8 Search Results for "Setzer, Anton"

Document
DOI: 10.4230/LIPIcs.CALCO.2025.7
A Coinductive Representation of Computable Functions

Authors: Alvin Tang and Dirk Pattinson

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract
We investigate a representation of computable functions as total functions over 2^∞, the set of finite and infinite sequences over {0,1}. In this model, infinite sequences are interpreted as non-terminating computations whilst finite sequences represent the sum of their digits. We introduce a new definition principle, function space corecursion, that simultaneously generalises minimisation and primitive recursion. This defines the class of computable corecursive functions that is closed under composition and function space corecursion. We prove computable corecursive functions represent all partial recursive functions, and show that all computable corecursive functions are indeed computable by translation into the untyped λ-calculus.
Cite as

Alvin Tang and Dirk Pattinson. A Coinductive Representation of Computable Functions. In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{tang_et_al:LIPIcs.CALCO.2025.7,
  author =	{Tang, Alvin and Pattinson, Dirk},
  title =	{{A Coinductive Representation of Computable Functions}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.7},
  URN =		{urn:nbn:de:0030-drops-235662},
  doi =		{10.4230/LIPIcs.CALCO.2025.7},
  annote =	{Keywords: Computability, Coinduction}
}
Document
DOI: 10.4230/OASIcs.FMBC.2025.10
Program Logics for Ledgers

Authors: Orestis Melkonian, Wouter Swierstra, and James Chapman

Published in: OASIcs, Volume 129, 6th International Workshop on Formal Methods for Blockchains (FMBC 2025)

Abstract
Distributed ledgers nowadays manage substantial monetary funds in the form of cryptocurrencies such as Bitcoin, Ethereum, and Cardano. For such ledgers to be safe, operations that add new entries must be cryptographically sound - but it is less clear how to reason effectively about such ever-growing linear data structures. This paper demonstrates how distributed ledgers may be viewed as computer programs, that, when executed, transfer funds between various parties. As a result, familiar program logics, such as Hoare logic, are applied in a novel setting. Borrowing ideas from concurrent separation logic, this enables modular reasoning principles over arbitrary fragments of any ledger. All of our results have been mechanised in the Agda proof assistant.
Cite as

Orestis Melkonian, Wouter Swierstra, and James Chapman. Program Logics for Ledgers. In 6th International Workshop on Formal Methods for Blockchains (FMBC 2025). Open Access Series in Informatics (OASIcs), Volume 129, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{melkonian_et_al:OASIcs.FMBC.2025.10,
  author =	{Melkonian, Orestis and Swierstra, Wouter and Chapman, James},
  title =	{{Program Logics for Ledgers}},
  booktitle =	{6th International Workshop on Formal Methods for Blockchains (FMBC 2025)},
  pages =	{10:1--10:22},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-371-3},
  ISSN =	{2190-6807},
  year =	{2025},
  volume =	{129},
  editor =	{Marmsoler, Diego and Xu, Meng},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.FMBC.2025.10},
  URN =		{urn:nbn:de:0030-drops-230370},
  doi =		{10.4230/OASIcs.FMBC.2025.10},
  annote =	{Keywords: blockchain, distributed ledgers, UTxO separation logic, program semantics, formal verification, Agda}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2023.5
Consistent Ultrafinitist Logic

Authors: Michał J. Gajda

Published in: LIPIcs, Volume 303, 29th International Conference on Types for Proofs and Programs (TYPES 2023)

Abstract
Ultrafinitism postulates that we can only compute on relatively short objects, and numbers beyond a certain value are not available. This approach would also forbid many forms of infinitary reasoning and allow removing certain paradoxes stemming from enumeration theorems. For a computational application of ultrafinitist logic, we need more than a proof system, but a logical framework to express both proofs, programs, and theorems in a single framework. We present its inference rules, reduction relation, and self-encoding to allow direct proving of the properties of ultrafinitist logic within itself. We also provide a justification why it can express all bounded Turing programs, and thus serve as a "logic of computability".
Cite as

