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Coinduction up-to in a fibrational setting

Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot
2014 Proceedings of the Joint Meeting of the Twenty-Third EACSL Annual Conference on Computer Science Logic (CSL) and the Twenty-Ninth Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) - CSL-LICS '14  
We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs.  ...  This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras.  ...  Acknowledgments We are grateful to the anonymous reviewers for their constructive comments, and in particular to the one who noticed that our no-  ... 
doi:10.1145/2603088.2603149 dblp:conf/csl/BonchiPPR14 fatcat:b7mgfjhc5fgmreoiq6wz5ifaey

Coinduction in Flow: The Later Modality in Fibrations

Henning Basold, Michael Wagner
2019 Conference on Algebra and Coalgebra in Computer Science  
The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations.  ...  As such incremental methods are valuable in any theory that is based on coiteration or coinduction schemes, we set out in this paper to replace invariant guessing by a general iterative programming and  ...  [16] from the codomain fibration Set → → Set of sets to arbitrary fibrations.  ... 
doi:10.4230/lipics.calco.2019.8 dblp:conf/calco/Basold19 fatcat:de4sngban5ggll2hp2ifba3shm

A Fibrational Method of Indexed Coinductive Data Types

2020 Tehnički Vjesnik  
As a fundamental issue in type theory, indexed coinductive data types (ICDT, for short) is of crucial importance, which is essentially semantic computing problem in programming.  ...  Our work is based on fibration; a general math setting that can compute semantics automatically rather than depend on particular computing environments and syntactic forms of ICDT.  ...  The fibered terminal object of predicate fibration Pre in Example 1 is a function map of all elements in the set X to a singleton set.  ... 
doi:10.17559/tv-20191130092745 fatcat:kvzljd7ewbaalk4w7hlindrpty

Indexed Induction and Coinduction, Fibrationally

Neil Ghani, Patricia Johann, Clement Fumex, Bartek Klin
2013 Logical Methods in Computer Science  
role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction.  ...  This paper extends the fibrational approach to induction and coinduction pioneered by Hermida and Jacobs, and developed by the current authors, in two key directions.  ...  defined up to isomorphism.  ... 
doi:10.2168/lmcs-9(3:6)2013 fatcat:77pcj77n5zfshpvcwstsrvx5zq

Indexed Induction and Coinduction, Fibrationally [chapter]

Clément Fumex, Neil Ghani, Patricia Johann
2011 Lecture Notes in Computer Science  
role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction.  ...  role in the theory of coinduction dual to that played by a comprehension category with unit (CCU) in the theory of induction.  ...  defined up to isomorphism.  ... 
doi:10.1007/978-3-642-22944-2_13 fatcat:lons4fr3xnfndmrzhi7roncwaa

Coinduction in Uniform: Foundations for Corecursive Proof Search with Horn Clauses [chapter]

Henning Basold, Ekaterina Komendantskaya, Yue Li
2019 Lecture Notes in Computer Science  
Operational semantics of coinductive Horn clause resolution is cast in terms of coinductive uniform proofs; its constructive content is exposed via soundness relative to an intuitionistic first-order logic  ...  with recursion controlled by the later modality; and soundness of both proof systems is proven relative to a novel coalgebraic description of complete Herbrand models.  ...  We would like to thank Damien Pous and the anonymous reviewers for their valuable feedback.  ... 
doi:10.1007/978-3-030-17184-1_28 fatcat:7azord5jkjhlvdxyir54hjsdqq

Breaking the Loop: Recursive Proofs for Coinductive Predicates in Fibrations [article]

Henning Basold
2018 arXiv   pre-print
Such recursive proofs allow one to discover proof goals in the construction of a proof of a coinductive predicate, while still allowing the use of up-to techniques.  ...  In this paper, we develop a framework for recursive proofs of coinductive predicates that are given through fibrational predicate liftings.  ...  Given a set A, we define a functor F and a lifting G A to the family fibration Fam(Set) → Set as follows.  ... 
arXiv:1802.07143v1 fatcat:xfonsdxujrd4hb62mraecf2dnq

Dependent Inductive and Coinductive Types are Fibrational Dialgebras

Henning Basold
2015 Electronic Proceedings in Theoretical Computer Science  
In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types.  ...  I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors.  ...  A partial stream is a stream together with a, possibly infinite, depth up to which it is defined.  ... 
doi:10.4204/eptcs.191.3 fatcat:v75s7jhz7nacdeo4mhywykcxnu

