Rewriting-related Techniques and Applications

We investigate solving methods for equational and membership constraints, generalization techniques in various theories, and calculi for conditional rule-based transformations. The developed algorithms and procedures are useful for equational reasoning, program analysis, and declarative programming. Our research interests also include rewriting-based deduction, models of computation, and computer security.

Rewriting has been playing an important role in symbolic computation and automated deduction. We are interested in fundamental algorithmic problems for rewriting-based techniques as well as in applications of rewriting-related frameworks in programming and reasoning.

Unification and matching are fundamental computational techniques for automated reasoning, term rewriting, and declarative programming. Essentially, they can be seen as methods for solving equations between terms. The difficulty of an unification problem depends on many parameters, e.g., the background equational theory, the order of the language, the presence of sorts, the representation of objects, etc. We have been investigating this problem under various combinations of those parameters, developing solving procedures, proving their properties, providing prototype implementations, identifying well-behaved special cases, and studying practical applications. Some of those algorithms have been included in the mathematical assistant system Theorema and in the rule-based programming tool PρLog. See also the Web page of the SToUT project.

Anti-unification is yet another fundamental technique originating from inductive reasoning. Its goal is to find an object which generalizes the given objects: retaining similarities between them as much as possible, and abstracting over the differences by variables. Unification and anti-unification are considered as dual methods in the following sense: solutions of an unification problem give most general common instance of the given expressions, while anti-unification computes their least general common generalization. We are interested in designing new anti-unification algorithms for various first- and higher-order theories, proving their properties, investigating complexity, providing implementations, and applying them in software code clone detection, program synthesis, and natural language processing. We have been developing an open-source online library of unification and anti-unification algorithms. See also the Web page of the GALA project.

We have contributed in the development of a calculus for conditional sequence transformations by strategies, called ρLog, which forms the theoretical basis of the rule-based programming tool PρLog. It deals with term sequences (also called hedges), transforming them by conditional rules. Transformations are nondeterministic and may yield several results. Strategies provide a control on rule applications in a declarative way. The transformation rules are combined with logic programming. See more details on the PρLog Web page.

Our other interests include equational reasoning, pattern calculi, and runtime verification. See the publication list below for relevant references.

Members

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Software

Unification and Anti-Unification Algorithm Library

This library contains unification, matching, and anti-unification algorithms in various theories developed at RISC. Unification with sequence variables. Context sequence matching. Rigid anti-unification for unranked terms and hedges and its experimental extension with commutative symbols. Unranked second-order anti-unification and its ...

Authors: Teimuraz Kutsia, Alexander Baumgartner, David Cerna
MoreSoftware Website

Publications

2021

[Dundua]

Variadic equational matching in associative and commutative theories

Besik Dundua, Temur Kutsia, Mircea Marin

Journal of Symbolic Computation 106, pp. 78-109. 2021. Elsevier, ISSN 0747-7171. [url] [pdf]
[bib]
@article{RISC6260,
author = {Besik Dundua and Temur Kutsia and Mircea Marin},
title = {{Variadic equational matching in associative and commutative theories}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {106},
pages = {78--109},
publisher = {Elsevier},
isbn_issn = {ISSN 0747-7171},
year = {2021},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1016/j.jsc.2021.01.001}
}
[Pau]

Proximity-Based Unification and Matching for Full Fuzzy Signatures

Temur Kutsia, Cleo Pau

Technical report no. 21-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2021. [pdf]
[bib]
@techreport{RISC6296,
author = {Temur Kutsia and Cleo Pau},
title = {{Proximity-Based Unification and Matching for Full Fuzzy Signatures}},
language = {english},
number = {21-08},
year = {2021},
length = {15},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Pau]

Generalization Algorithms with Proximity Relations in Full Fuzzy Signatures

Temur Kutsia, Cleo Pau

Technical report no. 21-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2021. [pdf]
[bib]
@techreport{RISC6297,
author = {Temur Kutsia and Cleo Pau},
title = {{Generalization Algorithms with Proximity Relations in Full Fuzzy Signatures}},
language = {english},
number = {21-09},
year = {2021},
length = {15},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

