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A Formalisation of the Myhill-Nerode Theorem Based on Regular Expressions

Chunhan Wu, Xingyuan Zhang, Christian Urban
2014 Journal of automated reasoning  
We show in this paper that a central result from formal language theory-the Myhill-Nerode Theorem-can be recreated using only regular expressions.  ...  Its type system is based on the Simple Theory of Types by Church (1940) .  ...  Acknowledgements: We are grateful for the comments we received from Larry Paulson. Tobias Nipkow made us aware of the properties in Theorem 4 and Tjark Weber helped us  ... 
doi:10.1007/s10817-013-9297-2 fatcat:yuhvxmyeprhh5nafmdxvt2xrqm

A Formalisation of the Myhill-Nerode Theorem Based on Regular Expressions (Proof Pearl) [chapter]

Chunhan Wu, Xingyuan Zhang, Christian Urban
2011 Lecture Notes in Computer Science  
We show in this paper that a central result from formal language theory-the Myhill-Nerode theorem-can be recreated using only regular expressions.  ...  There are numerous textbooks on regular languages. Nearly all of them introduce the subject by describing finite automata and only mentioning on the side a connection with regular expressions.  ...  Contributions: There is an extensive literature on regular languages. To our best knowledge, our proof of the Myhill-Nerode theorem is the first that is based on regular expressions, only.  ... 
doi:10.1007/978-3-642-22863-6_25 fatcat:bceexo2b5nd4nkbrep5dcjasyq

Algebra automata I: Parallel programming as a prolegomena to the categorical approach

M.A. Arbib, Y. Give'on
1968 Information and Control  
Our intuition will be based on the notion of a device which can execute highly parallel, but loop-free, programs.  ...  It is clear that aJ and X: do not depend on the choices of t or t~. within an equivalence class. (2.14) We call a: We may now generalize the Myhill eqni~alenee (see Rabin and Scott, 1959 and Myhill, 1957  ... 
doi:10.1016/s0019-9958(68)90374-4 fatcat:lyicij34djbgfnjnosyhhjy55e

On the Fine-Structure of Regular Algebra

Simon Foster, Georg Struth
2014 Journal of automated reasoning  
Regular algebra is the algebra of regular expressions as induced by regular language identity.  ...  As a case study in formalised mathematics, our investigations also shed some light on the power of theorem proving technology for reasoning with algebras and their models, including proof automation and  ...  We would like to thank Christian Urban and Tobias Nipkow for information on the implementation of regular languages in Isabelle, and the anonymous referees for comments and suggestions that helped us to  ... 
doi:10.1007/s10817-014-9318-9 fatcat:tt7toa7wwbdghgymeqowvzxkdu

Extensions of ω-Regular Languages [article]

Mikołaj Bojańczyk, Edon Kelmendi, Rafał Stefański, Georg Zetzsche
2020 arXiv   pre-print
We consider extensions of monadic second order logic over ω-words, which are obtained by adding one language that is not ω-regular.  ...  A corollary is that the ω-regular languages are the only decidable Boolean-closed full trio over ω-words.  ...  By the Myhill-Nerode theorem, ∼ has infinite index, i.e. infinitely many equivalence classes.  ... 
arXiv:2002.09393v1 fatcat:ief6tkybxnetdjtas4wjripiaa

On Computability, Learnability and Extractability of Finite State Machines from Recurrent Neural Networks [article]

Reda Marzouk
2020 arXiv   pre-print
With respect to the former, the long-standing clustering hypothesis of RNN hidden state space when trained to recognize regular languages was explored, and new insights into this hypothesis through the  ...  With regard to computability, new computational results on the distance and the equivalence problem between RNNs trained as language models and different types of weighted finite state machines were given  ...  (Myhill-Nerode Theorem) L is a regular language if and only if the number of equivalence classes of its Nerode relation is finite.  ... 
arXiv:2009.06398v1 fatcat:qow5uzwcezd7vafuobfocs4dye

Context-Free Language Theory Formalization [article]

Marcus Vinícius Midena Ramos, Ruy J. G. B. de Queiroz
2015 arXiv   pre-print
Proof assistants are software-based tools that are used in the mechanization of proof construction and validation in mathematics and computer science, and also in certified program development.  ...  Grammar simplification is a subject of high importance in computer language processing technology as well as in formal language theory, and the formalization refers to the fact that general context-free  ...  Wu, Zhang and Urban worked on a version of the Myhill-Nerode theorem that uses regular expressions instead of finite automata [?]. Tom Ridge [?]  ... 
arXiv:1505.00061v1 fatcat:kqugpa34jvcjlpdomfabtdl2aq

Three Learnable Models for the Description of Language [chapter]

