# Computer Algebra Algorithms for q-Series and Combinatorial Identities

### Project Lead

### Project Duration

01/10/1989 - 31/03/1993## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2021

[Ablinger]

### Extensions of the AZ-algorithm and the Package MultiIntegrate

#### J. Ablinger

Technical report no. 21-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2021. [doi] [pdf]@

author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

language = {english},

abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},

number = {21-02},

year = {2021},

month = {January},

keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},

length = {25},

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)}

}

**techreport**{RISC6272,author = {J. Ablinger},

title = {{Extensions of the AZ-algorithm and the Package MultiIntegrate}},

language = {english},

abstract = {We extend the (continuous) multivariate Almkvist-Zeilberger algorithm inorder to apply it for instance to special Feynman integrals emerging in renormalizable Quantum field Theories. We will consider multidimensional integrals overhyperexponential integrals and try to find closed form representations in terms ofnested sums and products or iterated integrals. In addition, if we fail to computea closed form solution in full generality, we may succeed in computing the firstcoeffcients of the Laurent series expansions of such integrals in terms of indefnitenested sums and products or iterated integrals. In this article we present the corresponding methods and algorithms. Our Mathematica package MultiIntegrate,can be considered as an enhanced implementation of the (continuous) multivariateAlmkvist Zeilberger algorithm to compute recurrences or differential equations forhyperexponential integrands and integrals. Together with the summation packageSigma and the package HarmonicSums our package provides methods to computeclosed form representations (or coeffcients of the Laurent series expansions) of multidimensional integrals over hyperexponential integrands in terms of nested sums oriterated integrals.},

number = {21-02},

year = {2021},

month = {January},

keywords = {multivariate Almkvist-Zeilberger algorithm, hyperexponential integrals, iterated integrals, nested sums},

length = {25},

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)}

}

[Cerna]

### A Special Case of Schematic Syntactic Unification

#### David M. Cerna

CAS ICS / RISC. Technical report, 2021. [pdf]@

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}

}

**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}

}

[Dundua]

### Variadic equational matching in associative and commutative theories

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

Journal of Symbolic Computation 106, pp. 78-109. 2021. Elsevier, ISSN 0747-7171. [doi] [pdf]@

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}

}

**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}

}

[Falkensteiner]

### On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry

#### S. Falkensteiner, C. Garay-Lopez, M. Haiech, M. P. Noordman, F. Boulier, Z. Toghani

Journal of Symbolic Computation, pp. 1-22. 2021. ISSN: 0747-7171. [url]@

author = {S. Falkensteiner and C. Garay-Lopez and M. Haiech and M. P. Noordman and F. Boulier and Z. Toghani},

title = {{On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {1--22},

isbn_issn = {ISSN: 0747-7171},

year = {2021},

refereed = {yes},

keywords = {Differential Algebra, Tropical Differential Algebraic Geometry, Power Series Solutions, Newton Polyhedra, Arc Spaces, Tropical Differential Equations, Initial forms of Differential Polynomials},

length = {22},

url = {http://hal.archives-ouvertes.fr/hal-03122437v1/document}

}

**article**{RISC6335,author = {S. Falkensteiner and C. Garay-Lopez and M. Haiech and M. P. Noordman and F. Boulier and Z. Toghani},

title = {{On Initials and the Fundamental Theorem of Tropical Partial Differential Geometry}},

language = {english},

journal = {Journal of Symbolic Computation},

pages = {1--22},

isbn_issn = {ISSN: 0747-7171},

year = {2021},

refereed = {yes},

keywords = {Differential Algebra, Tropical Differential Algebraic Geometry, Power Series Solutions, Newton Polyhedra, Arc Spaces, Tropical Differential Equations, Initial forms of Differential Polynomials},

length = {22},

url = {http://hal.archives-ouvertes.fr/hal-03122437v1/document}

}

[Grasegger]

