We present Alaska's ranked-choice elections as a case study in Formally Verified Democracy, a program to establish election results through formally checked proofs. We formalize an interpretation of Alaska's tabulation rules in Lean, verify its implementation, and prove election-specific results. Across 115 ranked-choice contests in Alaska since 2022, we identify 75 ballot records across 10 contests that were counted differently from this interpretation by a program reproducing the published results, without changing any winners. We trace these differences to overvotes and duplicate rankings and examine their legal justification, showing how formal verification can expose consequential choices in translating election law into software.
This note documents the specification of normal forms in cubical type theory. The definition is already present in the proof of normalization for cubical type theory, but we present it in a more traditional style explicitly for reference.
Even a carefully proofread paper can leave small gaps in its proofs or its stated assumptions. We introduce PaperFormalizer, a Lean-based agentic pipeline that helps authors find and close such gaps in their own papers and turn each paper into a Lean-verified version. Starting from the LaTeX source, PaperFormalizer transcribes the paper into a dependency graph of statements, audits each statement in natural language for faithfulness, ambiguity and correctness, and then formalizes and proves the statements in Lean against Mathlib, optional domain libraries and the author's personal toolkit. Only the author knows what a statement is meant to say and how a formalization gap in a result is best fixed. The pipeline therefore stops and asks the author whenever a statement admits more than one reading, presents every repair, in natural language and TeX symbols, as a proposal that the author can accept, decline or redirect, and leaves to the author every decision that could potentially change what the paper claims. The output is an annotated and revised version of the paper, a report of the changes, and Lean code in which each declaration is linked to the statement it formalizes. The author's toolkit grows as more papers are formalized, and authors can share their verified papers so that researchers who build on them can reuse their Lean code. By making verified papers practical to produce, PaperFormalizer aims to make research that rests on mathematical proofs more robust. PaperFormalizer runs inside a coding agent (Claude Code or Codex) and is available at https://github.com/Jiyuan-Tan/PaperFormalizer. Papers with Lean code shared by users are available at https://paperformalizer.org.
The genetic code exhibits redundancy, meaning that multiple distinct nucleotide sequences can encode the same amino acid sequence, a property mathematically characterized as non-injectivity. While this phenomenon is well studied in molecular biology, its rigorous verification within a computational framework remains limited. In this work, we present a machine-verified model of the central dogma of molecular biology using the Lean 4 interactive theorem prover. We formalize nucleic acid bases and amino acids as inductive types and define transcription and translation as functions mapping DNA sequences to RNA and subsequently to polypeptides. Within this framework, we prove key properties of these biological processes, including the redundancy of the genetic code, injectivity of transcription, and the non-injectivity of translation. Additionally, we formalize auxiliary processes such as DNA replication and mutation, and verify properties such as length conservation under transcription and point mutations. Finally, we demonstrate how Lean 4 helps prevent semantic errors in our code, showing its effectiveness as a tool for advancing reliable, bug-free scientific computing, particularly within the life sciences.
Benchmarks for verifiable code generation usually test a proof against one fixed implementation. We ask whether the same proof remains valid after a program refactoring that preserves behavior. We introduce RefactorProof, which derives 458 Lean-certified variants from the 185 VERINA tasks that compile in our pinned environment while holding the specification and evaluated proof fixed byte-for-byte. VERINA supplies reference proofs for 46 tasks, all from its basic split; the primary model-versus-reference comparisons further match on 21-25 short, non-recursive tasks. Structural refactorings expose substantial brittleness across proof sources: task-normalized structural PSR is 4.3% for the supplied references and 13.3-26.8% for the specialized provers. On the matched tasks, specialized prover proofs nevertheless have higher overall unchanged-proof survival than the supplied reference artifacts: +15.6 points for DeepSeek-Prover-V2-7B, +18.8 for Goedel-Prover-V2-32B, and +20.3 for Goedel-Prover-V2-8B; all three 95% CIs exclude zero. The ordering persists after excluding universally survived T1 variants, under family-balanced aggregation, and under alternative deployable valid-proof selection rules. However, unchanged-proof failure is not the same as high repair cost: predefined one-identifier edits repair 51/54 eligible reference structural failures and 58/85 eligible prover failures. RefactorProof therefore measures unchanged-proof stability under program refactoring, not the cost of maintaining a proof after it breaks.
