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1,409 records · Page 5Linked to original sources
arXiv2026

Polynomial Invariants for Probabilistic Transition Systems with Unbounded Support

We study the synthesis of polynomial invariants for probabilistic transition systems (PTS) based on martingale theory. We present tractable methods to verify that such polynomials are indeed invariants, in the sense that their expected value upon termination is the same as their value at the start of the computation. We do this by applying the Optional Stopping Theorem (OST) in the form of a specific precondition. This precondition requires the existence of an integrable dominating function for the martingale expression, which implies uniform integrability; we refer to this condition as dui. For linear PTS we simplify the dui property to proving finiteness of the expected value of an expression depending on the update matrix, the degree of the martingale expression, and the stopping time. Specifically, if all random samples have finite moments and we can verify a moment bound on the runtime of a linear loop, then we can automatically synthesise polynomial loop invariants that satisfy the OST. Notably, dui allows for the sampled distributions to have unbounded support, which is a novel contribution to the field.

cs.LO↗
arXiv2026

MathAdv: What Theorem Provers Know, Reason, Formalize, and Generalize

Formal theorem proving enables machine-verifiable evaluation of mathematical reasoning, yet existing benchmarks often emphasize aggregate proof accuracy, concentrate on a narrow range of mathematics, and provide limited evidence of robustness to equivalent reformulations. We introduce MathAdv, a diagnostic benchmark spanning 13 domains across undergraduate- and graduate-level mathematics. Alongside Lean 4 theorem proving, MathAdv provides up to three auxiliary tasks: multiple-choice questions that probe mathematical knowledge, fill-in-the-blank problems that isolate informal reasoning, and expert-crafted transformations that test robustness to problem presentation. Our evaluation of contemporary theorem provers yields four findings: formalization remains a major bottleneck; performance varies substantially across mathematical domains; natural-language guidance helps general-purpose LLMs but can hinder proof-specialized models; and mathematically equivalent reformulations expose substantial robustness limitations. Together, these results show how component-wise evaluation can reveal model capabilities and failure modes that aggregate theorem-proving accuracy obscures. The dataset and evaluation scripts are available at https://github.com/margotyjx/MathAdv.git.

cs.CL↗
arXiv2026

Fast Constraint Extraction for Corrective Control under STL Specifications via Logical Dependency Tracking

Ensuring the satisfaction of Signal Temporal Logic (STL) specifications under uncertainty is challenging, as reachability-based monitoring provides guarantees but does not indicate how to restore satisfaction when it becomes indeterminate. A key difficulty is identifying which uncertain components actually affect global satisfaction, especially for nested formulas. This paper introduces a logical dependency tracking framework that propagates uncertainty through the STL structure and captures the causal contribution of reachable sets to satisfaction. By associating markers to uncertain predicates and propagating them via three-valued semantics, we extract in milliseconds a compact Disjunctive Normal Form (DNF) of sufficient constraints, avoiding combinatorial enumeration. As an application, we formulate control correction as a minimum-effort optimization problem. Using zonotopic reachability, the derived constraints are enforced via linear programming, yielding corrections that guarantee STL satisfaction under bounded uncertainty and provide certified probabilistic bounds in the stochastic case. We demonstrate the approach on a nonlinear system with nested STL specifications, showing that dependency tracking enables efficient and formally guaranteed correction. The tracking implementation is available at https://github.com/Antoine-Bst/STL-Three-Valued-Clause-Filtering/.

cs.LO↗
arXiv2026

Zero-Knowledge Model Checking

We introduce a technology to formally verify that a software system satisfies a temporal specification of functional correctness, without revealing the system itself. Our method combines a deductive approach to model checking to obtain a formal certificate of correctness for the system, with zero-knowledge proofs to convince an external verifier that the system -- kept secret -- complies with its specification of correctness -- made public. We consider proof certificates represented as ranking functions, and introduce both an explicit-state and a symbolic scheme for model checking in zero knowledge. Our explicit-state scheme assumes systems represented as transition graphs. We use polynomial commitments to convince the verifier that the public proof certificates correspond to the secret transition relation. Our symbolic scheme assumes systems specified as linear guarded commands. We apply Farkas' lemma to obtain a witness for the validity of the ranking function and employ sigma protocols with folding to efficiently convince the verifier of the witness's existence. We built a prototype to demonstrate the practical efficacy of our two schemes on linear temporal logic verification examples. Our technology enables formal verification in domains where both the safety and the confidentiality of the system under analysis are critical.