### Combinatorics of Bricard's octahedra

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

Comptes Rendus. Mathématique 359(1), pp. 7-38. 2021. Académie des sciences, Paris, ISSN 1631-073X. [doi]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard's octahedra}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {359},

number = {1},

pages = {7--38},

publisher = {Académie des sciences, Paris},

isbn_issn = {ISSN 1631-073X},

year = {2021},

refereed = {yes},

length = {32},

url = {https://doi.org/10.5802/crmath.132}

}

**article**{RISC6288,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{Combinatorics of Bricard's octahedra}},

language = {english},

journal = {Comptes Rendus. Mathématique},

volume = {359},

number = {1},

pages = {7--38},

publisher = {Académie des sciences, Paris},

isbn_issn = {ISSN 1631-073X},

year = {2021},

refereed = {yes},

length = {32},

url = {https://doi.org/10.5802/crmath.132}

}

[Grasegger]

### On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere

#### M. Gallet, G. Grasegger, J. Legerský, J. Schicho

SIAM Journal on Discrete Mathematics 35(1), pp. 325-361. 2021. ISSN 0895-4801. [doi]@

author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},

language = {english},

journal = {SIAM Journal on Discrete Mathematics},

volume = {35},

number = {1},

pages = {325--361},

isbn_issn = {ISSN 0895-4801},

year = {2021},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1137/19M1289467}

}

**article**{RISC6290,author = {M. Gallet and G. Grasegger and J. Legerský and J. Schicho},

title = {{On the Existence of Paradoxical Motions of Generically Rigid Graphs on the Sphere}},

language = {english},

journal = {SIAM Journal on Discrete Mathematics},

volume = {35},

number = {1},

pages = {325--361},

isbn_issn = {ISSN 0895-4801},

year = {2021},

refereed = {yes},

length = {37},

url = {https://doi.org/10.1137/19M1289467}

}

[Hemmecke]

### Construction of Modular Function Bases for $Gamma_0(121)$ related to $p(11n+6)$

#### Ralf Hemmecke, Peter Paule, Silviu Radu

Integral Transforms and Special Functions 32(5-8), pp. 512-527. 2021. Taylor & Francis, 1065-2469. [doi]@

author = {Ralf Hemmecke and Peter Paule and Silviu Radu},

title = {{Construction of Modular Function Bases for $Gamma_0(121)$ related to $p(11n+6)$}},

language = {english},

abstract = {Motivated by arithmetic properties of partitionnumbers $p(n)$, our goal is to find algorithmicallya Ramanujan type identity of the form$sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is apolynomial in products of the form$e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$with $alpha=0,1,2$. To this end we multiply theleft side by an appropriate factor such the resultis a modular function for $Gamma_0(121)$ havingonly poles at infinity. It turns out thatpolynomials in the $e_alpha$ do not generate thefull space of such functions, so we were led tomodify our goal. More concretely, we give threedifferent ways to construct the space of modularfunctions for $Gamma_0(121)$ having only poles atinfinity. This in turn leads to three differentrepresentations of $R$ not solely in terms of the$e_alpha$ but, for example, by using as generatorsalso other functions like the modular invariant $j$.},

journal = {Integral Transforms and Special Functions},

volume = {32},

number = {5-8},

pages = {512--527},

publisher = {Taylor & Francis},

isbn_issn = {1065-2469},

year = {2021},

refereed = {yes},

keywords = {Ramanujan identities, bases for modular functions, integral bases},

sponsor = {FWF (SFB F50-06)},

length = {16},

url = {https://doi.org/10.1080/10652469.2020.1806261}

}

**article**{RISC6342,author = {Ralf Hemmecke and Peter Paule and Silviu Radu},

title = {{Construction of Modular Function Bases for $Gamma_0(121)$ related to $p(11n+6)$}},

language = {english},

abstract = {Motivated by arithmetic properties of partitionnumbers $p(n)$, our goal is to find algorithmicallya Ramanujan type identity of the form$sum_{n=0}^{infty}p(11n+6)q^n=R$, where $R$ is apolynomial in products of the form$e_alpha:=prod_{n=1}^{infty}(1-q^{11^alpha n})$with $alpha=0,1,2$. To this end we multiply theleft side by an appropriate factor such the resultis a modular function for $Gamma_0(121)$ havingonly poles at infinity. It turns out thatpolynomials in the $e_alpha$ do not generate thefull space of such functions, so we were led tomodify our goal. More concretely, we give threedifferent ways to construct the space of modularfunctions for $Gamma_0(121)$ having only poles atinfinity. This in turn leads to three differentrepresentations of $R$ not solely in terms of the$e_alpha$ but, for example, by using as generatorsalso other functions like the modular invariant $j$.},