Gödel's ontological argument is modal by design. Possibility and necessity are among its central concerns, and connected interpretation decisions -- extensions or intensions for the positive properties, constant or varying quantifier domains, the choice of modal logic, etc. -- subtly shape the argument and, in particular, whether the modal collapse is implied or not. It is tempting to abstract the modality away and to read the positive properties as merely forming an ultrafilter -- which, however, may replace Gödel's argument with a perhaps more manageable alternative that may have little to do with Gödel's original intention. In this spirit, Odifreddi and Gomes argue that modal collapse is an intrinsic structural feature of any theory that characterises positive properties as an ultrafilter and identifies God as its principal generator. This comment shows that the structural claim is false. There is an explicit S5 model, with constant domains, in which positivity is at every world a modal ultrafilter principally generated by a unique and necessarily existing Godlike individual, and in which the modal collapse nevertheless fails -- already for an atomic formula. What drives the collapse is not the ultrafilter structure but the rigidity of positivity -- itself a thoroughly modal condition: given only that positivity is a modal ultrafilter, modal collapse and rigidity are equivalent. Rigidity is thus exactly what the collapse adds to the ultrafilter structure, and the disagreement is whether it is to be counted part of that structure at all. Two further claims of Odifreddi and Gomes are clarified: one on the derivability of Gödel's Theorem IV, one on the extensionality of positivity. All results are machine-checked in Isabelle/HOL, which distinguishes kernel-checked proofs from model-finder output.
Dylan Waldner, Yiannis Kantaros, Guido Governatori, Risto Miikkulainen, Amir Banifatemi
原文摘要
As robotic systems grow more general, legal norms are needed to integrate them into society. This paper extends the isomorphism problem of aligning legal source texts with their encodings, and measures two key challenges to robot normative control: (1) the grounding isomorphism gap, where perception error grounds false atoms for legal reasoning, and (2) the ontological isomorphism gap, where one legal conclusion admits many faithful translations into planning constraints. The paper introduces a legal planning stack that employs Defeasible Deontic Logic (DDL) to constrain a motion planner. The stack leverages learned world models to plan and to provide legal context, enabling ex ante governance that intervenes before an illegal action is executed. It was deployed on a simulated robot arm pushing a cube across a 3x3 grid. The findings were (1) the legislated agent abided substantially more often than the non-legislated one, and modeling perception uncertainty lifted abidance even further, (2) the legal reasoning ran efficiently at runtime and its verdicts were auditable, and (3) the stack adapted to exogenous signals and endogenous rule changes. Both gaps were measured: (4) world model and probe error corrupted the factual input for the DDL reasoner, and (5) a single law admitted several faithful metric interpretations yielding drastically different abidance. Thus, ex ante legislation functions as intended, and closing these gaps with a standardized mapping from the law to runtime constraints and improved fact grounding from perception will yield robust laws that align robot behavior with society's norms. Project page: https://dylanwaldner-cail.github.io/Legislated-Planner/.
We present count-catcher, a tool that turns a CPOG certificate of a count by the exact model counter d4 into a Lean theorem that a CNF formula defined in Lean has exactly N models over V variables. With an encoding lemma, this theorem becomes a theorem about the number of combinatorial objects. A CPOG certificate consists of a Boolean circuit equivalent to the formula and a proof of equivalence. count-catcher checks in Lean that the formula implies the circuit (forward check) and the circuit implies the formula (reverse check). For the forward check, lrat-catcher streams a refutation through the LRAT checker of Lean's standard library. The reverse check and the validity check of the circuit are computations proved correct in Lean. Lean runs all three checks as compiled code, so the theorem depends on Lean's kernel, compiler and runtime, as does CPOG's verified checker. We combine the theorem with results from Mathlib, for instance with Burnside's lemma to count graphs up to isomorphism. We prove the number of Latin squares of order 8 as a weighted sum of hundreds of certified counts from about eight terabytes of certificates. We turn about half of the 2022 Model Counting Competition formulas into theorems, with the three checks taking about a quarter of the time of CPOG's verified checker. We found that CPOG's unverified checker reported wrong counts, and removed its verified checker's dependence on two unproved lemmas. End to end, a theorem costs about the same as a checked count.