cs.CR↗
arXiv2026

Automatic constraints with few subpowers and graphoid recognition

Finite automata can describe relations of unbounded arity that are exponentially larger than their descriptions. We prove that constraint satisfaction for such relations is solvable in polynomial time whenever their length slices are preserved by a common fixed edge operation on a finite domain. The algorithm computes compact representations of the complete solution relation and its projections. Its main ingredient is a polynomial-time compilation of nondeterministic finite automata into the fork witnesses and small projections required by the few-subpowers algorithm. In the Mal'tsev case, a direct proof is polynomial also when the domain and operation table are supplied as input, answering the Mal'tsev tractability question for automatic constraint satisfaction. We also characterize all invariant relations of a family of 3-edge algebras with neither Mal'tsev nor near-unanimity terms. Their normal forms combine Boolean activity constraints with affine value spaces and yield canonical quadratic-bit representations constructible from NFAs or arbitrary generators. For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations. The quadratic boundary bounds are optimal in the worst case. A fixed three-state example separates polynomial-time recognition from hard exact counting.

cs.LO↗
arXiv2026

Target Discounted Sum Problem on Markov Chains with Applications to Markov Decision Processes

The discounted sum is a way to aggregate a sequence of weights from a finite alphabet $Σ$, i.e., for a discount factor $λ$, the discounted sum of a sequence $w_0 w_1 w_2 \cdots$ over $Σ$ is $\sum_{i \in \mathbb{N}} w_i λ^i$. The target discounted-sum problem, which is currently open, asks, given $λ,Σ$ and a target $t$, whether there exists an infinite sequence over $Σ$ whose discounted sum is equal to $t$. We study and solve a probabilistic variant of this problem, i.e., the target discounted-sum problem on Markov chains. To do this, we prove that the event consisting of paths whose discounted sum is equal to the target and has infinitely many distinct suffix sums has probability zero. This structural property allows us to solve the target discounted-sum problem on Markov chains using an automata-theoretic technique. We apply our technical results to Markov decision processes with target discounted-sum objectives: we show that the infimum value and the finite-memory supremum value are computable in pseudo-polynomial time and are attained by deterministic finite-memory strategies.

cs.LO↗
arXiv2026

Factorized Boolean representations for efficient quantum synthesis

Quantum algorithms promise advantages beyond classical reach, but running them on error-corrected hardware requires translating Boolean specifications into reversible circuits, and the resources that translation demands determine what is executable. Established methods minimize a Boolean expression and map it to a circuit, assuming the minimized form is best. Here we show that minimized expressions retain algebraic structure minimization cannot reach, arising from containment and complementary-polarity relationships among their terms, and that extracting it yields circuits cheaper to execute despite having more operations. The decisive quantity is not a circuit's operation count but the control count of its widest operation, a superlinear cost; extracting shared factors trades a few wide operations for many narrow ones and reduces qubit count. Across benchmarks and oracles from quantum search and factoring algorithms, at the representation level the transformation never increases either cost measure, a guarantee from its construction. Translation to an executable circuit returns part of that advantage, since auxiliary lines must be uncomputed, yet the factorized circuit still left a leading circuit-level optimizer reaching lower final counts, and faster, than unaided. The representation of a computation is therefore itself a resource, optimizable before compilation and distinct from both logic minimization and circuit-level optimization.

quant-ph↗
arXiv2026

The Complexity of Coverability-Like Problems in Elementary Object Systems: Data-Nets to the Rescue