journal = {Integral Transforms and Special Functions},

volume = {32},

number = {5-8},

pages = {512--527},

publisher = {Taylor & Francis},

isbn_issn = {1065-2469},

year = {2021},

refereed = {yes},

keywords = {Ramanujan identities, bases for modular functions, integral bases},

sponsor = {FWF (SFB F50-06)},

length = {16},

url = {https://doi.org/10.1080/10652469.2020.1806261}

}

[Jebelean]

### 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]@

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)}

}

**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)}

}

[Jimenez Pastor]

### On C2-Finite Sequences

#### Antonio Jiménez-Pastor, Philipp Nuspl, Veronika Pillwein

In: Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation, Frédéric Chyzak, George Labahn (ed.), ISSAC '21 , pp. 217-224. 2021. Association for Computing Machinery, New York, NY, USA, ISBN 9781450383820. [doi]@

author = {Antonio Jiménez-Pastor and Philipp Nuspl and Veronika Pillwein},

title = {{On C2-Finite Sequences}},

booktitle = {{Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {Holonomic sequences are widely studied as many objects interesting to mathematiciansand computer scientists are in this class. In the univariate case, these are the sequencessatisfying linear recurrences with polynomial coefficients and also referred to asD-finite sequences. A subclass are C-finite sequences satisfying a linear recurrencewith constant coefficients.We investigate the set of sequences which satisfy linearrecurrence equations with coefficients that are C-finite sequences. These sequencesare a natural generalization of holonomic sequences. In this paper, we show that C2-finitesequences form a difference ring and provide methods to compute in this ring.},

series = {ISSAC '21},

pages = {217--224},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {ISBN 9781450383820},

year = {2021},

editor = {Frédéric Chyzak and George Labahn},

refereed = {yes},

keywords = {holonomic sequences, algorithms, closure properties, difference equations},

length = {8},

url = {https://doi.org/10.1145/3452143.3465529}

}

**inproceedings**{RISC6348,author = {Antonio Jiménez-Pastor and Philipp Nuspl and Veronika Pillwein},

title = {{On C2-Finite Sequences}},

booktitle = {{Proceedings of the 2021 on International Symposium on Symbolic and Algebraic Computation}},

language = {english},

abstract = {Holonomic sequences are widely studied as many objects interesting to mathematiciansand computer scientists are in this class. In the univariate case, these are the sequencessatisfying linear recurrences with polynomial coefficients and also referred to asD-finite sequences. A subclass are C-finite sequences satisfying a linear recurrencewith constant coefficients.We investigate the set of sequences which satisfy linearrecurrence equations with coefficients that are C-finite sequences. These sequencesare a natural generalization of holonomic sequences. In this paper, we show that C2-finitesequences form a difference ring and provide methods to compute in this ring.},

series = {ISSAC '21},

pages = {217--224},

publisher = {Association for Computing Machinery},

address = {New York, NY, USA},

isbn_issn = {ISBN 9781450383820},

year = {2021},

editor = {Frédéric Chyzak and George Labahn},

refereed = {yes},

keywords = {holonomic sequences, algorithms, closure properties, difference equations},

length = {8},

url = {https://doi.org/10.1145/3452143.3465529}

}

[Kutsia]

### 9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), short and work-in-progress papers

#### Temur Kutsia (editor)

Technical report no. 21-16 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). August 2021. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Temur Kutsia (editor)},

title = {{9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), short and work-in-progress papers}},

language = {english},

abstract = {This collection contains short and work-in-progress papers presented at the 9th International Symposium on Symbolic Computation in Software Science, SCSS 2021.},

number = {21-16},

year = {2021},

month = {August},

keywords = {Symbolic computation, software science, artificial intelligence},

length = {56},

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)}

}

**techreport**{RISC6359,author = {Temur Kutsia (editor)},

title = {{9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), short and work-in-progress papers}},

language = {english},

abstract = {This collection contains short and work-in-progress papers presented at the 9th International Symposium on Symbolic Computation in Software Science, SCSS 2021.},

number = {21-16},

year = {2021},

month = {August},

keywords = {Symbolic computation, software science, artificial intelligence},

length = {56},

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)}

}

[Legersky]

