## 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

### 2019

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

#### David M. Cerna

In: T.B.D, T.B.D (ed.), Proceedings of T.B.D, pp. 1-17. 2019. T.B.D. [pdf]**inproceedings**{RISC5841,

author = {David M. Cerna},

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

booktitle = {{T.B.D}},

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.},

pages = {1--17},

isbn_issn = {T.B.D},

year = {2019},

editor = {T.B.D},

refereed = {yes},

length = {17},

conferencename = {T.B.D}

}

### A Generic Framework for Higher-Order Generalizations

#### David M. Cerna, Temur Kutsia

Technical report no. 19-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. 2019. [pdf]**techreport**{RISC5883,

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

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

language = {english},

number = {19-01},

year = {2019},

length = {21},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### The Castle Game

#### David M. Cerna

2019. [pdf]**techreport**{RISC5886,

author = {David M. Cerna},

title = {{The Castle Game}},

language = {english},

abstract = {A description of a game for teaching certain aspects of first-order logic based on the Drink's Paradox. },

year = {2019},

length = {3},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Manual for AXolotl

#### David M. Cerna

2019. [pdf] [jar]**techreport**{RISC5887,

author = {David M. Cerna},

title = {{Manual for AXolotl}},

language = {english},

abstract = {In this document we outline how to play our preliminary version of \textbf{AX}olotl. We present a sequence of graphics illustrating the step by step process of playing the game. },

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}

}

### 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. Submitted for review. [pdf]**techreport**{RISC5530,

author = {David M. Cerna 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 = {Submitted for review},

institution = {RISC},

length = {27}

}

### Primitive Recursive Proof Systems for Arithmetic

#### David M. Cerna

RISC. Technical report, January 2018. In revision. [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}

}

### Experiments with Automatic Proofs in Elementary Analysis

#### T. Jebelean

Technical report no. 18-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. April 2018. Work in progress. [pdf] [zip] [pdf]**techreport**{RISC5692,

author = {T. Jebelean},

title = {{Experiments with Automatic Proofs in Elementary Analysis}},

language = {english},

abstract = {Combining methods from satisfiability checking with methods from symbolic computation promises to solve challenging problems in various areas of theory and application. We look at the basically equivalent problem of proving statements directly in a non--clausal setting, when additional information on the underlying domain is available in form of specific properties and algorithms.We demonstrate on some concrete examples several heuristic techniques for the automation of natural--style proving of statements from elementary analysis. The purpose of this work in progress is to generate proofs similar to those produced by humans, by combining automated reasoning methods with techniques from computer algebra.Our techniques include: the S-decomposition method for formulae with alternating quantifiers, quantifier elimination by cylindrical algebraic decomposition, analysis of terms behavior in zero, bounding the bounds, rewriting of expressions involving absolute value, algebraic manipulations, and identification of equal terms under unknown functions.These techniques are being implemented in the {\em Theorema} system and are able to construct automatically natural--style proofs for numerous examples including: convergence of sequences, limits and continuity of functions, uniform continuity, and other.},

number = {18-06},

year = {2018},

month = {April},

keywords = {Satisfiability Checking, Natural-style Proofs, Symbolic Computation},

sponsor = {European project ``Satisfiability Checking and Symbolic Computation'' (H2020-FETOPN-2015-CSA 712689)},

length = {33},

type = {Work in progress},

type = {RISC Report Series},

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

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}

### Idempotent Generalization is Infinitary

#### David M. Cerna and Temur Kutsia

RISC. Technical report, RISC Report, 2018. [pdf]**techreport**{RISC5529,

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

title = {{Idempotent Generalization is Infinitary}},

language = {english},

abstract = {Let §\mathbf{I}_{S}$ be an equational theory s.t. for each $f\in S$, $f(x,x)=x$. Such an equational theory is said to be {\em idempotent}. It is known that the anti-unification problem (AUP) $f(a,b) \triangleq g(a,b)$ modulo $\mathbf{I}_{\lbrace f,g \rbrace}$ admits infinitely many least-general generalizers (lggs)~\cite{LPottier1989}. We show that, modulo $\mathbf{I}_{\lbrace f\rbrace}$, $f(a,f(a,b)) \triangleq f(b,f(a,b))$ admits infinitely many lggs.},

