## Members

## Bruno Buchberger: Founding Chairman 1987-1998

## Michal Buran

## David Cerna

## Zoltán Kovács: joint PostDoc GeoGebra+RISC

## Teimuraz Kutsia

## Nikolaj Popov

## Markus Rosenkranz

## Wolfgang Schreiner

## Wolfgang Windsteiger

## Ongoing Projects

### Symbolic Techniques for Quantitative Extensions of Equality [SQUEE]

## 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 ...

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. ...

## Publications

### 2023

### Is ChatGPT Smarter Than Master’s Applicants?

#### Bruno Buchberger

Technical report no. 23-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2023. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6684,

author = {Bruno Buchberger},

title = {{Is ChatGPT Smarter Than Master’s Applicants?}},

language = {English},

abstract = {During the selection procedure for a particular informatics fellowship program sponsored by Upper Austrian companies, I ask the applicants a couple of simple technical questions about programming, etc., in a Zoom meeting. I put the same questions to the dialogue system ChatGPT, [ChatGPT]. The result surprised me: Nearly all answers of ChatGPT were totally correct and nicely explained. Also, in the dialogues to clarify some critical points in the answers, the explanations by ChatGPT were amazingly clear and goal-oriented.In comparison: I tried out the same questions in the personal Zoom interviews with approximately 30 applicants from five countries. Only the top three candidates (with a GPA of 1.0, i.e., the highest possible GPA in their bachelor’s study) performed approximately equally well in the interview. All the others performed (far) worse than ChatGPT. And, of course, all answers from ChatGPT came within 1 to 10 seconds, whereas most of the human applicants' answers needed lengthy and arduous dialogues.I am particularly impressed by the ability of ChatGPT to extract meaningful and well-structured programs from problem specifications in natural language. In this experiment, I also added some questions that ask for proofs for simple statements in natural language, which I do not ask in the student's interviews. The performance of ChatGPT was quite impressive as far as formalization and propositional logic are concerned. In examples where predicate logic reasoning is necessary, the ChatGPT answers are not (yet?) perfect. I am pleased to see that ChatGPT tries to present the proofs in a “natural style” This is something that I had as one of my main goals when I initiated the Theorema project in 1995. I think we already achieved this in the early stage of Theorema, and we performed this slightly better and more systematically than ChatGPT does.I also tried to develop a natural language input facility for Theorema in 2017, i.e., a tool to formalize natural language statements in predicate logic. However, I could not continue this research for a couple of reasons. Now I see that ChatGPT achieved this goal. Thus, I think that the following combination of methods could result in a significant leap forward:- the “natural style” proving methods that we developed within Theorema (for the automated generation of programs from specifications, the automated verification of programs in the frame of knowledge, and the automated proof of theorems in theories), in particular, my “Lazy Thinking Method” for algorithm synthesis from specifications- and the natural language formalization techniques of ChatGPT.I propose this as a research project topic and invite colleagues and students to contact me and join me in this effort: Buchberger.bruno@gmail.com.},

number = {23-04},

year = {2023},

month = {January},

keywords = {ChatGPT, automated programming, program synthesis, automated proving, formalization of natural language, master's screening},

length = {30},

license = {CC BY 4.0 International},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### Concrete Abstractions

#### Wolfgang Schreiner

Texts & Monographs in Symbolic Computation 1st edition, 2023. Springer, Cham, Switzerland, Hardcover ISBN 978-3-031-24933-4, Softcover ISBN 978-3-031-24936-5, eBook ISBN 978-3-031-24934-1. [doi]**book**{RISC6698,

author = {Wolfgang Schreiner},

title = {{Concrete Abstractions}},

language = {english},

series = {Texts & Monographs in Symbolic Computation},

publisher = {Springer},

address = {Cham, Switzerland},

isbn_issn = {Hardcover ISBN 978-3-031-24933-4, Softcover ISBN 978-3-031-24936-5, eBook ISBN 978-3-031-24934-1},

year = {2023},

edition = {1st},

translation = {0},

keywords = {logic in computer science, model checking, formal modeling and reasoning, program specification and verification, discrete structures and algorithms, nondeterminism and concurrency},