### On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2

#### E. Bartzos, I.Z. Emiris, J. Legerský, E. Tsigaridas

Journal of Symbolic Computation 102, pp. 189-208. 2021. ISSN 0747-7171. [doi]@

author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {102},

pages = {189--208},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1016/j.jsc.2019.10.015}

}

**article**{RISC5992,author = {E. Bartzos and I.Z. Emiris and J. Legerský and E. Tsigaridas},

title = {{On the maximal number of real embeddings of minimally rigid graphs in R2, R3 and S2}},

language = {english},

journal = {Journal of Symbolic Computation},

volume = {102},

pages = {189--208},

isbn_issn = {ISSN 0747-7171},

year = {2021},

refereed = {yes},

length = {20},

url = {https://doi.org/10.1016/j.jsc.2019.10.015}

}

[Maletzky]

### A generic and executable formalization of signature-based Gröbner basis algorithms

#### Alexander Maletzky

J. Symb. Comput. 106, pp. 23-47. 2021. Elsevier, ISSN 0747-7171. arXiv:2012.02239 [cs.SC], https://doi.org/10.1016/j.jsc.2020.12.001. [url]@

author = {Alexander Maletzky},

title = {{A generic and executable formalization of signature-based Gröbner basis algorithms}},

language = {english},

journal = {J. Symb. Comput.},

volume = {106},

pages = {23--47},

publisher = {Elsevier},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:2012.02239 [cs.SC], https://doi.org/10.1016/j.jsc.2020.12.001},

refereed = {yes},

length = {25},

url = {https://arxiv.org/abs/2012.02239}

}

**article**{RISC6225,author = {Alexander Maletzky},

title = {{A generic and executable formalization of signature-based Gröbner basis algorithms}},

language = {english},

journal = {J. Symb. Comput.},

volume = {106},

pages = {23--47},

publisher = {Elsevier},

isbn_issn = {ISSN 0747-7171},

year = {2021},

note = {arXiv:2012.02239 [cs.SC], https://doi.org/10.1016/j.jsc.2020.12.001},

refereed = {yes},

length = {25},

url = {https://arxiv.org/abs/2012.02239}

}

[Mitteramskogler]

### General solutions of first-order algebraic ODEs in simple constant extensions

#### Johann J. Mitteramskogler, Franz Winkler

Technical report no. 21-18 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). September 2021. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Johann J. Mitteramskogler and Franz Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

abstract = {If a first-order algebraic ODE is defined over a certain differential field, then the most elementary solution class, in which one can hope to find a general solution, is given by the adjunction of a single arbitrary constant to this field. Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial. As for the opposite direction, we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field. These automorphisms are determined by linear rational functions, i.e. Möbius transformations. Intrinsic properties of rational parametrizations, in combination with the particular shape of such automorphisms, lead to a number of necessary conditions on the existence of general solutions in this solution class. Furthermore, the desired linear rational function can be determined by solving a simple differential system over the ODE’s field of definition. All results are derived in a purely algebraic fashion and apply to any differential field of characteristic zero with arbitrary derivative operator.},

number = {21-18},

year = {2021},

month = {September},

keywords = {Algebraic ordinary differential equation, general solution, algebraic curve, rational parametrization},

length = {17},

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)}

}

**techreport**{RISC6364,author = {Johann J. Mitteramskogler and Franz Winkler},

title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},

language = {english},

abstract = {If a first-order algebraic ODE is defined over a certain differential field, then the most elementary solution class, in which one can hope to find a general solution, is given by the adjunction of a single arbitrary constant to this field. Solutions of this type give rise to a particular kind of generic point—a rational parametrization—of an algebraic curve which is associated in a natural way to the ODE’s defining polynomial. As for the opposite direction, we show that a suitable rational parametrization of the associated curve can be extended to a general solution of the ODE if and only if one can find a certain automorphism of the solution field. These automorphisms are determined by linear rational functions, i.e. Möbius transformations. Intrinsic properties of rational parametrizations, in combination with the particular shape of such automorphisms, lead to a number of necessary conditions on the existence of general solutions in this solution class. Furthermore, the desired linear rational function can be determined by solving a simple differential system over the ODE’s field of definition. All results are derived in a purely algebraic fashion and apply to any differential field of characteristic zero with arbitrary derivative operator.},

