9 Search Results for "Benzmüller, Christoph"

Document
DOI: 10.4230/LIPIcs.CSL.2026.18
A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light

Authors: Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi

Published in: LIPIcs, Volume 363, 34th EACSL Annual Conference on Computer Science Logic (CSL 2026)

Abstract
We extend the existing HOL Light Library for Modal Systems (HOLMS) to support a modular implementation of modal reasoning within the HOL Light proof assistant. We deeply embed axiomatic calculi and relational semantics for seven normal modal logics (K, T, B, K4, S4, S5, GL) and formalise modal adequacy theorems for these systems. We then leverage those formalisations to implement a mechanism for automated reasoning via proof-search in the associated labelled sequent calculi, which we shallowly embed in HOL Light’s goal-stack mechanism. This way, we equip the general-purpose proof assistant with (semi)decision procedures for these logics that, in case of failure to construct a proof for the input formula, return a certified countermodel within the appropriate class for the logic under consideration. On the methodological side, we propose a precise measure of the modularity of our approach by systematically adopting Christopher Strachey’s distinction between ad hoc and parametric polymorphism throughout the library.
Cite as

Antonella Bilotta, Marco Maggesi, and Cosimo Perini Brogi. A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 18:1-18:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bilotta_et_al:LIPIcs.CSL.2026.18,
  author =	{Bilotta, Antonella and Maggesi, Marco and Perini Brogi, Cosimo},
  title =	{{A Modular Framework for Proof-Search via Formalised Modal Completeness in HOL Light}},
  booktitle =	{34th EACSL Annual Conference on Computer Science Logic (CSL 2026)},
  pages =	{18:1--18:29},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-411-6},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{363},
  editor =	{Guerrini, Stefano and K\"{o}nig, Barbara},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2026.18},
  URN =		{urn:nbn:de:0030-drops-254427},
  doi =		{10.4230/LIPIcs.CSL.2026.18},
  annote =	{Keywords: Modal logic, HOL Light, Labelled sequent calculi, Logical verification, Interactive theorem proving, Automated proof-search}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.35
Verifying an Efficient Algorithm for Computing Bernoulli Numbers

Authors: Manuel Eberl and Peter Lammich

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
The Bernoulli numbers B_k are a sequence of rational numbers that is ubiquitous in mathematics, but difficult to compute efficiently (compared to e.g. approximating π). In 2008, Harvey gave the currently fastest known practical way for computing them: his algorithm computes B_k mod p in time O(p log^{1 + o(1)} p). By doing this for O(k) many small primes p in parallel and then combining the results with the Chinese Remainder Theorem, one recovers the value of B_k as a rational number in O(k² log^{2 + o(1)} k) time. One advantage of this approach is that the expensive part of the algorithm is highly parallelisable and has very low memory requirements. This algorithm still holds the world record with its computation of B_{10⁸}. We give a verified efficient LLVM implementation of this algorithm. This was achieved by formalising the necessary mathematical background theory in Isabelle/HOL, proving an abstract version of the algorithm correct, and refining this abstract version down to LLVM using Lammich’s Isabelle-LLVM framework, including many low-level optimisations. The performance of the resulting LLVM code is comparable with Harvey’s original unverified and hand-optimised C++ implementation.
Cite as

Manuel Eberl and Peter Lammich. Verifying an Efficient Algorithm for Computing Bernoulli Numbers. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 35:1-35:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{eberl_et_al:LIPIcs.ITP.2025.35,
  author =	{Eberl, Manuel and Lammich, Peter},
  title =	{{Verifying an Efficient Algorithm for Computing Bernoulli Numbers}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{35:1--35:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.35},
  URN =		{urn:nbn:de:0030-drops-246331},
  doi =		{10.4230/LIPIcs.ITP.2025.35},
  annote =	{Keywords: Bernoulli numbers, LLVM, verification, Isabelle, Chinese remainder theorem, modular arithmetic, Montgomery arithmetic}
}
Document
DOI: 10.4230/LIPIcs.ITP.2025.26
Improving the SMT Proof Reconstruction Pipeline in Isabelle/HOL

Authors: Hanna Lachnitt, Mathias Fleury, Haniel Barbosa, Jibiana Jakpor, Bruno Andreotti, Andrew Reynolds, Hans-Jörg Schurr, Clark Barrett, and Cesare Tinelli