Alexander Clark
2010 Lecture Notes in Computer Science  
One way to build learnable representations is by making them objective or empiricist: the structure of the representation should be based on the structure of the language.  ...  define a residuated lattice structure from the Galois connection between strings and contexts, which we call the syntactic concept lattice, and base a representation on this, which allows us to define  ...  In this case we base the representation on the right congruence classes, as used in the Myhill-Nerode theorem.  ... 
doi:10.1007/978-3-642-13089-2_2 fatcat:r5y7op6cn5gk5nvpu3owjbiqiy

Run-Time Security Traceability for Evolving Systems

A. Bauer, J. Jurjens, Y. Yu
2010 Computer journal  
This part of our approach relies on a technique also known as run-time verification.  ...  The proposed method has been applied to the Java-based implementation JESSIE of the Internet security protocol SSL, in which a security weakness was detected and fixed using our approach.  ...  Acknowledgements Discussions with Martin Leucker about a draft of this paper are gratefully acknowledged, as well as constructive comments by the reviewers which helped improving the presentation significantly  ... 
doi:10.1093/comjnl/bxq042 fatcat:5atpkvz7d5gufpvphwjno65lsa

Cone Types, Automata, and Regular Partitions in Coxeter Groups [article]

James Parkinson, Yeeka Yau
2021 arXiv   pre-print
In this article we introduce the notion of a regular partition of a Coxeter group.  ...  We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces.  ...  By the Myhill-Nerode Theorem there exists a unique (up to isomorphism) minimal (with respect to the number of states) automaton A(W, S) recognising L(W, S).  ... 
arXiv:2107.09962v3 fatcat:sjl6ceywyvdejil3fxvowfu63u

Recursion Schemes, Collapsible Pushdown Automata and Higher-Order Model Checking [chapter]

Luke Ong
2013 Lecture Notes in Computer Science  
We study and compare the expressive power of the two models and the algorithmic properties of infinite structures such as trees and graphs generated by them.  ...  A central theme of the work is the fruitful interplay of ideas between the neighbouring fields of semantics and algorithmic verification.  ...  Acknowledgements Much of my work presented here is based on collaboration. I am grateful to all my collaborators for their help. Thanks are also due to EPSRC for financial support.  ... 
doi:10.1007/978-3-642-37064-9_3 fatcat:vmzogjyskfahbeklhyssyajvuy

On the origins of bisimulation and coinduction

Davide Sangiorgi
2009 ACM Transactions on Programming Languages and Systems  
ACKNOWLEDGMENT I am very grateful to the following people who helped me to find relevant papers and materials or helped me in tracing back bits of history: P. Aczel, G. Boudol, J. van  ...  is the Myhill-Nerode theorem [Nerode 1958] ).  ...  The main topic of that paper is a different one, namely omega-regular languages (extensions of regular languages containing also infinite sequences) and operators for fair concurrency.  ... 
doi:10.1145/1516507.1516510 fatcat:m5uvypq4xnbqxclecr3eadpwsu

Languages recognised by finite semigroups, and their generalisations to objects such as trees and graphs, with an emphasis on definability in monadic second-order logic [article]

Mikołaj Bojańczyk
2020 arXiv   pre-print
These are lecture notes on the algebraic approach to regular languages. The classical algebraic approach is for finite words; it uses semigroups instead of automata.  ...  The purpose of this book is to describe the algebraic approach in a way that covers these extensions.  ...  deterministic finite automaton from the Myhill-Nerode theorem.  ... 
arXiv:2008.11635v1 fatcat:r6jidclh2zdtznsfsfsyfs3eam

A Partial Order Semantics Approach to the Clock Explosion Problem of Timed Automata [chapter]

D. Lugiez, P. Niebert, S. Zennou
2004 Lecture Notes in Computer Science  
The latter approach suffers from splitting of symbolic states depending on the order of transition occurrences, even if these transitions concern unrelated components in a parallel system.  ...  We propose a new approach for the symbolic exploration of timed automata that solves a particular aspect of the combinatory explosion occurring in the widely used clock zone automata (e.g.  ...  Acknowledgements We thank Victor Braberman, Sergio Yovine, Stavros Tripakis, Oded Maler, Eugene Asarin, Yasmina Abdeddaim, Bengt Johnsson and Rom Langerak for discussions about the challenging topic.  ... 
doi:10.1007/978-3-540-24730-2_24 fatcat:aybtd43ui5cgvhfvzykd2fa6yi

Fast Computations on Ordered Nominal Sets [article]

David Venhoek, Joshua Moerman, Jurriaan Rot
2020 arXiv   pre-print
We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon.  ...  In contrast to previous approaches, we work directly at the level of orbits, which allows for an accurate complexity analysis. The approach is implemented as the library ONS (Ordered Nominal Sets).  ...  At last, we want to thank the anonymous reviewers of ICTAC 2018 and this extended paper for their comments.  ... 
arXiv:1902.08414v2 fatcat:htbgousrenh5pfdmlpcdhng2we
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