number = {21-18},

year = {2021},

month = {September},

keywords = {Algebraic ordinary differential equation, general solution, algebraic curve, rational parametrization},

length = {17},

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)}

}

[Pau]

### Proximity-Based Unification and Matching for Full Fuzzy Signatures

#### Temur Kutsia, Cleo Pau

Technical report no. 21-08 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). April 2021. [doi] [pdf]@

author = {Temur Kutsia and Cleo Pau},

title = {{Proximity-Based Unification and Matching for Full Fuzzy Signatures}},

language = {english},

abstract = {Proximity relations are binary fuzzy relations, which are reflexiveand symmetric, but not transitive. We propose proximity-based unification and matching algorithms in fuzzy languages whose signaturestolerate mismatches in function symbol names, arity, and in the arguments order (so called full fuzzy signatures). This work generalizeson the one hand, proximity-based unification to full fuzzy signatures,and on the other hand, similarity-based unification over a full fuzzysignature by extending similarity to proximity.},

number = {21-08},

year = {2021},

month = {April},

keywords = {Fuzzy proximity relations, Unification Matching, Arity mismatch},

length = {15},

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)}

}

**techreport**{RISC6296,author = {Temur Kutsia and Cleo Pau},

title = {{Proximity-Based Unification and Matching for Full Fuzzy Signatures}},

language = {english},

abstract = {Proximity relations are binary fuzzy relations, which are reflexiveand symmetric, but not transitive. We propose proximity-based unification and matching algorithms in fuzzy languages whose signaturestolerate mismatches in function symbol names, arity, and in the arguments order (so called full fuzzy signatures). This work generalizeson the one hand, proximity-based unification to full fuzzy signatures,and on the other hand, similarity-based unification over a full fuzzysignature by extending similarity to proximity.},

number = {21-08},

year = {2021},

month = {April},

keywords = {Fuzzy proximity relations, Unification Matching, Arity mismatch},

length = {15},

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)}

}

[Pau]

### Generalization Algorithms with Proximity Relations in Full Fuzzy Signatures

#### Temur Kutsia, Cleo Pau

Technical report no. 21-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). April 2021. [doi] [pdf]@

author = {Temur Kutsia and Cleo Pau},

title = {{Generalization Algorithms with Proximity Relations in Full Fuzzy Signatures}},

language = {english},

abstract = {Anti-unification aims at computing generalizations for given terms,retaining their common structure and abstracting differences by variables. We study anti-unification for full fuzzy signatures, where thenotion of common structure is relaxed into a ``proximal'' one with re-spect to a given proximity relation. Mismatches between both symbolnames and their arities are permitted. We develop algorithms for different cases of the problem and study their properties.},

number = {21-09},

year = {2021},

month = {April},

keywords = {Fuzzy proximity relations, Generalization, Anti-unification, Arity mismatch},

length = {15},

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)}

}

**techreport**{RISC6297,author = {Temur Kutsia and Cleo Pau},

title = {{Generalization Algorithms with Proximity Relations in Full Fuzzy Signatures}},

language = {english},

abstract = {Anti-unification aims at computing generalizations for given terms,retaining their common structure and abstracting differences by variables. We study anti-unification for full fuzzy signatures, where thenotion of common structure is relaxed into a ``proximal'' one with re-spect to a given proximity relation. Mismatches between both symbolnames and their arities are permitted. We develop algorithms for different cases of the problem and study their properties.},

number = {21-09},

year = {2021},

month = {April},

keywords = {Fuzzy proximity relations, Generalization, Anti-unification, Arity mismatch},

length = {15},

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)}

}

[Pau]

