arXiv subjects

Josef Urban

Publications and source records attributed to Josef Urban.

At least 19 recordsLinked to original sources

arXiv2026

Munkres' General Topology Autoformalized in Isabelle/HOL

We describe an experiment in LLM-assisted autoformalization that produced over 85,000 lines of Isabelle/HOL code covering all 39 sections of Munkres' Topology (general topology, Chapters 2--8), from topological spaces through dimension theory. The LLM-based coding agents (initially ChatGPT 5.2 and then Claude Opus 4.6) used 24 active days for that. The formalization is complete: all 806 formal results are fully proved with zero sorry's. Proved results include the Tychonoff theorem, the Baire category theorem, the Nagata--Smirnov and Smirnov metrization theorems, the Stone--\v{C}ech compactification, Ascoli's theorem, the space-filling curve, and others. The methodology is based on a "sorry-first" declarative proof workflow combined with bulk use of sledgehammer - two of Isabelle major strengths. This leads to relatively fast autoformalization progress. We analyze the resulting formalization in detail, analyze the human--LLM interaction patterns from the session log, and briefly compare with related autoformalization efforts in Megalodon, HOL Light, and Naproche. The results indicate that LLM-assisted formalization of standard mathematical textbooks in Isabelle/HOL is quite feasible, cheap and fast, even if some human supervision is useful.

cs.AI↗

arXiv2026

Agent Hunt: Bounty Based Collaborative Autoformalization With LLM Agents

We describe an experiment in large-scale autoformalization of algebraic topology in an Interactive Theorem Proving (ITP) environment, where the workload is distributed among multiple LLM-based coding agents. Rather than relying on static central planning, we implement a simulated bounty-based marketplace in which agents dynamically propose new lemmas (formal statements), attach bounties to them, and compete to discharge these proof obligations and claim the bounties. The agents interact directly with the interactive proof system: they can invoke tactics, inspect proof states and goals, analyze tactic successes and failures, and iteratively refine their proof scripts. In addition to constructing proofs, agents may introduce new formal definitions and intermediate lemmas to structure the development. All accepted proofs are ultimately checked and verified by the underlying proof assistant. This setting explores collaborative, decentralized proof search and theory building, and the use of market-inspired mechanisms to scale autoformalization in ITP.

cs.LO↗

arXiv2026

130k Lines of Formal Topology in Two Weeks: Simple and Cheap Autoformalization for Everyone?

This is a brief description of a project that has already autoformalized a large portion of the general topology from the Munkres textbook (which has in total 241 pages in 7 chapters and 39 sections). The project has been running since November 21, 2025 and has as of January 4, 2026, produced 160k lines of formalized topology. Most of it (about 130k lines) have been done in two weeks,from December 22 to January 4, for an LLM subscription cost of about \$100. This includes a 3k-line proof of Urysohn's lemma, a 2k-line proof of Urysohn's Metrization theorem, over 10k-line proof of the Tietze extension theorem, and many more (in total over 1.5k lemmas/theorems). The approach is quite simple and cheap: build a long-running feedback loop between an LLM and a reasonably fast proof checker equipped with a core foundational library. The LLM is now instantiated as ChatGPT (mostly 5.2) or Claude Sonnet (4.5) run through the respective Codex or Claude Code command line interfaces. The proof checker is Chad Brown's higher-order set theory system Megalodon, and the core library is Brown's formalization of basic set theory and surreal numbers (including reals, etc). The rest is some prompt engineering and technical choices which we describe here. Based on the fast progress, low cost, virtually unknown ITP/library, and the simple setup available to everyone, we believe that (auto)formalization may become quite easy and ubiquitous in 2026, regardless of which proof assistant is used.

cs.LO↗

arXiv2025

Hammering Higher Order Set Theory

We use automated theorem provers to significantly shorten a formal development in higher order set theory. The development includes many standard theorems such as the fundamental theorem of arithmetic and irrationality of square root of two. Higher order automated theorem provers are particularly useful here, since the underlying framework of higher order set theory coincides with the classical extensional higher order logic of (most) higher order automated theorem provers, so no significant translation or encoding is required. Additionally, many subgoals are first order and so first order automated provers often suffice. We compare the performance of different provers on the subgoals generated from the development. We also discuss possibilities for proof reconstruction, i.e., obtaining formal proof terms when an automated theorem prover claims to have proven the subgoal.