2020

[Cerna]

Idempotent Anti-unification

David Cerna, Temur Kutsia

ACM Transactions on Computational Logic (TOCL) 21(2), pp. 10:1-10:32. 2020. ACM Press, ISSN 1529-3785. [url] [pdf]
[bib]
@article{RISC6023,
author = {David Cerna and Temur Kutsia},
title = {{Idempotent Anti-unification}},
language = {english},
journal = {ACM Transactions on Computational Logic (TOCL)},
volume = {21},
number = {2},
pages = {10:1--10:32},
publisher = {ACM Press},
isbn_issn = {ISSN 1529-3785},
year = {2020},
refereed = {yes},
length = {32},
url = {https://doi.org/10.1145/3359060}
}
[Cerna]

Unital Anti-Unification: Type and Algorithms

David M. Cerna, Temur Kutsia

In: Proceedings of the 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020), Zena M. Ariola (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 167, pp. 26:1-26:20. 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, ISBN 978-3-95977-155-9, ISSN 1868-8969. [url]
[bib]
@inproceedings{RISC6132,
author = {David M. Cerna and Temur Kutsia},
title = {{Unital Anti-Unification: Type and Algorithms}},
booktitle = {{Proceedings of the 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}},
language = {english},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {167},
pages = {26:1--26:20},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},
isbn_issn = {ISBN 978-3-95977-155-9, ISSN 1868-8969},
year = {2020},
editor = {Zena M. Ariola},
refereed = {yes},
length = {20},
url = {https://drops.dagstuhl.de/opus/volltexte/2020/12352/}
}
[Cerna]

Higher-order pattern generalization modulo equational theories

David M. Cerna, Temur Kutsia

Mathematical Structures in Computer Science 30(6), pp. 627-663. 2020. ISSN 0960-1295. [url]
[bib]
@article{RISC6226,
author = {David M. Cerna and Temur Kutsia},
title = {{Higher-order pattern generalization modulo equational theories}},
language = {english},
journal = {Mathematical Structures in Computer Science},
volume = {30},
number = {6},
pages = {627--663},
isbn_issn = {ISSN 0960-1295},
year = {2020},
refereed = {yes},
length = {37},
url = {https://www.cambridge.org/core/services/aop-cambridge-core/content/view/88E26F155F0FD02B3EDD648971D9AD1B/S0960129520000110a.pdf/higher-order-pattern-generalization-modulo-equational-theories.pdf}
}
[Cerna]

Anti-unification and the Theory of Semirings

David M. Cerna

Theor. Comput. Sci. 848, pp. 133-139. 2020. RISC / CAS ICS, 0304-3975. [url]
[bib]
@article{RISC6234,
author = {David M. Cerna},
title = {{Anti-unification and the Theory of Semirings}},
language = {english},
abstract = {It was recently shown that anti-unification over an equational theory consisting of only unitequations (more than one) is nullary. Such pure theories are artificial and are of little effect onpractical aspects of anti-unification. In this work, we extend these nullarity results to the theory ofsemirings, a heavily studied theory with many practical applications. Furthermore, our argumentholds over semirings with commutative multiplication and/or idempotent addition. We also cover afew open questions discussed in previous work.},
journal = {Theor. Comput. Sci.},
volume = {848},
pages = {133--139},
isbn_issn = {0304-3975},
year = {2020},
refereed = {yes},
institution = {RISC / CAS ICS},
length = {7},
url = {https://doi.org/10.1016/j.tcs.2020.10.020}
}
[Cerna]

Schematic Refutations of Formula Schemata

David Cerna and Alexander Leitsch and Anela Lolic

Journal of Automated Reasoning, pp. -. 2020. 1573-0670. [url]
[bib]
@article{RISC6235,
author = {David Cerna and Alexander Leitsch and Anela Lolic},
title = {{Schematic Refutations of Formula Schemata}},
language = {english},
abstract = {Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.},
journal = {Journal of Automated Reasoning},
pages = {--},
isbn_issn = {1573-0670},
year = {2020},
refereed = {yes},
length = {47},
url = {https://doi.org/10.1007/s10817-020-09583-8}
}
[Cerna]