length = {270},

url = {https://doi.org/10.1007/978-3-031-24934-1}

}

### Formalisation of Relational Algebra and a SQL-like Language with the RISCAL Model Checker

#### Joachim Borya

Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. Bachelor Thesis. May 2023. Also available as RISC Report 23-06. [doi] [pdf]**misc**{RISC6707,

author = {Joachim Borya},

title = {{Formalisation of Relational Algebra and a SQL-like Language with the RISCAL Model Checker}},

language = {english},

abstract = {The relational database model is based on the mathematical concept of relational algebra.Query languages have been developed to make data available quickly without creatingdedicated access procedures that depend on the internal representation of the data. SQL(structured query language) can be seen as a quasi-standard for this. This thesis dealswith the formalization and verification of relational algebra and a small but elementarysubset of SQL with the help of the RISCAL model checker, a software tool for the formalspecification and verification of mathematical theories and algorithms.},

year = {2023},

month = {May},

note = {Also available as RISC Report 23-06},

translation = {0},

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

keywords = {formal methods, program verification, model checking, automated theorem proving},

length = {77},

url = {https://doi.org/10.35011/risc.23-06}

}

### Model Checking Concurrent Systems Under Fairness Constraints in RISCAL

#### Ágoston Sütő

Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. Master Thesis. May 2023. Also available as RISC Report 23-07. Master's thesis. [doi] [pdf]**misc**{RISC6709,

author = {Ágoston Sütő},

title = {{Model Checking Concurrent Systems Under Fairness Constraints in RISCAL}},

language = {english},

abstract = {Model checking is a method for verifying that a program satisfies certain desirable properties formalised using mathematical logic. It is a rigorous method, similar to theorem proving, but it is generally applied when theorem proving would be too difficult due to the complexity of the algorithm, such as in concurrent systems. Model checking is used in the software industry. RISCAL (RISC Algorithm Language) is a language and software system that can be used to describe algorithms over a finite domain, specify their behaviour and then validate the specification. While it mainly focuses on deterministic algorithms, it has limited support for non-deterministic systems as well.The thesis extends the support for non-deterministic systems in RISCAL by allowing the user to specify complex properties about their behaviour in the language of Linear Temporal Logic (LTL) and then to validate them. The core contribution is a model checker implemented in Java using the so-called automaton-based explicit state model checking approach. The software is capable of verifying certain properties that could not be handled by a well-known model checker used in the industry. While in most cases it has underperformed its competitors, our implementation is promising, especially when it comes to properties with certain side conditions, called fairness constraints. The majority of the thesis is be concerned with the theoretical aspects of the automaton-based model checking approach, which is followed by a description of the implementation and various benchmarks.},

year = {2023},

month = {May},

note = {Also available as RISC Report 23-07},

translation = {0},

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

keywords = {formal methods, model checking, concurrent systems, nondeterminism, linear temporal logic},

sponsor = {Supported by Aktion Österreich–Slowakei project grant Nr. 2019-10-15-003 “Semantic Modeling of Component-Based Program Systems”},

length = {102},

url = {https://doi.org/10.35011/risc.23-07},

type = {Master's thesis}

}

### 2022

### Learning Higher-Order Programs without Meta-Interpretive Learning

#### Stanislav Purgal and David Cerna and Cezary Kalisyk

In: Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22, Lud De Raedt (ed.), Proceedings of International Joint Conference on Artificial Intelligence, Main Track , pp. 2726-2733. july 2022. International Joint Conferences on Artificial Intelligence Organization, 10.24963/ijcai.2022/378. [doi]**inproceedings**{RISC6646,

author = {Stanislav Purgal and David Cerna and Cezary Kalisyk},

title = {{Learning Higher-Order Programs without Meta-Interpretive Learning}},

booktitle = {{Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence, IJCAI-22}},

language = {english},

abstract = {Learning complex programs through textit{inductive logic programming} (ILP) remains a formidable challenge. Existing higher-order enabled ILP systems show improved accuracy and learning performance, though remain hampered by the limitations of the underlying learning mechanism. Experimental results show that our extension of the versatile textit{Learning From Failures} paradigm by higher-order definitions significantly improves learning performance without the burdensome human guidance required by existing systems. Furthermore, we provide a theoretical framework capturing the class of higher-order definitions handled by our extension.},

