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Applications of Metric Coinduction

Dexter Kozen, Nicholas Ruozzi, Till Mossakowski
2009 Logical Methods in Computer Science  
Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations.  ...  This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets.  ...  Any views and conclusions expressed herein are those of the authors and should not be interpreted as representing the official policies or endorsements of the National Science Foundation or the United  ... 
doi:10.2168/lmcs-5(3:10)2009 fatcat:kyxacru7yvhrjav2s3fdife63u

Applications of Metric Coinduction [chapter]

Dexter Kozen, Nicholas Ruozzi
Algebra and Coalgebra in Computer Science  
Metric coinduction is a form of coinduction that can be used to establish properties of objects constructed as a limit of finite approximations.  ...  This paper examines the application of this principle in a variety of areas, including infinite streams, Markov chains, Markov decision processes, and non-well-founded sets.  ...  Any views and conclusions expressed herein are those of the authors and should not be interpreted as representing the official policies or endorsements of the National Science Foundation or the United  ... 
doi:10.1007/978-3-540-73859-6_22 dblp:conf/calco/KozenR07 fatcat:qiak6kqtdraunb7quyemxetvuy

Infinitary Rewriting Coinductively

Jörg Endrullis, Andrew Polonsky, Marc Herbstritt
2013 Types for Proofs and Programs  
This approach avoids the notions of ordinals and metric convergence which have appeared in the earlier definitions of the concept.  ...  We provide a coinductive definition of strongly convergent reductions between infinite lambda terms.  ...  Then r −→ −→ −→ a u by coinduction. Now s −→ −→ −→ λx.u by the abstraction constructor of −→ −→ −→ a . Application: t β = t 1 t 2 .  ... 
doi:10.4230/lipics.types.2011.16 dblp:conf/types/EndrullisP11 fatcat:duk5p7ipf5a6lnmo4jmviurcby

A Coinductive Treatment of Infinitary Rewriting [article]

Joerg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, Alexandra Silva
2014 arXiv   pre-print
We introduce a coinductive definition of infinitary term rewriting.  ...  While the idea of a coinductive treatment of infinitary rewriting is not new, all previous approaches were limited to reductions of length at most omega.  ...  Acknowledgments We thank Patrick Bahr and Jeroen Ketema for fruitful discussions and comments to earlier versions of this paper.  ... 
arXiv:1306.6224v2 fatcat:afihp67ovvdhzaaf42ncwmlbpi

Long-Term Values in Markov Decision Processes, (Co)Algebraically [chapter]

Frank M. V. Feys, Helle Hvid Hansen, Lawrence S. Moss
2018 Lecture Notes in Computer Science  
We also explore boundedness features of the Kantorovich lifting of the distribution monad to metric spaces.  ...  The main contributions of this paper are (i) to give a coinductive explanation of policy improvement using a new proof principle, based on Banach's Fixpoint Theorem, that we call contraction coinduction  ...  We believe contraction (co)induction should have applications far beyond the topic of MDPs.  ... 
doi:10.1007/978-3-030-00389-0_6 fatcat:75sk42pkp5fzpnlbyolsu575em

CertRL: Formalizing Convergence Proofs for Value and Policy Iteration in Coq [article]

Koundinya Vajjha, Avraham Shinnar, Vasily Pestun, Barry Trager, Nathan Fulton
2020 arXiv   pre-print
The CertRL development exemplifies how the Giry monad and mechanized metric coinduction streamline optimality proofs for reinforcement learning algorithms.  ...  The desire to use reinforcement learning in safety-critical settings inspires a recent line of work on formally constrained reinforcement learning; however, these methods place the implementation of the  ...  Metric and Contraction Coinduction Our formalization uses metric coinduction to establish convergence properties for infinite sequences.  ... 
arXiv:2009.11403v2 fatcat:ab5oua6w5fdtllvkyfxrowneo4

Structural Induction and Coinduction in a Fibrational Setting

Claudio Hermida, Bart Jacobs
1998 Information and Computation  
We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively de ned types.  ...  of quotient types, the coinduction principle for terminal coalgebras is admissible.  ...  Applications of coinduction principles occur prominently in 15] (internal full abstraction for the lazy lambda calculus) and in 38] (adequacy and strong extensionality for operational semantics).  ... 
doi:10.1006/inco.1998.2725 fatcat:qwvx6yx3j5evdn7zrtdb6rox3i

Coinductive Big-Step Operational Semantics [chapter]

