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Decidability of the restriction equational theory in the partial lambda calculus

Ramón Pino Perez
1989 Theoretical Computer Science  
In his thesis about partiality and A-calculus Eugenio Moggi raised the following conjecture: the equational theory Tr (see definition below) corresponding to the restriction in the partial A-calculus is  ...  Testing the truth of this conjecture is very important because one of the results in Moggi's thesis is the confluence of the partial A-calculus modulo the theory Tr .  ...  In his thesis, Moggi showed the confluence of the partial A-calculus modulo the equational theory T[ (see definition below) corresponding to the restriction, and he conjectured that the theory Tr was aecidable  ... 
doi:10.1016/0304-3975(89)90029-7 fatcat:py7fdpvwx5a2vcnbt5rgdmdexe

Page 5566 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
03825 Pino Perez, Ramén Decidability of the restriction equational theory in the partial lambda calculus.  ...  It does not seem possible to prove the decidability of 7; by finding a canonical form for a term of the partial A-calculus by orienting the restriction equations (making them into a rewriting system) in  ... 

The exp-log normal form of types [article]

Danko Ilik
2016 arXiv   pre-print
We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specialized version of itself, while preserving the completeness of equality  ...  We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coq-implemented converter into/from our new term calculus.  ...  Terms of the {→, +, ×}-typed lambda calculus and axioms of the equational theory = βη between typed terms. is decidable, but without a practically implementable algorithm in sight (Ilik 2014) .  ... 
arXiv:1502.04634v3 fatcat:6fx7n53vrbejflrzqgywihisji

Lambda theories of effective lambda models [article]

Chantal Berline, Antonio Salibra
2007 arXiv   pre-print
A longstanding open problem is whether there exists a non-syntactical model of untyped lambda-calculus whose theory is exactly the least equational lambda-theory (=Lb).  ...  In this paper we make use of the Visser topology for investigating the more general question of whether the equational (resp. order) theory of a non syntactical model M, say Eq(M) (resp.  ...  The authors do not know any example of unsolvable satisfying the above condition.  ... 
arXiv:math/0701684v2 fatcat:uo7ia755zjgzdf6t4hvnxn6p24

Page 6058 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
The theory of LLAs can be regarded as axiomatizing the equations that hold between the contexts of the lambda calculus, as opposed to lambda terms with free variables.  ...  The present paper gives several properties of LLAs and of lambda-calculus. The lattice of subvarieties of LLAs is proved to be isomorphic to the lattice of lambda theories.  ... 

Page 6734 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
It is shown that the equational theory of the standard model of partial equivalence relations (PERs) is the minimal extensional theory, and is at the same time the largest consistent theory.  ...  The theories of standard models of the untyped lambda calcu- lus can be nicely characterised [see H. P.  ... 

Problems in rewriting III [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1995 Lecture Notes in Computer Science  
Acknowledgements We thank all the individuals who contributed questions, updates and solutions. Many thanks to Stefan Blom for substantial help producing this list.  ...  One of the outstanding open problems in typed lambda calculi is the following: Given a term in ordinary untyped Iambda calculus, is it decidable whether it can be typed in the second-order A2 calculus?  ...  Is satisfiability of equations in the theory of distributivity (unification modulo modulo one right-and one leftdistributivity axiom) decidable?  ... 
doi:10.1007/3-540-59200-8_82 fatcat:in5aspimbrbvngnrhzcm7tfwei

A Terminating and Confluent Linear Lambda Calculus [chapter]

Yo Ohta, Masahito Hasegawa
2006 Lecture Notes in Computer Science  
Thus it provides a simple decision method for the equational theory of the linear lambda calculus.  ...  As an application we prove the strong normalization of the simply typed computational lambda calculus by giving a reduction-preserving translation into the linear lambda calculus.  ...  This work was partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 17700013.  ... 
doi:10.1007/11805618_13 fatcat:esgqd53m2zfnbj43pxsosx5fyy

Page 569 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
W. (1-IL) Products in PER: an elementary treatment of the semantics of the polymorphic lambda calculus. Category theory at work (Bremen, 1990), 325-340, Res. Exp. Math., 18, Heldermann, Berlin, 1991.  ...  This paper is an exposition of the interpretation of the polymor- phic lambda-calculus in the category PER of partial equivalence relations on the natural numbers (the morphisms being (equiva- lence classes  ... 

Aspects of categorical recursion theory [article]

Pieter Hofstra, Philip Scott
2020 arXiv   pre-print
We present a survey of some developments in the general area of category-theoretic approaches to the theory of computation, with a focus on topics and ideas particularly close to the interests of Jim Lambek  ...  Conclusion We hope that we have shown in this -admittedly biased-overview of categorical recursion theory how various of Lambek's seminal ideas have initiated and inspired numerous strands of research  ...  We also hope to have conveyed to the reader that there are still many interesting unanswered questions and relatively unexplored facets of categorical recursion theory that deserve further investigation  ... 
arXiv:2001.05778v1 fatcat:orhmpipltngtdcnkuq2y7rvqaa

Page 2110 of Mathematical Reviews Vol. , Issue 80F [page]

1980 Mathematical Reviews  
., 1961; MR 24 #A2627], (2) a set theory in which the axiom schema of separation is subjected to an unusual restriction. Ayda I.  ...  It is then shown that the (equational) theory obtained from the A-calculus by equating all unsolvable terms has a unique Hilbert-Post-complete extension and this is just Th(D,,), independently of the original  ... 

Page 4529 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
x [x = t(X,y)] for every roncircular set of equations ¥ = f(x,y). The decidability of both theories is well known (see the first two works cited above for FT, and the third for RT).  ...  Even the decidability of more practically relevant cases of the same theories in finite signatures is known [M. J. Maher, op. cit.].  ... 

Open problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1991 Lecture Notes in Computer Science  
Acknowledgements We thank all the individuals mentioned above who contributed questions, F.-J. de Vries for helping prepare this list, and Ron Book, for going out of his way t o m a k e it possible.  ...  One of the outstanding open problems in typed lambda calculi is the following: Given a term in ordinary untyped lambda calculus, is it decidable whether it can be typed in the second-order 2 calculus  ...  Is satis ability of equations in the theory of distributivity uni cation modulo a distributivity axiom decidable? Problem 39.  ... 
doi:10.1007/3-540-53904-2_120 fatcat:bsnbx2vghrc2tmv6xcywwijbjq

Page 9529 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Summary: “It is shown that unifiability of terms in the simply typed lambda calculus with # and y rules becomes decidable if there is a bound on the number of bound variables and lambdas in a unifier in  ...  (with Kts — ft, for example) and applying the Church- Rosser theorem and the uniqueness of normal forms, but using a reduction calculus that “is no [longer] an equational calculus in the strict sense”  ... 

More problems in rewriting [chapter]

Nachum Dershowitz, Jean-Pierre Jouannaud, Jan Willem Klop
1993 Lecture Notes in Computer Science  
Which rewrite systems can be directly defined in lambda calculus?  ...  Is there an equational theory for which solvability of unification problems with linear constant restrictions is decidable, but solvability of unification problems with arbitrary constant restrictions  ... 
doi:10.1007/3-540-56868-9_39 fatcat:cibthhywyrgmvcsv73tryp3v2e
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