## RISC Reports Series

[Schneider]

### The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Technical report no. 22-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6487,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms}},
language = {english},
abstract = {We calculate the unpolarized and polarized two--loop massless off--shell operator matrix elements in QCD to $O(ep)$ in the dimensional parameter in an automated way. Here we use the method of arbitrary high Mellin moments and difference ring theory, based on integration-by-parts relations. This method also constitutes one way to compute the QCD anomalous dimensions. The presented higher order contributions to these operator matrix elements occur as building blocks in the corresponding higher order calculations upto four--loop order. All contributing quantities can be expressed in terms of harmonic sums in Mellin--$N$ space or by harmonic polylogarithms in $z$--space. We also perform comparisons to the literature. },
number = {22-01},
year = {2022},
month = {February},
keywords = {QCD, Operator Matrix Element, 2-loop Feynman diagrams, computer algebra, large moment method},
length = {101},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Nuspl]

### Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences

#### P. Nuspl, V. Pillwein

Technical report no. 22-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2022. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6479,
author = {P. Nuspl and V. Pillwein},
title = {{Simple $C^2$-finite Sequences: a Computable Generalization of $C$-finite Sequences}},
language = {english},
abstract = {The class of $C^2$-finite sequences is a natural generalization of holonomic sequences and consists of sequences satisfying a linear recurrence with C-finite coefficients, i.e., coefficients satisfying a linear recurrence with constant coefficients themselves. Recently, we investigated computational properties of $C^2$-finite sequences: we showed that these sequences form a difference ring and provided methods to compute in this ring.From an algorithmic point of view, some of these results were not as far reaching as we hoped for. In this paper, we define the class of simple $C^2$-finite sequences and show that it satisfies the same computational properties, but does not share the same technical issues. In particular, we are able to derive bounds for the asymptotic behavior, can compute closure properties more efficiently, and have a characterization via the generating function.},
number = {22-02},
year = {2022},
month = {February},
keywords = {difference equations, holonomic sequences, closure properties, generating functions, algorithms},
length = {16},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

#### J. Blümlein, C. Schneider

Technical report no. 22-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2022. arXiv:2203.13015 [hep-th]. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6495,
author = {J. Blümlein and C. Schneider},
title = {{The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals}},
language = {english},
abstract = {The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.},
number = {22-03},
year = {2022},
month = {March},
note = {arXiv:2203.13015 [hep-th]},
keywords = {Feynman integrals, computer algebra, special functions, linear differential equations, linear difference integrals},
length = {40},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Pau]

### 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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Nuspl]

### A comparison of algorithms for proving positivity of linearly recurrent sequences

#### P. Nuspl, V. Pillwein

Technical report no. 22-05 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{RISC6514,
author = {P. Nuspl and V. Pillwein},
title = {{A comparison of algorithms for proving positivity of linearly recurrent sequences}},
language = {english},
abstract = {Deciding positivity for recursively defined sequences based on only the recursive description as input is usually a non-trivial task. Even in the case of $C$-finite sequences, i.e., sequences satisfying a linear recurrence with constant coefficients, this is only known to be decidable for orders up to five. In this paper, we discuss several methods for proving positivity of $C$-finite sequences and compare their effectiveness on input from the Online Encyclopedia of Integer Sequences (OEIS).},
number = {22-05},
year = {2022},
month = {May},
keywords = {Difference equations Inequalities Holonomic sequences},
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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Nuspl]

### $C$-finite and $C^2$-finite Sequences in SageMath

#### P. Nuspl

Technical report no. 22-06 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{RISC6516,
author = {P. Nuspl},
title = {{$C$-finite and $C^2$-finite Sequences in SageMath}},
language = {english},
abstract = {We present the SageMath package rec_sequences which provides methods to compute with sequences satisfying linear recurrences. The package can be used to show inequalities of $C$-finite sequences, i.e., sequences satisfying a linear recurrence relation with constant coefficients. Furthermore, it provides functionality to compute in the $C^2$-finite sequence ring, i.e., to compute closure properties of sequences satisfying a linear recurrence with $C$-finite coefficients.},
number = {22-06},
year = {2022},
month = {June},
keywords = {Difference equations, Closure properties, Inequalities},
length = {4},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schreiner]

### 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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}

[Schneider]