Xavier Leroy
2006 Lecture Notes in Computer Science  
We show applications to proofs of type soundness and to proofs of semantic preservation for compilers.  ...  This paper illustrates the use of coinductive definitions and proofs in big-step operational semantics, enabling the latter to describe diverging evaluations in addition to terminating evaluations.  ...  We thank the anonymous reviewers and the participants of the 22nd meeting of IFIP Working Group 2.8 (Functional Programming) for their feedback.  ... 
doi:10.1007/11693024_5 fatcat:4bz7q4skfngjbfg6cttuzxohda

Coinduction Plain and Simple [article]

François Bry
2020 arXiv   pre-print
Coinduction refers to both a technique for the definition of infinite streams, so-called codata, and a technique for proving the equality of coinductively specified codata.  ...  The article finally suggests in its conclusion extensions of functional and logic programming with limited and decidable forms of the generalized coinduction proof principle.  ...  "Foundations of Composite Event Recognition" (February 914, 2020, https://www.dagstuhl.de/20071), for their invitation to present an early version of the work reported about in this article at the seminar  ... 
arXiv:2007.09909v2 fatcat:l5euzvldcrgmdl2wpbtrll7kpq

A Coinductive Framework for Infinitary Rewriting and Equational Reasoning (Extended Version) [article]

Jörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, Alexandra Silva
2015 arXiv   pre-print
We present a coinductive framework for defining infinitary analogues of equational reasoning and rewriting in a uniform way.  ...  The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence.  ...  Acknowledgments We thank Patrick Bahr, Jeroen Ketema, and Vincent van Oostrom for fruitful discussions and comments on earlier versions of this paper.  ... 
arXiv:1505.01128v1 fatcat:mdhz2sza7bfd7ctbvmkc7lcvxi

Coinductive Uniform Proofs [article]

Ekaterina Komendantskaya, Yue Li
2019 arXiv   pre-print
To this aim, we propose a coinductive extension of Miller et al framework of uniform proofs and prove its soundness relative to coinductive models of Horn clause logic.  ...  Both instances of coinductive reasoning appeared in the literature before, but a systematic analysis of these two kinds of proofs and of their relation was lacking.  ...  Instead of guarding proof terms, we use rules of Figure 4 to guard applications of the coinductive hypotheses.  ... 
arXiv:1903.07371v1 fatcat:wbq75w5wd5dfxefhevozlz7frm

On the foundations of final coalgebra semantics: non-well-founded sets, partial orders, metric spaces

DANIELE TURI, JAN RUTTEN
1998 Mathematical Structures in Computer Science  
and metric spaces.  ...  The aim of the present survey is to show that the elementary categorical notion of a final coalgebra is a suitable foundation for such a coinduction principle.  ...  as, for example, applicative bisimulation and trace equivalencecf.  ... 
doi:10.1017/s0960129598002588 fatcat:yszyhcmjsjeqbgzj73exutmbem

On Equivalences, Metrics, and Polynomial Time [chapter]

Alberto Cappai, Ugo Dal Lago
2015 Lecture Notes in Computer Science  
In particular, we show how notions of context equivalence and context metric can both be characterized by way of traces when defined on linear contexts.  ...  We also hint at what happens if a more general notion of a context is used.  ...  We then turn our attention to the problem of characterizing the obtained notion of equivalence by way of linear tests, giving a positive answer to that by way of a notion of parametrized trace metric.  ... 
doi:10.1007/978-3-319-22177-9_24 fatcat:4wzjikkktjhddbinn4qx5tndzy

Coinductive Formal Reasoning in Exact Real Arithmetic

Milad Niqui, Martín Escardó
2008 Logical Methods in Computer Science  
on a representation of real numbers by coinductive streams.  ...  The algorithms work on coinductive stream of Möbius maps and form the basis of the Edalat--Potts exact real arithmetic.  ...  that of CIC to be applicable to the normalisation algorithm.  ... 
doi:10.2168/lmcs-4(3:6)2008 fatcat:x5mygwzslvg4ron4e7hjmkg7yi

Contractive Functions on Infinite Data Structures

Venanzio Capretta, Graham Hutton, Mauro Jaskelioff
2016 Proceedings of the 28th Symposium on the Implementation and Application of Functional Programming Languages - IFL 2016  
Coinductive data structures, such as streams or infinite lists, have many applications in functional programming and type theory, and are naturally defined using recursive equations.  ...  Metric Spaces and Banach's theorem Banach's theorem was originally discovered as a useful tool to prove the unique existence of solutions to differential equations  ...  With different metrics, we could have different sets of contraction; it is in fact possible to define a distance that makes the above definition satisfy the conditions for application of Banach's theorem  ... 
doi:10.1145/3064899.3064900 dblp:conf/ifl/CaprettaHJ16 fatcat:pl5qdyphhbeexbfh7pde4hwkoi
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