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DOI: 10.4230/LIPIcs.FSCD.2024.16

On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems

Authors: Franz Baader and Jürgen Giesl

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system R, a term s, and a natural number n, decide whether there is a term t of size ≤ n reachable from s using the rules of R. We investigate the complexity of this problem depending on how termination of R can be established. We show that the problem is NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases.

Cite as

Franz Baader and Jürgen Giesl. On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{baader_et_al:LIPIcs.FSCD.2024.16,
  author =	{Baader, Franz and Giesl, J\"{u}rgen},
  title =	{{On the Complexity of the Small Term Reachability Problem for Terminating Term Rewriting Systems}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{16:1--16:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.16},
  URN =		{urn:nbn:de:0030-drops-203454},
  doi =		{10.4230/LIPIcs.FSCD.2024.16},
  annote =	{Keywords: Rewriting, Termination, Confluence, Creating small terms, Derivational complexity, Description Logics, Proof rewriting}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.21

Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti

Authors: Thiago Felicissimo and Théo Winterhalter

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Proof assistants such as Coq implement a type theory featuring three important features: impredicativity, cumulativity and product covariance. This combination has proven difficult to be expressed in the logical framework Dedukti, and previous attempts have failed in providing an encoding that is proven confluent, sound and conservative. In this work we solve this longstanding open problem by providing an encoding of these three features that we prove to be confluent, sound and to satisfy a restricted (but, we argue, strong enough) form of conservativity. Our proof of confluence is a contribution by itself, and combines various criteria and proof techniques from rewriting theory. Our proof of soundness also contributes a new strategy in which the result is shown in terms of an inverse translation function, fixing a common flaw made in some previous encoding attempts.

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Thiago Felicissimo and Théo Winterhalter. Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{felicissimo_et_al:LIPIcs.FSCD.2024.21,
  author =	{Felicissimo, Thiago and Winterhalter, Th\'{e}o},
  title =	{{Impredicativity, Cumulativity and Product Covariance in the Logical Framework Dedukti}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{21:1--21:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.21},
  URN =		{urn:nbn:de:0030-drops-203503},
  doi =		{10.4230/LIPIcs.FSCD.2024.21},
  annote =	{Keywords: Dedukti, Rewriting, Confluence, Dependent types, Cumulativity, Universes}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.27

A Verified Algorithm for Deciding Pattern Completeness

Authors: René Thiemann and Akihisa Yamada

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Pattern completeness is the property that the left-hand sides of a functional program cover all cases w.r.t. pattern matching. In the context of term rewriting a related notion is quasi-reducibility, a prerequisite if one wants to perform ground confluence proofs by rewriting induction. In order to certify such confluence proofs, we develop a novel algorithm that decides pattern completeness and that can be used to ensure quasi-reducibility. One of the advantages of the proposed algorithm is its simple structure: it is similar to that of a regular matching algorithm and, unlike an existing decision procedure for quasi-reducibility, it avoids enumerating all terms up to a given depth. Despite the simple structure, proving the correctness of the algorithm is not immediate. Therefore we formalize the algorithm and verify its correctness using the proof assistant Isabelle/HOL. To this end, we not only verify some auxiliary algorithms, but also design an Isabelle library on sorted term rewriting. Moreover, we export the verified code in Haskell and experimentally evaluate its performance. We observe that our algorithm significantly outperforms existing algorithms, even including the pattern completeness check of the GHC Haskell compiler.

Cite as

René Thiemann and Akihisa Yamada. A Verified Algorithm for Deciding Pattern Completeness. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{thiemann_et_al:LIPIcs.FSCD.2024.27,
  author =	{Thiemann, Ren\'{e} and Yamada, Akihisa},
  title =	{{A Verified Algorithm for Deciding Pattern Completeness}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{27:1--27:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.27},
  URN =		{urn:nbn:de:0030-drops-203566},
  doi =		{10.4230/LIPIcs.FSCD.2024.27},
  annote =	{Keywords: Isabelle/HOL, pattern matching, term rewriting}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.31

Equational Theories and Validity for Logically Constrained Term Rewriting

Authors: Takahito Aoto, Naoki Nishida, and Jonas Schöpf

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

Logically constrained term rewriting is a relatively new formalism where rules are equipped with constraints over some arbitrary theory. Although there are many recent advances with respect to rewriting induction, completion, complexity analysis and confluence analysis for logically constrained term rewriting, these works solely focus on the syntactic side of the formalism lacking detailed investigations on semantics. In this paper, we investigate a semantic side of logically constrained term rewriting. To this end, we first define constrained equations, constrained equational theories and validity of the former based on the latter. After presenting the relationship of validity and conversion of rewriting, we then construct a sound inference system to prove validity of constrained equations in constrained equational theories. Finally, we give an algebraic semantics, which enables one to establish invalidity of constrained equations in constrained equational theories. This algebraic semantics derives a new notion of consistency for constrained equational theories.