### Proximity-Based Unification and Matching for Fully Fuzzy Signatures

#### Cleo Pau, Temur Kutsia

In: Proceedings of the 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, , pp. 1-6. 2021. IEEE, isbn 978-1-6654-4407-1. [doi] [pdf]@

author = {Cleo Pau and Temur Kutsia},

title = {{Proximity-Based Unification and Matching for Fully Fuzzy Signatures}},

booktitle = {{Proceedings of the 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021}},

language = {english},

pages = {1--6},

publisher = {IEEE},

isbn_issn = {isbn 978-1-6654-4407-1},

year = {2021},

editor = { },

refereed = {yes},

length = {6},

url = {https://doi.org/10.1109/FUZZ45933.2021.9494438}

}

**inproceedings**{RISC6369,author = {Cleo Pau and Temur Kutsia},

title = {{Proximity-Based Unification and Matching for Fully Fuzzy Signatures}},

booktitle = {{Proceedings of the 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021}},

language = {english},

pages = {1--6},

publisher = {IEEE},

isbn_issn = {isbn 978-1-6654-4407-1},

year = {2021},

editor = { },

refereed = {yes},

length = {6},

url = {https://doi.org/10.1109/FUZZ45933.2021.9494438}

}

[Paule]

### An Invitation to Analytic Combinatorics

#### Peter Paule (ed.), Stephen Melczer

Texts and Monographs in Symbolic Computation 1st edition, 2021. Springer, 978-3-030-67080-1.@

author = {Peter Paule (ed.) and Stephen Melczer},

title = {{An Invitation to Analytic Combinatorics}},

language = {english},

series = {Texts and Monographs in Symbolic Computation},

publisher = {Springer},

isbn_issn = {978-3-030-67080-1},

year = {2021},

edition = {1st},

translation = {0},

length = {405}

}

**book**{RISC6277,author = {Peter Paule (ed.) and Stephen Melczer},

title = {{An Invitation to Analytic Combinatorics}},

language = {english},

series = {Texts and Monographs in Symbolic Computation},

publisher = {Springer},

isbn_issn = {978-3-030-67080-1},

year = {2021},

edition = {1st},

translation = {0},

length = {405}

}

[Paule]

### Contiguous Relations and Creative Telescoping

#### Peter Paule

In: Anti-Differentiation and the Calculation of Feynman Amplitudes, J. Bluemlein and C. Schneider (ed.), Texts and Monographs in Symbolic Computation , pp. -. 2021. Springer, ISBN 978-3-030-80218-9. To appear. [pdf]@

author = {Peter Paule},

title = {{Contiguous Relations and Creative Telescoping}},

booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},

language = {english},

abstract = {This article presents an algorithmic theory of contiguous relations.Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non- terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.},

series = {Texts and Monographs in Symbolic Computation},

pages = {--},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-80218-9},

year = {2021},

note = {To appear},

editor = {J. Bluemlein and C. Schneider},

refereed = {yes},

length = {61}

}

**incollection**{RISC6366,author = {Peter Paule},

title = {{Contiguous Relations and Creative Telescoping}},

booktitle = {{Anti-Differentiation and the Calculation of Feynman Amplitudes}},

language = {english},

abstract = {This article presents an algorithmic theory of contiguous relations.Contiguous relations, first studied by Gauß, are a fundamental concept within the theory of hypergeometric series. In contrast to Takayama’s approach, which for elimination uses non-commutative Gröbner bases, our framework is based on parameterized telescoping and can be viewed as an extension of Zeilberger’s creative telescoping paradigm based on Gosper’s algorithm. The wide range of applications include elementary algorithmic explanations of the existence of classical formulas for non- terminating hypergeometric series such as Gauß, Pfaff-Saalschütz, or Dixon summation. The method can be used to derive new theorems, like a non-terminating extension of a classical recurrence established by Wilson between terminating 4F3-series. Moreover, our setting helps to explain the non-minimal order phenomenon of Zeilberger’s algorithm.},

series = {Texts and Monographs in Symbolic Computation},

pages = {--},

publisher = {Springer},

isbn_issn = {ISBN 978-3-030-80218-9},

year = {2021},

note = {To appear},

editor = {J. Bluemlein and C. Schneider},

refereed = {yes},

length = {61}

}

[Reichl]

