## Members

## David Cerna

## Teimuraz Kutsia

## Ongoing Projects

### Combinatorics and Codes for Information Security [SBA-K1]

### Generalization ALgorithms and Applications [GALA]

## Software

PρLog (pronounced Pē-rō-log) is an experimental tool that extends logic programming with strategic conditional transformation rules, combining Prolog with ρLog calculus. It deals with term sequences (also called hedges), transforming them by conditional rules. Transformations are nondeterministic and may yield ...

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 ...

## Publications

### 2020

### Unital Anti-Unification: Type and Algorithms

#### David M. Cerna , Temur Kutsia

Technical report no. 20-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. RISC Report, Febrary 2020. [pdf]**techreport**{RISC6080,

author = {David M. Cerna and Temur Kutsia},

title = {{Unital Anti-Unification: Type and Algorithms}},

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. },

number = {20-02},

year = {2020},

month = {Febrary},

howpublished = {RISC Report},

keywords = {Anti-unification, tree grammars, unital theories, collapse theories},

length = {19},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### 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]**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}

}

### 2019

### Higher-Order Pattern Generalization Modulo Equational Theories

#### David M. Cerna and Temur Kutsia

2019. [pdf]**techreport**{RISC5918,

author = {David M. Cerna and Temur Kutsia},

title = {{Higher-Order Pattern Generalization Modulo Equational Theories}},

language = {english},

abstract = {In this paper we address Three problems related to unital anti-unification. First, we develop a generalalgorithm based on a tree grammar representation of the set of computed generalizations. Secondlywe show that restricting the algorithm to computing linear generalizations only or to term signaturescontaining a single unital function results in a procedure which is minimal complete and Finitary.Thirdly, we show that when the term signature contains two unital functions, unital anti-unification isNullary.The algorithm does not depend on the number of idempotent function symbols in the input terms. Thelanguage generated by the grammar is the minimal complete set of generalizations of the givenanti-unification problem, which implies that idempotent anti-unification is infinitary.},

year = {2019},

length = {40},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### A Generic Framework for Higher-Order Generalizations

#### David M. Cerna, Temur Kutsia

In: Proceedings of the 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019), Herman Geuvers (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 131, pp. 10:1-10:19. 2019. Schloss Dagstuhl, ISSN 1868-8969. [url]**inproceedings**{RISC5947,

author = {David M. Cerna and Temur Kutsia},

title = {{A Generic Framework for Higher-Order Generalizations}},

booktitle = {{Proceedings of the 4th International Conference on Formal Structures for Computation and Deduction (FSCD 2019)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {131},

pages = {10:1--10:19},

publisher = {Schloss Dagstuhl},

isbn_issn = {ISSN 1868-8969},

year = {2019},

editor = {Herman Geuvers},

refereed = {yes},

length = {19},

url = {http://dx.doi.org/10.4230/LIPIcs.FSCD.2019.10}

}

### On the Complexity of Unsatisfiable Primitive Recursively defined $\Sigma_1$-Sentences

#### David M. Cerna

2019. [pdf]**techreport**{RISC5981,

author = {David M. Cerna},

title = {{On the Complexity of Unsatisfiable Primitive Recursively defined $\Sigma_1$-Sentences}},

language = {english},

abstract = {We introduce a measure of complexity based on formula occurrence within instance proofs of an inductive statement. Our measure is closely related to {\em Herbrand Sequent length}, but instead of capturing the number of necessary term instantiations, it captures the finite representational difficulty of a recursive sequence of proofs. We restrict ourselves to a class of unsatisfiable primitive recursively defined negation normal form first-order sentences, referred to as {\em abstract sentences}, which capture many problems of interest; for example, variants of the {\em infinitary pigeonhole principle}. This class of sentences has been particularly useful for inductive formal proof analysis and proof transformation. Together our complexity measure and abstract sentences allow use to capture a notion of {\em tractability} for state-of-the-art approaches to inductive theorem proving, in particular {\em loop discovery} and {\em tree grammar} based inductive theorem provers. We provide a complexity analysis of an important abstract sentence, and discuss the analysis of a few related sentences, based on the infinitary pigeonhole principle which we conjecture represent the upper limits of tractability and foundation of intractability with respect to the current approaches.},

year = {2019},

length = {17},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Idempotent Anti-unification

