Automated Reasoning

At RISC we are devoted to the foundations, the design, and the implementation of software to generate mathematical proofs.

The main enterprise in the area of automated reasoning at RISC is the Theorema project. Theorema has been initiated around 1995 by Bruno Buchberger, who has led the project ever since. Some activities in the area of formal methods are tightly related to what we do in automated reasoning.

Software

Theorema

A Mathematical Assistant System implemented in Mathematica

The present prototype version of the Theorema software system is implemented in Mathematica . The system consists of a general higher-order predicate logic prover and a collection of special provers that call each other depending on the particular proof situations. ...

MoreSoftware Website

Publications

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]
[bib]
@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]
[bib]
@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}
}

Idempotent Anti-unification

David M. Cerna Temur Kutsia

RISC. Technical report, Feb. 2018. Submitted for review. [pdf]
[bib]
@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 and Anela Lolic

RISC. Technical report, January 2018. In revision. [pdf]
[bib]
@techreport{RISC5528,
author = {David M. Cerna and Anela Lolic},
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 = {16}
}

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]
[bib]
@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]
[bib]
@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]
[bib]
@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]
[bib]
@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]
[bib]
@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]
[bib]
@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]
[bib]
@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}
}

WebEx: Web Exercises for RISCAL

Wolfgang Schreiner

Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria. Technical report, October 2018. [pdf]
[bib]
@techreport{RISC5793,
author = {Wolfgang Schreiner},
title = {{WebEx: Web Exercises for RISCAL}},
language = {english},
abstract = {We report on a software framework "WebEx" for developing web-basedstudent exercises whose correctness is checked with the help of the RISCAL (RISCAlgorithm Language) software. This framework allows to generate from anappropriately annotated RISCAL specification file an HTML file that serves asthe frontend to a remote execution service. Student input (RISCAL fragments) aretransmitted to this execution service which generates from the annotatedspecification file and the input a plain RISCAL specification on which RISCAL isinvoked (in a non-interactive mode); the success status of the execution and theproduced output are reported back to the web interface. For each successfulexercise the server produces a digitally signed certificate that is returned tothe user who may submit this certificate as a proof of successful completion ofthe exercise (which may be subsequently automatically checked). Furthermore eachannotated RISCAL specification may serve as a template that may be instantiatedwith other data to produce a set of exercise instances. The WebEx software ismostly independent of RISCAL; it may be also used to provide a web front end forother scientific software of a similar nature.},
year = {2018},
month = {October},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria},
keywords = {formal methods, logic, computer-supported education},
sponsor = {Austrian OEAD WTZ program and the Slovak SRDA agency contract SK 14/2018 SemTech and Johannes Kepler University Linz, Linz Institute of Technology (LIT), Project LOGTECHEDU},
length = {35}
}

2017

Unranked Second-Order Anti-Unification

Alexander Baumgartner, Temur Kutsia

Information and Computation 255(2), pp. 262-286. 2017. ISSN: 0890-5401. [url]
[bib]
@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]
[bib]
@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.
[bib]
@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}
}

Satisfiability Checking and Symbolic Computation

Abraham Erika , Abbott John , Becker Bernd , Bigatti Anna Maria , Brain Martin , Buchberger Bruno , Cimatti Alessandro , Davenport James , England Matthew , Fontaine Pascal , Forrest Stephen , Griggio Alberto , Kröning Daniel , Seiler Werner M. , Sturm

ACM Communications in Computer Algebra 50(4), pp. 145-147. 2017. 0.
[bib]
@article{RISC5587,
author = { Abraham Erika and Abbott John and Becker Bernd and Bigatti Anna Maria and Brain Martin and Buchberger Bruno and Cimatti Alessandro and Davenport James and England Matthew and Fontaine Pascal and Forrest Stephen and Griggio Alberto and Kröning Daniel and Seiler Werner M. and Sturm },
title = {{Satisfiability Checking and Symbolic Computation}},
language = {english},
journal = {ACM Communications in Computer Algebra},
volume = {50},
number = {4},
pages = {145--147},
isbn_issn = {0},
year = {2017},
refereed = {no},
length = {3}
}