### Semantic Evaluation versus SMT Solving in the RISCAL Model Checker

#### Wolfgang Schreiner, Franz-Xaver Reichl

Technical report no. 21-11 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). June 2021. Licensed under CC BY 4.0 International. [doi] [pdf]@

author = {Wolfgang Schreiner and Franz-Xaver Reichl},

title = {{Semantic Evaluation versus SMT Solving in the RISCAL Model Checker}},

language = {english},

abstract = {In this paper, we compare two alternative mechanisms for deciding the validity of first-order formulas over finite domains supported by the mathematical model checker RISCAL: first, the original built-in approach of “semantic evaluation” (based on an implementation of the denotational semantics of the RISCAL language) and, second, the later implemented approach of SMT solving (based on satisfiability preserving translations of RISCAL formulas to formulas in the SMT-LIB logic QF_UFBV, respectively to quantified SMT-LIB bitvector formulas). After a short presentation of the two approaches and a discussion oftheir fundamental pros and cons, we quantitatively evaluate them, both by a set of artificial benchmarks and by a set of benchmarks taken from real-life applications of RISCAL; for this, we apply the state-of-the-art SMT solvers Boolector, CVC4, Yices, and Z3. Our benchmarks demonstrate that (while SMT solving generally vastly outperforms semantic evaluation), the various SMT solvers exhibit great performance differences. More important, our investigations also identify some classes of formulas where semantic evaluation is able to compete with (or even outperform) satisfiability solving, outlining some room forimprovements in the translation of RISCAL formulas to SMT-LIB formulas as well as in the current SMT technology.},

number = {21-11},

year = {2021},

month = {June},

keywords = {model checking, satisfiability solving, formal specification, formal verficiation},

sponsor = {JKU Linz Institute of Technology (LIT) Project LOGTECHEDU, Aktion Österreich- Slowakei Project 2019-10-15-003, Austrian Science Fund (FWF) grant W1255.},

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)}

}

**techreport**{RISC6328,author = {Wolfgang Schreiner and Franz-Xaver Reichl},

title = {{Semantic Evaluation versus SMT Solving in the RISCAL Model Checker}},

language = {english},

abstract = {In this paper, we compare two alternative mechanisms for deciding the validity of first-order formulas over finite domains supported by the mathematical model checker RISCAL: first, the original built-in approach of “semantic evaluation” (based on an implementation of the denotational semantics of the RISCAL language) and, second, the later implemented approach of SMT solving (based on satisfiability preserving translations of RISCAL formulas to formulas in the SMT-LIB logic QF_UFBV, respectively to quantified SMT-LIB bitvector formulas). After a short presentation of the two approaches and a discussion oftheir fundamental pros and cons, we quantitatively evaluate them, both by a set of artificial benchmarks and by a set of benchmarks taken from real-life applications of RISCAL; for this, we apply the state-of-the-art SMT solvers Boolector, CVC4, Yices, and Z3. Our benchmarks demonstrate that (while SMT solving generally vastly outperforms semantic evaluation), the various SMT solvers exhibit great performance differences. More important, our investigations also identify some classes of formulas where semantic evaluation is able to compete with (or even outperform) satisfiability solving, outlining some room forimprovements in the translation of RISCAL formulas to SMT-LIB formulas as well as in the current SMT technology.},

number = {21-11},

year = {2021},

month = {June},

keywords = {model checking, satisfiability solving, formal specification, formal verficiation},

sponsor = {JKU Linz Institute of Technology (LIT) Project LOGTECHEDU, Aktion Österreich- Slowakei Project 2019-10-15-003, Austrian Science Fund (FWF) grant W1255.},

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)}

}

[Reichl]