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Takahito Aoto, Naoki Nishida, and Jonas Schöpf. Equational Theories and Validity for Logically Constrained Term Rewriting. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aoto_et_al:LIPIcs.FSCD.2024.31,
  author =	{Aoto, Takahito and Nishida, Naoki and Sch\"{o}pf, Jonas},
  title =	{{Equational Theories and Validity for Logically Constrained Term Rewriting}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{31:1--31:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.31},
  URN =		{urn:nbn:de:0030-drops-203607},
  doi =		{10.4230/LIPIcs.FSCD.2024.31},
  annote =	{Keywords: constrained equation, constrained equational theory, logically constrained term rewriting, algebraic semantics, consistency}
}

Document

DOI: 10.4230/LIPIcs.FSCD.2024.32

Termination of Generalized Term Rewriting Systems

Authors: Salvador Lucas

Published in: LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

Abstract

We investigate termination of Generalized Term Rewriting Systems (GTRS), which extend Conditional Term Rewriting Systems by considering replacement restrictions on selected arguments of function symbols, as in Context-Sensitive Rewriting, and conditional rewriting rules whose conditional part may include not only a mix of the usual (reachability, joinability,...) conditions, but also atoms defined by a set of definite Horn clauses. GTRS can be used to prove confluence and termination of Generalized Rewrite Theories and Maude programs. We have characterized confluence of terminating GTRS as the joinability of a finite set of conditional pairs. Since termination of GTRS is underexplored to date, this paper introduces a Dependency Pair Framework which is well-suited to automatically (dis)prove termination of GTRS.

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Salvador Lucas. Termination of Generalized Term Rewriting Systems. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{lucas:LIPIcs.FSCD.2024.32,
  author =	{Lucas, Salvador},
  title =	{{Termination of Generalized Term Rewriting Systems}},
  booktitle =	{9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)},
  pages =	{32:1--32:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-323-2},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{299},
  editor =	{Rehof, Jakob},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.32},
  URN =		{urn:nbn:de:0030-drops-203616},
  doi =		{10.4230/LIPIcs.FSCD.2024.32},
  annote =	{Keywords: Program Analysis, Reduction-Based Systems, Termination}
}

Document

Track B: Automata, Logic, Semantics, and Theory of Programming

DOI: 10.4230/LIPIcs.ICALP.2024.139

T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations

Authors: Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract

We introduce a powerful termination algorithm for structurally recursive functions that improves on the core ideas behind lexicographic termination algorithms for functional programs. The algorithm generates linear-lexicographic combinations of primitive measure functions measuring the recursive structure of terms. We introduce a measure language that enables the simplification and comparison of measures and we prove meta-theoretic properties of our measure language. Moreover, we demonstrate our algorithm, on an untyped first-order functional language and prove its soundness and that it runs in polynomial time. We also provide a Haskell implementation. As part of this work, we also show how to solve the maximisation of negative vector-components as a linear program.

Cite as

Raphael Douglas Giles, Vincent Jackson, and Christine Rizkallah. T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 139:1-139:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{giles_et_al:LIPIcs.ICALP.2024.139,
  author =	{Giles, Raphael Douglas and Jackson, Vincent and Rizkallah, Christine},
  title =	{{T-Rex: Termination of Recursive Functions Using Lexicographic Linear Combinations}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{139:1--139:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.139},
  URN =		{urn:nbn:de:0030-drops-202827},
  doi =		{10.4230/LIPIcs.ICALP.2024.139},
  annote =	{Keywords: Termination, Recursive functions}
}

