Permutations in Coinductive Graph Representation

Picard, Célia; Matthes, Ralph

doi:10.1007/978-3-642-32784-1_12

Cited by 7 publications

(5 citation statements)

References 14 publications

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“…We would have to give coinductive specifications and corecursively create their proofs, which would be a major challenge in Coq (cf. the experience of the second author with coinductive rose trees in Coq [16] where the restrictive guardedness criterion of Coq had to be circumvented in particular for corecursive constructions).…”

Section: Final Remarksmentioning

confidence: 99%

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Certification of Breadth-First Algorithms by Extraction

Larchey-Wendling¹,

Matthes²

2019

Lecture Notes in Computer Science

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By using pointers, breadth-first algorithms are very easy to implement efficiently in imperative languages. Implementing them with the same bounds on execution time in purely functional style can be challenging, as explained in Okasaki's paper at ICFP 2000 that even restricts the problem to binary trees but considers numbering instead of just traversal. Okasaki's solution is modular and factors out the problem of implementing queues (FIFOs) with worst-case constant time operations. We certify those FIFO-based breadth-first algorithms on binary trees by extracting them from fully specified Coq terms, given an axiomatic description of FIFOs. In addition, we axiomatically characterize the strict and total order on branches that captures the nature of breadthfirst traversal and propose alternative characterizations of breadth-first traversal of forests. We also propose efficient certified implementations of FIFOs by extraction, either with pairs of lists (with amortized constant time operations) or triples of lazy lists (with worst-case constant time operations), thus getting from extraction certified breadth-first algorithms with the optimal bounds on execution time. Keywords: Breadth-first algorithms • Queues in functional programming • Correctness by extraction • Coq The first author got financial support by the TICAMORE joint ANR-FWF project. The second author got financial support by the COST action CA15123 EUTYPES.

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Section: Final Remarksmentioning

confidence: 99%

“…tactic outputs terms like exist s O ? s where is recovered by unification, and O ?s is left open to be solved later by the user 16. The constructor exist packs the pair (s, O ?…”

mentioning

confidence: 99%

Certification of Breadth-First Algorithms by Extraction

Larchey-Wendling¹,

Matthes²

2019

Lecture Notes in Computer Science

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“…This means, in particular, that this identification is used recursively when considering bisimilarity (anyway recursively modulo αequivalence). This approach is convenient for a mathematical treatment but would be less so for a formalization on a computer: It has been shown by Picard and the second author [PM12] that bisimulation up to permutations in unbounded lists of children can be managed in a coinductive type even with the interactive proof assistant Coq, but it did not seem feasible to abstract away from the number of occurrences of an alternative (which is the meaning of idempotence of + in presence of symmetry), where multiplicity depends on the very same notion of equivalence that is undecidable in general.…”

Section: σ Systemmentioning

confidence: 99%

A Coinductive Approach to Proof Search through Typed Lambda-Calculi

Santo,

Matthes,

Pinto

2016

Preprint

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In reductive proof search, proofs are naturally generalized by solutions, comprising all (possibly infinite) structures generated by locally correct, bottom-up application of inference rules. We propose a rather natural extension of the Curry-Howard paradigm of representation, from proofs to solutions: to represent solutions by (possibly infinite) terms of the coinductive variant of the typed lambda-calculus that represents proofs. We take this as a starting point for a new, comprehensive approach to proof search; our case study is proof search in the sequent calculus LJT for intuitionistic implication logic. A second, finitary representation is proposed, where the lambda-calculus that represents proofs is extended with a formal greatest fixed point. In the latter system, fixed-point variables enjoy a relaxed form of binding that allows the detection of cycles through the type system. Formal sums are used in both representations to express alternatives in the search process, so that not only individual solutions but actually solution spaces are expressed. Moreover, formal sums are used in the coinductive syntax to define "co-contraction" (contraction bottom-up). A semantics is defined assigning a coinductive λ-term to each finitary term, making use of co-contraction as a semantical match to the relaxed form of binding of fixed-point variables present in the finitary system. The main result is the existence of an equivalent finitary representation for any given solution space expressed coinductively.

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“…In contrast to our graph model, their graph model is based on graph isomorphism, and order between outgoing edges are not considered. Picard and Matthes (2012) deal with node-labeled graphs using coinductive data types in Coq proof assistant. Although outgoing edges of their graphs are also ordered and graph equivalence based on bisimulation is considered, ε-edges are not considered.…”

Section: Related Workmentioning

confidence: 99%

Structural recursion for querying ordered graphs

et al. 2013

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Structural recursion, in the form of, for example, folds on lists and catamorphisms on algebraic data structures including trees, plays an important role in functional programming, by providing a systematic way for constructing and manipulating functional programs. It is, however, a challenge to define structural recursions for graph data structures, the most ubiquitous sort of data in computing. This is because unlike lists and trees, graphs are essentially not inductive and cannot be formalized as an initial algebra in general. In this paper, we borrow from the database community the idea of structural recursion on how to restrict recursions on infinite unordered regular trees so that they preserve the finiteness property and become terminating, which are desirable properties for query languages. We propose a new graph transformation language called λFG for transforming and querying ordered graphs, based on the well-defined bisimulation relation on ordered graphs with special ε-edges. The language λFG is a higher order graph transformation language that extends the simply typed lambda calculus with graph constructors and more powerful structural recursions, which is extended for transformations on the sibling dimension. It not only gives a general framework for manipulating graphs and reasoning about them, but also provides a solution to the open problem of how to define a structural recursion on ordered graphs, with the help of the bisimilarity for ordered graphs with ε-edges.

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Permutations in Coinductive Graph Representation

Cited by 7 publications

References 14 publications

Certification of Breadth-First Algorithms by Extraction

Certification of Breadth-First Algorithms by Extraction

A Coinductive Approach to Proof Search through Typed Lambda-Calculi

Structural recursion for querying ordered graphs

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