Michał J. Gajda. Consistent Ultrafinitist Logic. In 29th International Conference on Types for Proofs and Programs (TYPES 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 303, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{gajda:LIPIcs.TYPES.2023.5,
  author =	{Gajda, Micha{\l} J.},
  title =	{{Consistent Ultrafinitist Logic}},
  booktitle =	{29th International Conference on Types for Proofs and Programs (TYPES 2023)},
  pages =	{5:1--5:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-332-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{303},
  editor =	{Kesner, Delia and Reyes, Eduardo Hermo and van den Berg, Benno},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2023.5},
  URN =		{urn:nbn:de:0030-drops-204833},
  doi =		{10.4230/LIPIcs.TYPES.2023.5},
  annote =	{Keywords: ultrafinitism, bounded Turing completeness, logic of computability, decidable logic, explicit complexity, strict finitism}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2021.1
Verification of Bitcoin Script in Agda Using Weakest Preconditions for Access Control

Authors: Fahad F. Alhabardi, Arnold Beckmann, Bogdan Lazar, and Anton Setzer

Published in: LIPIcs, Volume 239, 27th International Conference on Types for Proofs and Programs (TYPES 2021)

Abstract
This paper contributes to the verification of programs written in Bitcoin’s smart contract language script in the interactive theorem prover Agda. It focuses on the security property of access control for script programs that govern the distribution of Bitcoins. It advocates that weakest preconditions in the context of Hoare triples are the appropriate notion for verifying access control. It aims at obtaining human-readable descriptions of weakest preconditions in order to close the validation gap between user requirements and formal specification of smart contracts. As examples for the proposed approach, the paper focuses on two standard script programs that govern the distribution of Bitcoins, Pay to Public Key Hash (P2PKH) and Pay to Multisig (P2MS). The paper introduces an operational semantics of the script commands used in P2PKH and P2MS, which is formalised in the Agda proof assistant and reasoned about using Hoare triples. Two methodologies for obtaining human-readable descriptions of weakest preconditions are discussed: (1) a step-by-step approach, which works backwards instruction by instruction through a script, sometimes grouping several instructions together; (2) symbolic execution of the code and translation into a nested case distinction, which allows to read off weakest preconditions as the disjunction of conjunctions of conditions along accepting paths. A syntax for equational reasoning with Hoare Triples is defined in order to formalise those approaches in Agda.
Cite as

Fahad F. Alhabardi, Arnold Beckmann, Bogdan Lazar, and Anton Setzer. Verification of Bitcoin Script in Agda Using Weakest Preconditions for Access Control. In 27th International Conference on Types for Proofs and Programs (TYPES 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 239, pp. 1:1-1:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{alhabardi_et_al:LIPIcs.TYPES.2021.1,
  author =	{Alhabardi, Fahad F. and Beckmann, Arnold and Lazar, Bogdan and Setzer, Anton},
  title =	{{Verification of Bitcoin Script in Agda Using Weakest Preconditions for Access Control}},
  booktitle =	{27th International Conference on Types for Proofs and Programs (TYPES 2021)},
  pages =	{1:1--1:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-254-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{239},
  editor =	{Basold, Henning and Cockx, Jesper and Ghilezan, Silvia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2021.1},
  URN =		{urn:nbn:de:0030-drops-167704},
  doi =		{10.4230/LIPIcs.TYPES.2021.1},
  annote =	{Keywords: Blockchain, Cryptocurrency, Bitcoin, Agda, Verification, Hoare logic, Bitcoin Script, P2PKH, P2MS, Access control, Weakest precondition, Predicate transformer semantics, Provable correctness, Symbolic execution, Smart contracts}
}
Document
DOI: 10.4230/LIPIcs.CSL.2022.14
Realising Intensional S4 and GL Modalities

