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The Difference λ-Calculus: A Language for Difference Categories

Mario Alvarez-Picallo, C.-H. Luke Ong, Zena M. Ariola
2020 International Conference on Formal Structures for Computation and Deduction  
In this work, we construct a simply-typed calculus in the spirit of the differential λ-calculus equipped with syntactic "infinitesimals" and show how its models correspond to difference λ-categories, a  ...  Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with  ...  The exact same result holds for the difference λ-calculus and difference λ-categories.  ... 
doi:10.4230/lipics.fscd.2020.32 dblp:conf/fscd/Alvarez-Picallo20 fatcat:evasmukchveobbwbip2tdqbo5e

The Difference Lambda-Calculus: A Language for Difference Categories [article]

Mario Alvarez-Picallo, C.-H. Luke Ong
2020 arXiv   pre-print
Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with  ...  In this work, we construct a simply-typed calculus in the spirit of the differential lambda-calculus equipped with syntactic infinitesimals and show how its models correspond to difference lambda-categories  ...  ]) • x, u , v, 0 Difference λ-Categories In order to give a semantics for the differential λ-calculus, it does not suffice to ask for a Cartesian differential category equipped with exponentials -the  ... 
arXiv:2011.14476v1 fatcat:m5jt5is6svgkll2xtibxsqzkny

Models for the computational λ-calculus

John Power
2001 Electronical Notes in Theoretical Computer Science  
, the evident two maps from A ⊗ B to A ⊗ B may differ.  ...  We consider several different sound and complete classes of models for the computational -calculus, explain the definitions, and outline why one might be interested in the various classes.  ...  Modelling the simply typed λ-calculus with indexed categories highlights different features of λ-calculus to modelling it with cartesian closed categories.  ... 
doi:10.1016/s1571-0661(05)80056-8 fatcat:bw23ucxjd5bbhf5lsl5scrgn4u

Metacircularity in the polymorphic λ-calculus

Frank Pfenning, Peter Lee
1991 Theoretical Computer Science  
Lee, Metacircularity in the polymorphic A-calculus, Theoretical Computer Science 89 (1991) 137-159. *  ...  Acknowledgment The authors would like to thank Christine Paulin-Mohring for valuable criticism of an earlier draft and also Ken Cline, Scott Dietzen, Jean Gallier, Robert Harper,  ...  Instead, we will try to give some informal criteria for judging the degree of reflexivity of a language, the basic one being the ability of a language to serve as its own metalanguage.  ... 
doi:10.1016/0304-3975(90)90109-u fatcat:oopib6b3djbzvpp3ra6yvvqjym

A λ-Calculus for Resource Separation [chapter]

Robert Atkey
2004 Lecture Notes in Computer Science  
We present a typed λ-calculus for recording resource separation constraints between terms.  ...  The calculus contains a novel way of manipulating nested multi-place contexts augmented with constraints, allowing a concise presentation of the typing rules.  ...  Acknowledgements I would like to thank David Aspinall and Michal Konečný for helpful discussions on the work presented here.  ... 
doi:10.1007/978-3-540-27836-8_16 fatcat:qoxavcux4jekzmpnjiynwbzfku

RPO, Second-Order Contexts, and λ-Calculus [chapter]

Pietro Di Gianantonio, Furio Honsell, Marina Lenisa
2008 Lecture Notes in Computer Science  
We show that, contrary to process calculi, one can deal directly with the λ-calculus syntax and apply Leifer-Milner technique to a category of contexts, provided that we work in the framework of weak bisimilarities  ...  We apply Leifer-Milner RPO approach to the λ-calculus, endowed with lazy and call by value reduction strategies.  ...  The formal definition is the following Definition 17 (Category of Second-order Term Contexts). Let Σ be a signature for a term language.  ... 
doi:10.1007/978-3-540-78499-9_24 fatcat:pf7no7ubovesxag35ylphb3p5y

Semantics of a Relational λ-Calculus (Extended Version) [article]

Pablo Barenbaum, Federico Lochbaum, Mariana Milicich
2021 arXiv   pre-print
We extend the λ-calculus with constructs suitable for relational and functional-logic programming: non-deterministic choice, fresh variable introduction, and unification of expressions.  ...  We study a simply typed version of the system. Moreover, a denotational semantics for the calculus is proposed and reduction is shown to be sound with respect to the denotational semantics.  ...  The goal of this paper is to provide a foundation for functional-logic programming by extending the λ-calculus with relational constructs.  ... 
arXiv:2009.10929v4 fatcat:fkdh5ukucva37dnawzcnpnqsde

A colored version of the λ-calculus [chapter]

Dieter Hutter, Michael Kohlhase
1997 Lecture Notes in Computer Science  
Rippling is a technique developed for inductive theorem proving which uses syntactic differences of terms to guide the proof search.  ...  Another application is in the construction of natural language semantics, where the color annotations rule out linguistically invalid readings that are possible using standard higher-order unification.  ...  Clearly the colored λ-calculus is a generalization of the simply typed λcalculus, since we can always restrict the supply of colors to a single color constant.  ... 
doi:10.1007/3-540-63104-6_30 fatcat:dnwzng3kjbfb5pkzgdyjku2umu