A Mobile Application for Self-Guided Study of Formal Reasoning

David M. Cerna and Rafael P. D. Kiesel and Alexandra Dzhiganskaya

In: Proceedings 8th International Workshop on Theorem Proving Components for Educational Software, ThEdu@CADE 2019, Natal, Brazil, 25th August 2019, Pedro Quaresma and Walther Neuper and Jo{ {a}}o Marcos (ed.), EPTCS 313, pp. 35-53. 2020. 2075-2180. [url]
[bib]
@inproceedings{RISC6236,
author = {David M. Cerna and Rafael P. D. Kiesel and Alexandra Dzhiganskaya},
title = {{A Mobile Application for Self-Guided Study of Formal Reasoning}},
booktitle = {{Proceedings 8th International Workshop on Theorem Proving Components for Educational Software, ThEdu@CADE 2019, Natal, Brazil, 25th August 2019}},
language = {english},
abstract = {In this work, we introduce AXolotl, a self-study aid designed to guide students through the basics of formal reasoning and term manipulation. Unlike most of the existing study aids for formal reasoning, AXolotl is an Android-based application with a simple touch-based interface. Part of the design goal was to minimize the possibility of user errors which distract from the learning process. Such as typos or inconsistent application of the provided rules. The system includes a zoomable proof viewer which displays the progress made so far and allows for storage of the completed proofs as a JPEG or LaTeX file. The software is available on the google play store and comes with a small library of problems. Additional problems may be opened in AXolotl using a simple input language. Currently, AXolotl supports problems that can be solved using rules which transform a single expression into a set of expressions. This covers educational scenarios found in our first-semester introduction to logic course and helps bridge the gap between propositional and first-order reasoning. Future developments will include rewrite rules which take a set of expressions and return a set of expressions, as well as a quantified first-order extension.},
series = {EPTCS},
volume = {313},
pages = {35--53},
isbn_issn = {2075-2180},
year = {2020},
editor = {Pedro Quaresma and Walther Neuper and Jo{~{a}}o Marcos},
refereed = {yes},
length = {19},
url = {https://doi.org/10.4204/EPTCS.313.3}
}
[Cerna]

Unital Anti-Unification: Type and Algorithms

David M. Cerna , Temur Kutsia

In: 5th International Conference on Formal Structures for Computation and Deduction, {FSCD} 2020, June 29-July 6, 2020, Paris, France (Virtual Conference, Zena M. Ariola (ed.), Proceedings of FSCD, LIPICS 167/20-02, pp. 1-20. 2020. 1868-8969. [url]
[bib]
@inproceedings{RISC6237,
author = {David M. Cerna and Temur Kutsia},
title = {{Unital Anti-Unification: Type and Algorithms}},
booktitle = {{5th International Conference on Formal Structures for Computation and Deduction, {FSCD} 2020, June 29-July 6, 2020, Paris, France (Virtual Conference}},
language = {english},
abstract = {Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete and return tree grammars from which set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions. },
series = {LIPICS},
volume = {167},
number = {20-02},
pages = {1--20},
isbn_issn = {1868-8969},
year = {2020},
editor = {Zena M. Ariola},
refereed = {yes},
keywords = {Anti-unification, tree grammars, unital theories, collapse theories},
length = {19},
conferencename = {FSCD},
url = {https://doi.org/10.4230/LIPIcs.FSCD.2020.26}
}
[Dramnesc]

Implementation of Deletion Algorithms on Lists and Binary Trees in Theorema

Isabela Dramnesc, Tudor Jebelean

Technical report no. 20-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. April 2020. [pdf]
[bib]
@techreport{RISC6094,
author = {Isabela Dramnesc and Tudor Jebelean},
title = {{Implementation of Deletion Algorithms on Lists and Binary Trees in Theorema}},
language = {english},
number = {20-04},
year = {2020},
month = {April},
length = {25},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Dramnesc]

