August 2024

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3 Reads

Journal of Symbolic Computation

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August 2024

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3 Reads

Journal of Symbolic Computation

August 2023

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25 Reads

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2 Citations

International Journal of Approximate Reasoning

January 2023

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34 Reads

Transactions of the American Mathematical Society

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

January 2023

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18 Reads

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, and (ii) proposing many new programs for each OEIS sequence by the trained neural machine translator. The algorithm discovers on its own programs for more than 78000 OEIS sequences, sometimes developing unusual programming methods. We analyze its behavior and the invented programs in several experiments.

October 2022

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13 Reads

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.

May 2022

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24 Reads

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 methods are trained in several iterations over hundreds of thousands untyped and typed first-order problems extracted from Isabelle. Our final best single-strategy ENIGMA and premise selection system improves the best previous version of E by 25.3% in 15 seconds, outperforming also all other previous ATP and SMT systems.

September 2021

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15 Reads

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4 Citations

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 .

August 2021

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19 Reads

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1 Citation

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 a graph neural network (GNN) that chooses the next given clause based on its evaluation in the context of previously selected clauses. In this work, we describe several algorithms and experiments with ENIGMA, advancing the idea of contextual evaluation based on learning important components of the graph of clauses.

August 2021

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13 Reads

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4 Citations

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 related to research in human decision-making under uncertainty, and in particular the probability matching theory. Our main result shows that a proper entropy regularization, i.e., training the GNN not to be overconfident, greatly improves plCoP ’s performance on a large mathematical corpus.

July 2021

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42 Reads

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 a graph neural network (GNN) that chooses the next given clause based on its evaluation in the context of previously selected clauses. In this work, we describe several algorithms and experiments with ENIGMA, advancing the idea of contextual evaluation based on learning important components of the graph of clauses.

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... 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]. ...

- Citing Chapter
August 2021

... 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). ...

- Citing Chapter
September 2021

... This conjecture states that such CSPs are in P or NP-complete, and even provides a mathematical condition to describe the boundary between the cases in P and the NP-complete cases. This condition has numerous equivalent characterisations [6,7,9], but despite recent progress [35] the tractability conjecture for MMSNP 2 is still wide open. In contrast, the P versus NP-complete dichotomy is true for MMSNP [25,34] (using the complexity dichotomy for finite-domain CSPs [20,43]), and even the tractability conjecture has been verified in this case [12]. ...

- Citing Article
February 2021

Journal of Mathematical Logic

... 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]). ...

- Citing Article
- Full-text available
February 2021

Journal of Symbolic Logic

... Step (27) follows from (26) and (15), step (28) from (26) and (16). ...

Reference:

Investigations into Proof Structures

- Citing Chapter
June 2020

... 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. ...

- Citing Article
May 2020

Transactions of the American Mathematical Society

... , 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. ...

- Citing Article
October 2019

International Journal of Algebra and Computation

... 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. ...

- Citing Conference Paper
June 2019

... The structure C 3 is a model-complete core, i.e., for every finite F ⊆ V (C 3 ) and every endomorphism e of C 3 there exists an automorphism a of C 3 such that e| F = a| F . 2 Proof. Every structure A with a homogeneous expansion whose relations are existentially positively and universally negatively definable in A is a model-complete core [4,Theorem 4.5.1]. ...

Reference:

The Generic Circular Triangle-Free Graph

- Citing Conference Paper
June 2017

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

- Citing Article
- Full-text available
November 2019

Algebra universalis