Coinductive predicates and final sequences in a fibration

ICHIRO HASUO, TOSHIKI KATAOKA, KENTA CHO
2017 Mathematical Structures in Computer Science  
In its course, we develop a relevant categorical infrastructure that relates fibrations and locally presentable categories, a combination that does not seem to be studied a lot.  ...  In this paper, we follow the spirit of a seminal work by Worrell and study final sequences in a fibration.  ...  Acknowledgments Thanks are due to Kazuyuki Asada, Keisuke Nakano, Keiko Nakata, Ana Sokolova, and the participants of Dagstuhl Seminar 12411 "Coalgebraic Log  ... 
doi:10.1017/s0960129517000056 fatcat:4a3jwmzx7bgs3blxxvlsnzmrwy

A general account of coinduction up-to

Filippo Bonchi, Daniela Petrişan, Damien Pous, Jurriaan Rot
2016 Acta Informatica  
We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs.  ...  This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modeled as coalgebras.  ...  Proof Recall that the canonical relation lifting Rel(G)(R) of a relation i : R → X × X is obtained via the (epi,mono)-factorisation in (27) .  ... 
doi:10.1007/s00236-016-0271-4 fatcat:hogjqmrjmzhfromkx5sybovzkq

Expressivity of Quantitative Modal Logics: Categorical Foundations via Codensity and Approximation [article]

Yuichi Komorida, Shin-ya Katsumata, Clemens Kupke, Jurriaan Rot, Ichiro Hasuo
2021 arXiv   pre-print
Recently, with the growing diversity of systems to be reasoned about (probabilistic, cyber-physical, etc.), the focus shifted to quantitative settings which resulted in a number of expressivity results  ...  A modal logic that is strong enough to fully characterize the behavior of a system is called expressive.  ...  Let S be an expressivity situation as in Def. III.1 and x : X → BX be a Bcoalgebra. Assume that the chain (5) in E X stabilizes after ω steps.  ... 
arXiv:2105.10164v1 fatcat:vnu7fq3tprbilmty6njig35pky

Expressivity of Quantitative Modal Logics : Categorical Foundations via Codensity and Approximation

Yuichi Komorida, Shin-ya Katsumata, Clemens Kupke, Jurriaan Rot, Ichiro Hasuo
2021 2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
Recently, with the growing diversity of systems to be reasoned about (probabilistic, cyber-physical, etc.), the focus shifted to quantitative settings which resulted in a number of expressivity results  ...  A modal logic that is strong enough to fully characterize the behavior of a system is called expressive.  ...  In a fibration, these fiber categories E C are patched up to form a single total category E, a formulation that accommodates many structural reasoning principles.  ... 
doi:10.1109/lics52264.2021.9470656 fatcat:yfyv6dqxrnaexpkdlyz6ftlcsi

Semantic Behavior of Indexed Co-inductive Data Type in Fibrational Setting

Miao De-cheng, Xi Jian-qing, Jiang Chang-jin, Liang Yong-lin
2016 International Journal of u- and e- Service, Science and Technology  
In this paper, we applying Fibrations theory to research indexed co-inductive data type, by taking indexed co-inductive data type to be object set in base category, taking their semantic behavior to be  ...  Our main work was that we firstly made some basic logical structures of indexed co-inductive data type over a fibration such as truth and quotient functor; using endo-functor in base categories and their  ...  For categoryC , let bj O C to be set of objects and or M C to be set of morphisms. We introduce some basic concepts in this paper.  ... 
doi:10.14257/ijunesst.2016.9.6.24 fatcat:5m5527cewjg7ddewhs3qmk6ctu

Exploiting Parallelism in Coalgebraic Logic Programming

Ekaterina Komendantskaya, Martin Schmidt, Jónathan Heras
2014 Electronical Notes in Theoretical Computer Science  
We present a parallel implementation of Coalgebraic Logic Programming (CoALP) in the programming language Go.  ...  CoALP was initially introduced to reflect coalgebraic semantics of logic programming, with coalgebraic derivation algorithm featuring both corecursion and parallelism.  ...  A clause C * i is a variant of the clause C i if C * i = C i θ, with θ being a variable renaming substitution such that variables in C * i do not appear in the derivation up to G i−1 .  ... 
doi:10.1016/j.entcs.2014.02.007 fatcat:eznl7fyu55eori4psc7r3avxom

Fibrational Bisimulations and Quantitative Reasoning [chapter]

David Sprunger, Shin-ya Katsumata, Jérémy Dubut, Ichiro Hasuo
2018 Lecture Notes in Computer Science  
Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions.  ...  We use coalgebras as system models and fibrations for organizing predicatesfollowing the seminal work by Hermida and Jacobs-but our focus is on the structural aspect of fibrational frameworks.  ...  A concrete goal in this section is to establish some general conditions under which a BVal coinductive invariant has an ERel coinductive invariant at its kernel.  ... 
doi:10.1007/978-3-030-00389-0_11 fatcat:hwm2ozmnbzcvdf5gdetl6zx75y
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