### First-Order Logic in Finite Domains: Where Semantic Evaluation Competes with SMT Solving

#### Wolfgang Schreiner, Franz-Xaver Reichl

In: Proceedings of the 9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), Hagenberg, Austria, September 8-10, 2021, Temur Kutsia (ed.), Electronic Proceedings in Theoretical Computer Science 342, pp. 99-113. September 2021. ISSN 2075-2180. [doi]@

author = {Wolfgang Schreiner and Franz-Xaver Reichl},

title = {{First-Order Logic in Finite Domains: Where Semantic Evaluation Competes with SMT Solving}},

booktitle = {{Proceedings of the 9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), Hagenberg, Austria, September 8-10, 2021}},

language = {english},

abstract = {In this paper, we compare two alternative mechanisms for deciding the validity of first-order formulas over finite domains supported by the mathematical model checker RISCAL: first, the original approach of semantic evaluation (based on an implementation of the denotational semantics of the RISCAL language) and, second, the later approach of SMT solving (based on satisfiability preserving translations of RISCAL formulas to SMT-LIB formulas as inputs for SMT solvers). After a short presentation of the two approaches and a discussion of their fundamental pros and cons, we quantitatively evaluate them, both by a set of artificial benchmarks and by a set of benchmarks taken from real-life applications of RISCAL; for this, we apply the state-of-the-art SMT solvers Boolector, CVC4, Yices, and Z3. Our benchmarks demonstrate that (while SMT solving generally vastly outperforms semantic evaluation), the various SMT solvers exhibit great performance differences. More important, we identify classes of formulas where semantic evaluation is able to compete with (or even outperform) satisfiability solving, outlining some room for improvements in the translation of RISCAL formulas to SMT-LIB formulas as well as in the current SMT technology.},

series = {Electronic Proceedings in Theoretical Computer Science},

volume = {342},

pages = {99--113},

isbn_issn = {ISSN 2075-2180},

year = {2021},

month = {September},

editor = {Temur Kutsia},

refereed = {yes},

keywords = {model checking, SMT solving, first-order logic, automated reasoning, formal methods},

sponsor = {e JKU Linz LIT Project LOGTECHEDU, Aktion Österreich-Slowakei Project 2019-10-15-003, Austrian Science Fund (FWF) grant W1255},

length = {15},

url = {https://doi.org/10.4204/EPTCS.342.9}

}

**inproceedings**{RISC6361,author = {Wolfgang Schreiner and Franz-Xaver Reichl},

title = {{First-Order Logic in Finite Domains: Where Semantic Evaluation Competes with SMT Solving}},

booktitle = {{Proceedings of the 9th International Symposium on Symbolic Computation in Software Science (SCSS 2021), Hagenberg, Austria, September 8-10, 2021}},

language = {english},

abstract = {In this paper, we compare two alternative mechanisms for deciding the validity of first-order formulas over finite domains supported by the mathematical model checker RISCAL: first, the original approach of semantic evaluation (based on an implementation of the denotational semantics of the RISCAL language) and, second, the later approach of SMT solving (based on satisfiability preserving translations of RISCAL formulas to SMT-LIB formulas as inputs for SMT solvers). After a short presentation of the two approaches and a discussion of their fundamental pros and cons, we quantitatively evaluate them, both by a set of artificial benchmarks and by a set of benchmarks taken from real-life applications of RISCAL; for this, we apply the state-of-the-art SMT solvers Boolector, CVC4, Yices, and Z3. Our benchmarks demonstrate that (while SMT solving generally vastly outperforms semantic evaluation), the various SMT solvers exhibit great performance differences. More important, we identify classes of formulas where semantic evaluation is able to compete with (or even outperform) satisfiability solving, outlining some room for improvements in the translation of RISCAL formulas to SMT-LIB formulas as well as in the current SMT technology.},

series = {Electronic Proceedings in Theoretical Computer Science},

volume = {342},

pages = {99--113},

isbn_issn = {ISSN 2075-2180},

year = {2021},

month = {September},

editor = {Temur Kutsia},

refereed = {yes},

keywords = {model checking, SMT solving, first-order logic, automated reasoning, formal methods},

sponsor = {e JKU Linz LIT Project LOGTECHEDU, Aktion Österreich-Slowakei Project 2019-10-15-003, Austrian Science Fund (FWF) grant W1255},

length = {15},

url = {https://doi.org/10.4204/EPTCS.342.9}

}