### The Polarized Transition Matrix Element $A_{g, q}(N)$ of the Variable Flavor Number Scheme at $O(\alpha_s^3)$

#### A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, K. Schönwald, and C. Schneider

Technical report no. 21-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2021. Available also under arXiv:2101.05733 [hep-ph]. [doi] [pdf]
@techreport{RISC6250,
author = {A. Behring and J. Blümlein and A. De Freitas and A. von Manteuffel and K. Schönwald and and C. Schneider},
title = {{The Polarized Transition Matrix Element $A_{g,q}(N)$ of the Variable Flavor Number Scheme at $O(\alpha_s^3)$}},
language = {english},
abstract = {We calculate the polarized massive operator matrix element $A_{gq}^3(N)$ to 3-loop order inQuantum Chromodynamics analytically at general values of the Mellin variable $N$ both inthe single- and double-mass case in the Larin scheme. It is a transition function requiredin the variable flavor number scheme at $O(\alpha_s^3)$. We also present the results in momentumfraction space.},
number = {21-01},
year = {2021},
month = {January},
note = {Available also under arXiv:2101.05733 [hep-ph]},
keywords = {polarized massive operator matrix element, symbolic summation, harmonic sums, harmonic polylogarithm},
length = {25},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[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]
@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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation

#### C. Schneider

Technical report no. 21-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2021. [doi] [pdf]
@techreport{RISC6280,
author = {C. Schneider},
title = {{Term Algebras, Canonical Representations and Difference Ring Theory for Symbolic Summation}},
language = {english},
abstract = {A general overview of the existing difference ring theory for symbolicsummation is given. Special emphasis is put on the user interface: the translationand back translation of the corresponding representations within the term algebra andthe formal difference ring setting. In particular, canonical (unique) representationsand their refinements in the introduced term algebra are explored by utilizing theavailable difference ring theory. Based on that, precise input-output specifications ofthe available tools of the summation package Sigma are provided.},
number = {21-03},
year = {2021},
month = {February},
keywords = {term algebra, canonical representations, difference rings, indefinite nested sums, symbolic summation},
length = {55},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### Solving linear difference equations with coefficients in rings with idempotent representations

#### J. Ablinger, C. Schneider

Technical report no. 21-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2021. [doi] [pdf]
@techreport{RISC6284,
author = {J. Ablinger and C. Schneider},
title = {{Solving linear difference equations with coefficients in rings with idempotent representations}},
language = {english},
abstract = {We introduce a general reduction strategy that enables one to searchfor solutions of parameterized linear difference equations in difference rings. Here we assume that the ring itself can be decomposedby a direct sum of integral domains (using idempotent elements)that enjoys certain technical features and that the coeicients ofthe difference equation are not degenerated. Using this mechanismwe can reduce the problem to ind solutions in a ring (with zero-divisors) to search solutions in several copies of integral domains.Utilizing existing solvers in this integral domain setting, we obtaina general solver where the components of the linear differenceequations and the solutions can be taken from difference rings thatare built e.g., by $R\Pi\Sigma$-extensions over $\Pi\Sigma$-fields. This class of difference rings contains, e.g., nested sums and products, products overroots of unity and nested sums defined over such objects.},
number = {21-04},
year = {2021},
month = {February},
keywords = {linear difference equations, difference rings, idempotent elements},
length = {8},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### Iterated integrals over letters induced by quadratic forms

#### J. Ablinger, J. Blümlein, C. Schneider

Technical report no. 21-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2021. [doi] [pdf]
@techreport{RISC6289,
author = {J. Ablinger and J. Blümlein and C. Schneider},
title = {{Iterated integrals over letters induced by quadratic forms}},
language = {english},
abstract = {An automated treatment of iterated integrals based on letters induced by real-valuedquadratic forms and Kummer–Poincaré letters is presented. These quantities emerge inanalytic single and multi–scale Feynman diagram calculations. To compactify representations, one wishes to apply general properties of these quantities in computer-algebraicimplementations. We provide the reduction to basis representations, expansions, analyticcontinuation and numerical evaluation of these quantities.},
number = {21-05},
year = {2021},
month = {March},
keywords = {iterated integrals, quadratic forms, Kummer–Poincare letters},
length = {14},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering

