Miroslav Olšák - Academia.edu (original) (raw)

Papers by Miroslav Olšák

Nedávný výzkum v oblasti problému splnitelnosti podmínek vedl k užitečným nástrojům v uni- verzál... more Nedávný výzkum v oblasti problému splnitelnosti podmínek vedl k užitečným nástrojům v uni- verzální algebře a pro studium výpočetní složitosti. Tento výzkum byl zaměřen zejména na konečné re- lační struktury a tím pádem na konečné algebry. Práce zobecňuje tyto předchozí výsledky na nekonečné algebry. Ukážeme, že ačkoli Maltsevská podmínka t(p, i, s, i) = t(s, p, i, s) obecně necharakterizuje Tay- lorovské algebry (algebry splňující netriviální idempotentní Maltsevskou podmínku) jako v konečném případě, existuje jiná silná Maltsevská podmínka, která je charakterizuje, a t(p, i, s, i) = t(s, p, i, s) charakterizuje jinou širokou třídu algeber. Také najdeme (slabou) Maltsevskou podmínku pro SD(∧) algebry (algebry splňující idempotentní Maltssevskou podmínku, kterou nelze splnit v modulech). Vedle Maltsevskych podmínek zkoumáme smyčková lemmata. Speciálně dokážeme známé konečné smyčkové lemma pomocí dvou různých (nekonečných) přístupů.The recent research on constraint satisfaction probl...

Lecture Notes in Computer Science, 2020

Domain of mathematical logic in computers is dominated by automated theorem provers (ATP) and int... more 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 create a simple human-friendly logical system for mathematical problem solving. We picked the case study of Euclidean geometry as it can be easily visualized, has simple logic, and yet potentially offers many high-school problems of various difficulty levels. To make the environment user friendly, we abandoned strict logic required by ITPs, allowing to infer topological facts from pictures. We present our system for Euclidean geometry, together with a graphical application GeoLogic, similar to GeoGebra, which allows users to interactively study and prove properties about the geometrical setup.

International Journal of Algebra and Computation, 2019

We prove that every strongly connected (not necessarily finite) digraph of algebraic length 1, wh... more We prove that every strongly connected (not necessarily finite) digraph of algebraic length 1, which is compatible with an operation [Formula: see text] satisfying a non-trivial identity of the form [Formula: see text], has a loop.

arXiv (Cornell University), Mar 12, 2023

As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically prov... more As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60 % of the Mizar theorems in the hammer setting. We also automatically prove 75 % of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically.

arXiv (Cornell University), Jan 26, 2023

arXiv (Cornell University), Oct 7, 2022

The appearance of strong CDCL-based propositional (SAT) solvers has greatly advanced several area... more 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, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is choosing the right instances from the typically infinite Herbrand universe. In this work, we develop the first machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on an invariant graph neural network (GNN) that learns from problems and their solutions independently of symbol names (addressing the abundance of skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then trained on a corpus of mathematical problems and their instantiation-based proofs, and its performance is evaluated in several ways. We show that the trained system achieves high accuracy in predicting the right instances, and that it is capable of solving many problems by educated guessing when combined with a ground solver. To our knowledge, this is the first convincing use of machine learning in synthesizing relevant elements from arbitrary Herbrand universes.

arXiv (Cornell University), May 19, 2018

We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding i... more We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding its connection-style proof search. Instead, it runs many Monte-Carlo simulations guided by reinforcement learning from previous proof attempts. We produce several versions of the prover, parameterized by different learning and guiding algorithms. The strongest version of the system is trained on a large corpus of mathematical problems and evaluated on previously unseen problems. The trained system solves within the same number of inferences over 40% more problems than a baseline prover, which is an unusually high improvement in this hard AI domain. To our knowledge this is the first time reinforcement learning has been convincingly applied to solving general mathematical problems on a large scale.

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

is relevant (in a dichotomy conjecture for innite-domain constraint satisfaction problems).', in ... more is relevant (in a dichotomy conjecture for innite-domain constraint satisfaction problems).', in 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). .