An overview of PρLog

Besik Dundua, Temur Kutsia, Klaus Reisenberger-Hagmayr

In: Proceedings of the 19th International Symposium on Practical Aspects of Declarative Languages, PADL 2017, Y. Lierler and W. Taha (ed.), Lecture Notes in Computer Science 10137, pp. 34-49. 2017. Springer, ISBN 978-3-319-51675-2. [pdf]
[bib]
@inproceedings{RISC5416,
author = {Besik Dundua and Temur Kutsia and Klaus Reisenberger-Hagmayr},
title = {{An overview of PρLog}},
booktitle = {{ Proceedings of the 19th International Symposium on Practical Aspects of Declarative Languages, PADL 2017}},
language = {english},
series = {Lecture Notes in Computer Science},
volume = {10137},
pages = {34--49},
publisher = {Springer},
isbn_issn = {ISBN 978-3-319-51675-2},
year = {2017},
editor = {Y. Lierler and W. Taha},
refereed = {yes},
length = {16}
}

Rewriting Logic from a ρLog Point of View

Mauricio Ayala-Rincon, Besik Dundua, Temur Kutsia, Mircea Marin

In: Proceedings of the 12th Workshop on Logical and Semantic Frameworks with Applications (LSFA 2017) , Sandra Alves and Renata Wasserman (ed.), pp. -. 2017. [pdf]
[bib]
@inproceedings{RISC5473,
author = {Mauricio Ayala-Rincon and Besik Dundua and Temur Kutsia and Mircea Marin},
title = {{Rewriting Logic from a ρLog Point of View}},
booktitle = {{Proceedings of the 12th Workshop on Logical and Semantic Frameworks with Applications (LSFA 2017) }},
language = {english},
pages = {--},
isbn_issn = { },
year = {2017},
editor = {Sandra Alves and Renata Wasserman},
refereed = {yes},
length = {16}
}

Pattern-Based Calculi with Finitary Matching

Sandra Alves, Besik Dundua, Mario Florido, Temur Kutsia

Logic Journal of the IGPL, pp. -. 2017. Oxford University Press, ISSN 1367-0751. To appear. [url]
[bib]
@article{RISC5517,
author = {Sandra Alves and Besik Dundua and Mario Florido and Temur Kutsia},
title = {{Pattern-Based Calculi with Finitary Matching}},
language = {english},
journal = {Logic Journal of the IGPL},
pages = {--},
publisher = {Oxford University Press},
isbn_issn = {ISSN 1367-0751},
year = {2017},
note = {To appear},
refereed = {yes},
length = {41},
url = {https://doi.org/10.1093/JIGPAL/jzx059}
}

Adapting Cylindrical Algebraic Decomposition for Proof Specific Tasks

E. Abraham, T. Jebelean

In: ICAI 2017: 10th International Conference on Applied Informatics, G. Kusper (ed.), Proceedings of ICAI 2017: 10th International Conference on Applied Informatics, pp. 0-0. 2017. ISBN 0. In print. [url] [pdf]
[bib]
@inproceedings{RISC5546,
author = {E. Abraham and T. Jebelean},
title = {{Adapting Cylindrical Algebraic Decomposition for Proof Specific Tasks}},
booktitle = {{ICAI 2017: 10th International Conference on Applied Informatics}},
language = {English},
abstract = {We develop a case study on using quantifier elimination and cylindrical algebraic decomposition for the purpose of finding specific terms for the automation of proving mathematical properties in elementary analysis.Real--life proofs in specific mathematical domains are difficult to automate because of the high number of assumptions necessary for the prover to succeed. In particular, in elementary analysis, it is allmost impossible to find automatically the appropriate terms for the instantiation of universal assumptions and for witnessing the existential goals. We aim at developing such proofs in natural style, in the frame of the{\em Theorema} system.Finding such terms is actually possible by using quantifier elimination (QE) based on cylindrical algebraic decomposition (CAD). However, the current straightforward approach lacks in efficiency, because several redundant calls may be necessary, and because the nature of the problem is slightly different.We study some natural--style proofs, the necessary special terms, and the corresponding usage of the QE/CAD, and identify specific techniques for adapting these algorithms in order to increase their efficiency.The experiments are performed partially in {\em Mathematica} and partially in {\tt SMT-RAT}, the latter used as ``white--box'', which allows to inspect the intermediate results and to adapt certain parts of the algorithms.},
pages = {0--0},
isbn_issn = {ISBN 0},
year = {2017},
note = {In print},
editor = {G. Kusper},
refereed = {yes},
length = {0},
conferencename = {ICAI 2017: 10th International Conference on Applied Informatics},
url = {https://icai.uni-eszterhazy.hu/}
}

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