Hoang-Loc Cao, Truong Thanh Hung Nguyen, Phuc Truong Loc Nguyen, Vo Thanh Khang Nguyen, Phuc Ho, Hung Nguyen, Duc Pham, Long Tran-Thanh 等 9 位作者
原文摘要
In multi-agent debates, matching reports can arise for two different reasons: agents may repeat the same evidence (replication), or they may provide separate evidential roots that support the same claim (corroboration). A judge must distinguish these two cases. Counting report occurrences gives extra weight to relays, paraphrases, alias resubmissions, and excerpt splits, while merging all matching content avoids this only by also removing genuine corroboration. We formalize this tension as a minimal judgment game in which acquiring new evidence is costly, but replication is cheap. We show that root replication invariance (RRI) is exactly threshold-uniform replication immunity. Under RRI, replication-only deviations cannot change the verdict, and positive-cost replication is strictly dominated, whereas occurrence-sensitive and content-keyed judges admit profitable deviations. We also establish an information boundary: if a judge's observation interface cannot distinguish a replica from a matched fresh root, no judge can both ignore the replica and respond strictly to the fresh root. To preserve this distinction, we introduce the lineage quotient, which removes wording, identity, occurrence multiplicity, and arbitrary root names while keeping distinct roots separate. On orbit-connected domains, it is the coarsest representation sufficient for every deterministic judge invariant to replication and root-label renaming. We further lift this construction to quantitative bipolar argumentation frameworks (QBAFs), characterize strict fresh-root response, prove its propagation along sign-consistent paths under DF-QuAD, and derive a tight closed-form response modulus. Exhaustive audits over 28,224 direct-star profiles, 2,052 multi-layer DAGs, with matched interventions on 277 AVeriTeC-derived profiles, show the predicted separation between replication and corroboration under a supplied partition.
The validity problem for first-order logic is a well-known undecidable problem. The undecidability also holds even for $\mathsf{FO}3$ and (equational formulas of) the calculus of relations. In this paper we tighten these undecidability results to the following: (1) $\mathsf{FO}3$ with just one binary relation is undecidable even without equality; and (2) the calculus of relations with just one character and with only composition, union, and complement is undecidable. Additionally we prove that the finite validity problem is also undecidable for the above two classes.
Alfredo Burrieza, Fernando Soler-Toscano, Antonio Yuste-Ginel
原文摘要
We propose a novel modal study of bisimulations. Our contribution is threefold. First, we extend the basic modal language with a new modality $\nbi$, whose intended meaning is universal quantification over all states that are bisimilar to the current one under a given bisimulation. We show that bisimulations are definable in this object language via frame validity (for the forth and back conditions) and model validity (for the atomic harmony condition). Additionally, some other related notions (refinements, simulations, bisimulations for a restricted set of variables, and $n$-bisimulations) are also shown to be definable. Second, we provide a sound and complete axiomatisation of the class of all pairs of Kripke models that are bisimulation-related. Third, we show that the satisfiability problem of this logic is decidable and PSPACE-complete via a translation to standard modal logic $K$ under a simple frame condition. All our results are encoded and verified by Isabelle/HOL.
This paper investigates the proof-theoretic foundations of the principle of reiteration (R) in natural deduction, arguing that its legitimacy originates in the structural preservation of formulae within sequent calculus rather than in an independent logical rule. By examining singlesuccedent and multisuccedent sequent calculus, we demonstrate how logical operations transform principal formulae while preserving unaffected antecedent and succedent contexts. We subsequently analyse the schematic formulation of reiteration and compare natural deduction derivations with corresponding sequent calculus derivations. This comparison identifies two fundamental functions of reiteration, namely the reuse of previously employed formulae and the restoration of previously derived conclusions. We conclude that reiteration introduces no independent inferential content but explicitly represents the structural continuity of inferential dependencies already included into sequent calculus.