Algebra Universalis, Nov 29, 2019

We prove that the existence of a term s satisfying s(x, y, y, z, z, x) = s(y, x, z, y, x, z) is t... more We prove that the existence of a term s satisfying s(x, y, y, z, z, x) = s(y, x, z, y, x, z) is the weakest non-trivial strong Maltsev condition given by a single identity.

Lecture Notes in Computer Science, 2021

We describe a purely image-based method for finding geometric constructions with a ruler and comp... more 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-theart 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 of geometric construction problems from the first six level packs of Euclidea with an average 92% accuracy. When evaluated on new kinds of problems, the method can solve 31 of the 68 kinds of Euclidea problems. We believe that this is the first time that purely image-based learning has been trained to solve geometric construction problems of this difficulty.

ArXiv, 2020

Automated reasoning and theorem proving have recently become major challenges for machine learnin... more 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 theorem proving are however yet unable to handle many important transformations. In particular, embedding previously unseen labels, that often arise in definitional encodings and in Skolemization, has been very weak so far. Similar problems appear when transferring knowledge between known symbols. We propose a novel encoding of formulas that extends existing graph neural network models. This encoding represents symbols only by nodes in the graph, without giving the network any knowledge of the original labels. We provide additional links between such nodes that allow the network to recover the meaning and therefore correctly embed such nodes...

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (inf... more 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 modelcomplete core of the template has a pseudo-Siggers polymorphism, and NPcomplete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivia...

The recent research on constraint satisfaction problems (CSPs) on fixed finite templates provided... more The recent research on constraint satisfaction problems (CSPs) on fixed finite templates provided useful tools for computational complexity and universal algebra. However, the research mainly focused on finite relational structures, and consequently, finite algebras. We pursue a generalization of these tools and results into the domain of infinite algebras. In particular, we show that despite the fact that the Maltsev condition s(r, a, r, e) = s(a, r, e, a) does not characterize Taylor algebras (i.e., algebras that satisfy a nontrivial idempotent Maltsev condition) in general, as it does in the finite case, there is another strong Maltsev condition characterizing Taylor algebras, and s(r, a, r, e) = s(a, r, e, a) characterizes another interesting broad class of algebras. We also provide a (weak) Maltsev condition for SD(∧) algebras (i.e., algebras that satisfy an idempotent Maltsev condition not satisfiable in a module). Beyond Maltsev conditions, we study loop lemmata and, in parti...

A PCSP is a combination of two CSPs defined by two similar templates; the computational question ... more 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 of Brakensiek and Guruswami [SODA'18] shows that Boolean PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric and allow for negation of variables''. In this paper we remove the "allow for negation of variables'' assumption from the theorem. The "symmetric" assumption means that changing the order of variables in a constraint does not change its satisfiability. The "negation of variables" means that both of the templates share a relation which can be used to effectively negate Boolean variables. The main result of this paper establishes dichotomy for all the symm...

arXiv: Logic, 2018

We prove that the existence of a term sss satisfying s(r,a,r,e)=s(a,r,e,a)s(r,a,r,e) = s(a,r,e,a)s(r,a,r,e)=s(a,r,e,a) in a general algeb... more We prove that the existence of a term sss satisfying s(r,a,r,e)=s(a,r,e,a)s(r,a,r,e) = s(a,r,e,a)s(r,a,r,e)=s(a,r,e,a) in a general algebraic structure is equivalent to an existence of a term ttt satisfying t(x,x,y,y,z,z)=t(y,z,z,x,x,y)t(x,x,y,y,z,z)=t(y,z,z,x,x,y)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 is compatible with an operation ttt satisfying an identity of the from t(ldots)=t(ldots)t(\ldots)=t(\ldots)t(ldots)=t(ldots), has a loop.

ArXiv, 2019

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (inf... more 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 conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-triv...

In this work we study how to learn good algorithms for selecting reasoning steps in theorem provi... more 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 learning algorithm – a graph neural network (GNN) – into the plCoP theorem prover. Then we use it to observe the system’s behavior in a reinforcement learning setting, i.e., when learning inference guidance from successful Monte-Carlo tree searches on many problems. Despite its better pattern matching capability, the GNN initially performs worse than a simpler previously used learning algorithm. We observe that the simpler algorithm is less confident, i.e., its recommendations have higher entropy. This leads us to explore how the entropy of the inference selection implemented via the neural network influences the proof search. This is relat...