series = {Main Track},

pages = {2726--2733},

publisher = {International Joint Conferences on Artificial Intelligence Organization},

isbn_issn = {10.24963/ijcai.2022/378},

year = {2022},

month = {july},

editor = {Lud De Raedt},

refereed = {yes},

keywords = {Inductive Logic Programming, Higher order definitions},

length = {8},

conferencename = {International Joint Conference on Artificial Intelligence},

url = {https://doi.org/10.35011/risc.21-22}

}

### Unranked Nominal Unification

#### Besik Dundua, Temur Kutsia, Mikheil Rukhaia

In: Proceedings of TbiLLC 2019 - 13th International Tbilisi Symposium on Logic, Language, and Computation, Aybüke Özgün and Yulia Zinova (ed.), Proceedings of 13th International Tbilisi Symposium on Logic, Language, and Computation, Lecture Notes in Computer Science 13206, pp. 279-296. 2022. Springer, ISBN 978-3-030-98478-6. [doi] [pdf]**inproceedings**{RISC6498,

author = {Besik Dundua and Temur Kutsia and Mikheil Rukhaia},

title = {{Unranked Nominal Unification}},

booktitle = {{Proceedings of TbiLLC 2019 - 13th International Tbilisi Symposium on Logic, Language, and Computation}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {13206},

pages = {279--296},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-98478-6},

year = {2022},

editor = {Aybüke Özgün and Yulia Zinova},

refereed = {yes},

length = {17},

conferencename = {13th International Tbilisi Symposium on Logic, Language, and Computation},

url = {https://doi.org/10.1007/978-3-030-98479-3_14}

}

### Nominal Unification and Matching of Higher Order Expressions with Recursive Let

#### Manfred Schmidt-Schauß, Temur Kutsia, Jordi Levy, Mateu Villaret, Yunus Kutz

Fundamenta Informaticae 185(3), pp. 247-283. 2022. IOS Press, ISSN 1875-8681. [url]**article**{RISC6506,

author = {Manfred Schmidt-Schauß and Temur Kutsia and Jordi Levy and Mateu Villaret and Yunus Kutz},

title = {{Nominal Unification and Matching of Higher Order Expressions with Recursive Let}},

language = {english},

journal = {Fundamenta Informaticae},

volume = {185},

number = {3},

pages = {247--283},

publisher = {IOS Press},

isbn_issn = {ISSN 1875-8681},

year = {2022},

refereed = {yes},

length = {37},

url = {https://arxiv.org/abs/2102.08146v4}

}

### Work-in-progress papers presented at the 15th Conference on Intelligent Computer Mathematics, CICM 2022 (Informal Proceedings)

#### Kevin Buzzard and Temur Kutsia

, 2022. [url] [pdf]**techreport**{RISC6584,

author = {Kevin Buzzard and Temur Kutsia},

title = {{Work-in-progress papers presented at the 15th Conference on Intelligent Computer Mathematics, CICM 2022 (Informal Proceedings)}},

language = {english},

year = {2022},

institution = { },

length = {77},

url = {https://cicm-conference.org/2022/cicm.php}

}

### Matching and Generalization Modulo Proximity and Tolerance Relations

#### Temur Kutsia, Cleo Pau

In: Proceedings of TbiLLC 2019 - 13th International Tbilisi Symposium on Logic, Language, and Computation, Aybüke Özgün and Yulia Zinova (ed.), Lecture Notes in Computer Science 13206, pp. 323-342. 2022. Springer, ISBN 978-3-030-98478-6. [doi] [pdf]**inproceedings**{RISC6491,

author = {Temur Kutsia and Cleo Pau},

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

booktitle = {{Proceedings of TbiLLC 2019 - 13th International Tbilisi Symposium on Logic, Language, and Computation}},

language = {english},

series = {Lecture Notes in Computer Science},

volume = {13206},

pages = {323--342},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-98478-6},

year = {2022},

editor = {Aybüke Özgün and Yulia Zinova},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1007/978-3-030-98479-3_16}

