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Coinductive Foundations of Infinitary Rewriting and Infinitary Equational Logic [article]

Jörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, Alexandra Silva
2017 arXiv   pre-print
We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform, coinductive way.  ...  The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence.  ...  We thank Patrick Bahr, Jeroen Ketema, and Vincent van Oostrom for fruitful discussions and comments on earlier versions of this paper.  ... 
arXiv:1706.00677v1 fatcat:kfi6xpcfrzafnh6wikznzgusim

A new coinductive confluence proof for infinitary lambda calculus

Łukasz Czajka
2018 Logical Methods in Computer Science  
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.  ...  We do not want to do this here, because this paper is about a coinductive confluence proof for infinitary lambda calculus, not about foundations of coinduction.  ... 
doi:10.23638/lmcs-16(1:31)2020 fatcat:oz5iah3r3baqfe73zacq4lobki

A new coinductive confluence proof for infinitary lambda calculus [article]

Łukasz Czajka
2020 arXiv   pre-print
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  The technique of the proof is new, i.e., it is not merely a coinductive reformulation of any earlier proofs. We formalised the proof in the Coq proof assistant.  ...  We do not want to do this here, because this paper is about a coinductive confluence proof for infinitary lambda calculus, not about foundations of coinduction.  ... 
arXiv:1808.05481v4 fatcat:kyvjdffsc5g2xotcmjfwssif4y

Encoding many-valued logic in λ-calculus [article]

Fer-Jan de Vries
2021 arXiv   pre-print
We will extend the well-known Church encoding of Boolean logic into λ-calculus to an encoding of McCarthy's 3-valued logic into a suitable infinitary extension of λ-calculus that identifies all unsolvables  ...  By way of motivation we consider Russell's paradox, exploiting the fact that the same encoding allows us also to calculate truth values of infinite closed propositions in this infinitary setting.  ...  Acknowledgements We would like to thank the editor for her encouragement and infinite patience, and the referees for careful reading of our paper and their constructive comments that helped to improve  ... 
arXiv:1810.07667v6 fatcat:zwi276dxgbb6xgo2piwixflej4

Relational semantics of linear logic and higher-order model-checking [article]

Charles Grellois, Paul-André Melliès
2015 arXiv   pre-print
We briefly explain in the end of the paper how his analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order  ...  In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic.  ...  In this alternative relational semantics, there is a coinductive fixpoint operator Y satisfying the equations of a Conway operator, and thus providing an interpretation of the λY -calculus.  ... 
arXiv:1501.04789v3 fatcat:4yd3ryhkpzgzfhpy52fctlmglq

Coinduction: an elementary approach [article]

Łukasz Czajka
2019 arXiv   pre-print
The main aim of this paper is to promote a certain style of doing coinductive proofs, similar to inductive proofs as commonly done by mathematicians.  ...  In this way, we provide an elementary explanation of how to interpret coinduction in set theory.  ...  See also [59, 12, 30] for an overview of various results in infinitary lambda-calculus and infinitary rewriting.  ... 
arXiv:1501.04354v8 fatcat:swuyyztqujhexdhbi3bgimjnb4

Infinitary Rewriting: Foundations Revisited

Stefan Kahrs, Marc Herbstritt
2010 International Conference on Rewriting Techniques and Applications  
This connection is the original reason for the study of these properties in rewriting. For infinitary rewriting there is barely an established notion of an equational theory.  ...  Infinitary Term Rewriting allows to express infinitary terms and infinitary reductions that converge to them.  ...  In the presence of infinite terms, ordinary congruence relations fail to capture what is needed for equational reasoning in infinitary rewriting as equivalence closure is an inductive concept, not a coinductive  ... 
doi:10.4230/lipics.rta.2010.161 dblp:conf/rta/Kahrs10 fatcat:77adngv2yzfchfkl4oyoqzcqfe

Circular Coinduction: A Proof Theoretical Foundation [chapter]

