How to Reason Coinductively Informally

Setzer, Anton

doi:10.1007/978-3-319-29198-7_12

Cited by 4 publications

(7 citation statements)

References 24 publications

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“…This approach has been further developed by Geuvers [Geu92], Howard [How96], Greiner [Gre92], Mendler [Men91]. It has been promoted for the use in MLTT by the second author in several talks and in [Set12,Set16], and by Granström [Gra08], and McBride [McB09]. See as well the work by Abbott, Altenkirch, and Ghani on containers [AAG03], and by Basold and Geuvers [BG16].…”

Section: Codata Types and Coalgebrasmentioning

confidence: 99%

“…Instead of defining Stream by its introduction rule, it is defined by its elimination rules Elements of Stream are terms such that head and tail applied to them return elements of N and Stream, respectively. A model of coalgebras as sets of natural numbers can for instance be found in [Set16].…”

Section: Codata Types and Coalgebrasmentioning

confidence: 99%

“…For coalgebras we have the principles of primitive corecursion and coiteration which are the dual of primitive recursion and iteration for algebraic data types. A detailed discussion of these concepts and why they are dual can for instance be found in [Set16]. When we define a function f : A → Stream by primitive corecursion, we have the choice of defining tail(f (a)) = f (a ) or tail(f (a)) = s for some given stream s. Coiteration restricts this choice by demanding that tail(f (a)) always needs to be equal to f (a ) for some a .…”

Section: Extension Of Theorem 9 To Coiterationmentioning

confidence: 99%

“…If one has uniqueness, one can derive the existence and uniqueness of functions defined by primitive corecursion. See[Set16] for a proof that for strictly positive coalgebras uniqueness of the functions defined by coiteration and by primitive corecursion are both equivalent to having a final coalgebra.…”

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Undecidability of Equality for Codata Types

Berger

Setzer

2018

Coalgebraic Methods in Computer Science

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Decidability of type checking for dependently typed languages usually requires a decidable equality on types. Since bisimilarity on (weakly final) coalgebras such as streams is undecidable, one cannot use it as the equality in type checking (which is definitional or judgemental equality). Instead languages based on dependent types with decidable type checking such as Coq or Agda use intensional equality for type checking. Two streams are definitionally equal if the underlying terms reduce to the same normal form, i.e. if the underlying programs are syntactically equivalent. For reasoning about equality of streams one introduces bisimilarity as a propositional rather than judgemental equality.In this paper we show that it is not possible to strengthen intensional equality in a decidable way while having the property that the equality respects one step expansion, which means that a stream with head n and tail s is equal to cons(n, s). This property, which would be very useful in type checking, would not necessarily imply that bisimilar streams are equal, and we prove that there exist equalities with this properties which are not bisimilarity. Whereas a proof that bisimilarity on streams is undecidable is straightforward, proving that respecting one step expansion makes equality undecidable is much more involved and relies on an inseparability result for sets of codes for Turing machines. We prove this theorem both for streams with primitive corecursion and with coiteration as introduction rule.Therefore, pattern matching on streams is, understood literally, not a valid principle, since it assumes that every stream is equal to a stream of the form cons(n, s). We relate this problem to the subject reduction problem found when adding pattern matching on coalgebras to Coq and Agda. We discuss how this was solved in Agda by defining coalgebras by their elimination rule and replacing pattern matching on coalgebras by copattern matching, and how this relates to the approach in Agda which uses the type of delayed computations.

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Section: Codata Types and Coalgebrasmentioning

confidence: 99%

Section: Codata Types and Coalgebrasmentioning

confidence: 99%

Section: Extension Of Theorem 9 To Coiterationmentioning

confidence: 99%

mentioning

confidence: 99%

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Undecidability of Equality for Codata Types

Berger

Setzer

2018

Coalgebraic Methods in Computer Science

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“…Note that Object s is given as a ΠΣ-types rather than ΣΠ-types. In general, polynomial functors can be reduced to ΣΠ-types (see, e.g.,Setzer (2016)). The coalgebras corresponding to ΣΠ-functors are Petersson-Synek trees(Petersson & Synek, 1989), which are trees with nodes indexed over some set I, and each node having a label and a branching degree (where the set of labels, the branching degree, and the index set of the subtrees depend on the index of the root).…”

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confidence: 99%

Interactive programming in Agda – Objects and graphical user interfaces

Abel¹,

Adelsberger²,

Setzer³

2017

J. Funct. Prog.

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We develop a methodology for writing interactive and object-based programs (in the sense of Wegner) in dependently typed functional programming languages. The methodology is implemented in the ooAgda library. ooAgda provides a syntax similar to the one used in object-oriented programming languages, thanks to Agda's copattern matching facility. The library allows for the development of graphical user interfaces (GUIs), including the use of action listeners.Our notion of interactive programs is based on the IO monad defined by Hancock and Setzer, which is a coinductive data type. We use a sized coinductive type which allows us to write corecursive programs in a modular way. Objects are server-side interactive programs that respond to method calls by giving answers and changing their state. We introduce two kinds of objects: simple objects and IO objects. Methods in simple objects are pure, while method calls in IO objects allow for interactions before returning their result. Our approach also allows us to extend interfaces and objects by additional methods.We refine our approach to state-dependent interactive programs and objects through which we can avoid exceptions. For example, with a state-dependent stack object, we can statically disable the pop method for empty stacks. As an example, we develop the implementation of recursive functions using a safe stack. Using a coinductive notion of object bisimilarity, we verify basic correctness properties of stack objects and show the equivalence of different stack implementations. Finally, we give a proof of concept that our interaction model allows to write GUI programs in a natural way: we present a simple drawing program, and a program which allows the users to move a small spaceship using a button.

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Formal Languages, Formally and Coinductively

Traytel¹

2017

Logical Methods in Computer Science ; Volume 13

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Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs of classic results in language theory. In this article, we study this representation in the Isabelle proof assistant. We define regular operations on infinite tries and prove the axioms of Kleene algebra for those operations. Thereby, we exercise corecursion and coinduction and confirm the coinductive view being profitable in formalizations, as it improves over the set-of-words view with respect to proof automation. LOGICAL METHODS lI N COMPUTER SCIENCE

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How to Reason Coinductively Informally

Cited by 4 publications

References 24 publications

Undecidability of Equality for Codata Types

Undecidability of Equality for Codata Types

Interactive programming in Agda – Objects and graphical user interfaces

Formal Languages, Formally and Coinductively

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