}

### A framework for approximate generalization in quantitative theories

#### Temur Kutsia, Cleo Pau

Technical report no. 22-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). May 2022. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6505,

author = {Temur Kutsia and Cleo Pau},

title = {{A framework for approximate generalization in quantitative theories}},

language = {english},

abstract = {Anti-unification aims at computing generalizations for given terms, retaining their common structure and abstracting differences by variables. We study quantitative anti-unification where the notion of the common structure is relaxed into "proximal'' up to the given degree with respect to the given fuzzy proximity relation. Proximal symbols may have different names and arities. We develop a generic set of rules for computing minimal complete sets of approximate generalizations and study their properties. Depending on the characterizations of proximities between symbols and the desired forms of solutions, these rules give rise to different versions of concrete algorithms.},

number = {22-04},

year = {2022},

month = {May},

keywords = {Generalization, anti-unification, quantiative theories, fuzzy proximity relations},

length = {22},

license = {CC BY 4.0 International},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### A Framework for Approximate Generalization in Quantitative Theories

#### Temur Kutsia and Cleo Pau

In: Automated Reasoning, Jasmin Blanchette, Laura Kovács, and Dirk Pattinson (ed.), Proceedings of 11th International Joint Conference, IJCAR 2022, Haifa, Israel, August 8-10, 2022, Lecture Notes in Artificial Intelligence 13385, pp. 578-596. 2022. Springer, ISBN 978-3-031-10768-9. [doi]**inproceedings**{RISC6631,

author = {Temur Kutsia and Cleo Pau},

title = {{A Framework for Approximate Generalization in Quantitative Theories}},

booktitle = {{Automated Reasoning}},

language = {english},

series = {Lecture Notes in Artificial Intelligence},

volume = {13385},

pages = {578--596},

publisher = {Springer},

isbn_issn = {ISBN 978-3-031-10768-9},

year = {2022},

editor = {Jasmin Blanchette and Laura Kovács and and Dirk Pattinson},

refereed = {yes},

sponsor = {FWF},

length = {19},

conferencename = {11th International Joint Conference, IJCAR 2022, Haifa, Israel, August 8-10, 2022},

url = {https://doi.org/10.1007/978-3-031-10769-6}

}

### The RISCTP Theorem Proving Interface - Tutorial and Reference Manual (Version 1.0.*)

#### Wolfgang Schreiner

Technical report no. 22-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2022. Licensed under CC BY 4.0 International. [doi] [pdf]**techreport**{RISC6517,

author = {Wolfgang Schreiner},

title = {{The RISCTP Theorem Proving Interface - Tutorial and Reference Manual (Version 1.0.*)}},

language = {english},

abstract = {This report documents the RISCTP theorem proving interface. RISCTP consists of alanguage for specifying proof problems and of an associated software for solving theseproblems. The RISCTP language is a typed variant of first-order logic whose level ofabstraction is between that of higher level formal specification languages (such as thelanguage of the RISCAL model checker) and lower level theorem proving languages (such asthe language SMT-LIB supported by various satisfiability modulo theories solvers such as Z3).Thus the RISCTP language can serve as an intermediate layer that simplifies the connectionof specification and verification systems to theorem provers; in fact, it was developed toequip the RISCAL model checker with theorem proving capabilities. The RISCTP softwareis implemented in Java with an API that enables the implementation of such connections;however, RISCTP also provides a text-based frontend that allows its use as a theorem proveron its own. RISCTP already implements a backend that translates a proving problem intoSMT-LIB and solves it by the "black box" application of Z3; in the future, RISCTP shall alsoprovide built-in proving capabilities with greater transparency.},

number = {22-07},

year = {2022},

month = {June},

keywords = {automated reasoning, theorem proving, model checking, first-order logic, RISCAL, SMT-LIB, Z3},

length = {31},

license = {CC BY 4.0 International},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### A Temporal Logic Extension of the RISCAL Model Checker