#### David Cerna, Temur Kutsia

ACM Transactions on Computational Logic (TOCL) 21(2), pp. 10:1-10:32. 2019. ACM Press, ISSN 1529-3785. [url] [pdf]**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 = {2019},

refereed = {yes},

length = {32},

url = {https://doi.org/10.1145/3359060}

}

### Variadic Equational Matching

#### Besik Dundua, Temur Kutsia, Mircea Marin

In: Intelligent Computer Mathematics - 12th International Conference, CICM 2019, Cezary Kaliszyk, Edwin Brady, Andrea Kohlhase, Claudio Sacerdoti Coen (ed.), Lecture Notes in Computer Science 11617, pp. 77-92. 2019. Springer, ISBN 978-3-030-23249-8. [pdf]**inproceedings**{RISC5948,

author = {Besik Dundua and Temur Kutsia and Mircea Marin},

title = {{Variadic Equational Matching}},

booktitle = {{Intelligent Computer Mathematics - 12th International Conference, CICM 2019}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {11617},

pages = {77--92},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-23249-8},

year = {2019},

editor = {Cezary Kaliszyk and Edwin Brady and Andrea Kohlhase and Claudio Sacerdoti Coen},

refereed = {yes},

length = {16}

}

### A Rule-based Approach to the Decidability of Safety of ABACα

#### Mircea Marin, Temur Kutsia, Besik Dundua

In: Proceedings of the 24th ACM Symposium on Access Control Models and Technologies, SACMAT 2019, Florian Kerschbaum, Atefeh Mashatan, Jianwei Niu, Adam J. Lee (ed.), pp. 173-178. 2019. ACM, ISBN 978-1-4503-6753-0. [url] [pdf]**inproceedings**{RISC5955,

author = {Mircea Marin and Temur Kutsia and Besik Dundua},

title = {{A Rule-based Approach to the Decidability of Safety of ABACα}},

booktitle = {{Proceedings of the 24th ACM Symposium on Access Control Models and Technologies, SACMAT 2019}},

language = {english},

pages = {173--178},

publisher = {ACM},

isbn_issn = {ISBN 978-1-4503-6753-0},

year = {2019},

editor = {Florian Kerschbaum and Atefeh Mashatan and Jianwei Niu and Adam J. Lee},

refereed = {yes},

length = {6},

url = {https://doi.org/10.1145/3322431.3325416}

}

### Computing All Maximal Clique Partitions in a Graph

#### Temur Kutsia, Cleo Pau

Technical report no. 19-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]**techreport**{RISC5939,

author = {Temur Kutsia and Cleo Pau},

title = {{Computing All Maximal Clique Partitions in a Graph}},

language = {english},

number = {19-04},

year = {2019},

length = {9},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Solving Proximity Constraints

#### Temur Kutsia, Cleo Pau

Technical report no. 19-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]**techreport**{RISC5950,

author = {Temur Kutsia and Cleo Pau},

title = {{Solving Proximity Constraints}},

language = {english},

number = {19-06},

year = {2019},

length = {22},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Matching and Generalization Modulo Proximity and Tolerance

#### Temur Kutsia, Cleo Pau

Technical report no. 19-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]**techreport**{RISC5953,

author = {Temur Kutsia and Cleo Pau},

title = {{Matching and Generalization Modulo Proximity and Tolerance}},

language = {english},

number = {19-07},

year = {2019},

length = {5},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### McCarthy-Kleene fuzzy automata and MSO logics

