Giulio Guerrieri

A Semantical and Operational Account of Call-by-Value Solvability

In Plotkin’s call-by-value lambda-calculus, solvable terms are characterized syntactically by means of call-by-name reductions and there is no neat semantical characterization of such terms. Preserving confluence, we extend Plotkin’s original reduction without adding extra syntactical constructors, and we get a call-by-value operational characterization of solvable terms. Moreover, we give a semantical characterization of solvable terms in a relational model, based on Linear Logic, satisfying the Taylor expansion formula. As a technical tool, we also use a resource-sensitive calculus (with tests) in which the elements of the model are definable.

Open Call-by-Value

The elegant theory of the call-by-value lambda-calculus relies on weak evaluation and closed terms, that are natural hypotheses in the study of programming languages. To model proof assistants, however, strong evaluation and open terms are required, and it is well known that the operational semantics of call-by-value becomes problematic in this case. Here we study the intermediate setting—that we call Open Call-by-Value—of weak evaluation with open terms, on top of which Gregoire and Leroy designed the abstract machine of Coq. Various calculi for Open Call-by-Value already exist, each one with its pros and cons. This paper presents a detailed comparative study of the operational semantics of four of them, coming from different areas such as the study of abstract machines, denotational semantics, linear logic proof nets, and sequent calculus. We show that these calculi are all equivalent from a termination point of view, justifying the slogan Open Call-by-Value.

The Bang Calculus: an untyped lambda-calculus generalizing call-by-name and call-by-value

We introduce and study the Bang Calculus, an untyped functional calculus in which the promotion operation of Linear Logic is made explicit and where application is a bilinear operation. This calculus, which can be understood as an untyped version of Call-By-Push-Value, subsumes both Call-By-Name and Call-By-Value lambda-calculi, factorizing the Girards translations of these calculi in Linear Logic. We build a denotational model of the Bang Calculus based on the relational interpretation of Linear Logic and prove an adequacy theorem by means of a resource Bang Calculus whose design is based on Differential Linear Logic.

Head reduction and normalization in a call-by-value lambda-calculus

Recently, a standardization theorem has been proven for a variant of Plotkins call-by-value lambda-calculus extended by means of two commutation rules (sigma-reductions): this result was based on a partitioning between head and internal reductions. We study the head normalization for this call-by-value calculus with sigma-reductions and we relate it to the weak evaluation of original Plotkins call-by-value lambda-calculus. We give also a (non-deterministic) normalization strategy for the call-by-value lambda-calculus with sigma-reductions.

Computing connected proof(-structure)s from their Taylor expansion

We show that every connected Multiplicative Exponential Linear Logic (MELL) proof-structure (with or without cuts) is uniquely determined by a well-chosen element of its Taylor expansion: the one obtained by taking two copies of the content of each box. As a consequence, the relational model is injective with respect to connected MELL proof-structures.

Standardization and Conservativity of a Refined Call-by-Value lambda-Calculus

We study an extension of Plotkins call-by-value lambda-calculus via two commutation rules (sigma-reductions). These commutation rules are sufficient to remove harmful call-by-value normal forms from the calculus, so that it enjoys elegant characterizations of many semantic properties. We prove that this extended calculus is a conservative refinement of Plotkins one. In particular, the notions of solvability and potential valuability for this calculus coincide with those for Plotkins call-by-value lambda-calculus. The proof rests on a standardization theorem proved by generalizing Takahashis approach of parallel reductions to our set of reduction rules. The standardization is weak (i.e. redexes are not fully sequentialized) because of overlapping interferences between reductions.

Standardization of a Call-By-Value Lambda-Calculus

We study an extension of Plotkins call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this extended calculus provides elegant characterizations of many semantic properties, as for example solvability. We prove a standardization theorem for this calculus by generalizing Takahashis approach of parallel reductions. The standardization property allows us to prove that our calculus is conservative with respect to the Plotkins one. In particular, we show that the notion of solvability for this calculus coincides with that for Plotkins call-by-value lambda-calculus.