In this paper, we construct an interpretation-preserving functor from a category of concurrent games to the category of Scott domains and Scott-continuous functions. We give a concrete description of this functor, extending earlier results on the relational collapse of game semantics. The crux is an intricate combinatorial lemma allowing us to synchronize states of strategies which reach the same resources, but with different multiplicity. Putting this together with the previously established relational collapse, this provides a new proof of the qualitative-quantitative correspondence first established by Ehrhard in his celebrated extensional collapse theorem. Whereas Ehrhard's proof is indirect and rests on an abstract realizability construction, our result gives a concrete, combinatorial description of the extraction of quantitative information from a qualitative model.
Papers in statistical theory rest on long proofs that few readers check line by line. We present KAPPA (Kit of Agents for Proof-checking Papers with a proof Assistant), open-source software that organizes language model agents, run by Claude Code, to translate the proofs of a LaTeX paper into the proof assistant Lean 4 with its library Mathlib and check them there, result by result. Each numbered result receives one of four verdicts: verified unconditionally; verified conditionally, with every cited or unproved input stated as a named hypothesis; issue recorded, with its place in the paper and the side at fault; or unfinished, with the reason. The kit follows the paper's own proof step by step, rather than finding a new one. Separate agents write and review, each statement is compared with the paper before it is proved and fingerprinted afterwards, and every verdict rests on evidence that can be rerun without a language model. The kit runs unattended for hours, waits out usage limits, re-checks only what a revision changed and reports its cost by role, model and result. We describe the design, walk through a complete run, and apply the kit to four example papers with answer keys and to three published statistical papers, checked blind in their versions from before public corrections. In all three, the kit found the published error at the step that the correction names, together with further problems that we confirmed by hand. Given the same papers directly, the same language model working with Lean classified fewer results correctly on the longest paper and reported some statements narrower than the paper's as proved; no verdict of the kit claimed more than its Lean evidence showed.
We investigate the eliminability of the absoluteness operator Delta in Goedel logics. While Delta is not definable from the standard connectives and disrupts important proof-theoretic properties, we show that it becomes eliminable at the propositional level under a non-standard semantics in which all propositional atoms (except the truth constant 'True') are interpreted strictly below 1. Under this semantics, every formula containing Delta is equivalent to a disjunction of chain formulas, yielding a Delta-free normal form (standard and non-standard semantics coincide w.r.t. valid formulas without Delta). We further analyze the situation in the first-order setting, where Delta-elimination fails in general due to recursion-theoretic and topological constraints, but can be recovered under witnessed semantics.
Adrienne Lancelot, Giulio Manzonetto, Guy McCusker, Gabriele Vanoni
原文摘要
The relational semantics of linear logic is a powerful framework for defining resource-aware models of the $λ$-calculus. However, its quantitative aspects are not reflected in the preorders and equational theories induced by these models. Indeed, they can be characterized in terms of (in)equalities between Böhm trees up to extensionality, which are qualitative in nature. We employ the recently introduced checkers calculus to define a quantitative contextual preorder on $λ$-terms, and demonstrate that it coincides with the preorder associated to the relational semantics.
Ranking functions are a fundamental technique for proving loop termination, but constructing them manually is challenging and requires substantial expertise. We propose RAINFALL, a neuro-symbolic framework which follows the generate-verify-refine paradigm and tightly integrates LLMs with symbolic verification. RAINFALL formulates ranking function synthesis as a code-completion task by inserting placeholders into the input program and constraining the LLM to fill four predefined slots for each loop. The completed program is then checked by a symbolic verifier. If verification succeeds, a valid ranking function is obtained; otherwise, the verifier's diagnostics are translated into natural-language feedback that guides iterative refinement of the LLM's output. We evaluate RAINFALL on two public benchmark suites comprising 358 terminating C programs. Using MiniMax V2.7, RAINFALL synthesizes verifiable ranking functions for 323 programs (90.22%), outperforming all promising baselines. In particular, it solves 12 more programs than PROTON 2.1, the best-performing baseline and winner of the SV-COMP 2026 Termination category. Furthermore, combining RAINFALL with PROTON 2.1 increases the success rate to 96.65% (346/358 programs). We also conduct ablation study confirming individual contributions of placeholder contracts, few-shot examples, symbolic verification, and diagnostic information feedback. Experiments across LLMs further confirm the effectiveness of RAINFALL. Our results suggest a generic, sound framework for ranking-function synthesis and, beyond termination, LLM-assisted synthesis of formally checkable liveness certificates, both of which represent new directions of software engineering research.