cs.LO↗

arXiv2025

Exploring Formal Math on the Blockchain: An Explorer for Proofgold

Proofgold is a blockchain that supports formalized mathematics alongside standard cryptocurrency functionality. It incorporates logical constructs into the blockchain, including declarations of formal theories, definitions, propositions and proofs. It also supports placing and collecting bounties on proving these propositions, incentivizing the development of the formal libraries contained in Proofgold. In this paper, we present a web-based blockchain explorer for Proofgold. The system exposes not only the usual transactional data but also the formal mathematical components embedded in the chain and allows some interaction with them. The explorer allows users to inspect blocks, transactions, and addresses, as well as formal objects: theories, definitions, theorems and their proofs. We also support the submission of transactions to the blockchain using our interface. We describe the system architecture and its integration with the Proofgold Lava software, highlighting how the explorer supports navigation of formal content and facilitates mathematical knowledge management in a decentralized setting, as well as a number of formalizations in category theory done in the system.

cs.LO↗

arXiv2025

Payment Channels with Proofs

The fundamental building blocks of the Bitcoin lightning network are bidirectional payment channels. We describe an extension of payment channels in the Proofgold network which allow the two parties to bet on whether a proposition will be proven by a certain time. These provide the foundation for a Proofgold lightning network that would allow parties to request proofs (by betting there will be no proof by a certain time) and other parties to provide proofs (and be rewarded by betting there will be a proof). The bets may also provide a way to approximate the probability that a certain proposition is provable (in the given amount of time). We describe the implementation of payment channels supporting proofs in Proofgold and discuss a potential lightning network that could be built as a result. One application of such lightning network would be a large decentralized infrastructure for fast collaborative formalization projects.

cs.LO↗

arXiv2025

Learning Conjecturing from Scratch

We develop a self-learning approach for conjecturing of induction predicates on a dataset of 16197 problems derived from the OEIS. These problems are hard for today's SMT and ATP systems because they require a combination of inductive and arithmetical reasoning. Starting from scratch, our approach consists of a feedback loop that iterates between (i) training a neural translator to learn the correspondence between the problems solved so far and the induction predicates useful for them, (ii) using the trained neural system to generate many new induction predicates for the problems, (iii) fast runs of the z3 prover attempting to prove the problems using the generated predicates, (iv) using heuristics such as predicate size and solution speed on the proved problems to choose the best predicates for the next iteration of training. The algorithm discovers on its own many interesting induction predicates, ultimately solving 5565 problems, compared to 2265 problems solved by CVC5, Vampire or Z3 in 60 seconds.

cs.AI↗

arXiv2024

Machine Learning for Quantifier Selection in cvc5

In this work we considerably improve the state-of-the-art SMT solving on first-order quantified problems by efficient machine learning guidance of quantifier selection. Quantifiers represent a significant challenge for SMT and are technically a source of undecidability. In our approach, we train an efficient machine learning model that informs the solver which quantifiers should be instantiated and which not. Each quantifier may be instantiated multiple times and the set of the active quantifiers changes as the solving progresses. Therefore, we invoke the ML predictor many times, during the whole run of the solver. To make this efficient, we use fast ML models based on gradient boosting decision trees. We integrate our approach into the state-of-the-art cvc5 SMT solver and show a considerable increase of the system's holdout-set performance after training it on a large set of first-order problems collected from the Mizar Mathematical Library.

cs.AI↗

arXiv2024

Solving Hard Mizar Problems with Instantiation and Strategy Invention

In this work, we prove over 3000 previously ATP-unproved Mizar/MPTP problems by using several ATP and AI methods, raising the number of ATP-solved Mizar problems from 75\% to above 80\%. First, we start to experiment with the cvc5 SMT solver which uses several instantiation-based heuristics that differ from the superposition-based systems, that were previously applied to Mizar,and add many new solutions. Then we use automated strategy invention to develop cvc5 strategies that largely improve cvc5's performance on the hard problems. In particular, the best invented strategy solves over 14\% more problems than the best previously available cvc5 strategy. We also show that different clausification methods have a high impact on such instantiation-based methods, again producing many new solutions. In total, the methods solve 3021 (21.3\%) of the 14163 previously unsolved hard Mizar problems. This is a new milestone over the Mizar large-theory benchmark and a large strengthening of the hammer methods for Mizar.