The Logic of the Partial λ-Calculus with Equality [chapter]

Lutz Schröder
2004 Lecture Notes in Computer Science  
We investigate the logical aspects of the partial λ-calculus with equality, exploiting an equivalence between partial λ-theories and partial cartesian closed categories (pcccs) established here.  ...  The partial λ-calculus with equality provides a full-blown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter  ...  Acknowledgements The author wishes to thank Till Mossakowski and Christoph Lüth for useful comments and discussions.  ... 
doi:10.1007/978-3-540-30124-0_30 fatcat:aac5w3m7bjdc3mk7kxloqwda6q

The call-by-value λ-calculus: a semantic investigation

ALBERTO PRAVATO, SIMONA RONCHI della ROCCA, LUCA ROVERSI
1999 Mathematical Structures in Computer Science  
In particular, the properties a category Cbv must enjoy to describe a model of call-by-value λ-calculus are given.  ...  This paper is about a categorical approach for modeling the pure (i.e., without constants) call-by-value λ-calculus, defined by Plotkin as a restriction of the call-by-name λ-calculus.  ...  We would like to thank Jean-Yves Girard, Furio Honsell, Yves Lafont, and Eugenio Moggi for very stimulating and useful discussions about the subject of this work.  ... 
doi:10.1017/s0960129598002722 fatcat:sqhheps35rcinmt24hhevj3udi

Game Semantics for the Pure Lazy λ-Calculus [chapter]

Pietro Di Gianantonio
2001 Lecture Notes in Computer Science  
In this paper we present a fully abstract game model for the pure lazy λ-calculus, i.e. the lazy λ-calculus without constants.  ...  In order to obtain this result we introduce a new category of games, the monotonic games, whose main characteristic consists in having an order relation on moves.  ...  Introduction The aim of this paper is to present a fully abstract model, based on game semantics, for the lazy λ-calculus.  ... 
doi:10.1007/3-540-45413-6_12 fatcat:xh3tea2rxfeofh4p7ofrtow6je

The typed polymorphic label-selective λ-calculus

Jacques Garrigue, Hassan Aït-Kaci
1994 Proceedings of the 21st ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '94  
For the latter calculus, we establish the existence of principal types and we give an algorithm to compute them.  ...  Thanks to the fact that label-selective -calculus is a conservative extension of -calculus by adding numeric labels to stand for argument positions, its polymorphic typing provides us with a keyword argument-passing  ...  We construct the substitution by taking for each equation = , solved, () = . is a most general unifier of the final system, and, as a consequence, if we suppress definitions for all variables introduced  ... 
doi:10.1145/174675.174434 dblp:conf/popl/GarrigueA94 fatcat:yjchufqfwvclzlkihrahmddgne

Universal Semantics for the Stochastic λ-Calculus

Pedro H. Azevedo de Amorim, Dexter Kozen, Radu Mardare, Prakash Panangaden, Michael Roberts
2021 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
We define sound and adequate denotational and operational semantics for the stochastic lambda calculus.  ...  Special thanks to Dana Scott for many inspiring conversations. Panangaden is funded by NSERC (Canada).  ...  One approach that is radically different from other approaches is that of [1] , which involves Boolean-valued models for the stochastic λ-calculus.  ... 
doi:10.1109/lics52264.2021.9470747 fatcat:szvvbfhplrcwdbcjkipk4g5u64

Using λ-calculus to represent meanings in logic grammars

David Scott Warren
1983 Proceedings of the 21st annual meeting on Association for Computational Linguistics -  
This 'programmed' representation motivates a suggestion, discussed in the third section, for an extension to Prolog so that the language itself would include a mechanism for handling the ~-formulas directly  ...  This paper descrlbes how meanings are represented in a semantic grammar for a fragment of English in the logic programming language Prolog. The conventions of Definite Clause Grammars are used.  ...  Consider mow the meaning structure Dahl uses for phrases of a category other than sentence, a noun phrase, for example.  ... 
doi:10.3115/981311.981321 dblp:conf/acl/Warren83 fatcat:el7sp32tifgdlolnpuppecrumq

A quantitative model for simply typed λ-calculus

Martin Hofmann, Jérémy Ledent
2021 Mathematical Structures in Computer Science  
We use a simplified version of the framework of resource monoids, introduced by Dal Lago and Hofmann, to interpret simply typed λ-calculus with constants zero and successor.  ...  Our use of resource monoids differs from the other instances found in the literature, in that it measures the size of λ-terms rather than time complexity.  ...  We also thank the anonymous referees for their valuable comments and pointers to the literature. Financial support.  ... 
doi:10.1017/s0960129521000256 fatcat:u4fhg3asz5gehhpbif6wansjlq
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