Authors: Liang-Ting Chen and Hsiang-Shang Ko

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract
There have been investigations into type-theoretic foundations for metaprogramming, notably Davies and Pfenning’s (2001) treatment in S4 modal logic, where code evaluating to values of type A is given the modal type Code A (□A in the original paper). Recently Kavvos (2017) extended PCF with Code A and intensional recursion, understood as the deductive form of the GL (Gödel-Löb) axiom in provability logic, but the resulting type system is logically inconsistent. Inspired by staged computation, we observe that a term of type Code A is, in general, code to be evaluated in a next stage, whereas S4 modal type theory is a special case where code can be evaluated in the current stage, and the two types of code should be discriminated. Consequently, we use two separate modalities ⊠ and □ to model S4 and GL respectively in a unified categorical framework while retaining logical consistency. Following Kavvos’ (2017) novel approach to the semantics of intensionality, we interpret the two modalities in the P-category of assemblies and trackable maps. For the GL modality □ in particular, we use guarded type theory to articulate what it means by a “next” stage and to model intensional recursion by guarded recursion together with Kleene’s second recursion theorem. Besides validating the S4 and GL axioms, our model better captures the essence of intensionality by refuting congruence (so that two extensionally equal terms may not be intensionally equal) and internal quoting (both A → □A and A → ⊠A). Our results are developed in (guarded) homotopy type theory and formalised in Agda.
Cite as

Liang-Ting Chen and Hsiang-Shang Ko. Realising Intensional S4 and GL Modalities. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chen_et_al:LIPIcs.CSL.2022.14,
  author =	{Chen, Liang-Ting and Ko, Hsiang-Shang},
  title =	{{Realising Intensional S4 and GL Modalities}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.14},
  URN =		{urn:nbn:de:0030-drops-157341},
  doi =		{10.4230/LIPIcs.CSL.2022.14},
  annote =	{Keywords: provability, guarded recursion, realisability, modal types, metaprogramming}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2018.1
Martin Hofmann’s Case for Non-Strictly Positive Data Types

Authors: Ulrich Berger, Ralph Matthes, and Anton Setzer

Published in: LIPIcs, Volume 130, 24th International Conference on Types for Proofs and Programs (TYPES 2018)

Abstract
We describe the breadth-first traversal algorithm by Martin Hofmann that uses a non-strictly positive data type and carry out a simple verification in an extensional setting. Termination is shown by implementing the algorithm in the strongly normalising extension of system F by Mendler-style recursion. We then analyze the same algorithm by alternative verifications first in an intensional setting using a non-strictly positive inductive definition (not just a non-strictly positive data type), and subsequently by two different algebraic reductions. The verification approaches are compared in terms of notions of simulation and should elucidate the somewhat mysterious algorithm and thus make a case for other uses of non-strictly positive data types. Except for the termination proof, which cannot be formalised in Coq, all proofs were formalised in Coq and some of the algorithms were implemented in Agda and Haskell.
Cite as

Ulrich Berger, Ralph Matthes, and Anton Setzer. Martin Hofmann’s Case for Non-Strictly Positive Data Types. In 24th International Conference on Types for Proofs and Programs (TYPES 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 130, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{berger_et_al:LIPIcs.TYPES.2018.1,
  author =	{Berger, Ulrich and Matthes, Ralph and Setzer, Anton},
  title =	{{Martin Hofmann’s Case for Non-Strictly Positive Data Types}},
  booktitle =	{24th International Conference on Types for Proofs and Programs (TYPES 2018)},
  pages =	{1:1--1:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-106-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{130},
  editor =	{Dybjer, Peter and Esp{\'\i}rito Santo, Jos\'{e} and Pinto, Lu{\'\i}s},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2018.1},
  URN =		{urn:nbn:de:0030-drops-114052},
  doi =		{10.4230/LIPIcs.TYPES.2018.1},
  annote =	{Keywords: non strictly-positive data types, breadth-first traversal, program verification, Mendler-style recursion, System F, theorem proving, Coq, Agda, Haskell}
}
Document
DOI: 10.4230/LIPIcs.TYPES.2016.12
Defining Trace Semantics for CSP-Agda

Authors: Bashar Igried and Anton Setzer

Published in: LIPIcs, Volume 97, 22nd International Conference on Types for Proofs and Programs (TYPES 2016)