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
Sledgehammer is a tool that increases the level of automation in the Isabelle/HOL proof assistant by asking external automatic theorem provers (ATPs), including SMT solvers, to prove the current goal. When the external ATP succeeds it must provide enough evidence that the goal holds for Isabelle to be able to reprove it internally based on that evidence. In particular, Isabelle can do this by replaying fine-grained proof certificates from proof-producing SMT solvers as long as they are expressed in the Alethe format, which until now was supported only by the veriT SMT solver. We report on our experience adding proof reconstruction support for the cvc5 SMT solver in Isabelle by extending cvc5 to produce proofs in the Alethe format and then adapting Isabelle to reconstruct those proofs. We discuss several difficulties and pitfalls we encountered and describe a set of tools and techniques we developed to improve the process. A notable outcome of this effort is that Isabelle can now be used as an independent proof checker for SMT problems written in the SMT-LIB standard. We evaluate cvc5’s integration on a set of SMT-LIB benchmarks originating from Isabelle as well as on a set of Isabelle proofs. Our results confirm that this integration complements and improves Sledgehammer’s capabilities.
Cite as

Hanna Lachnitt, Mathias Fleury, Haniel Barbosa, Jibiana Jakpor, Bruno Andreotti, Andrew Reynolds, Hans-Jörg Schurr, Clark Barrett, and Cesare Tinelli. Improving the SMT Proof Reconstruction Pipeline in Isabelle/HOL. In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 26:1-26:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lachnitt_et_al:LIPIcs.ITP.2025.26,
  author =	{Lachnitt, Hanna and Fleury, Mathias and Barbosa, Haniel and Jakpor, Jibiana and Andreotti, Bruno and Reynolds, Andrew and Schurr, Hans-J\"{o}rg and Barrett, Clark and Tinelli, Cesare},
  title =	{{Improving the SMT Proof Reconstruction Pipeline in Isabelle/HOL}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{26:1--26:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.26},
  URN =		{urn:nbn:de:0030-drops-246243},
  doi =		{10.4230/LIPIcs.ITP.2025.26},
  annote =	{Keywords: interactive theorem proving, proof assistants, Isabelle/HOL, SMT, certification, proof certificates, proof reconstruction, proof automation}
}
Document
Short Paper
DOI: 10.4230/LIPIcs.ITP.2025.38
Sledgehammering Without ATPs (Short Paper)

Authors: Martin Desharnais and Jasmin Blanchette

Published in: LIPIcs, Volume 352, 16th International Conference on Interactive Theorem Proving (ITP 2025)

Abstract
We describe an alternative architecture for "hammers," inspired by Magnushammer, in which proofs are found by the proof assistant’s built-in automation instead of by external automatic theorem provers (ATPs). We implemented this approach in Isabelle’s Sledgehammer and evaluated it. The new ATP-free approach nicely complements the traditional Sledgehammer. The two approaches in combination solve more goals than the traditional ATP-based approach alone.
Cite as

Martin Desharnais and Jasmin Blanchette. Sledgehammering Without ATPs (Short Paper). In 16th International Conference on Interactive Theorem Proving (ITP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 352, pp. 38:1-38:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{desharnais_et_al:LIPIcs.ITP.2025.38,
  author =	{Desharnais, Martin and Blanchette, Jasmin},
  title =	{{Sledgehammering Without ATPs}},
  booktitle =	{16th International Conference on Interactive Theorem Proving (ITP 2025)},
  pages =	{38:1--38:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-396-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{352},
  editor =	{Forster, Yannick and Keller, Chantal},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2025.38},
  URN =		{urn:nbn:de:0030-drops-246366},
  doi =		{10.4230/LIPIcs.ITP.2025.38},
  annote =	{Keywords: Interactive theorem proving, proof assistants, proof automation}
}
Document
DOI: 10.4230/LIPIcs.CP.2025.38
Reducing Quantum Circuit Synthesis to #SAT

Authors: Dekel Zak, Jingyi Mei, Jean-Marie Lagniez, and Alfons Laarman

Published in: LIPIcs, Volume 340, 31st International Conference on Principles and Practice of Constraint Programming (CP 2025)