Sean Kauffman, Kim Guldstrand Larsen, Martin Zimmermann
原文摘要
Nfer is a Runtime Verification language for the analysis of event traces that applies rules to create hierarchies of time intervals. This work examines the complexity of the evaluation and satisfiability problems for the data-free fragment of nfer. The evaluation problem asks whether a given interval is generated by applying rules to a known input, while the satisfiability problem asks if an input exists that will generate a given interval. By excluding data from the language, we obtain polynomial-time algorithms for the evaluation problem and for satisfiability when only considering inclusive rules. Furthermore, we show decidability for the satisfiability problem for cycle-free specifications with a NExpTime lower bound and undecidability for satisfiability of full data-free nfer.
Decisiveness is central to reachability algorithms for infinite Markov chains, but is a qualitative notion: it guarantees that path-enumeration algorithms terminate, without an a priori bound on the horizon. We study decisiveness quantitatively of Markov chains, and apply our results to the approximate quantitative coverability problem for effective probabilistic well-structured transition systems (pWSTS). We introduce truncation sets, which represent the horizons sufficient to approximate the reachability probability within a precision $\varepsilon$. These sets reformulate decisiveness quantitatively, and their elements can be computed when it is decisive. To obtain such horizons, before any exploration, we introduce geometrically reachable sets, for which the probability of not reaching them within n steps decreases exponentially in n, up to a prefactor depending on the initial state. Horizons valid for all initial states exist iff this prefactor can be chosen uniformly for boundedly coarse chains. For such pWSTS, including probabilistic vector addition systems and Galton-Watson processes, we obtain a priori global horizons growing at most logarithmically in $1/\varepsilon$. In the non-uniform case, we give a drift condition ensuring the existence of a finite geometrically reachable set. Under this condition, we derive a priori horizons growing at most logarithmically in $1/\varepsilon$ and in the weight w(x) measuring the size of the initial state x. In both cases, we compute a horizon function whose evaluation on any initial state and precision requires only a constant number of elementary operations $(+, -, \times, / \text{ and } \log)$. For probabilistic lossy channel systems, this yields, to our knowledge, the first explicit a priori horizons from arbitrary initial states, and it extends to $ν$-pLCS, a new infinitely branching variant with probabilistic emissions.
We give a categorical formulation of Kraus' "magic trick" for recovering information from truncated types. Rather than type theory, we work in Van den Berg-Moerdijk path categories with a univalent universe, and rather than propositional truncation we work with arbitrary cofibrations, which includes truncation as a special case. We show, using Kraus' argument that any cofibration with homogeneous domain is a monomorphism. We give some simple concrete examples in groupoids to illustrate the interaction between homogeneous types, cofibrations and univalent fibrations.
Samir Datta, Asif Khan, Felix Tschirbs, Nils Vortmeier, Thomas Zeume
原文摘要
Consider two planar graphs which are subject to edge insertions and deletions. We show that whether the two graphs are isomorphic can be maintained with first-order logic formulas and auxiliary data of polynomial size. This places the dynamic planar graph isomorphism problem into the dynamic descriptive complexity class DynFO. As a consequence, there is a dynamic constant-time parallel algorithm with polynomial-size auxiliary data which maintains whether two dynamic planar graphs are isomorphic.
Linear Temporal Logic over Finite Traces (LTLf) is widely used in Artificial Intelligence to specify desired temporal behavior and determine whether finite traces satisfy or violate it. Explaining these outcomes is essential for debugging and trust. However, existing approaches typically capture either sufficient reasons or counterfactual causes, without clarifying their relationship. We introduce a native characterization of abductive (AXps) and contrastive (CXps) explanations directly over finite traces, based on locking and unlocking time-indexed events. We prove its equivalence to the classifier-based formulation, and show that CXps coincide with a recently proposed temporal causality semantics for LTLf. This characterization enables an efficient implementation based on the enumeration of minimal unsatisfiable cores and minimal correction sets of ASP programs. Experiments on real-world event logs demonstrate the effectiveness of the approach.