year = {2018},

howpublished = {RISC Report},

institution = {RISC},

length = {1}

}

### Gröbner Bases of Modules and Faugère's F4 Algorithm in Isabelle/HOL

#### A. Maletzky, F. Immler

In: Intelligent Computer Mathematics (Proceedings of CICM 2018, Hagenberg, Austria, August 13-17), Florian Rabe and William Farmer and Grant Passmore and Abdou Youssef (ed.), Proceedings of CICM 2018, Lecture Notes in Computer Science 11006, pp. 178-193. 2018. Springer, ISBN 978-3-319-96811-7. The final publication is available at Springer via https://doi.org/10.1007/978-3-319-96812-4_16. [url]**inproceedings**{RISC5733,

author = {A. Maletzky and F. Immler},

title = {{Gröbner Bases of Modules and Faugère's F4 Algorithm in Isabelle/HOL}},

booktitle = {{Intelligent Computer Mathematics (Proceedings of CICM 2018, Hagenberg, Austria, August 13-17)}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {11006},

pages = {178--193},

publisher = {Springer},

isbn_issn = {ISBN 978-3-319-96811-7},

year = {2018},

note = {The final publication is available at Springer via https://doi.org/10.1007/978-3-319-96812-4_16},

editor = {Florian Rabe and William Farmer and Grant Passmore and Abdou Youssef},

refereed = {yes},

length = {16},

conferencename = {CICM 2018},

url = {https://doi.org/10.1007/978-3-319-96812-4_16}

}

### Gröbner Bases of Modules and Faugère's F4 Algorithm in Isabelle/HOL (extended version)

#### A. Maletzky, F. Immler

RISC, JKU Linz. Technical report, May 2018. arXiv:1805.00304 [cs.LO]. [url]**techreport**{RISC5734,

author = {A. Maletzky and F. Immler},

title = {{Gröbner Bases of Modules and Faugère's F4 Algorithm in Isabelle/HOL (extended version)}},

language = {english},

year = {2018},

month = {May},

note = {arXiv:1805.00304 [cs.LO]},

institution = {RISC, JKU Linz},

length = {28},

url = {https://arxiv.org/abs/1805.00304}

}

### Proximity-Based Generalization

#### Temur Kutsia, Cleo Pau

In: 32nd International Workshop on Unification, UNIF 2018, Mauricio Ayala-Rincon and Philippe Balbiani (ed.), pp. - . 2018. [pdf]**inproceedings**{RISC5713,

author = {Temur Kutsia and Cleo Pau},

title = {{Proximity-Based Generalization}},

booktitle = {{32nd International Workshop on Unification, UNIF 2018}},

language = {english},

pages = { -- },

isbn_issn = { },

year = {2018},

editor = {Mauricio Ayala-Rincon and Philippe Balbiani},

refereed = {yes},

length = {0}

}

### 2017

### Unranked Second-Order Anti-Unification

#### Alexander Baumgartner, Temur Kutsia

Information and Computation 255(2), pp. 262-286. 2017. ISSN: 0890-5401. [url]**article**{RISC5192,

author = {Alexander Baumgartner and Temur Kutsia},

title = {{Unranked Second-Order Anti-Unification}},

language = {english},

journal = {Information and Computation},

volume = {255},

number = {2},

pages = {262--286},

isbn_issn = {ISSN: 0890-5401},

year = {2017},

refereed = {yes},

length = {25},

url = {http://doi.org/10.1016/j.ic.2017.01.005}

}

### Higher-Order Pattern Anti-Unification in Linear Time

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

Journal of Automated Reasoning 58(2), pp. 293-310. 2017. ISSN 0168-7433. [url]**article**{RISC5337,

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

title = {{Higher-Order Pattern Anti-Unification in Linear Time}},

language = {english},

journal = { Journal of Automated Reasoning},

volume = {58},

number = {2},

pages = {293--310},

isbn_issn = {ISSN 0168-7433},

year = {2017},

refereed = {yes},

length = {18},

url = {http://dx.doi.org/10.1007/s10817-016-9383-3}

}

### Gröbner Bases Computation by Triangularizing Macaulay Matrices

#### Bruno Buchberger

Advanced Studies in Pure Mathematics (The 50th Anniversary of Gröbner Bases) 75, pp. 1-9. 2017. Mathematical Society of Japan, 0.**article**{RISC5586,

author = {Bruno Buchberger},

title = {{Gröbner Bases Computation by Triangularizing Macaulay Matrices}},

language = {english},

journal = {Advanced Studies in Pure Mathematics (The 50th Anniversary of Gröbner Bases)},

volume = {75},

pages = {1--9},

publisher = {Mathematical Society of Japan},

isbn_issn = {0},

year = {2017},

refereed = {no},

length = {9}

}