Synthesis of Delete on Lists and Binary Trees Using Multisets in Theorema

Isabela Dramnesc, Tudor Jebelean

Technical report no. 20-15 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. September 2020. [pdf]
[bib]
@techreport{RISC6206,
author = {Isabela Dramnesc and Tudor Jebelean},
title = {{Synthesis of Delete on Lists and Binary Trees Using Multisets in Theorema}},
language = {English},
abstract = {We demonstrate the deduction based synthesis of element deletion algorithms for[sorted] lists and [sorted] binary trees, by first developing the necessary theory which ismulti–type: basic ordered elements, multisets, lists, and trees. The generated algorithms are“pattern matching”, namely sets of conditional equalities, and we also demonstrate theirtransformation into functional algorithms and, for lists, into tail recursive algorithms. Thiswork constitutes a case study first in theory exploration and second in automated synthesisand transformation of algorithms synthesized on the basis of natural style proofs, whichallows to investigate the heuristics of theory construction on multiple types, as well as thenatural style inferences and strategies for constructing human readable synthesis proofs. Theexperiments are performed in the frame of the Theorema system.},
number = {20-15},
year = {2020},
month = {September},
keywords = {automated reasoning; algorithm synthesis; lists; binary trees; multisets; Theorema},
length = {22},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Dundua]

Constraint Solving over Multiple Similarity Relations

Besik Dundua, Temur Kutsia, Mircea Marin, Cleopatra Pau

In: Proceedings of the 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020), Zena M. Ariola (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 167, pp. 30:1-30:19. 2020. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, ISBN 978-3-95977-155-9, ISSN 1868-8969. [url]
[bib]
@inproceedings{RISC6133,
author = {Besik Dundua and Temur Kutsia and Mircea Marin and Cleopatra Pau},
title = {{Constraint Solving over Multiple Similarity Relations}},
booktitle = {{Proceedings of the 5th International Conference on Formal Structures for Computation and Deduction (FSCD 2020)}},
language = {english},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
volume = {167},
pages = {30:1--30:19},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum für Informatik},
isbn_issn = {ISBN 978-3-95977-155-9, ISSN 1868-8969},
year = {2020},
editor = {Zena M. Ariola},
refereed = {yes},
length = {19},
url = {https://doi.org/10.4230/LIPIcs.FSCD.2020.30}
}
[Dundua]

Extending the 𝜌Log Calculus with Proximity Relations

Besik Dundua, Temur Kutsia, Mircea Marin, Cleo Pau

In: AMINSE 2019: Applications of Mathematics and Informatics in Natural Sciences and Engineering, George Jaiani, David Natroshvili (ed.), Springer Proceedings in Mathematics & Statistics 334, pp. 83-100. 2020. Springer, ISBN 978-3-030-56356-1, 978-3-030-56355-4. [url] [pdf]
[bib]
@incollection{RISC6227,
author = {Besik Dundua and Temur Kutsia and Mircea Marin and Cleo Pau},
title = {{Extending the 𝜌Log Calculus with Proximity Relations}},
booktitle = {{AMINSE 2019: Applications of Mathematics and Informatics in Natural Sciences and Engineering}},
language = {english},
series = {Springer Proceedings in Mathematics & Statistics},
volume = {334},
pages = {83--100},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-56356-1, 978-3-030-56355-4},
year = {2020},
editor = {George Jaiani and David Natroshvili},
refereed = {yes},
length = {18},
url = {https://doi.org/10.1007/978-3-030-56356-1_6}
}
[Dundua]

Specification and Analysis of ABAC Policies in a Rule-Based Framework

Besik Dundua, Temur Kutsia, Mircea Marin, Mikheil Rukhaia

In: AMINSE 2019: Applications of Mathematics and Informatics in Natural Sciences and Engineering, George Jaiani, David Natroshvili (ed.), Springer Proceedings in Mathematics & Statistics 334, pp. 101-116. 2020. Springer, ISBN 978-3-030-56356-1, 978-3-030-56355-4. [url] [pdf]
[bib]
@incollection{RISC6228,
author = {Besik Dundua and Temur Kutsia and Mircea Marin and Mikheil Rukhaia},
title = {{Specification and Analysis of ABAC Policies in a Rule-Based Framework}},
booktitle = {{AMINSE 2019: Applications of Mathematics and Informatics in Natural Sciences and Engineering}},
language = {english},
series = {Springer Proceedings in Mathematics & Statistics},
volume = {334},
pages = {101--116},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-56356-1, 978-3-030-56355-4},
year = {2020},
editor = {George Jaiani and David Natroshvili},
refereed = {yes},
length = {16},
url = {https://doi.org/10.1007/978-3-030-56356-1_7}
}
[Dundua]