Document

RANDOM

DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.35

An Embarrassingly Parallel Optimal-Space Cardinality Estimation Algorithm

Authors: Emin Karayel

Published in: LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

Abstract

In 2020 Błasiok (ACM Trans. Algorithms 16(2) 3:1-3:28) constructed an optimal space streaming algorithm for the cardinality estimation problem with the space complexity of O(ε^{-2} ln(δ^{-1}) + ln n) where ε, δ and n denote the relative accuracy, failure probability and universe size, respectively. However, his solution requires the stream to be processed sequentially. On the other hand, there are algorithms that admit a merge operation; they can be used in a distributed setting, allowing parallel processing of sections of the stream, and are highly relevant for large-scale distributed applications. The best-known such algorithm, unfortunately, has a space complexity exceeding Ω(ln(δ^{-1}) (ε^{-2} ln ln n + ln n)). This work presents a new algorithm that improves on the solution by Błasiok, preserving its space complexity, but with the benefit that it admits such a merge operation, thus providing an optimal solution for the problem for both sequential and parallel applications. Orthogonally, the new algorithm also improves algorithmically on Błasiok’s solution (even in the sequential setting) by reducing its implementation complexity and requiring fewer distinct pseudo-random objects.

Cite as

Emin Karayel. An Embarrassingly Parallel Optimal-Space Cardinality Estimation Algorithm. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 35:1-35:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{karayel:LIPIcs.APPROX/RANDOM.2023.35,
  author =	{Karayel, Emin},
  title =	{{An Embarrassingly Parallel Optimal-Space Cardinality Estimation Algorithm}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{35:1--35:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.35},
  URN =		{urn:nbn:de:0030-drops-188607},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.35},
  annote =	{Keywords: Distinct Elements, Distributed Algorithms, Randomized Algorithms, Expander Graphs, Derandomization, Sketching}
}

Document

DOI: 10.4230/LIPIcs.ITP.2023.4

Fast, Verified Computation for Candle

Authors: Oskar Abrahamsson and Magnus O. Myreen

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract

This paper describes how we have added an efficient function for computation to the kernel of the Candle interactive theorem prover. Candle is a CakeML port of HOL Light which we have, in prior work, proved sound w.r.t. the inference rules of the higher-order logic. This paper extends the original implementation and soundness proof with a new kernel function for fast computation. Experiments show that the new computation function is able to speed up certain evaluation proofs by several orders of magnitude.

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Oskar Abrahamsson and Magnus O. Myreen. Fast, Verified Computation for Candle. In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{abrahamsson_et_al:LIPIcs.ITP.2023.4,
  author =	{Abrahamsson, Oskar and Myreen, Magnus O.},
  title =	{{Fast, Verified Computation for Candle}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{4:1--4:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.4},
  URN =		{urn:nbn:de:0030-drops-183797},
  doi =		{10.4230/LIPIcs.ITP.2023.4},
  annote =	{Keywords: Prover soundness, Higher-order logic, Interactive theorem proving}
}

Document

DOI: 10.4230/LIPIcs.ITP.2023.29

Real-Time Double-Ended Queue Verified (Proof Pearl)

Authors: Balazs Toth and Tobias Nipkow

Published in: LIPIcs, Volume 268, 14th International Conference on Interactive Theorem Proving (ITP 2023)

Abstract

We present the first verification of the real-time doubled-ended queue by Chuang and Goldberg where all operations take constant time. The main contributions are the full system invariant, the precise definition of all abstraction functions, the structure of the proof and the main lemmas.

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Balazs Toth and Tobias Nipkow. Real-Time Double-Ended Queue Verified (Proof Pearl). In 14th International Conference on Interactive Theorem Proving (ITP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 268, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{toth_et_al:LIPIcs.ITP.2023.29,
  author =	{Toth, Balazs and Nipkow, Tobias},
  title =	{{Real-Time Double-Ended Queue Verified (Proof Pearl)}},
  booktitle =	{14th International Conference on Interactive Theorem Proving (ITP 2023)},
  pages =	{29:1--29:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-284-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{268},
  editor =	{Naumowicz, Adam and Thiemann, Ren\'{e}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2023.29},
  URN =		{urn:nbn:de:0030-drops-184044},
  doi =		{10.4230/LIPIcs.ITP.2023.29},
  annote =	{Keywords: Double-ended queue, data structures, verification, Isabelle}
}

Document

DOI: 10.4230/LIPIcs.ITP.2022.21

Formalization of Randomized Approximation Algorithms for Frequency Moments

Authors: Emin Karayel

Published in: LIPIcs, Volume 237, 13th International Conference on Interactive Theorem Proving (ITP 2022)