cs.AI↗

arXiv2024

Learning Guided Automated Reasoning: A Brief Survey

Automated theorem provers and formal proof assistants are general reasoning systems that are in theory capable of proving arbitrarily hard theorems, thus solving arbitrary problems reducible to mathematics and logical reasoning. In practice, such systems however face large combinatorial explosion, and therefore include many heuristics and choice points that considerably influence their performance. This is an opportunity for trained machine learning predictors, which can guide the work of such reasoning systems. Conversely, deductive search supported by the notion of logically valid proof allows one to train machine learning systems on large reasoning corpora. Such bodies of proof are usually correct by construction and when combined with more and more precise trained guidance they can be boostrapped into very large corpora, with increasingly long reasoning chains and possibly novel proof ideas. In this paper we provide an overview of several automated reasoning and theorem proving domains and the learning and AI methods that have been so far developed for them. These include premise selection, proof guidance in several settings, AI systems and feedback loops iterating between reasoning and learning, and symbolic classification problems.

cs.AI↗

arXiv2023

Translating SUMO-K to Higher-Order Set Theory

We describe a translation from a fragment of SUMO (SUMO-K) into higher-order set theory. The translation provides a formal semantics for portions of SUMO which are beyond first-order and which have previously only had an informal interpretation. It also for the first time embeds a large common-sense ontology into a very secure interactive theorem proving system. We further extend our previous work in finding contradictions in SUMO from first order constructs to include a portion of SUMO's higher order constructs. Finally, using the translation, we can create problems that can be proven using higher-order interactive and automated theorem provers. This is tested in several systems and can be used to form a corpus of higher-order common-sense reasoning problems.

cs.AI↗

arXiv2023

A Mathematical Benchmark for Inductive Theorem Provers

We present a benchmark of 29687 problems derived from the On-Line Encyclopedia of Integer Sequences (OEIS). Each problem expresses the equivalence of two syntactically different programs generating the same OEIS sequence. Such programs were conjectured by a learning-guided synthesis system using a language with looping operators. The operators implement recursion, and thus many of the proofs require induction on natural numbers. The benchmark contains problems of varying difficulty from a wide area of mathematical domains. We believe that these characteristics will make it an effective judge for the progress of inductive theorem provers in this domain for years to come.

cs.LO↗

arXiv2023

MizAR 60 for Mizar 50

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.

cs.AI↗

arXiv2023

Alien Coding

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.

cs.AI↗

arXiv2022

Learning Program Synthesis for Integer Sequences from Scratch

We present a self-learning approach for synthesizing programs from integer sequences. Our method relies on a tree search guided by a learned policy. Our system is tested on the On-Line Encyclopedia of Integer Sequences. There, it discovers, on its own, solutions for 27987 sequences starting from basic operators and without human-written training examples.

cs.AI↗

arXiv2022

Machine Learning Meets The Herbrand Universe

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.

cs.LG↗

arXiv2022

Six Insights into 6G: Orientation and Input for Developing Your Strategic 6G Research Plan

This paper is a summary of the findings from a series of workshops which were held by Thinknet 6G and MUENCHNER KREIS in 2021, with the goal to provide orientation and input for developing a strategic 6G research plan. The topics selected for the workshops are aspects of 6G that we expect will have a significant impact on other industries and on society: - 6G as both a communication infrastructure and a sensing infrastructure - The extensive use of artificial intelligence in 6G - The security and resilience of 6G This paper does not go into the technical details of how to develop and implement 6G. Rather, it provides input from experts from both the wireless industry as well as from other sectors about (mostly) non-technical topics that will need to be addressed in parallel with the technical developments, such as new use cases, regulation, communication with the public, and cross-industry cooperation. We have identified six areas that will have a significant impact on the development and use of 6G, and that organizations must consider as they begin their plans and designs for 6G. Based on these six impact areas and on the discussion in the workshops, we compiled a list of the top 10 recommendations for specific areas where organizations should place their focus when developing their strategic plan for 6G. In addition, for our readers who are involved in 6G research, be it at a university, at a research institute or in industrial research, we also included a summary of the top 10 areas that require additional research, again based on the input received in the workshops. A version of this paper is also available at www.thinknet-6g.de. If you had a copy of the preview version of this paper, the text is exactly the same. Only the layout and graphics have changed.

cs.NI↗

arXiv2022

The Isabelle ENIGMA

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.

cs.AI↗