# Miroslav Olšák's research while affiliated with University of Innsbruck and other places

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## Publications (33)

We correct the funding information for the article mentioned in the title.

We introduce a self-learning algorithm for synthesizing programs for OEIS sequences. The algorithm starts from scratch initially generating programs at random. Then it runs many iterations of a self-learning loop that interleaves (i) training neural machine translation to learn the correspondence between sequences and the programs discovered so far...

The appearance of strong CDCL-based propositional (SAT) solvers has greatly advanced several areas of automated reasoning (AR). One of the directions in AR is thus to apply SAT solvers to expressive formalisms such as first-order logic, for which large corpora of general mathematical problems exist today. This is possible due to Herbrand's theorem,...

We significantly improve the performance of the E automated theorem prover on the Isabelle Sledgehammer problems by combining learning and theorem proving in several ways. In particular, we develop targeted versions of the ENIGMA guidance for the Isabelle problems, targeted versions of neural premise selection, and targeted strategies for E. The me...

We describe several additions to the ENIGMA system that guides clause selection in the E automated theorem prover. First, we significantly speed up its neural guidance by adding server-based GPU evaluation. The second addition is motivated by fast weight-based rejection filters that are currently used in systems like E and Prover9. Such systems can...

Saturation-style automated theorem provers (ATPs) based on the given clause procedure are today the strongest general reasoners for classical first-order logic. The clause selection heuristics in such systems are, however, often evaluating clauses in isolation, ignoring other clauses. This has changed recently by equipping the E/ENIGMA system with...

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art l...

Saturation-style automated theorem provers (ATPs) based on the given clause procedure are today the strongest general reasoners for classical first-order logic. The clause selection heuristics in such systems are, however, often evaluating clauses in isolation, ignoring other clauses. This has changed recently by equipping the E/ENIGMA system with...

We describe several additions to the ENIGMA system that guides clause selection in the E automated theorem prover. First, we significantly speed up its neural guidance by adding server-based GPU evaluation. The second addition is motivated by fast weight-based rejection filters that are currently used in systems like E and Prover9. Such systems can...

We describe a purely image-based method for finding geometric constructions with a ruler and compass in the Euclidea geometric game. The method is based on adapting the Mask R-CNN state-of-the-art visual recognition neural architecture and adding a tree-based search procedure to it. In a supervised setting, the method learns to solve all 68 kinds o...

We describe a purely image-based method for finding geometric constructions with a ruler and compass in the Euclidea geometric game. The method is based on adapting the Mask R-CNN state-of-the-art image processing neural architecture and adding a tree-based search procedure to it. In a supervised setting, the method learns to solve all 68 kinds of...

In this work we study how to learn good algorithms for selecting reasoning steps in theorem proving. We explore this in the connection tableau calculus implemented by leanCoP where the partial tableau provides a clean and compact notion of a state to which a limited number of inferences can be applied. We start by incorporating a state-of-the-art l...

Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condi...

Domain of mathematical logic in computers is dominated by automated theorem provers (ATP) and interactive theorem provers (ITP). Both of these are hard to access by AI from the human-imitation approach: ATPs often use human-unfriendly logical foundations while ITPs are meant for formalizing existing proofs rather than problem solving. We aim to cre...

We describe an implementation of gradient boosting and neural guidance of saturation-style automated theorem provers that does not depend on consistent symbol names across problems. For the gradient-boosting guidance, we manually create abstracted features by considering arity-based encodings of formulas. For the neural guidance, we use symbol-inde...

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to the dic...

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and is NP-complete otherwise.
One of the important questions related to the...

Domain of mathematical logic in computers is dominated by automated theorem provers (ATP) and interactive theorem provers (ITP). Both of these are hard to access by AI from the human-imitation approach: ATPs often use human-unfriendly logical foundations while ITPs are meant for formalizing existing proofs rather than problem solving. We aim to cre...

We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed closed walks of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of weakest non-trivial idempotent equations, and that a finite strongly connected digraph of algebraic length 1 compatible...

We describe an implementation of gradient boosting and neural guidance of saturation-style automated theorem provers that does not depend on consistent symbol names across problems. For the gradient-boosting guidance, we manually create abstracted features by considering arity-based encodings of formulas. For the neural guidance, we use symbol-inde...

We prove that a weakest non-trivial strong Maltsev condition given by a single identity of the form t(variables)=t(variables)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{docume...

Automated reasoning and theorem proving have recently become major challenges for machine learning. In other domains, representations that are able to abstract over unimportant transformations, such as abstraction over translations and rotations in vision, are becoming more common. Standard methods of embedding mathematical formulas for learning th...

We prove that every strongly connected (not necessarily finite) digraph of algebraic length 1, which is compatible with an operation t satisfying a non-trivial identity of the form t(…)≈t(…), has a loop.

A PCSP is a combination of two CSPs defined by two similar templates; the computational question is to distinguish a YES instance of the first one from a NO instance of the second. The computational complexity of many PCSPs remains unknown. Even the case of Boolean templates (solved for CSP by Schaefer [STOC'78]) remains wide open. The main result...