Journal of Mathematical Logic, 2021

Automated Reasoning, 2020

We describe an implementation of gradient boosting and neural guidance of saturation-style automa... more 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-independent graph neural networks (GNNs) and their embedding of the terms and clauses. The two methods are efficiently implemented in the E prover and its ENIGMA learning-guided framework. To provide competitive real-time performance of the GNNs, we have developed a new context-based approach to evaluation of generated clauses in E. Clauses are evaluated jointly in larger batches and with respect to a large number of already selected clauses (context) by the GNN that estimates their collectively most useful subset in several rounds of message passing. This means that approximative inference rounds done by the GNN are efficiently interleaved with precise symbolic inference rounds done inside E. The methods are evaluated on the MPTP large-theory benchmark and shown to achieve comparable realtime performance to state-of-the-art symbol-based methods. The methods also show high complementarity, solving a large number of hard Mizar problems.

Frontiers of Combining Systems, 2021

We describe several additions to the ENIGMA system that guides clause selection in the E automate... more 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 be made more intelligent by instead training fast versions of ENIGMA that implement more intelligent pre-filtering. This results in combinations of trainable fast and slow thinking that improves over both the fast-only and slow-only methods. The third addition is based on "judging the children by their parents", i.e., possibly rejecting an inference before it produces a clause. This is motivated by standard evolutionary mechanisms, where there is always a cost to producing all possible offsprings in the current population. This saves time by not evaluating all clauses by more expensive methods and provides a complementary view of the generated clauses. The methods are evaluated on a large benchmark coming from the Mizar Mathematical Library, showing good improvements over the state of the art.

Nedávný výzkum v oblasti problému splnitelnosti podmínek vedl k užitečným nástrojům v uni- verzál... more Nedávný výzkum v oblasti problému splnitelnosti podmínek vedl k užitečným nástrojům v uni- verzální algebře a pro studium výpočetní složitosti. Tento výzkum byl zaměřen zejména na konečné re- lační struktury a tím pádem na konečné algebry. Práce zobecňuje tyto předchozí výsledky na nekonečné algebry. Ukážeme, že ačkoli Maltsevská podmínka t(p, i, s, i) = t(s, p, i, s) obecně necharakterizuje Tay- lorovské algebry (algebry splňující netriviální idempotentní Maltsevskou podmínku) jako v konečném případě, existuje jiná silná Maltsevská podmínka, která je charakterizuje, a t(p, i, s, i) = t(s, p, i, s) charakterizuje jinou širokou třídu algeber. Také najdeme (slabou) Maltsevskou podmínku pro SD(∧) algebry (algebry splňující idempotentní Maltssevskou podmínku, kterou nelze splnit v modulech). Vedle Maltsevskych podmínek zkoumáme smyčková lemmata. Speciálně dokážeme známé konečné smyčkové lemma pomocí dvou různých (nekonečných) přístupů.The recent research on constraint satisfaction probl...

Lecture Notes in Computer Science, 2020

Domain of mathematical logic in computers is dominated by automated theorem provers (ATP) and int... more 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 create a simple human-friendly logical system for mathematical problem solving. We picked the case study of Euclidean geometry as it can be easily visualized, has simple logic, and yet potentially offers many high-school problems of various difficulty levels. To make the environment user friendly, we abandoned strict logic required by ITPs, allowing to infer topological facts from pictures. We present our system for Euclidean geometry, together with a graphical application GeoLogic, similar to GeoGebra, which allows users to interactively study and prove properties about the geometrical setup.

