Abstract We use SMT technology to address a class of problems involving uninterpreted functions and nonlinear real arithmetic. In particular, we focus on problems commonly found in mathematical competitions, such as the International Mathematical Olympiad (IMO), where the task is to determine all solutions to constraints on an uninterpreted function. Although these problems require only high-school-level mathematics, state-of-the-art SMT solvers often struggle with them. We propose several techniques to improve SMT performance in this setting.

We present AlphaGeometry2, a significantly improved version of AlphaGeometry introduced in Nature, 625 (7995):476, 2024, which has now surpassed an average gold medalist in solving Olympiad geometry problems. To achieve this, we first extend the original AlphaGeometry language to tackle problems involving movements of objects, and problems containing linear equations of angles, ratios, and distances. This, together with support for non-constructive problems, has markedly improved the coverage rate of the AlphaGeometry language on International Math Olympiads 2000-2024 geometry problems from 66% to 88%. The search process of AlphaGeometry2 has also been greatly improved through the use of Gemini architecture for better language modeling, and a novel knowledge-sharing mechanism that enables effective communication between search trees. Together with further enhancements to the symbolic engine and synthetic data generation, we have significantly boosted the overall solving rate of AlphaGeometry to 84% on all geometry problems over the last 25 years, compared to 54% previously. AlphaGeometry2 was also part of the system that achieved the silver-medal standard at IMO 2024 https://dpmd.ai/imo-silver. Finally, we report progress towards using AlphaGeometry2 as a part of a fully automated system that reliably solves geometry problems from natural language input. Code: https://github.com/google-deepmind/alphageometry2.

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In proof assistants, the physical proximity between two formal mathematical concepts is a strong predictor of their mutual relevance. Furthermore, lemmas with close proximity regularly exhibit similar proof structures. We show that this locality property can be exploited through online learning techniques to obtain solving agents that far surpass offline learners when asked to prove theorems in an unseen mathematical setting. We extensively benchmark two such online solvers implemented in the Tactician platform for the Coq proof assistant: First, Tactician’s online $k$-nearest neighbor solver, which can learn from recent proofs, shows a $1.72\times$ improvement in theorems proved over an offline equivalent. Second, we introduce a graph neural network, Graph2Tac, with a novel approach to build hierarchical representations for new definitions. Graph2Tac’s online definition task realizes a $1.5\times$ improvement in theorems solved over an offline baseline. The $k$-NN and Graph2Tac solvers rely on orthogonal online data, making them highly complementary. Their combination improves $1.27\times$ over their individual performances. Both solvers outperform all other general purpose provers for Coq, including CoqHammer, Proverbot9001, and a transformer baseline by at least $1.48\times$ and are available for practical use by end-users.

The notion of 𝛼-equivalence between 𝜆-terms is commonly used to identify terms that are considered equal. However, due to the primitive treatment of free variables, this notion falls short when comparing subterms occurring within a larger context. Depending on the usage of the Barendregt convention (choosing different variable names for all involved binders), it will equate either too few or too many subterms. We introduce a formal notion of context-sensitive 𝛼-equivalence, where two open terms can be compared within a context that resolves their free variables. We show that this equivalence coincides exactly with the notion of bisimulation equivalence. Furthermore, we present an efficient O ( n log n ) runtime hashing scheme that identifies 𝜆-terms modulo context-sensitive 𝛼 -equivalence, generalizing over traditional bisimulation partitioning algorithms and improving upon a previously established O ( n log 2 n ) bound for a hashing modulo ordinary 𝛼-equivalence byMaziarz et al [ 21 ]. Hashing 𝜆-terms is useful in many applications that require common subterm elimination and structure sharing. We hav employed the algorithm to obtain a large-scale, densely packed, interconnected graph of mathematical knowledge from the Coq proof assistant for machine learning purposes.

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.

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.

The ability to form complex plans based on raw visual input is a litmus test for current capabilities of artificial intelligence, as it requires a seamless combination of visual processing and abstract algorithmic execution, two traditionally separate areas of computer science. A recent surge of interest in this field brought advances that yield good performance in tasks ranging from arcade games to continuous control; these methods however do not come without significant issues, such as limited generalization capabilities and difficulties when dealing with combinatorially hard planning instances. Our contribution is two-fold: (i) we present a method that learns to represent its environment as a latent graph and leverages state reidentification to reduce the complexity of finding a good policy from exponential to linear (ii) we introduce a set of lightweight environments with an underlying discrete combinatorial structure in which planning is challenging even for humans. Moreover, we show that our methods achieves strong empirical generalization to variations in the environment, even across highly disadvantaged regimes, such as “one-shot” planning, or in an offline RL paradigm which only provides low-quality trajectories.

Abstract 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 condition.

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 irrespective of the given labels. We test the proposed encoding in an automated theorem prover based on the tableaux connection calculus, and show that it improves on the best characterizations used so far. The encoding is further evaluated on the premise selection task and a newly introduced symbol guessing task, and shown to correctly predict 65% of the symbol names.

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 symmetric boolean templates. The tractability case of our theorem and the theorem of Brakensiek and Guruswami are almost identical. The main difference, and the main contribution of this work, is the new reason for hardness and the reasoning proving the split.

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 nontrivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of nontrivial height 1 identities differ for ω -categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

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.

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 linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial. We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that ω-categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core. Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions. We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset. Finally, we provide a new and short proof, in the language of monoids, of the theorem stating that every ω-categorical structure is homomorphically equivalent to a model-complete core.