Host-based intrusion detectors order events, window audit streams and sequence attack stages by their timestamps. Every conclusion they draw therefore rests on assumptions about clocks, delivery, entity identity and the integrity of the evidence, and most of these assumptions go unstated. This Systematization of Knowledge (SoK) makes them explicit. Drawing on temporal databases, stream processing and runtime verification, we distill five components of a time model, together with a checklist of evidence assumptions and four levels of guarantee, and walk through a running campaign with them. A documented search yields fifteen families of detection and investigation. For papers selected by screening role and period, plus named systems and landmarks, we check the stated assumptions in the full texts, and we assess eleven temporal knowledge representation and reasoning (KRR) formalisms and a learned baseline on nine attack patterns. Some early alert correlators stated clock and delivery assumptions, and forensics measures clock error. Current provenance-based detectors state no clock bound. Four audited papers describe input-ordering conditions or conventions, but none specifies conditions for conclusion validity across a stated class of delivery failures. In the selected provenance-based systems, campaign composition uses system-specific procedures or objectives, learned state, causal graph queries or language models, and beyond CEP-based query systems, we found no temporal rule language or stream reasoner from KRR applied to host audit data. In 28 of 38 audited provenance-era detector papers, capture or stored evidence is trusted or its manipulation is placed out of scope, and none says which conclusions survive deleted, delayed or forged records. Five open challenges follow from these gaps.
Hans van Ditmarsch, Djanira Gomes, Rojo Randrianomentsoa, Thomas Studer
原文摘要
We present the directed simplicial complex: this is an extension of a simplicial complex with a direction on its faces. This enables us to model a local notion of belief in a three-valued semantics, in which formulas may be true, false, or undefined. Of particular interest is a class of directed simplicial complexes called cups, in which any maximal set of vertices of different colors is a facet. Cups permit a more concise relational representation. We give examples and applications of our logical semantics. We then provide a sound and complete axiomatization for the class of directed simplicial complexes and extend it to an axiomatization for the class of cups. We include a detailed technical comparison with other works on belief on complexes. Our work generalizes a prior axiomatization for knowledge on (undirected) impure simplicial complexes.
Verifying quantum programs today requires specialized proof systems built on interactive theorem provers such as Rocq, Lean, Isabelle, or Why3, and even modest quantum algorithms cost substantial human effort to verify. Because quantum programs cannot be effectively checked by runtime testing, formal verification is the only viable path to correctness, making the cost of these bespoke systems the chief barrier to scalable quantum software assurance. We show that verifying a quantum program should be no harder than verifying a classical array program, a paradigm commonly used in parallel programming: the Qafny quantum proof system translates into Hoare logic with array theory completely (every Qafny proof becomes a valid Hoare triple) and soundly (a valid translated triple certifies only true quantum specifications), and we mechanize the completeness direction in Rocq. The embedding determinizes quantum proof obligations and exposes the full machinery of classical automated verifiers (e.g. Dafny and SMT-based separation logic) to quantum code, removing the need for dedicated quantum proof system infrastructure. In particular, the embedding supplies the correctness argument that the existing compilation of Qafny into Dafny lacks.
What does a recursive definition mean when the logic is grounded, so a sentence is true only if a terminating process makes it so? Kripke's answer for the special case of truth was the least fixed point. This paper takes that answer as a rule. Reflective Grounded Arithmetic (RGA) already writes any recursive definition as one self-referential term, paradoxes included. Inductive Grounded Arithmetic (IGA) adds the elimination rule for every positive one, so that accessibility, the winning positions of a game, and grounded truth itself each come with their own induction principle. The rule is sound for the omega model and the system is consistent, both machine-checked in Isabelle/HOL. With the rule, IGA proves Goodstein's theorem outright, in a logic without implication at an undecided hypothesis, and the Kirby-Paris hydra game for primitive recursive strategies, both with every step of the arithmetic a grounded derivation. It proves the Cantor normal forms well-founded, from which the well-foundedness of the standard $\varepsilon_0$ notation system in HOL follows, and it certifies the Veblen notation as an ordinal below every $\varphi(n,0)$. It defines Kripke truth for RGA's own language as one fixed point that agrees with the omega model exactly. Two equivalences locate what the classical world has and the grounded one does not. IGA proves the whole Veblen notation well-founded exactly when it admits bivalence of accessibility for that order, and bivalence of accessibility for an order is sound exactly when the order is well-founded; so $Γ_0$ lies at most one instance of excluded middle away, at the accessibility predicate of the notation, and that instance, the well-foundedness sentence and one membership claim in a primitive-recursive form of Kleene's $\mathcal O$ are interderivable over IGA.