International Journal of Algebra and Computation, 2019

We prove that every strongly connected (not necessarily finite) digraph of algebraic length 1, wh... more We prove that every strongly connected (not necessarily finite) digraph of algebraic length 1, which is compatible with an operation [Formula: see text] satisfying a non-trivial identity of the form [Formula: see text], has a loop.

arXiv (Cornell University), Mar 12, 2023

As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically prov... more As a present to Mizar on its 50th anniversary, we develop an AI/TP system that automatically proves about 60 % of the Mizar theorems in the hammer setting. We also automatically prove 75 % of the Mizar theorems when the automated provers are helped by using only the premises used in the human-written Mizar proofs. We describe the methods and large-scale experiments leading to these results. This includes in particular the E and Vampire provers, their ENIGMA and Deepire learning modifications, a number of learning-based premise selection methods, and the incremental loop that interleaves growing a corpus of millions of ATP proofs with training increasingly strong AI/TP systems on them. We also present a selection of Mizar problems that were proved automatically.

arXiv (Cornell University), Jan 26, 2023

arXiv (Cornell University), Oct 7, 2022

The appearance of strong CDCL-based propositional (SAT) solvers has greatly advanced several area... more 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, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is choosing the right instances from the typically infinite Herbrand universe. In this work, we develop the first machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on an invariant graph neural network (GNN) that learns from problems and their solutions independently of symbol names (addressing the abundance of skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then trained on a corpus of mathematical problems and their instantiation-based proofs, and its performance is evaluated in several ways. We show that the trained system achieves high accuracy in predicting the right instances, and that it is capable of solving many problems by educated guessing when combined with a ground solver. To our knowledge, this is the first convincing use of machine learning in synthesizing relevant elements from arbitrary Herbrand universes.

arXiv (Cornell University), May 19, 2018

We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding i... more We introduce a theorem proving algorithm that uses practically no domain heuristics for guiding its connection-style proof search. Instead, it runs many Monte-Carlo simulations guided by reinforcement learning from previous proof attempts. We produce several versions of the prover, parameterized by different learning and guiding algorithms. The strongest version of the system is trained on a large corpus of mathematical problems and evaluated on previously unseen problems. The trained system solves within the same number of inferences over 40% more problems than a baseline prover, which is an unusually high improvement in this hard AI domain. To our knowledge this is the first time reinforcement learning has been convincingly applied to solving general mathematical problems on a large scale.

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

is relevant (in a dichotomy conjecture for innite-domain constraint satisfaction problems).', in ... more is relevant (in a dichotomy conjecture for innite-domain constraint satisfaction problems).', in 2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS). .

Algebra Universalis, Nov 29, 2019

We prove that the existence of a term s satisfying s(x, y, y, z, z, x) = s(y, x, z, y, x, z) is t... more We prove that the existence of a term s satisfying s(x, y, y, z, z, x) = s(y, x, z, y, x, z) is the weakest non-trivial strong Maltsev condition given by a single identity.

Lecture Notes in Computer Science, 2021

We describe a purely image-based method for finding geometric constructions with a ruler and comp... more 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-theart 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 of geometric construction problems from the first six level packs of Euclidea with an average 92% accuracy. When evaluated on new kinds of problems, the method can solve 31 of the 68 kinds of Euclidea problems. We believe that this is the first time that purely image-based learning has been trained to solve geometric construction problems of this difficulty.

ArXiv, 2020

Automated reasoning and theorem proving have recently become major challenges for machine learnin... more 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 theorem proving are however yet unable to handle many important transformations. In particular, embedding previously unseen labels, that often arise in definitional encodings and in Skolemization, has been very weak so far. Similar problems appear when transferring knowledge between known symbols. We propose a novel encoding of formulas that extends existing graph neural network models. This encoding represents symbols only by nodes in the graph, without giving the network any knowledge of the original labels. We provide additional links between such nodes that allow the network to recover the meaning and therefore correctly embed such nodes...

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (inf... more 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 modelcomplete core of the template has a pseudo-Siggers polymorphism, and NPcomplete otherwise. One of the important questions related to this conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-trivia...

The recent research on constraint satisfaction problems (CSPs) on fixed finite templates provided... more The recent research on constraint satisfaction problems (CSPs) on fixed finite templates provided useful tools for computational complexity and universal algebra. However, the research mainly focused on finite relational structures, and consequently, finite algebras. We pursue a generalization of these tools and results into the domain of infinite algebras. In particular, we show that despite the fact that the Maltsev condition s(r, a, r, e) = s(a, r, e, a) does not characterize Taylor algebras (i.e., algebras that satisfy a nontrivial idempotent Maltsev condition) in general, as it does in the finite case, there is another strong Maltsev condition characterizing Taylor algebras, and s(r, a, r, e) = s(a, r, e, a) characterizes another interesting broad class of algebras. We also provide a (weak) Maltsev condition for SD(∧) algebras (i.e., algebras that satisfy an idempotent Maltsev condition not satisfiable in a module). Beyond Maltsev conditions, we study loop lemmata and, in parti...