Abstract
Quantum circuit synthesis is the task of decomposing a given quantum operator into a sequence of elementary quantum gates. Since the finite target gate set cannot exactly implement any given operator, approximation is often necessary. Model counting, or #SAT, has recently been demonstrated as a promising new approach for tackling core problems in quantum circuit analysis. In this work, we show for the first time that the universal quantum circuit synthesis problem can be reduced to maximum model counting. We formulate a #SAT encoding for exact and approximate depth-optimal quantum circuit synthesis into the Clifford+T gate set. We evaluate our method with an open-source implementation that uses the maximum model counter d4Max as a backend. For this purpose, we extended d4Max with support for complex and negative weights to represent amplitudes. Experimental results show that existing classical tools have potential for the quantum circuit synthesis problem.
Cite as

Dekel Zak, Jingyi Mei, Jean-Marie Lagniez, and Alfons Laarman. Reducing Quantum Circuit Synthesis to #SAT. In 31st International Conference on Principles and Practice of Constraint Programming (CP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 340, pp. 38:1-38:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{zak_et_al:LIPIcs.CP.2025.38,
  author =	{Zak, Dekel and Mei, Jingyi and Lagniez, Jean-Marie and Laarman, Alfons},
  title =	{{Reducing Quantum Circuit Synthesis to #SAT}},
  booktitle =	{31st International Conference on Principles and Practice of Constraint Programming (CP 2025)},
  pages =	{38:1--38:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-380-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{340},
  editor =	{de la Banda, Maria Garcia},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CP.2025.38},
  URN =		{urn:nbn:de:0030-drops-238997},
  doi =		{10.4230/LIPIcs.CP.2025.38},
  annote =	{Keywords: Maximum weighted model counting, quantum circuit synthesis}
}
Document
DOI: 10.4230/LIPIcs.FSCD.2025.7
Combining Generalization Algorithms in Regular Collapse-Free Theories

Authors: Mauricio Ayala-Rincón, David M. Cerna, Temur Kutsia, and Christophe Ringeissen

Published in: LIPIcs, Volume 337, 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)

Abstract
We look at the generalization problem modulo some equational theories. This problem is dual to the unification problem: given two input terms, we want to find a common term whose respective two instances are equivalent to the original terms modulo the theory. There exist algorithms for finding generalizations over various equational theories. We focus on modular construction of equational generalization algorithms for the union of signature-disjoint theories. Specifically, we consider the class of regular and collapse-free theories, showing how to combine existing generalization algorithms to produce specific solutions in these cases. Additionally, we identify a class of theories that admit a generalization algorithm based on the application of axioms to resolve the problem. To define this class, we rely on the notion of syntactic theories, a concept originally introduced to develop unification procedures similar to the one known for syntactic unification. We demonstrate that syntactic theories are also helpful in developing generalization procedures similar to those used for syntactic generalization.
Cite as

Mauricio Ayala-Rincón, David M. Cerna, Temur Kutsia, and Christophe Ringeissen. Combining Generalization Algorithms in Regular Collapse-Free Theories. In 10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 337, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ayalarincon_et_al:LIPIcs.FSCD.2025.7,
  author =	{Ayala-Rinc\'{o}n, Mauricio and Cerna, David M. and Kutsia, Temur and Ringeissen, Christophe},
  title =	{{Combining Generalization Algorithms in Regular Collapse-Free Theories}},
  booktitle =	{10th International Conference on Formal Structures for Computation and Deduction (FSCD 2025)},
  pages =	{7:1--7:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-374-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{337},
  editor =	{Fern\'{a}ndez, Maribel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2025.7},
  URN =		{urn:nbn:de:0030-drops-236228},
  doi =		{10.4230/LIPIcs.FSCD.2025.7},
  annote =	{Keywords: Generalization, Anti-unification, Equational theories, Combination}
}
Document
DOI: 10.4230/LIPIcs.STACS.2025.19
The Complexity of Learning LTL, CTL and ATL Formulas

Authors: Benjamin Bordais, Daniel Neider, and Rajarshi Roy

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
We consider the problem of learning temporal logic formulas from examples of system behavior. Learning temporal properties has crystallized as an effective means to explain complex temporal behaviors. Several efficient algorithms have been designed for learning temporal formulas. However, the theoretical understanding of the complexity of the learning decision problems remains largely unexplored. To address this, we study the complexity of the passive learning problems of three prominent temporal logics, Linear Temporal Logic (LTL), Computation Tree Logic (CTL) and Alternating-time Temporal Logic (ATL) and several of their fragments. We show that learning formulas with unbounded occurrences of binary operators is NP-complete for all of these logics. On the other hand, when investigating the complexity of learning formulas with bounded occurrences of binary operators, we exhibit discrepancies between the complexity of learning LTL, CTL and ATL formulas (with a varying number of agents).
Cite as