The Minimum Equivalent Expression problem (MEE) asks, given a propositional formula $φ$ and an integer $k$, whether $φ$ is equivalent to a formula with at most $k$ occurrences of propositional variables. While the problem is clearly in $Σ_2^p$, its completeness under polynomial-time many-one reductions has remained open since at least 1979. Buchfuhrer and Umans showed in 2008 that it is $Σ_2^p$-complete under polynomial-time Turing reductions. In this note, we present a proof discovered by GPT-Sol-6.1 which establishes that MEE is $Σ_2^p$-complete also under polynomial-time many-one reductions.
Andrzej Indrzejczak, Tomasz Jarmu{ż}ek, Mateusz Klonowski, Michał Zawidzki
原文摘要
Non-Classical Logics. Theory and Applications (NCL) is an international conference aimed at presenting novel results and survey papers in the broad field of non-classical logics and their applications. The first two editions were held in Ł{ó}d{ź}, Poland, in September 2008 and 2009. Subsequent editions were organised in Toru{ń} (2010, 2012, 2015, 2018) and Ł{ó}d{ź} (2011, 2013, 2016, 2022, 2024). The tenth edition of the conference, organised by the University of Ł{ó}d{ź} in 2022, was the first to have its proceedings published in EPTCS. We are honoured and pleased to continue this practice and, for the third time, include all accepted long papers in an EPTCS volume. Furthermore, for the first time, we have also included all accepted extended abstracts in the volume.
Logically constrained term rewrite systems (LCTRSs) are a rewriting formalism that naturally supports built-in data structures, including integers and bit-vectors. The recent framework of existentially constrained terms and most general constrained rewriting on them (Takahata et al., 2025) has many advantages over the original approach of rewriting constrained terms. In this paper, we introduce partial constrained rewriting, a variant of rewriting existentially constrained terms whose underlying idea has already appeared implicitly in previous analyses and applications of LCTRSs. We examine the differences between these two notions of constrained rewriting. First, we establish a direct correspondence between them, leveraging subsumption and equivalence of constrained terms where appropriate. Then we give characterizations of each of them, using the interpretation of existentially constrained terms by instantiation. We further introduce the novel notion of value interpretation, that highlights subtle differences between partial and most general rewriting.
Skander Moalla, Ahmad Rammal, Niket Patel, Remi Munos, Julia Kempe
原文摘要
While progress in informal mathematics has come largely from scaling the chain-of-thought and agentic reasoning of frontier language models, formal theorem proving has increasingly relied on human-designed workflows around these models, such as planning, decomposition, and orchestrating specialized provers. This raises a bitter-lesson question: can a simple agentic loop that scales test-time compute match these workflows? The answer matters as formal verification must scale to keep up with frontier models producing research mathematics, yet new workflows rarely settle it, as their gains are often confounded with stronger models or more compute. We study this question with h1, a simple harness that gives a model an efficient interface to Lean 4 and keeps it working through verifier feedback and self-reflection, without imposing a decomposition or search structure. h1 reaches state-of-the-art results on PutnamBench, FATE-X, and FormalConjectures100-Solved, settles eight open problems in FormalConjectures, and, at matched models, matches or outperforms specialized workflows such as Goedel-Architect and AlphaProof Nexus, surprisingly showing that a single attempt (pass@1) can often scale to solve open problems. Through ablations and matched-model comparisons with Claude Code and AxProverBase, we find that simplicity alone is not enough: an efficient Lean interface largely determines cost and solve rate. We thus recommend that future work evaluate frontier models with efficient execution feedback before prescribing workflows, at the risk of bitter lessons.