Abstract
This article is based on the library CSP-Agda, which represents the process algebra CSP coinductively in the interactive theorem prover Agda. The intended application area of CSP-Agda is the proof of properties of safety critical systems (especially the railway domain). In CSP-Agda, CSP processes have been extended to monadic form, allowing the design of processes in a more modular way. In this article we extend the trace semantics of CSP to the monadic setting. We implement this semantics, together with the corresponding refinement and equality relation, formally in CSP-Agda. In order to demonstrate the proof capabilities of CSP-Agda, we prove in CSP-Agda selected algebraic laws of CSP based on the trace semantics. Because of the monadic settings, some adjustments need to be made to these laws. The examples covered in this article are the laws of refinement, commutativity of interleaving and parallel, and the monad laws for the monadic extension of CSP. All proofs and definitions have been type checked in Agda. Further proofs of algebraic laws will be available in the repository of CSP-Agda.
Cite as

Bashar Igried and Anton Setzer. Defining Trace Semantics for CSP-Agda. In 22nd International Conference on Types for Proofs and Programs (TYPES 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 97, pp. 12:1-12:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{igried_et_al:LIPIcs.TYPES.2016.12,
  author =	{Igried, Bashar and Setzer, Anton},
  title =	{{Defining Trace Semantics for CSP-Agda}},
  booktitle =	{22nd International Conference on Types for Proofs and Programs (TYPES 2016)},
  pages =	{12:1--12:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-065-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{97},
  editor =	{Ghilezan, Silvia and Geuvers, Herman and Ivetic, Jelena},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2016.12},
  URN =		{urn:nbn:de:0030-drops-98509},
  doi =		{10.4230/LIPIcs.TYPES.2016.12},
  annote =	{Keywords: Agda, CSP, Coalgebras, Coinductive Data Types, Dependent Type Theory, IO-Monad, Induction-Recursion, Interactive Program, Monad, Monadic Programming,}
}
Document
DOI: 10.4230/DagSemProc.04381.2
Interactive Programs and Weakly Final Coalgebras in Dependent Type Theory (Extended Version)

Authors: Anton Setzer and Peter Hancock

Published in: Dagstuhl Seminar Proceedings, Volume 4381, Dependently Typed Programming (2005)

Abstract
We reconsider the representation of interactive programs in dependent type theory that the authors proposed in earlier papers. Whereas in previous versions the type of interactive programs was introduced in an ad hoc way, it is here defined as a weakly final coalgebra for a general form of polynomial functor. The are two versions: in the first the interface with the real world is fixed, while in the second the potential interactions can depend on the history of previous interactions. The second version may be appropriate for working with specifications of interactive programs. We focus on command-response interfaces, and consider both client and server programs, that run on opposite sides such an interface. We give formation/introduction/elimination/equality rules for these coalgebras. These are explored in two dimensions: coiterative versus corecursive, and monadic versus non-monadic. We also comment upon the relationship of the corresponding rules with guarded induction. It turns out that the introduction rules are nothing but a slightly restricted form of guarded induction. However, the form in which we write guarded induction is not recursive equations (which would break normalisation – we show that type checking becomes undecidable), but instead involves an elimination operator in a crucial way.
Cite as

Anton Setzer and Peter Hancock. Interactive Programs and Weakly Final Coalgebras in Dependent Type Theory (Extended Version). In Dependently Typed Programming. Dagstuhl Seminar Proceedings, Volume 4381, pp. 1-30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2005)

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@InProceedings{setzer_et_al:DagSemProc.04381.2,
  author =	{Setzer, Anton and Hancock, Peter},
  title =	{{Interactive Programs and Weakly Final Coalgebras in Dependent Type Theory (Extended Version)}},
  booktitle =	{Dependently Typed Programming},
  pages =	{1--30},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2005},
  volume =	{4381},
  editor =	{Thorsten Altenkirch and Martin Hofmann and John Hughes},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04381.2},
  URN =		{urn:nbn:de:0030-drops-1768},
  doi =		{10.4230/DagSemProc.04381.2},
  annote =	{Keywords: Dependently types programming , interactive programs , coalgebras , weakly final coalgebras , coiteration , corecursion , monad}
}
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