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Efficient lambda encodings for Mendler-style coinductive types in Cedille

Christopher Jenkins, Aaron Stump, Larry Diehl
2020 Electronic Proceedings in Theoretical Computer Science  
In the calculus of dependent lambda eliminations (CDLE), it is possible to define inductive datatypes via lambda encodings that feature constant-time destructors and a course-of-values induction scheme  ...  The lambda encodings we present implementing coinductive types feature constant-time constructors and a course-of-values corecursion scheme.  ...  CDLE is an extension of impredicative Curry-style (i.e., extrinsically typed) CC that overcomes some traditional short-comings of lambda encodings in type theory (e.g., underivability of induction for  ... 
doi:10.4204/eptcs.317.5 fatcat:vxiyfwyfwzbrzkgecdankgreyq

Monotone recursive types and recursive data representations in Cedille

Christopher Jenkins, Aaron Stump
2021 Mathematical Structures in Computer Science  
As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding.  ...  consistent pure typed lambda calculus.  ...  So despite its significant overhead in space representation, the Parigot encoding appears to have a clear advantage over the Scott encoding in total typed lambda calculi.  ... 
doi:10.1017/s0960129521000402 fatcat:sq5rxjpb2jfzferkpqbbz4pl6y

Ranking/Unranking of Lambda Terms with Compressed de Bruijn Indices [chapter]

Paul Tarau
2015 Lecture Notes in Computer Science  
Our compressed terms facilitate derivation of size-proportionate ranking and unranking algorithms of lambda terms and their inferred simple types.  ...  Keywords: lambda calculus, de Bruijn indices, lambda term compression, combinatorics of lambda terms, ranking and unranking of lambda terms, bijective Gödel numberings of lambda terms.  ...  Acknowledgement We thank the anonymous referees of Calculemus'15 for their constructive criticisms and valuable suggestions that have helped improving the paper.  ... 
doi:10.1007/978-3-319-20615-8_8 fatcat:o6ix7yirerb6fa35gb46p7tsci

Page 501 of Mathematical Reviews Vol. , Issue 2004a [page]

2004 Mathematical Reviews  
It is usual in type theory to relate typability of lambda-terms to important notions of normalizability like solvability, weak- and strong-normalizability.  ...  of the type-free lambda-calculus and its models.  ... 

Modeling and Verifying Systems Using a Logic of Counter Arithmetic with Lambda Expressions and Uninterpreted Functions [chapter]

Randal E. Bryant, Shuvendu K. Lahiri, Sanjit A. Seshia
2002 Lecture Notes in Computer Science  
In this paper, we present the logic of Counter Arithmetic with Lambda Expressions and Uninterpreted Functions (CLU).  ...  We give theoretical and empirical evidence for the efficiency of our decision procedure.  ...  The third author was supported in part by a National Defense Science and Engineering Graduate Fellowship.  ... 
doi:10.1007/3-540-45657-0_7 fatcat:77cmw5bmkrasrpcxijkxnfwe4y

Mechanizing proofs with logical relations – Kripke-style

ANDREW CAVE, BRIGITTE PIENTKA
2018 Mathematical Structures in Computer Science  
The development of these proofs in Beluga relies on three key ingredients: (1) we encode lambda-terms together with their typing rules, operational semantics, algorithmic and declarative equality using  ...  In this paper, we describe two case studies using the proof environment Beluga: First, we explain the mechanization of the weak normalization proof for the simply typed lambda-calculus; second, we outline  ...  Encoding Inductive Proofs as Total Functions in Beluga Inductive proofs can be encoded as total functions using pattern matching in Beluga.  ... 
doi:10.1017/s0960129518000154 fatcat:uhyfqoacjfdido2zdg3pijayha

Monotone recursive types and recursive data representations in Cedille [article]

Christopher Jenkins, Aaron Stump
2021 arXiv   pre-print
As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding.  ...  consistent pure typed lambda calculus.  ...  The approach used for the Mendler encoding was then further refined by to enable efficient data accessors, resulting in the first-ever example of a lambda encoding in type theory with derivable induction  ... 
arXiv:2001.02828v2 fatcat:cohsc3e5vvcnhm6agsj7wu23qi

Model checking RAISE applicative specifications

Juan I. Perna, Chris George
2011 Formal Aspects of Computing  
In particular, the usage of Model Checking in formal languages has been reinforced in the last decades given the fact that specifications provide an abstraction of the problem under study, supplying a  ...  An outline of the main problems faced in the process and of the solutions to solve them are also presented.  ...  . • Curried functions are transformed on-the-fly into lambda functions. • Function-type declared values are declared to be of function type (using the total function type provided in SAL) and the value  ... 
doi:10.1007/s00165-011-0217-0 fatcat:wfcsb2sitvewbb7hpd2xhds27u