#### Wolfgang Schreiner, Ágoston Sütő

In: 2022 IEEE 16th International Scientific Conference on Informatics, Poprad, Slovakia, November 23-25, William Steingartner, Štefan Korečko, Anikó Szakál (ed.), pp. 267-272. 2022. IEEE, ISBN 979-8-3503-1034-4. [doi]**inproceedings**{RISC6633,

author = {Wolfgang Schreiner and Ágoston Sütő},

title = {{A Temporal Logic Extension of the RISCAL Model Checker}},

booktitle = {{2022 IEEE 16th International Scientific Conference on Informatics, Poprad, Slovakia, November 23-25}},

language = {english},

abstract = {We report on a new extension of the RISCAL model checker that allows to specify nondeterministic transition systems by formulas in linear temporal logic (LTL) and to verify them under fairness constraints. This extension is based on an automata-theoretic approach to explicit state model checking; the performance of its implementation is in some representative examples competitive with (in fact mostly superior to) that of TLA+, a widely known tool for system modeling and analysis. Thus, while RISCAL was originally developed for describing and analyzing mathematical theories and algorithms over discrete structures, these extensions make RISCAL also a competent checker for formal models of concurrent systems.},

pages = {267--272},

publisher = {IEEE},

isbn_issn = {ISBN 979-8-3503-1034-4},

year = {2022},

editor = {William Steingartner and Štefan Korečko and Anikó Szakál},

refereed = {yes},

keywords = {model checking, first-order logic, linear temporal logic, automata theory, formal specification and verification},

sponsor = {Supported by Aktion Österreich–Slowakei project grant Nr. 2019-10-15-003 “Semantic Modeling of Component-Based Program Systems”},

length = {6},

url = {https://doi.org/10.1109/Informatics57926.2022.10083433}

}

### Implementation Techniques for Mathematical Model Checking

#### Wolfgang Schreiner

In: SYNASC 2022, 24th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Hagenberg, Austria, September 12-15, Bruno Buchberger, Mircea Marin, Viorel Negru, Daniela Zaharie (ed.), pp. 12-15. 2022. IEEE, ISBN 978-1-6654-6545-8. [doi]**inproceedings**{RISC6634,

author = {Wolfgang Schreiner},

title = {{Implementation Techniques for Mathematical Model Checking}},

booktitle = {{SYNASC 2022, 24th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, Hagenberg, Austria, September 12-15}},

language = {english},

abstract = {We report on the various implementation techniques that the model checker RISCAL applies for the formal verification of mathematical algorithms and theorems in finite models of configurable sizes. Originally, RISCAL was based entirely on semantic evaluation where every syntactic phrase is translated to an executable version of its denotational semantics, which allows to execute algorithms and to evaluate first-order formulas. Later this was extended by a translation of formulas from the RISCAL language to an SMT-LIB logic, which enables their decision by the application of external SMT (satisfiability modulo theories) solvers. Subsequently, semantic evaluation was extended to nondeterministic/concurrent transition systems which facilitates the verification of invariance properties by state space exploration; this was recently generalized to an automata-based technique for verifying system specifications expressed in a LTL (linear temporal logic) extension of the RISCAL formula language. Recently, the checking capabilities of RISCAL have been complemented (via an embedding of the RISCTP theorem proving interface) by capabilities for verifying formulas in domains of arbitrary size with the help of external theorem provers. We briefly sketch these techniques and discuss their purpose and relationship within the general problem area of algorithm specification and verification.},

pages = {12--15},

publisher = {IEEE},

isbn_issn = {ISBN 978-1-6654-6545-8},

year = {2022},

editor = {Bruno Buchberger and Mircea Marin and Viorel Negru and Daniela Zaharie},

refereed = {yes},

keywords = {formal specification and verification, model checking, satisfiability solving, theorem proving, first-order logic, linear temporal logic},

sponsor = {Supported by Aktion Österreich–Slowakei project grant Nr. 2019-10-15-003 “Semantic Modeling of Component-Based Program Systems”},

length = {4},

url = {https://doi.org/10.1109/SYNASC57785.2022.00011}

}

### Learning to Reason Assisted by Automated Reasoning

#### W. Windsteiger

In: Intelligent Computer Mathematics: 15th International Conference, K. Buzzard and T. Kutsia (ed.), Proceedings of CICM 2022, Lecture Notes in Artificial Intelligence LNAI 13467, pp. 305-320. 2022. Springer, ISBN 978-3-031-16681-5. [doi] [pdf]**inproceedings**{RISC6530,