#### Manfred Droste, Temur Kutsia, George Rahonis, Wolfgang Schreiner

Information and Computation, pp. -. 2019. Elsevier, ISSN 0890-5401. In Press. [url]**article**{RISC6011,

author = {Manfred Droste and Temur Kutsia and George Rahonis and Wolfgang Schreiner},

title = {{McCarthy-Kleene fuzzy automata and MSO logics}},

language = {english},

abstract = {We introduce McCarthy-Kleene fuzzy automata (MK-fuzzy automata) over a bimonoid K which is related to the fuzzification of the McCarthy-Kleene logic. Our automata are inspired by, and intend to contribute to, practical applications being in development in a project on runtime network monitoring based on predicate logic. We investigate closure properties of the class of recognizable MK-fuzzy languages accepted by MK-fuzzy automata as well as of deterministically recognizable MK-fuzzy languages accepted by their deterministic counterparts. Moreover, we establish a Nivat-like result for recognizable MK-fuzzy languages. We introduce an MK-fuzzy MSO logic and show the expressive equivalence of a fragment of this logic with MK-fuzzy automata, i.e., a Büchi type theorem.},

journal = {Information and Computation},

pages = {--},

publisher = {Elsevier},

isbn_issn = {ISSN 0890-5401},

year = {2019},

note = {In Press},

refereed = {yes},

keywords = {Bimonoids, McCarthy-Kleene logic, MK-fuzzy automata, MK-fuzzy MSO logic},

sponsor = {Supported by the Austrian Research Promotion Agency (FFG) in the frame of the BRIDGE program 846003 “LogicGuard II”},

length = {23},

url = {https://doi.org/10.1016/j.ic.2019.104499}

}

### 2018

### Anti-Unification and Natural Language Processing

#### N. Amiridze, T. Kutsia

In: Fifth Workshop on Natural Language and Computer Science, NLCS’18, A. Asudeh, V. de Paiva, L. Moss (ed.), EasyChair preprints 203, pp. 1-12. 2018. [url] [pdf]**inproceedings**{RISC5707,

author = {N. Amiridze and T. Kutsia},

title = {{Anti-Unification and Natural Language Processing}},

booktitle = {{Fifth Workshop on Natural Language and Computer Science, NLCS’18}},

language = {english},

series = {EasyChair preprints},

number = {203},

pages = {1--12},

isbn_issn = { },

year = {2018},

editor = {A. Asudeh and V. de Paiva and L. Moss},

refereed = {yes},

length = {12},

url = {https://doi.org/10.29007/fkrh}

}

### Term-Graph Anti-Unification

#### Alexander Baumgartner, Temur Kutsia, Jordi Levy, Mateu Villaret

Technical report no. 18-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2018. [pdf]**techreport**{RISC5549,

author = {Alexander Baumgartner and Temur Kutsia and Jordi Levy and Mateu Villaret},

title = {{Term-Graph Anti-Unification}},

language = {english},

number = {18-02},

year = {2018},

length = {19},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Term-Graph Anti-Unification

#### Alexander Baumgartner, Temur Kutsia, Jordi Levy, Mateu Villaret

In: 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), Helene Kirchner (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 108, pp. 9:1-9:17. 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, ISBN 978-3-95977-077-4 ISSN 1868-8969. [url]**inproceedings**{RISC5764,

author = {Alexander Baumgartner and Temur Kutsia and Jordi Levy and Mateu Villaret},

title = {{Term-Graph Anti-Unification}},

booktitle = {{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {108},

pages = {9:1--9:17},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},

address = {Dagstuhl, Germany},

isbn_issn = {ISBN 978-3-95977-077-4 ISSN 1868-8969},

year = {2018},

editor = {Helene Kirchner},

refereed = {yes},

length = {17},

url = {http://drops.dagstuhl.de/opus/volltexte/2018/9179}

}

### Idempotent Anti-unification

#### David M. Cerna, Temur Kutsia

RISC. Technical report, Feb. 2018. to appear in TOCL. [pdf]**techreport**{RISC5530,

author = {David M. Cerna and Temur Kutsia},

title = {{Idempotent Anti-unification }},

language = {english},

abstract = {In this paper we address two problems related to idempotent anti-unification. First, we show thatthere exists an anti-unification problem with a single idempotent symbol which has an infiniteminimal complete set of generalizations. It means that anti-unification with a single idempotentsymbol has infinitary or nullary generalization type, similar to anti-unification with two idem-potent symbols, shown earlier by Loı̈c Pottier. Next, we develop an algorithm, which takes anarbitrary idempotent anti-unification problem and computes a representation of its solution set inthe form of a regular tree grammar. The algorithm does not depend on the number of idempotentfunction symbols in the input terms. The language generated by the grammar is the minimalcomplete set of generalizations of the given anti-unification problem, which implies that idem-potent anti-unification is infinitary.},