Grigore Roşu, Dorel Lucanu
2009 Lecture Notes in Computer Science  
of circular coinduction) If B is a behavioral specification and G is a set of equations such that B G is derivable using the Circular Coinduction Proof System, then B G .  ...  (soundness of circular coinduction) If B is a behavioral specification and G is a set of equations such that B G is derivable using the Circular Coinduction Proof System, then B G .  ... 
doi:10.1007/978-3-642-03741-2_10 fatcat:oweoi23a7va6bazeyp7dk7wyki

Proof terms for infinitary rewriting, progress report [article]

Carlos Lombardi, Alejandro Ríos, Roel de Vrijer
2014 arXiv   pre-print
This definition involves a variant of equational logic, adapted for dealing with infinite objects.  ...  Our main use of proof terms is in a definition of permutation equivalence for transfinite reductions, on the basis of permutation equations.  ...  This rule allows to incorporate the idea of limit into infinitary equational logic judgements.  ... 
arXiv:1402.2245v2 fatcat:hp27ys7rvffjpkjydjidyt3lr4

PhD Abstracts

GRAHAM HUTTON
2014 Journal of functional programming  
The Journal of Functional Programming would like to change that!  ...  Many students complete PhDs in functional programming each year, but there is currently no common location in which to promote and advertise the resulting work.  ...  limit inferior in a partially ordered set; 2. extending infinitary term rewriting to infinitary term graph rewriting.  ... 
doi:10.1017/s0956796814000215 fatcat:rs2j5wgm5ndf7ek2zls5pmg2w4

Type-based Enforcement of Infinitary Trace Properties for Java [article]

Serdar Erbatur, Ulrich Schöpp, Chuangjie Xu
2021 arXiv   pre-print
Our type and effect system can express properties of both finite and infinite traces and can compute information about the possible infinite traces of FJ programs.  ...  We formalize guidelines as sets of finite or infinite execution traces and develop a region-based type and effect system for FJ that can enforce such guidelines.  ...  ACKNOWLEDGMENTS We thank the anonymous reviewers for their valuable comments and helpful suggestions.  ... 
arXiv:2107.11280v1 fatcat:56jivlcdwngznnqdfbxjhm43da

Partial Order Infinitary Term Rewriting

Patrick Bahr, Christopher Lynch
2014 Logical Methods in Computer Science  
We study an alternative model of infinitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions.  ...  We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the partial order model restricted to total terms.  ...  In this paper we introduce a novel foundation of infinitary term rewriting based on the partially ordered set of partial terms [12] .  ... 
doi:10.2168/lmcs-10(2:6)2014 fatcat:ihj5prqx2zhilcz2vsiongiosu

Copatterns

Andreas Abel, Brigitte Pientka, David Thibodeau, Anton Setzer
2013 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '13  
We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.  ...  We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns.  ...  discussions on copatterns and the best way to integrate coinduction into Agda.  ... 
doi:10.1145/2429069.2429075 dblp:conf/popl/AbelPTS13 fatcat:7wevbhmshfewdncsc3tqo34cwm

Copatterns

Andreas Abel, Brigitte Pientka, David Thibodeau, Anton Setzer
2013 SIGPLAN notices  
We prove type soundness for our language and sketch how copatterns open new directions for solving problems in the interaction of coinductive and dependent types.  ...  We present a core language for programming with infinite structures by observations together with its operational semantics based on (co)pattern matching and describe coverage of copatterns.  ...  discussions on copatterns and the best way to integrate coinduction into Agda.  ... 
doi:10.1145/2480359.2429075 fatcat:zrbooidzpjfhro4qinjzcotaxa

Discriminating Lambda-Terms Using Clocked Boehm Trees

Jörg Endrullis, Dimitri Hendriks, Jan Willem Klop, Andrew Polonsky, Jean-Pierre Jouannaud
2014 Logical Methods in Computer Science  
Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but  ...  It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are  ...  and subsequent communication pertinent to Selinger's work concerning the existence of unorderable models of the λ-calculus.  ... 
doi:10.2168/lmcs-10(2:4)2014 fatcat:4yy27fuvt5dibaqrhn2m7tvsui
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