A PCSP is a combination of two CSPs defined by two similar templates; the computational question ... more 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 of Brakensiek and Guruswami [SODA'18] shows that Boolean PCSPs exhibit a dichotomy (PTIME vs. NPC) when "all the clauses are symmetric and allow for negation of variables''. In this paper we remove the "allow for negation of variables'' assumption from the theorem. The "symmetric" assumption means that changing the order of variables in a constraint does not change its satisfiability. The "negation of variables" means that both of the templates share a relation which can be used to effectively negate Boolean variables. The main result of this paper establishes dichotomy for all the symm...

arXiv: Logic, 2018

We prove that the existence of a term sss satisfying s(r,a,r,e)=s(a,r,e,a)s(r,a,r,e) = s(a,r,e,a)s(r,a,r,e)=s(a,r,e,a) in a general algeb... more We prove that the existence of a term sss satisfying s(r,a,r,e)=s(a,r,e,a)s(r,a,r,e) = s(a,r,e,a)s(r,a,r,e)=s(a,r,e,a) in a general algebraic structure is equivalent to an existence of a term ttt satisfying t(x,x,y,y,z,z)=t(y,z,z,x,x,y)t(x,x,y,y,z,z)=t(y,z,z,x,x,y)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 is compatible with an operation ttt satisfying an identity of the from t(ldots)=t(ldots)t(\ldots)=t(\ldots)t(ldots)=t(ldots), has a loop.

ArXiv, 2019

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (inf... more 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 conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each non-trivial set of height 1 identities a structure whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of non-triv...

In this work we study how to learn good algorithms for selecting reasoning steps in theorem provi... more 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 learning algorithm – a graph neural network (GNN) – into the plCoP theorem prover. Then we use it to observe the system’s behavior in a reinforcement learning setting, i.e., when learning inference guidance from successful Monte-Carlo tree searches on many problems. Despite its better pattern matching capability, the GNN initially performs worse than a simpler previously used learning algorithm. We observe that the simpler algorithm is less confident, i.e., its recommendations have higher entropy. This leads us to explore how the entropy of the inference selection implemented via the neural network influences the proof search. This is relat...

Journal of Mathematical Logic, 2021

Automated Reasoning, 2020

We describe an implementation of gradient boosting and neural guidance of saturation-style automa... more 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-independent graph neural networks (GNNs) and their embedding of the terms and clauses. The two methods are efficiently implemented in the E prover and its ENIGMA learning-guided framework. To provide competitive real-time performance of the GNNs, we have developed a new context-based approach to evaluation of generated clauses in E. Clauses are evaluated jointly in larger batches and with respect to a large number of already selected clauses (context) by the GNN that estimates their collectively most useful subset in several rounds of message passing. This means that approximative inference rounds done by the GNN are efficiently interleaved with precise symbolic inference rounds done inside E. The methods are evaluated on the MPTP large-theory benchmark and shown to achieve comparable realtime performance to state-of-the-art symbol-based methods. The methods also show high complementarity, solving a large number of hard Mizar problems.

Frontiers of Combining Systems, 2021

We describe several additions to the ENIGMA system that guides clause selection in the E automate... more 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 be made more intelligent by instead training fast versions of ENIGMA that implement more intelligent pre-filtering. This results in combinations of trainable fast and slow thinking that improves over both the fast-only and slow-only methods. The third addition is based on "judging the children by their parents", i.e., possibly rejecting an inference before it produces a clause. This is motivated by standard evolutionary mechanisms, where there is always a cost to producing all possible offsprings in the current population. This saves time by not evaluating all clauses by more expensive methods and provides a complementary view of the generated clauses. The methods are evaluated on a large benchmark coming from the Mizar Mathematical Library, showing good improvements over the state of the art.