author = {W. Windsteiger},

title = {{Learning to Reason Assisted by Automated Reasoning}},

booktitle = {{Intelligent Computer Mathematics: 15th International Conference}},

language = {english},

abstract = {We report on using logic software in a novel course-format for an undergraduate logic course for studentsin computer science or artificial intelligence. Although being designed as the students' basic introduction tothe field of logic, the course features a novel structure and it adds some modern content, such as SATand SMT solving, to the traditional and established topics, such as propositional logicand first order predicate logic. The novel course design is characterized by, among others, the integration ofexisting logic software into the teaching of logic.In this paper we focus on the module on first-order predicate logic and the use of the Theorema system as a proof-tutor for the students. We report on statistical evaluation of data collected over two consecutiveyears of teaching this course. On the one hand, we asked for feedback of students on howhelpful they felt the software support was. On the other hand, we evaluated their results in the exams during thecourse and their development over the entire teaching period. The performance in exams is then correlated withstudents'' own perception of the helpfulness of software.},

series = {Lecture Notes in Artificial Intelligence LNAI},

number = {13467},

pages = {305--320},

publisher = {Springer},

isbn_issn = {ISBN 978-3-031-16681-5},

year = {2022},

editor = {K. Buzzard and T. Kutsia},

refereed = {yes},

length = {16},

conferencename = {CICM 2022},

url = {https://doi.org/10.1007/978-3-031-16681-5_22}

}

### 2021

### A Special Case of Schematic Syntactic Unification

#### David M. Cerna

CAS ICS / RISC. Technical report, 2021. [pdf]**techreport**{RISC6349,

author = {David M. Cerna},

title = {{A Special Case of Schematic Syntactic Unification}},

language = {english},

abstract = {We present a unification problem based on first-order syntactic unification which ask whether every problemin a schematically-defined sequence of unification problems isunifiable, so called loop unification. Alternatively, our problemmay be formulated as a recursive procedure calling first-ordersyntactic unification on certain bindings occurring in the solvedform resulting from unification. Loop unification is closely relatedto Narrowing as the schematic constructions can be seen as arewrite rule applied during unification, and primal grammars, aswe deal with recursive term constructions. However, loop unifi-cation relaxes the restrictions put on variables as fresh as wellas used extra variables may be introduced by rewriting. In thiswork we consider an important special case, so called semiloopunification. We provide a sufficient condition for unifiability of theentire sequence based on the structure of a sufficiently long initialsegment. It remains an open question whether this conditionis also necessary for semiloop unification and how it may beextended to loop unification.},

year = {2021},

institution = {CAS ICS / RISC},

length = {8}

}

### AlCons: Deductive Synthesis of Sorting Algorithms in Theorema

#### I. Dramnesc, T. Jebelean

In: Theoretical Aspects of Computing - ICTAC 2021, A. Cerone, P. Csaba Ölveczky (ed.), Proceedings of 18th International Colloquium on Theoretical Aspects of Computing - ICTAC, Nur-Sultan, Kazakhstan, LNCS 12819, pp. 314-333. September 2021. Springer, 978-3-030-85315-0. [pdf]**inproceedings**{RISC6478,

author = {I. Dramnesc and T. Jebelean},

title = {{AlCons: Deductive Synthesis of Sorting Algorithms in Theorema}},

booktitle = {{Theoretical Aspects of Computing - ICTAC 2021}},

language = {english},

abstract = {We describe the principles and the implementation of AlCons (em Algorithm Constructor), a system for the automatic proof--based synthesis of sorting algorithms on lists and on binary trees, in the frame of the Theorema system. The core of the system is a dedicated prover based on specific inference rules and strategies for constructive proofs over the domains of lists and of binary trees, aimed at the automatic synthesis of sorting algorithms and their auxiliary functions from logical specifications. The specific distinctive feature of our approach is the use of multisets for expressing the fact that two lists (trees) have the same elements. This allows a more natural expression of the properties related to sorting, compared to the classical approach using the permutation relation (a list is a permutation of another). Moreover, the use of multisets leads to special inference rules and strategies which make the proofs more efficient, as for instance: expand/compress multiset terms and solve meta-variables using multiset equalities. Additionally we use a Noetherian induction strategy based on the relation induced by the strict inclusion of multisets, which facilitates the synthesis of arbitrary recursion structures, without having to indicate the recursion schemes in advance. The necessary auxiliary algorithms (like, e.g., for insertion and merging) are generated by the same principles from the synthesis conjectures that are automatically produced during the main proof, using a ``cascading" method, which in fact contributes to the automation of theory exploration. The prover is implemented in the frame of the Theorema system and works in natural style, while the generated algorithms can be immediately tested in the same system.},

series = {LNCS},

volume = {12819},

pages = {314--333},

publisher = {Springer},

isbn_issn = {978-3-030-85315-0},

year = {2021},

month = {September },

editor = {A. Cerone and P. Csaba Ölveczky},

refereed = {yes},

length = {20},

conferencename = {18th International Colloquium on Theoretical Aspects of Computing - ICTAC, Nur-Sultan, Kazakhstan}