#### J. Blümlein, A. De Freitas, M. Saragnese, K. Schönwald, C. Schneider

Technical report no. 21-06 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2021. [doi] [pdf]
@techreport{RISC6292,
author = {J. Blümlein and A. De Freitas and M. Saragnese and K. Schönwald and C. Schneider},
title = {{The Logarithmic Contributions to the Polarized $O(alpha_s^3)$ Asymptotic Massive Wilson Coefficients and Operator Matrix Elements in Deeply Inelastic Scattering}},
language = {english},
abstract = {We compute the logarithmic contributions to the polarized massive Wilson coefficients fordeep-inelastic scattering in the asymptotic region $Q^2\gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massiveoperator matrix elements needed in the variable flavor number scheme. The calculationis performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),PS}$ and $A_{qg,Q}^{(3),S}$the complete results are presented. The expressions are given in Mellin-$N$ space andin momentum fraction $z$-space.},
number = {21-06},
year = {2021},
month = {March},
keywords = {logarithmic contributions to the polarized massive Wilson coefficients, symbolic summation, harmonic sums, harmonic polylogarithm},
length = {86},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[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]
@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 StraÃŸe 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]
@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 StraÃŸe 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]
@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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[STUDENT]

### Refinement Types for Elm

#### Lucas Payr

Technical report no. 21-10 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). April 2021. Master Thesis. [doi] [pdf]
@techreport{RISC6303,
author = {Lucas Payr},
title = {{Refinement Types for Elm}},
language = {english},
abstract = {The aim of this thesis is to add refinement types to Elm. Elm is a pure functionalprogramming language that uses a Hindley-Miler type system. Refinement types aresubtypes of existing types. These subtypes are defined by a predicate that specifieswhich values are part of the subtypes. To extend a Hindley-Miler type system, onecan use so-called liquid types. These are special refinement types that come with analgorithm for type inference. This algorithm interacts with an SMT solver to solvesubtyping conditions. A type system using liquid types is not complete, meaning notevery valid program can be checked. Instead, liquid types are only defined for a subsetof expressions and only allow specific predicates. In this thesis, we give a formaldefinition of the Elm language and its type system. We extend the type system withliquid types and provide a subset of expressions and a subset of predicates such thatthe extended type system is sound. We use the free software system “K Framework”for rapid prototyping of the formal Elm type system. For rapid prototyping of thecore algorithm of the extended type system we implemented said algorithm in Elmand checked the subtyping condition in Microsoft’s SMT solver Z3.},
number = {21-10},
year = {2021},
month = {April},
note = {Master Thesis},
keywords = {formal semantics, formal type systems, programming languages, satisfiability solving},
length = {135},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[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]
@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 StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Winkler]

### My Life in Computer Algebra

#### F. Winkler

Technical report no. 21-12 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]
@techreport{RISC6329,
author = {F. Winkler},
title = {{My Life in Computer Algebra}},
language = {english},
abstract = {After having spent more than 40 years in Mathematics, and in particular in Computer Algebra, I recollect stages in my scientific career and I explain the connections between my different areas of interest.},
number = {21-12},
year = {2021},
month = {June},
keywords = {computer algebra, biography},
length = {19},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]

### The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements

#### J. Blümlein, P. Marquard, C. Schneider, K. Schönwald

Technical report no. 21-13 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). July 2021. Licensed under CC BY 4.0 International. [doi] [pdf]
@techreport{RISC6347,
author = {J. Blümlein and P. Marquard and C. Schneider and K. Schönwald},
title = {{The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements}},
language = {english},
abstract = {We calculate the unpolarized and polarized three--loop anomalous dimensions and splitting functions $P_{rm NS}^+, P_{rm NS}^-$ and $P_{rm NS}^{rm s}$ in QCD in the $overline{sf MS}$ scheme by using the traditional method of space--like off shell massless operator matrix elements. This is a gauge--dependent framework. For the first time we also calculate the three--loop anomalous dimensions $P_{rm NS}^{rm pm, tr}$ for transversity directly. We compare our results to the literature. },
number = {21-13},
year = {2021},
month = {July},
keywords = {particle physics, 3-loop anomalous dimensions, large moment method, recurrence guessing and solving},
length = {40},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger StraÃŸe 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}