Sequent Calculus in the Topos of Trees

Clouston, Ranald; Goré, Rajeev

doi:10.1007/978-3-662-46678-0_9

Cited by 8 publications

(11 citation statements)

References 34 publications

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“…Let us add that, while Gentzen-style systems are not our main interest here, the above quote from Clouston and Goré [CG15] hints at another motivation for studying constructive . Namely, even in the setups which make it a definable connective, it can still prove a more convenient primitive from a proof-theoretical point of view than P is.…”

Section: Modalities For Guarded (Co)recursionmentioning

confidence: 99%

Lewis meets Brouwer: Constructive strict implication

Litak¹,

Visser

2018

Indagationes Mathematicae

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C. I. Lewis invented modern modal logic as a theory of "strict implication" . Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than P and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction has been discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of intuitionistic in terms of preservativity in extensions of HA, i.e., Heyting's Arithmetic. 65The basic option is to read § § 2-4 to get the basics of motivational background, the Kripke semantics and an impression of possible reasoning systems. ? The reader who wants more solid treatment of Kripke semantics can extend the basic option with § 6. | The computer science package consists of the basic option and § 7.= The reader who wants to go somewhat more deeply into the history of the subject can extend the basic option with Appendix D. « The reader who wants to understand the basics of arithmetical interpretations can extend the basic option with § 5. » An extended package for arithmetical interpretations combines « with § 8.-The full arithmetical package extends » with Appendices A, B and C. The rise and fall of the house of Lewis "The error of philosophers"We are reflecting on L.E.J. Brouwer's heritage half a century after his passing. Given his negative views on the rôle of logic and formalisms in mathematics, it seems somewhat paradoxical that these days the name of intuitionism survives mostly in the context of intuitionistic logic. 3 One is reminded in this context of what Nietzsche called the error of philosophers:The philosopher believes that the value of his philosophy lies in the whole, in the structure. Posterity finds it in the stone with which he built and with which, from that time forth, men will build oftener and better-in other words, in the fact that the structure may be destroyed and yet have value as material. 4

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Section: Modalities For Guarded (Co)recursionmentioning

confidence: 99%

Lewis meets Brouwer: Constructive strict implication

Litak¹,

Visser

2018

Indagationes Mathematicae

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“…This modality is the essential ingredient that allows us to equip proofs with a controlled form of recursion. The later modality stems originally from provability logic, which characterises transitive, well-founded Kripke frames [29,73], and thus allows one to carry out induction without an explicit induction scheme [15]. Later, the later modality was picked up by the type-theoretic community to control recursion in coinductive programming [7,8,20,56,58], mostly with the intent to replace syntactic guardedness checks for coinductive de nitions by type-based checks of wellde nedness.…”

Section: Coinductive Uniform Proofs and Intuitionistic Logicmentioning

confidence: 99%

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

Basold

Komendantskaya

2019

Programming Languages and Systems

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We establish proof-theoretic, constructive and coalgebraic foundations for proof search in coinductive Horn clause theories. 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 rst-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 can also view Horn clauses coinductively. The greatest complete Herbrand model for a set P of Horn clauses is the largest set of nite and in nite ground atomic formulae coinductively entailed by P. For example, the greatest complete Herbrand model for the above two clauses is the set N ∞ = N ∪ {nat (s (s (· · · )))}, obtained by taking a backward closure of the above two inference rules on the set of all nite and in nite ground atomic formulae. The greatest Herbrand model is the largest set of nite ground atomic formulae coinductively entailed by P. In our example, it would be given by N already. Finally, one can also consider the least complete Hebrand model, which interprets entailment inductively but over potentially in nite terms. In the case of nat, this interpretation does not di er from N . However, nite paths in coinductive structures like transition systems, for example, require such semantics.The need for coinductive semantics of Horn clauses arises in several scenarios: the Horn clause theory may explicitely de ne a coinductive data structure or a coinductive relation. However, it may also happen that a Horn clause theory, which is not explicitly intended as coinductive, nevertheless gives rise to in nite inference by resolution and has an interesting coinductive model. This commonly happens in type inference. We will illustrate all these cases by means of examples.

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“…S is a presheaf category, and so a topos, and so its internal logic provides a model of higher-order logic with equality [32]. The internal logic of S has been explored elsewhere [7,15,31], but to motivate the results of this section we make some observations here.…”

Section: From Internal Logic To Programmentioning

confidence: 99%

“…The definition of the logic Lgλ from the previous section establishes its syntax, and semantics in the topos of trees, without giving much sense of how proofs might be constructed. Clouston and Goré [15] have provided a sound and complete sequent calculus, and hence decision procedure, for the fragment of the internal logic of S with propositional connectives and ⊲, but the full logic Lgλ is considerably more expressive than this; for example it is not decidable [37]. In this section we will establish some reasoning principles for Lgλ, which will assist us in the next section in constructing proofs about gλ-programs.…”

Section: 2mentioning

confidence: 99%

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

Clouston¹,

Bizjak²,

Grathwohl³

et al. 2017

Logical Methods in Computer Science

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Abstract. We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with Löb induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Rutten's behavioural differential equations.

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Sequent Calculus in the Topos of Trees

Cited by 8 publications

References 34 publications

Lewis meets Brouwer: Constructive strict implication

Lewis meets Brouwer: Constructive strict implication

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

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

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