Benjamin Bordais, Daniel Neider, and Rajarshi Roy. The Complexity of Learning LTL, CTL and ATL Formulas. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bordais_et_al:LIPIcs.STACS.2025.19,
  author =	{Bordais, Benjamin and Neider, Daniel and Roy, Rajarshi},
  title =	{{The Complexity of Learning LTL, CTL and ATL Formulas}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{19:1--19:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.19},
  URN =		{urn:nbn:de:0030-drops-228441},
  doi =		{10.4230/LIPIcs.STACS.2025.19},
  annote =	{Keywords: Temporal logic, passive learning, complexity}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2025.3
Algebras for Automata: Reasoning with Regularity (Invited Talk)

Authors: Anupam Das

Published in: LIPIcs, Volume 327, 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)

Abstract
In the second half of the 20th century various theories of regular expressions were proposed, eventually leading to the notion of a Kleene Algebra (KA). Kozen and Krob independently proved the completeness of KA for the model of regular languages, a now celebrated result that has been refined and generalised over the years. In recent years proof theoretic approaches to regular languages have been studied, providing alternative routes to metalogical results like completeness and decidability. In this talk I will present a new approach from a different starting point: finite state automata. A notation for non-deterministic finite automata is readily obtained via expressions with least fixed points, leading to a theory of right-linear algebras (RLA). RLA is strictly more general than KA, e.g. admitting ω-regular languages as a model too, and enjoys a simpler proof theory than KA. This allows us to recover (more general) metalogical results in a robust way, combining techniques from automata, games, and cyclic proofs. Finally I will discuss extensions of RLA by greatest fixed points, comprising a notation for parity automata, to reason about ω-regular languages too. This talk is based on joint works with Abhishek De [Anupam Das and Abhishek De, 2024a and 2024b].
Cite as

Anupam Das. Algebras for Automata: Reasoning with Regularity (Invited Talk). In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{das:LIPIcs.STACS.2025.3,
  author =	{Das, Anupam},
  title =	{{Algebras for Automata: Reasoning with Regularity}},
  booktitle =	{42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-365-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{327},
  editor =	{Beyersdorff, Olaf and Pilipczuk, Micha{\l} and Pimentel, Elaine and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2025.3},
  URN =		{urn:nbn:de:0030-drops-228281},
  doi =		{10.4230/LIPIcs.STACS.2025.3},
  annote =	{Keywords: Regular languages, Linear grammars, Proof theory, Cyclic proofs, Automata theory, Fixed points, Games}
}
Document
DOI: 10.4230/LIPIcs.ITP.2021.7
Value-Oriented Legal Argumentation in Isabelle/HOL

Authors: Christoph Benzmüller and David Fuenmayor

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
Literature in AI & Law contemplates argumentation in legal cases as an instance of theory construction. The task of a lawyer in a legal case is to construct a theory containing: (a) relevant generic facts about the world, (b) relevant legal rules such as precedents and statutes, and (c) contingent facts describing or interpreting the situation at hand. Lawyers then elaborate convincing arguments starting from these facts and rules, deriving into a positive decision in favour of their client, often employing sophisticated argumentation techniques involving such notions as burden of proof, stare decisis, legal balancing, etc. In this paper we exemplarily show how to harness Isabelle/HOL to model lawyer’s argumentation using value-oriented legal balancing, while drawing upon shallow embeddings of combinations of expressive modal logics in HOL. We highlight the essential role of model finders (Nitpick) and "hammers" (Sledgehammer) in assisting the task of legal theory construction and share some thoughts on the practicability of extending the catalogue of ITP applications towards legal informatics.
Cite as

Christoph Benzmüller and David Fuenmayor. Value-Oriented Legal Argumentation in Isabelle/HOL. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{benzmuller_et_al:LIPIcs.ITP.2021.7,
  author =	{Benzm\"{u}ller, Christoph and Fuenmayor, David},
  title =	{{Value-Oriented Legal Argumentation in Isabelle/HOL}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{7:1--7:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.7},
  URN =		{urn:nbn:de:0030-drops-139028},
  doi =		{10.4230/LIPIcs.ITP.2021.7},
  annote =	{Keywords: Higher order logic, preference logic, shallow embedding, legal reasoning}
}
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