}

### 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. [doi] [pdf]**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}

}

### A Heuristic Prover for Elementary Analysis in Theorema

#### Tudor Jebelean

Technical report no. 21-07 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). April 2021. [doi] [pdf]**techreport**{RISC6293,

author = {Tudor Jebelean},

title = {{A Heuristic Prover for Elementary Analysis in Theorema}},

language = {english},

abstract = {We present the application of certain heuristic techniques for the automation of proofs in elementary analysis. the techniques used are: the S-decomposition method for formulae with alternating quantifiers, quantifier elimination by cylindrical algebraic decomposition, analysis of terms behavior in zero, bounding the [Epsilon]-bounds, semantic simplification of expressions involving absolute value, polynomial arithmetic, usage of equal arguments to arbitrary functions, and automatic reordering of proof steps in order to check the admisibility of solutions to the metavariables. The proofs are very similar to those produced automatically, but they are edited for readability and aspect, and also for inserting the appropriate explanation about the use of the proof techniques. The proofs are: convergence of product of two sequences, continuity of the sum of two functions, uniform continuity of the sum of two functions, uniform continuity of the product of two functions, and continuity of the composition of functions.},

number = {21-07},

year = {2021},

month = {April},

keywords = {Theorema, S-decomposition, automated theorem proving},

length = {29},

type = {RISC Report Series},

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

address = {Altenberger Straße 69, 4040 Linz, Austria},

issn = {2791-4267 (online)}

}

### A Heuristic Prover for Elementary Analysis in Theorema

#### T. Jebelean

In: Intelligent Computer Mathematics, 14th International Conference, CICM 2021, Timisoara, Romania, F. Kamaredine, C. Sacerdoti Coen (ed.), Proceedings of Intelligent Computer Mathematics, 14th International Conference, CICM 2021, LNAI 12833, pp. 130-134. July 2021. Springer, 978-3-030-81096-2. [pdf]**inproceedings**{RISC6477,

author = {T. Jebelean},

title = {{A Heuristic Prover for Elementary Analysis in Theorema}},

booktitle = {{Intelligent Computer Mathematics, 14th International Conference, CICM 2021, Timisoara, Romania}},

language = {english},

abstract = { We present a plug-in to the Theorema system, which generates proofs similar to those produced by humans for theorems in elementary analysis and is based on heuristic techniques combining methods from automated reasoning and computer algebra. The prover is able to construct automatically natural-style proofs for various examples related to convergence of sequences as well as to limits, continuity, and uniform continuity of functions. Additionally to general inference rules for predicate logic, the techniques used are: the S-decomposition method for formulae with alternating quantifiers, use of Quantifier Elimination by Cylindrical Algebraic Decomposition, analysis of terms behavior in zero, bounding the epsilon-bounds, semantic simplification of expressions involving absolute value, polynomial arithmetic, usage of equal arguments to arbitrary functions, and automatic reordering of proof steps in order to check the admissibility of solutions to the metavariables. The problem of proving such theorems directly without using refutation and clausification is logically equivalent to the problem of satisfiability modulo the theory of real numbers, thus these techniques are relevant for SMT solving also.},

series = {LNAI},

volume = {12833},

pages = {130--134},

publisher = {Springer},

isbn_issn = {978-3-030-81096-2},

year = {2021},

month = {July},

editor = {F. Kamaredine and C. Sacerdoti Coen},

refereed = {yes},

keywords = {Satisfiability Checking, Natural-style Proofs, Computer Algebra, Symbolic Computation, Satisfiability Modulo Theories},

length = {5},

conferencename = {Intelligent Computer Mathematics, 14th International Conference, CICM 2021}

}