Abstract

In 1999 Alon et al. introduced the still active research topic of approximating the frequency moments of a data stream using randomized algorithms with minimal space usage. This includes the problem of estimating the cardinality of the stream elements - the zeroth frequency moment. Higher-order frequency moments provide information about the skew of the data stream which is, for example, critical information for parallel processing. (The k-th frequency moment of a data stream is the sum of the k-th powers of the occurrence counts of each element in the stream.) They introduce both lower bounds and upper bounds on the space complexity of the problems, which were later improved by newer publications. The algorithms have guaranteed success probabilities and accuracies without making any assumptions on the input distribution. They are an interesting use case for formal verification because their correctness proofs require a large body of deep results from algebra, analysis and probability theory. This work reports on the formal verification of three algorithms for the approximation of F₀, F₂ and F_k for k ≥ 3. The results include the identification of significantly simpler algorithms with the same runtime and space complexities as the previously known ones as well as the development of several reusable components, such as a formalization of universal hash families, amplification methods for randomized algorithms, a model for one-pass data stream algorithms or a generic flexible encoding library for the verification of space complexities.

Cite as

Emin Karayel. Formalization of Randomized Approximation Algorithms for Frequency Moments. In 13th International Conference on Interactive Theorem Proving (ITP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 237, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{karayel:LIPIcs.ITP.2022.21,
  author =	{Karayel, Emin},
  title =	{{Formalization of Randomized Approximation Algorithms for Frequency Moments}},
  booktitle =	{13th International Conference on Interactive Theorem Proving (ITP 2022)},
  pages =	{21:1--21:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-252-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{237},
  editor =	{Andronick, June and de Moura, Leonardo},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2022.21},
  URN =		{urn:nbn:de:0030-drops-167308},
  doi =		{10.4230/LIPIcs.ITP.2022.21},
  annote =	{Keywords: Formal Verification, Isabelle/HOL, Randomized Algorithms, Frequency Moments}
}

Document

DOI: 10.4230/LIPIcs.ITP.2019.11

A Verified and Compositional Translation of LTL to Deterministic Rabin Automata

Authors: Julian Brunner, Benedikt Seidl, and Salomon Sickert

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract

We present a formalisation of the unified translation approach from linear temporal logic (LTL) to omega-automata from [Javier Esparza et al., 2018]. This approach decomposes LTL formulas into "simple" languages and allows a clear separation of concerns: first, we formalise the purely logical result yielding this decomposition; second, we develop a generic, executable, and expressive automata library providing necessary operations on automata to re-combine the "simple" languages; third, we instantiate this generic theory to obtain a construction for deterministic Rabin automata (DRA). We extract from this particular instantiation an executable tool translating LTL to DRAs. To the best of our knowledge this is the first verified translation of LTL to DRAs that is proven to be double-exponential in the worst case which asymptotically matches the known lower bound.

Cite as

Julian Brunner, Benedikt Seidl, and Salomon Sickert. A Verified and Compositional Translation of LTL to Deterministic Rabin Automata. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 11:1-11:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{brunner_et_al:LIPIcs.ITP.2019.11,
  author =	{Brunner, Julian and Seidl, Benedikt and Sickert, Salomon},
  title =	{{A Verified and Compositional Translation of LTL to Deterministic Rabin Automata}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{11:1--11:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.11},
  URN =		{urn:nbn:de:0030-drops-110664},
  doi =		{10.4230/LIPIcs.ITP.2019.11},
  annote =	{Keywords: Automata Theory, Automata over Infinite Words, Deterministic Automata, Linear Temporal Logic, Model Checking, Verified Algorithms}
}

Document

DOI: 10.4230/LIPIcs.ITP.2019.23

Proof Pearl: Purely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra

Authors: Peter Lammich and Tobias Nipkow

Published in: LIPIcs, Volume 141, 10th International Conference on Interactive Theorem Proving (ITP 2019)

Abstract

The starting point of this paper is a new, purely functional, simple and efficient data structure combining a search tree and a priority queue, which we call a priority search tree. The salient feature of priority search trees is that they offer a decrease-key operation, something that is missing from other simple, purely functional priority queue implementations. As two applications of this data structure we verify purely functional, simple and efficient implementations of Prim’s and Dijkstra’s algorithms. This constitutes the first verification of an executable and even efficient version of Prim’s algorithm.