A Rule-Based System for Computation and Deduction in Mathematica

Mircea Marin, Besik Dundua, Temur Kutsia

In: WRLA 2020: Rewriting Logic and Its Applications, Santiago Escobar, Narciso Martí-Oliet (ed.), Lecture Notes in Computer Science 12328, pp. 57-74. 2020. Springer, ISBN 978-3-030-63595-4, 978-3-030-63594-7. [url] [pdf]
[bib]
@inproceedings{RISC6229,
author = {Mircea Marin and Besik Dundua and Temur Kutsia},
title = {{A Rule-Based System for Computation and Deduction in Mathematica}},
booktitle = {{WRLA 2020: Rewriting Logic and Its Applications}},
language = {english},
series = { Lecture Notes in Computer Science},
volume = {12328},
pages = {57--74},
publisher = {Springer},
isbn_issn = {ISBN 978-3-030-63595-4, 978-3-030-63594-7},
year = {2020},
editor = {Santiago Escobar and Narciso Martí-Oliet},
refereed = {yes},
length = {18},
url = {https://doi.org/10.1007/978-3-030-63595-4_4}
}
[Kutsia]

Unification modulo alpha-equivalence in a mathematical assistant system

Temur Kutsia

Technical report no. 20-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. [pdf]
[bib]
@techreport{RISC6074,
author = {Temur Kutsia},
title = {{Unification modulo alpha-equivalence in a mathematical assistant system}},
language = {english},
number = {20-01},
year = {2020},
length = {21},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Kutsia]

Constructing orthogonal designs in powers of two via symbolic computation and rewriting techniques

Ilias Kotsireas, Temur Kutsia, Dimitris Simos

Annals of Mathematics and Artificial Intelligence 88(1), pp. 213-236. 2020. ISSN 1573-7470. [url] [pdf]
[bib]
@article{RISC6116,
author = {Ilias Kotsireas and Temur Kutsia and Dimitris Simos},
title = {{ Constructing orthogonal designs in powers of two via symbolic computation and rewriting techniques}},
language = {english},
journal = {Annals of Mathematics and Artificial Intelligence},
volume = {88},
number = {1},
pages = {213--236},
isbn_issn = {ISSN 1573-7470},
year = {2020},
refereed = {yes},
length = {24},
url = {https://doi.org/10.1007/s10472-018-9607-9}
}
[Kutsia]

Proceedings of The 34th International Workshop on Unification, UNIF 2020

Temur Kutsia and Andrew M. Marshall (Editors)

Technical report no. 20-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2020. [pdf]
[bib]
@techreport{RISC6129,
author = {Temur Kutsia and Andrew M. Marshall (Editors)},
title = {{Proceedings of The 34th International Workshop on Unification, UNIF 2020}},
language = {english},
number = {20-10},
year = {2020},
length = {82},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Pau]

Proximity-based Unification with Arity Mismatch

Temur Kutsia, Cleo Pau

In: Proceedings of the 34th International Workshop on Unification, Temur Kutsia, Andrew M. Marshall (ed.), pp. 9:1-9:6. 2020.
[bib]
@inproceedings{RISC6295,
author = {Temur Kutsia and Cleo Pau},
title = {{Proximity-based Unification with Arity Mismatch}},
booktitle = {{Proceedings of the 34th International Workshop on Unification}},
language = {english},
pages = {9:1--9:6},
isbn_issn = { },
year = {2020},
editor = {Temur Kutsia and Andrew M. Marshall},
refereed = {yes},
length = {6}
}

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