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### Solvability in Resource Lambda-Calculus [chapter]

Michele Pagani, Simona Ronchi della Rocca
2010 Lecture Notes in Computer Science
Because of the non-determinism, different definitions of solvability are possible in the resource calculus.  ...  The resource calculus is an extension of the λ-calculus allowing to model resource consumption.  ...  In Section 5 there is the proof of the main theorem, showing all the characterizations of solvability. In Section 6 alternative notions of solvability are discussed. Resource Calculus Syntax.  ...

### A Semantical and Operational Account of Call-by-Value Solvability [chapter]

Alberto Carraro, Giulio Guerrieri
2014 Lecture Notes in Computer Science
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.  ...  As a technical tool, we also use a resource-sensitive calculus in which the elements of the model are definable.  ...  A relational model of (resource) CBV lambda-calculus In this section we present a relational model for both the λ σ v -calculus and the resource λ σ v -calculus.  ...

### Standardization in resource lambda-calculus

Maurizio Dominici, Simona Ronchi Della Rocca, Paolo Tranquilli
2012 Electronic Proceedings in Theoretical Computer Science
The resource calculus is an extension of the lambda-calculus allowing to model resource consumption.  ...  We prove that the non-deterministic reduction enjoys a notion of standardization, which is the natural extension with respect to the similar one in classical lambda-calculus.  ...  Introduction The resource calculus (Λ r ) is an extension of the λ -calculus allowing to model resource consumption.  ...

### Page 6020 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews
In the last section, the simply typed lambda calculus is studied. The standard set theoretical model of STT is given.  ...  In the second section, the untyped lambda calculus and the untyped universe are considered. Their connection with combinatory logic is shown.  ...

### Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion

Thomas Ehrhard, Antonio Bucciarelli, Alberto Carraro, Giulio Manzonetto, Pawel Urzyczyn
2012 Logical Methods in Computer Science
We study the semantics of a resource-sensitive extension of the lambda calculus in a canonical reflexive object of a category of sets and relations, a relational version of Scott's original model of the  ...  This calculus is related to Boudol's resource calculus and is derived from Ehrhard and Regnier's differential extension of Linear Logic and of the lambda calculus.  ...  FULL ABSTRACTION FOR RESOURCE CALCULI  ...

### What is a categorical model of the differential and the resource lambda-calculi? [article]

Manzonetto Giulio
2010 arXiv   pre-print
In this paper we provide an abstract model theory for the untyped differential lambda-calculus and the resource calculus.  ...  In particular we propose a general definition of model of these calculi, namely the notion of linear reflexive object in a Cartesian closed differential category.  ...  May and must solvability have been studied in [ 30] in the context of the resource calculus.  ...

### Resource Combinatory Algebras [chapter]

Alberto Carraro, Thomas Ehrhard, Antonino Salibra
2010 Lecture Notes in Computer Science
We initiate a purely algebraic study of Ehrhard and Regnier's resource λ-calculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambda-algebras and resource  ...  model of the classical λ-calculus raising from a resource lambda-algebra determines a λ-theory which equates all terms having the same Böhm tree.  ...  lambda calculus We will now briefly introduce the linear fragment of resource λ-calculus (rλcalculus, for short).  ...

### A Survey of Quantum Programming Languages: History, Methods, and Tools [article]

Donald A. Sofge
2008 arXiv   pre-print
Quantum computer programming is emerging as a new subject domain from multidisciplinary research in quantum computing, computer science, mathematics (especially quantum logic, lambda calculi, and linear  ...  It is intended to provide an extensive but non-exhaustive look at work leading up to the current state-of-the-art in quantum computer programming.  ...  In 1996 Maymin  proposed a quantum lambda calculus to investigate the Turing computability of quantum algorithms.  ...

### The Resource Lambda Calculus Is Short-Sighted in Its Relational Model [chapter]

Flavien Breuvart
2013 Lecture Notes in Computer Science
As a by-product we achieve a context lemma for the resource λ-calculus.  ...  That particular object of MRel is also a model of the resource λ-calculus, deriving from Ehrhard and Regnier's differential extension of Linear Logic and related to Boudol's λ-calculus with multiplicities  ...  Notation: We denote N A for the set of finite multisetes of elements in the set A. Syntax ∂λ-Calculus In this section we give some background on the ∂λ-calculus, a lambda calculus with resources.  ...

### What is a categorical model of the differential and the resource λ-calculi?

GIULIO MANZONETTO
2012 Mathematical Structures in Computer Science
Therefore the resource calculus can be interpreted by translation into every linear reflexive object living in a Cartesian closed differential category.  ...  Finally, we study the relationship between the differential λ-calculus and the resource calculus, a functional programming language combining the ideas behind the differential λ-calculus with those behind  ...  From the Resource to the Differential Lambda Calculus. . .  ...

### Essential and relational models

LUCA PAOLINI, MAURO PICCOLO, SIMONA RONCHI DELLA ROCCA
2015 Mathematical Structures in Computer Science
Intersection type assignment systems can be used as a general framework for building logical models of λ-calculus that allow to reason about the denotation of terms in a finitary way.  ...  A similar technique has been used in Pagani and Ronchi Della Rocca (2010b) for charactering solvability in the resource λ-calculus.  ...  Recently non idempotent intersection types have been used in Pagani and Ronchi Della Rocca (2010b,a) to characterize the solvability in the resource λ-calculus.  ...

### A Resource Aware Computational Interpretation for Herbelin's Syntax [chapter]

Delia Kesner, Daniel Ventura
2015 Lecture Notes in Computer Science
We investigate a new computational interpretation for an intuitionistic focused sequent calculus which is compatible with a resource aware semantics.  ...  For the sake of completeness, we also study typability (and the corresponding strong normalization characterization) in the reduction calculus obtained from the former one by projecting the explicit substitutions  ...  Non-idempotent types also appear in linearization of the lambda-calculus  , type inference [30, 38] , different characterizations of solvability  and verification of higher-order programs  ...

### An Infinitary Affine Lambda-Calculus Isomorphic to the Full Lambda-Calculus

Damiano Mazza
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science
We also show how this construction brings interesting insights on some standard rewriting properties of the lambda-calculus (finite developments, confluence, standardization, head normalization and solvability  ...  relation is exactly the full (non-affine) lambda-calculus.  ...  An example might be Ehrhard and Regnier's resource λ-calculus (which is strongly related to Boudol's λ-calculus with multiplicities).  ...

### Normalization, Taylor expansion and rigid approximation of λ-terms [article]

Federico Olimpieri
2020 arXiv   pre-print
The aim of this work is to characterize three fundamental normalization proprieties in lambda-calculus trough the Taylor expansion of λ-terms.  ...  The general proof strategy consists in stating the dependence of ordinary reduction strategies on their resource counterparts and in finding a convenient resource term in the Taylor expansion that behaves  ...  In particular we have that M N 1 · · · N n → * β λx.x. We say that a generic M ∈ Λ is solvable if there exists a closure of M that is solvable. Theorem 3.14. M is solvable iff M is head-normalisable.  ...

### The differential lambda-calculus

Thomas Ehrhard, Laurent Regnier
2003 Theoretical Computer Science
We state and prove some basic results (con uence, strong normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus  ...  We present an extension of the lambda-calculus with di erential constructions.  ...  thank especially one of the referees of this paper, who made many insightful comments on an earlier version of this work and derived us to clarify several delicate syntactical aspects of the di erential lambda-calculus  ...
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