Cite as

Peter Lammich and Tobias Nipkow. Proof Pearl: Purely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra. In 10th International Conference on Interactive Theorem Proving (ITP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 141, pp. 23:1-23:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{lammich_et_al:LIPIcs.ITP.2019.23,
  author =	{Lammich, Peter and Nipkow, Tobias},
  title =	{{Proof Pearl: Purely Functional, Simple and Efficient Priority Search Trees and Applications to Prim and Dijkstra}},
  booktitle =	{10th International Conference on Interactive Theorem Proving (ITP 2019)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-122-1},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{141},
  editor =	{Harrison, John and O'Leary, John and Tolmach, Andrew},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2019.23},
  URN =		{urn:nbn:de:0030-drops-110788},
  doi =		{10.4230/LIPIcs.ITP.2019.23},
  annote =	{Keywords: Priority queue, Dijkstra’s algorithm, Prim’s algorithm, verification, Isabelle}
}

Document

Invited Talk

DOI: 10.4230/LIPIcs.MFCS.2019.1

Trustworthy Graph Algorithms (Invited Talk)

Authors: Mohammad Abdulaziz, Kurt Mehlhorn, and Tobias Nipkow

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract

The goal of the LEDA project was to build an easy-to-use and extendable library of correct and efficient data structures, graph algorithms and geometric algorithms. We report on the use of formal program verification to achieve an even higher level of trustworthiness. Specifically, we report on an ongoing and largely finished verification of the blossom-shrinking algorithm for maximum cardinality matching.

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Mohammad Abdulaziz, Kurt Mehlhorn, and Tobias Nipkow. Trustworthy Graph Algorithms (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 1:1-1:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abdulaziz_et_al:LIPIcs.MFCS.2019.1,
  author =	{Abdulaziz, Mohammad and Mehlhorn, Kurt and Nipkow, Tobias},
  title =	{{Trustworthy Graph Algorithms}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{1:1--1:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.1},
  URN =		{urn:nbn:de:0030-drops-109456},
  doi =		{10.4230/LIPIcs.MFCS.2019.1},
  annote =	{Keywords: graph algorithms, formal correct proofs, Isabelle, LEDA, certifying algorithms}
}

Document

DOI: 10.4230/LIPIcs.TYPES.2017.5

Formalized Proof Systems for Propositional Logic

Authors: Julius Michaelis and Tobias Nipkow

Published in: LIPIcs, Volume 104, 23rd International Conference on Types for Proofs and Programs (TYPES 2017)

Abstract

We have formalized a range of proof systems for classical propositional logic (sequent calculus, natural deduction, Hilbert systems, resolution) in Isabelle/HOL and have proved the most important meta-theoretic results about semantics and proofs: compactness, soundness, completeness, translations between proof systems, cut-elimination, interpolation and model existence.

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Julius Michaelis and Tobias Nipkow. Formalized Proof Systems for Propositional Logic. In 23rd International Conference on Types for Proofs and Programs (TYPES 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 104, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{michaelis_et_al:LIPIcs.TYPES.2017.5,
  author =	{Michaelis, Julius and Nipkow, Tobias},
  title =	{{Formalized Proof Systems for Propositional Logic}},
  booktitle =	{23rd International Conference on Types for Proofs and Programs (TYPES 2017)},
  pages =	{5:1--5:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-071-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{104},
  editor =	{Abel, Andreas and Nordvall Forsberg, Fredrik and Kaposi, Ambrus},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TYPES.2017.5},
  URN =		{urn:nbn:de:0030-drops-100537},
  doi =		{10.4230/LIPIcs.TYPES.2017.5},
  annote =	{Keywords: formalization of logic, proof systems, sequent calculus, natural deduction, resolution}
}

Document

DOI: 10.4230/LIPIcs.FSTTCS.2016.49

Verified Analysis of List Update Algorithms

Authors: Maximilian P. L. Haslbeck and Tobias Nipkow

Published in: LIPIcs, Volume 65, 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)

Abstract

This paper presents a machine-verified analysis of a number of classical algorithms for the list update problem: 2-competitiveness of move-to-front, the lower bound of 2 for the competitiveness of deterministic list update algorithms and 1.6-competitiveness of the randomized COMB algorithm, the best randomized list update algorithm known to date. The analysis is verified with help of the theorem prover Isabelle; some low-level proofs could be automated.

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Maximilian P. L. Haslbeck and Tobias Nipkow. Verified Analysis of List Update Algorithms. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 49:1-49:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{haslbeck_et_al:LIPIcs.FSTTCS.2016.49,
  author =	{Haslbeck, Maximilian P. L. and Nipkow, Tobias},
  title =	{{Verified Analysis of List Update Algorithms}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{49:1--49:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Lal, Akash and Akshay, S. and Saurabh, Saket and Sen, Sandeep},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2016.49},
  URN =		{urn:nbn:de:0030-drops-68849},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.49},
  annote =	{Keywords: Program Verification, Algorithm Analysis, Online Algorithms}
}

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