Higher-Order Beta Matching with Solutions in Long Beta-Eta Normal Form

Higher-order matching is a special case of unification of simply-typed lambda-terms: in a matching equation, one of the two sides contains no unification variables. Loader has recently shown that higher-order matching up to beta equivalence is undecidable, but decidability of higher-order matching up to beta-eta equivalence is a long-standing open problem. We show that higher-order matching up to beta-eta equivalence is decidable if and only if a restricted form of higher-order matching up to beta equivalence is decidable: the restriction is that solutions must be in long beta-eta normal form

Statman\u27s 1-Section Theorem

Statman\u27s 1-Section Theorem [17] is an important but little-known result in the model theory of the simply-typed λ-calculus. The λ-Section Theorem states a necessary and sufficient condition on models of the simply-typed λ-calculus for determining whether βη-equational reasoning is complete for proving equations that hold in a model. We review the statement of the theorem, give a detailed proof, and discuss its significance

Syntactically and semantically regular languages of lambda-terms coincide through logical relations

A fundamental theme in automata theory is regular languages of words and trees, and their many equivalent definitions. Salvati has proposed a generalization to regular languages of simply typed $\lambda$-terms, defined using denotational semantics in finite sets. We provide here some evidence for its robustness. First, we give an equivalent syntactic characterization that naturally extends the seminal work of Hillebrand and Kanellakis connecting regular languages of words and syntactic $\lambda$-definability. Second, we show that any finitary extensional model of the simply typed $\lambda$-calculus, when used in Salvati's definition, recognizes exactly the same class of languages of $\lambda$-terms as the category of finite sets does. The proofs of these two results rely on logical relations and can be seen as instances of a more general construction of a categorical nature, inspired by previous categorical accounts of logical relations using the gluing construction.Comment: The proofs on "finitely pointable" CCCs in versions 1 and 2 were wrong; we now make slightly weaker claims on well-pointed locally finite CCCs. New in this version: added reference [3] and official DOI (proceedings of CSL 2024

Profinite lambda-terms and parametricity

Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type

Weak Typed Boehm Theorem on IMLL

In the Boehm theorem workshop on Crete island, Zoran Petric called Statman's ``Typical Ambiguity theorem'' typed Boehm theorem. Moreover, he gave a new proof of the theorem based on set-theoretical models of the simply typed lambda calculus. In this paper, we study the linear version of the typed Boehm theorem on a fragment of Intuitionistic Linear Logic. We show that in the multiplicative fragment of intuitionistic linear logic without the multiplicative unit 1 (for short IMLL) weak typed Boehm theorem holds. The system IMLL exactly corresponds to the linear lambda calculus without exponentials, additives and logical constants. The system IMLL also exactly corresponds to the free symmetric monoidal closed category without the unit object. As far as we know, our separation result is the first one with regard to these systems in a purely syntactical manner.Comment: a few minor correction

Equality between Functionals in the Presence of Coproducts

AbstractWe consider the lambda calculus obtained from the simply typed calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any set-theoretic model with an infinite base type

Type theory in a type theory with quotient inductive types

Type theory (with dependent types) was introduced by Per Martin-Löf with the intention of providing a foundation for constructive mathematics. A part of constructive mathematics is type theory itself, hence we should be able to say what type theory is using the formal language of type theory. In addition, metatheoretic properties of type theory such as normalisation should be provable in type theory. The usual way of defining type theory formally is by starting with an inductive definition of precontexts, pretypes and preterms and as a second step defining a ternary typing relation over these three components. Well-typed terms are those preterms for which there exists a precontext and pretype such that the relation holds. However, if we use the rich metalanguage of type theory to talk about type theory, we can define well-typed terms directly as an inductive family indexed over contexts and types. We believe that this latter approach is closer to the spirit of type theory where objects come intrinsically with their types. Internalising a type theory with dependent types is challenging because of the mutual definitions of types, terms, substitution of terms and the conversion relation. We use induction induction to express this mutual dependency. Furthermore, to reduce the type-theoretic boilerplate needed for reasoning in the syntax, we encode the conversion relation as the equality type of the syntax. We use equality constructors thus we define the syntax as a quotient inductive type (a special case of higher inductive types from homotopy type theory). We define the syntax of a basic type theory with dependent function space, a base type and a family over the base type as a quotient inductive inductive type. The definition of the syntax comes with a notion of model and an eliminator: whenever one is able to define a model, the eliminator provides a function from the syntax to the model. We show that this method of representing type theory is practically feasible by defining a number of models: the standard model, the logical predicate interpretation for parametricity (as a syntactic translation) and the proof-relevant presheaf logical predicate interpretation. By extending the latter with a quote function back into the syntax, we prove normalisation for type theory. This can be seen as a proof of normalisation by evaluation. Internalising the syntax of type theory is not only of theoretical interest. It opens the possibility of type-theoretic metaprogramming in a type-safe way. This could be used for generic programming in type theory and to implement extensions of type theory which are justified by models such as guarded type theory or homotopy type theory

Type theory in a type theory with quotient inductive types

Type theory (with dependent types) was introduced by Per Martin-Löf with the intention of providing a foundation for constructive mathematics. A part of constructive mathematics is type theory itself, hence we should be able to say what type theory is using the formal language of type theory. In addition, metatheoretic properties of type theory such as normalisation should be provable in type theory. The usual way of defining type theory formally is by starting with an inductive definition of precontexts, pretypes and preterms and as a second step defining a ternary typing relation over these three components. Well-typed terms are those preterms for which there exists a precontext and pretype such that the relation holds. However, if we use the rich metalanguage of type theory to talk about type theory, we can define well-typed terms directly as an inductive family indexed over contexts and types. We believe that this latter approach is closer to the spirit of type theory where objects come intrinsically with their types. Internalising a type theory with dependent types is challenging because of the mutual definitions of types, terms, substitution of terms and the conversion relation. We use induction induction to express this mutual dependency. Furthermore, to reduce the type-theoretic boilerplate needed for reasoning in the syntax, we encode the conversion relation as the equality type of the syntax. We use equality constructors thus we define the syntax as a quotient inductive type (a special case of higher inductive types from homotopy type theory). We define the syntax of a basic type theory with dependent function space, a base type and a family over the base type as a quotient inductive inductive type. The definition of the syntax comes with a notion of model and an eliminator: whenever one is able to define a model, the eliminator provides a function from the syntax to the model. We show that this method of representing type theory is practically feasible by defining a number of models: the standard model, the logical predicate interpretation for parametricity (as a syntactic translation) and the proof-relevant presheaf logical predicate interpretation. By extending the latter with a quote function back into the syntax, we prove normalisation for type theory. This can be seen as a proof of normalisation by evaluation. Internalising the syntax of type theory is not only of theoretical interest. It opens the possibility of type-theoretic metaprogramming in a type-safe way. This could be used for generic programming in type theory and to implement extensions of type theory which are justified by models such as guarded type theory or homotopy type theory