We prove that an idempotent operation generates a loop from a strongly connected digraph containing directed cycles of all lengths under very mild (local) algebraic assumptions. Using the result, we reprove the existence of a weakest non-trivial idempotent equations, and that a strongly connected digraph with algebraic length 1 compatible with a Ta...

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable when the model-complete core of the template has a pseudo-Siggers polymorphism, and NP-complete otherwise. One of the important questions related to this...

Meet semidistributive varieties are in a sense the last of the most important classes in universal algebra for which it is unknown whether it can be characterized by a strong Maltsev condition. We present a new, relatively simple Maltsev condition characterizing the meet-semidistributive varieties, and provide a candidate for a strong Maltsev condi...

We prove that the existence of a term $s$ satisfying $s(r,a,r,e) = s(a,r,e,a)$ in a general algebraic structure is equivalent to an existence of a term $t$ satisfying $t(x,x,y,y,z,z)=t(y,z,z,x,x,y)$. As a consequence of a general version of this theorem and previous results we get that each strongly connected digraph of algebraic length one, which...

There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial identity of height one modulo oute...

An equational condition is a set of equations in an algebraic language, and an algebraic structure satisfies such a condition if it possesses terms that meet the required equations. We find a single nontrivial equational condition which is implied by any nontrivial idempotent equational condition.

## Citations

... In general, through iterative deepening the emitted proofs tend to be short, which again is useful for further processing, including integration with other systems and presentation for humans. Implementations following the approach are typically manageable and small (with leanCoP outbidding all others [34]), making them attractive for adaptation to specific logics [31,32,33] and novel combinations with other techniques [21,56,57,37,38,14]. Another aspect of the approach with potential long-term relevance is its role as a foundation for systematic investigations of first-order ATP, as for example in [6,13,23,52]. ...

... This can be achieved using specialized programming languages that facilitate the simulation of logical and mathematical thinking using a computer, such as Coq (development team, 2022), Isabelle (Wenzel et al., 2008), HOL4 (Harrison, 1996), Lean (de Moura et al., 2015), Metamath (Megill and Wheeler, 2019) and Mizar (Grabowski et al., 2010). Work on automation of proof assistants and automated theorem provers such as E (Schulz, 2013), leanCoP (Otten, 2008), and Vampire (Kovács and Voronkov, 2013) has substantially benefited from integration with machine learning methods (Alemi et al., 2016;Goertzel et al., 2021;Li et al., 2021;Polu and Sutskever, 2020;Kaliszyk et al., 2018). ...

... in 2011 [26]. The modern formulation of the conjecture based on recent progress [4,3,5] is the following: Conjecture 1. Let A be a CSP template which is a first-order reduct of a finitely bounded homogeneous structure. ...

... If witnessed by idempotent operations (Definition 7.6), (m + n)-terms characterize the universalalgebraic property SD(∧) for general varieties (see Theorem 3.1 in [39]). ...

... In the realm of computer algebra for symbolic integration and differential-equation solving, there is good progress in approaching these problems by means of deep neural networks [20], although more remains to be done [21]. ML has been applied to improving the ATP inference engines in ITV [22,23,24,25,26] based on deep reinforcement learning, GPT-2, graph neural networks, and k-nearest neighbors respectively, as well as on co-training on multiple related tasks. Conversely, ITV can be used to verify ML theory (e.g. ...

... Let A be the Fraïssé limit of the superposition of all the classes C(A i , k(i)). Then A is ω-categorical and even has small orbit growth (see [21], or Lemma 5.6 in [30]). By definition, it is homogeneous, and as in Proposition 37, one sees that it is a model-complete core since the Fraïssé limit of each of the superposed classes is. ...

... , Σ Zp n . This was already known before, see for instance [Olš20], where it is further shown that Σ Zn ≤ Σ Zm if and only if every prime divisor of n is also a prime divisor of m. ...

... On the one hand, Theorem 2 still shows that polymorphisms do not satisfy any nontrivial idempotent identities. On the other hand, T 3,3 /G is linked and has no loop, but T 3,3 satisfies some non-trivial identities, see [21,Example 6.3]. This shows, e.g., that one cannot switch in Theorem 1 "A linked" to "A/G linked" or "A symmetric non-bipartite", and that parameters are necessary in Theorem 2. The entire section is devoted to the proof of Theorem 7. We fix a finite smooth digraph A = (A; →) and a subgroup H of the ranked automorphism group of (A; → 01 ) whose projection is G. ...

... Moreover, the orbit growth, i.e., the growth of the number of n-orbits as n increases, can be taken to be smaller than doubly exponential. It was shown in [27], [28] that if A is an ωcategorical model-complete core whose orbit growth is smaller than 2 2 n , then A pp-interprets K 3 with parameters if, and only if, there exists a finite subset of A on which Pol(A) does not satisfy any non-trivial minor condition. Thus, our structure in Theorem 4 locally admits polymorphisms satisfying nontrivial minor conditions while still avoiding pseudo-WNU polymorphisms. ...

... Definition 5.26 ( [Olš17a]). Fix a loopless graph G, with vertices 1 , . . . , n and edges (a 1 , b 1 ), . . . ...