year = {2018},

month = {Feb.},

note = {to appear in TOCL},

institution = {RISC},

length = {32}

}

### Primitive Recursive Proof Systems for Arithmetic

#### David M. Cerna

RISC. Technical report, January 2018. In revision. [pdf] [pdf]**techreport**{RISC5528,

author = {David M. Cerna},

title = {{Primitive Recursive Proof Systems for Arithmetic}},

language = {english},

abstract = {Peano arithmetic, as formalized by Gentzen, can be presented as an axiom extensionof the LK-calculus with equality and an additional inference rule formalizing induction.While this formalism was enough (with the addition of some meta-theoretic argumentation)to show the consistency of arithmetic, alternative formulations of induction such asthe infinitary ω-rule and cyclic reasoning provide insight into the structure of arithmeticproofs obfuscated by the inference rule formulation of induction. For example, questionsconcerning the elimination of cut, consistency, and proof shape are given more clarity. Thesame could be said for functional interpretations of arithmetic such as system T whichenumerates the recursive functions provably total by arithmetic. A key feature of thesevariations on the formalization of arithmetic is that they get somewhat closer to formalizingthe concept of induction directly using the inferences of the LK-calculus, albeit byadding extra machinery at the meta-level. In this work we present a recursive sequentcalculus for arithmetic which can be syntactically translated into Gentzen formalism ofarithmetic and allows proof normalization to the LK-calculus with equality.},

year = {2018},

month = {January },

note = {In revision},

institution = {RISC},

length = {20}

}

### Higher-Order Equational Pattern Anti-Unification

#### David M. Cerna, Temur Kutsia

In: 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018), Helene Kirchner (ed.), Leibniz International Proceedings in Informatics (LIPIcs) 108, pp. 12:1-12:17. 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, ISBN 978-3-95977-077-4 ISSN 1868-8969. [url]**inproceedings**{RISC5765,

author = {David M. Cerna and Temur Kutsia},

title = {{Higher-Order Equational Pattern Anti-Unification}},

booktitle = {{3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}},

language = {english},

series = {Leibniz International Proceedings in Informatics (LIPIcs)},

volume = {108},

pages = {12:1--12:17},

publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},

address = {Dagstuhl, Germany},

isbn_issn = {ISBN 978-3-95977-077-4 ISSN 1868-8969},

year = {2018},

editor = {Helene Kirchner},

refereed = {yes},

length = {17},

url = {http://drops.dagstuhl.de/opus/volltexte/2018/9182}

}

### Mechanical Synthesis of Sorting Algorithms for Binary Trees by Logic and Combinatorial Techniques

#### Isabela Dramnesc, Tudor Jebelean, Sorin Stratulat

Journal of Symbolic Computation 90, pp. 3-41. 2018. Elsevier, 07477171. [url]**article**{RISC5715,

author = {Isabela Dramnesc and Tudor Jebelean and Sorin Stratulat},

title = {{Mechanical Synthesis of Sorting Algorithms for Binary Trees by Logic and Combinatorial Techniques}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {90},

pages = {3--41},

publisher = {Elsevier},

isbn_issn = {07477171},

year = {2018},

refereed = {yes},

keywords = {algorithm synthesis ; automated reasoning ; natural--style proving},

length = {39},

url = {https://doi.org/10.1016/j.jsc.2018.04.002}

}

### Pattern-based calculi with finitary matching

#### Sandra Alves, Besik Dundua, Mário Florido, Temur Kutsia

Logic Journal of the IGPL 26(2), pp. 203-243. 2018. ISSN 1367-0751. [url]**article**{RISC5763,

author = {Sandra Alves and Besik Dundua and Mário Florido and Temur Kutsia},

title = {{Pattern-based calculi with finitary matching}},

language = {english},

journal = {Logic Journal of the IGPL},

volume = {26},

number = {2},

pages = {203--243},

isbn_issn = {ISSN 1367-0751},

year = {2018},

refereed = {yes},

length = {41},

url = {https://doi.org/10.1093/JIGPAL/jzx059}

}