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Efficient lambda encodings for Mendler-style coinductive types in Cedille

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 ...

##
###
Monotone recursive types and recursive data representations in Cedille

2021
*
Mathematical Structures in Computer Science
*

As applications, we use monotone recursive

doi:10.1017/s0960129521000402
fatcat:sq5rxjpb2jfzferkpqbbz4pl6y
*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. ...##
###
Ranking/Unranking of Lambda Terms with Compressed de Bruijn Indices
[chapter]

2015
*
Lecture Notes in Computer Science
*

Our compressed terms facilitate derivation

doi:10.1007/978-3-319-20615-8_8
fatcat:o6ix7yirerb6fa35gb46p7tsci
*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. ...##
###
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]

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. ...

##
###
Mechanizing proofs with logical relations – Kripke-style

2018
*
Mathematical Structures in Computer Science
*

The development

doi:10.1017/s0960129518000154
fatcat:uhyfqoacjfdido2zdg3pijayha
*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. ...##
###
Monotone recursive types and recursive data representations in Cedille
[article]

2021
*
arXiv
*
pre-print

As applications, we use monotone recursive

arXiv:2001.02828v2
fatcat:cohsc3e5vvcnhm6agsj7wu23qi
*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 ...##
###
Model checking RAISE applicative specifications

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 ...

##
###
Model Checking RAISE Applicative Specifications

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 ...

##
###
Matching Power
[chapter]

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. ...

##
###
Detection of First Order Axiomatic Theories
[chapter]

2013
*
Lecture Notes in Computer Science
*

Even detecting the presence

doi:10.1007/978-3-642-40885-4_16
fatcat:j2wxzydcnvaw3pc2a2fnxgme6i
*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. ...##
###
Amalia -- A Unified Platform for Parsing and Generation
[article]

1997
*
arXiv
*
pre-print

Contemporary linguistic

arXiv:cmp-lg/9709014v1
fatcat:gcwtwxrpobhtbhn5qfs7e4poi4
*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. ...##
###
On computing with types

2012
*
Proceedings of the 27th Annual ACM Symposium on Applied Computing - SAC '12
*

derive an

doi:10.1145/2245276.2232087
dblp:conf/sac/TarauH12
fatcat:w77o3haxijerzgwj2ajqnsaxma
*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. ...##
###
A Logic Programming Playground for Lambda Terms, Combinators, Types and Tree-based Arithmetic Computations
[article]

2015
*
arXiv
*
pre-print

We describe a Prolog-based combined

arXiv:1507.06944v1
fatcat:deve5flknzexdcrddhimayx6dy
*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 ...##
###
Quantum Calculi—From Theory to Language Design

2019
*
Applied Sciences
*

We also provide the higher-order

doi:10.3390/app9245472
fatcat:ysri3ams6jbsvaxbripdmq7izu
*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. ...
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