graph-theoretic representation of the search space (Wells and Yakobowski 2004). All these approaches have their merits, and making connections to other fields is one of them.…”

Inhabitation in Simply-Typed Lambda-Calculus through a Lambda-Calculus for Proof Search

“…We could add that such representation has a special status: it was derived as an inductive, finitary counterpart to the coinductive characterization of the search process; and the latter is a rather natural (we might say canonical) mathematical definition of the process which, in addition, may be seen as extending the Curry-Howard paradigm of representation, from proofs to runs of search processes (all this will be recalled in Section 2). Furthermore, the finitary representation stays within the methods of λ-calculus and type systems, which dispenses us from importing and adapting methods from other areas, like automata and language theory or games [TAH96,BS11,SDB15], or from creating new representations like in the proof-tree method [BD05,AB15].…”