Model Checking RAISE Applicative Specifications

Juan Ignacio Perna, Chris George
2007 Fifth IEEE International Conference on Software Engineering and Formal Methods (SEFM 2007)  
In particular, the usage of Model Checking in formal languages has been reinforced in the last decades given the fact that specifications provide an abstraction of the problem under study, supplying a  ...  An outline of the main problems faced in the process and of the solutions to solve them are also presented.  ...  . • Curried functions are transformed on-the-fly into lambda functions. • Function-type declared values are declared to be of function type (using the total function type provided in SAL) and the value  ... 
doi:10.1109/sefm.2007.25 dblp:conf/sefm/PernaG07 fatcat:a2iu76cdo5e4lhpe4l2qfppvvu

Matching Power [chapter]

Horatiu Cirstea, Claude Kirchner, Luigi Liquori
2001 Lecture Notes in Computer Science  
In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations.  ...  In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects.  ...  We thank the referees for their constructive remarks, Hubert Dubois and all the members of the ELAN group for their comments and interactions on the topics of the Rho Calculus.  ... 
doi:10.1007/3-540-45127-7_8 fatcat:grylwumtnzh3ffmhkhnoe32rfq

Detection of First Order Axiomatic Theories [chapter]

Guillaume Burel, Simon Cruanes
2013 Lecture Notes in Computer Science  
Even detecting the presence of a theory in a problem is generally solved in an ad-hoc fashion.  ...  We present here a generic way of describing and recognizing axiomatic theories in clausal form first-order logic with equality.  ...  We call lambda terms terms in which some variables are bound by a lambda-abstraction.  ... 
doi:10.1007/978-3-642-40885-4_16 fatcat:j2wxzydcnvaw3pc2a2fnxgme6i

Amalia -- A Unified Platform for Parsing and Generation [article]

Shuly Wintner Evgeniy Gabrilovich and Nissim Francez (Laboratory for Computational Linguistics, Technion, Israel)
1997 arXiv   pre-print
Contemporary linguistic theories (in particular, HPSG) are declarative in nature: they specify constraints on permissible structures, not how such structures are to be computed.  ...  However, practical implementations of such theories don't usually support bidirectional processing of grammars.  ...  In general, Amalia supports totally well-typed, possibly cyclic, non-disjunctive feature structures. Set values, as in ale, are not supported, but list values are.  ... 
arXiv:cmp-lg/9709014v1 fatcat:gcwtwxrpobhtbhn5qfs7e4poi4

On computing with types

Paul Tarau, David Haraburda
2012 Proceedings of the 27th Annual ACM Symposium on Applied Computing - SAC '12  
derive an efficiently testable total ordering on types, isomorphic to the ordering of natural numbers.  ...  We express in terms of binary trees seen as Gödel System T types (with the empty type as the only primitive type) arithmetic computations within time and space bounds comparable to binary arithmetic and  ...  that lead to the idea of using the free magma of binary trees as the key building block for our type classes.  ... 
doi:10.1145/2245276.2232087 dblp:conf/sac/TarauH12 fatcat:w77o3haxijerzgwj2ajqnsaxma

A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations [article]

Paul Tarau
2015 arXiv   pre-print
We describe a Prolog-based combined lambda term generator and type-inferrer for closed well-typed terms of a given size, in de Bruijn notation.  ...  Our compressed terms facilitate derivation of size-proportionate ranking and unranking algorithms of lambda terms and their inferred simple types.  ...  Generating terms in binary lambda calculus encoding Generating de Bruijn terms based on the size of their binary lambda calculus encoding (Wikipedia 2015) works by using a DCG mechanism to build the  ... 
arXiv:1507.06944v1 fatcat:deve5flknzexdcrddhimayx6dy

Quantum Calculi—From Theory to Language Design

Margherita Zorzi
2019 Applied Sciences  
We also provide the higher-order encoding in the functional languages qPCFand IQu of the well known Deutsch-Jozsa and Simon's algorithms.  ...  We explore the twofold perspective (theoretical and concrete) of the approach and we list the main problems one has to face in quantum language design.  ...  In IQu enjoys a simple type theory: classical types (for natural numbers, command and classical variables) are extended with two new types. The first one, cırc, is the type of quantum circuits.  ... 
doi:10.3390/app9245472 fatcat:ysri3ams6jbsvaxbripdmq7izu
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