Elementary Object Systems (EOSs) are a model in the nets-within-nets (NWNs) paradigm, where tokens in turn can host standard Petri nets. We study the complexity of coverability-like problems, including termination and boundedness, over EOSs. Since coverability and boundedness are undecidable in general on EOSs, we focus on the relevant fragment of conservative EOSs (cEOSs). Our technique interprets cEOSs into the framework of data nets, whose tokens carry data from an infinite domain, thus bridging the nesting and the data-aware paradigms. Specifically, we show that cEOS coverability-like problems are equivalent to the coverability-like problems over an interesting fragment, called channel-$ν$PNs (c-$ν$PNs), of data nets that extends $ν$PN (featuring globally fresh name creation) with restricted forms of transfers with renaming. c-$ν$PNs remain less expressive than Unordered Data Nets, which feature lossy name creation as well as powerful forms of whole-place operations and broadcasts. These reductions allow us to analyze cEOS coverability taking advantage of known results on data nets. We conclude that the complexity of cEOS coverability is double-Ackermanian, $\mathcal{F}_{ω2}$-complete, while termination and boundedness are non-primitive recursive.

cs.CC↗
arXiv2026

Granthi: Higher-Order Quantum Programming via Unitary Wiring

Many mainstream quantum programming languages confine higher-order structure to a classical host while restricting the quantum layer to first-order operations on qubits. This paper presents Granthi, a purely unitary higher-order quantum programming language built on three design commitments: quantum programs are first-class values that may be passed, returned, and coherently composed; additive structure is tag-preserving routing rather than observational branching, so control may remain in superposition; and programmer-facing finite label types with staged reversible-operation bindings provide domain-level control spaces without exposing tag management. These bindings are eliminated by elaboration before Source typing. Granthi deterministically normalizes each Source program to a canonical wiring form. Every well-typed Source program, including a term of function type, has a unitary boundary interpretation. Under backend correctness (BC), the reference compiler produces a unitary circuit realizing that interpretation. Granthi's currently supported executable fragment is implemented end-to-end: an OCaml DSL elaborates surface programs through a higher-order Core IR to executable quantum circuits via pytket. The language directly supports the pure-unitary quantum switch for explicitly supplied operations; closed instances compile to static circuits. It also supports interference on control-flow history and structured finite control, all within the purely unitary fragment

quant-ph↗
arXiv2026

Erased Postulates, Identity Types and Quotients

This text is concerned with the question of whether, in type theory with erasure annotations, one can postulate that some type is inhabited and still have a guarantee that a program will not get stuck. Previous work has provided such guarantees for consistent erased postulates, i.e. postulates that are restricted to be used in erased contexts. Here those guarantees are extended to type theory with identity types. Similar ideas provide a simple way to support quotient types: it is shown that one can let things like "the equivalence classes for two related values are equal" be erased postulates and have an eliminator that only computes for the equivalence class constructor, and still get a guarantee that programs will compute correctly. Another question is whether programs compute correctly if one is allowed to transport (cast) using erased identity proofs. It is shown that this is safe in the absence of quotients and postulates, and in the presence of quotients and erased postulates that can be implemented using equality reflection. However, unrestricted transports of this kind are not compatible with erased, postulated univalence. For that reason the text includes a study of the function []-cong, which encapsulates a limited form of transport for erased identity proofs. The text is accompanied by machine-checked Agda proofs.

cs.PL↗
arXiv2026

Antichains for Concurrent Parameterized Games (Long Version)

Concurrent parameterized games involve a fixed yet arbitrary number of players. They are described by finite arenas in which the edges are labeled with languages that describe the possible move combinations leading from one vertex to another (n players yield a word of length n). Previous work showed that, when edge labels are regular languages, one can decide whether a distinguished player, called Eve, has a strategy to ensure a reachability objective, against any strategy profile of her arbitrarily many opponents. This decision problem is known to be PSPACE-complete. A basic ingredient in the PSPACE-membership proof is the reduction to the exponential-size knowledge game, a 2-player game that reflects the knowledge Eve has on the number of opponents. In this paper, we provide a symbolic approach, based on antichains, to compute Eve's winning region in the knowledge game. In words, it gives the minimal knowledge Eve needs at every vertex to win the concurrent parameterized reachability game. More precisely, we propose two fixed-point algorithms that compute, as an antichain, the maximal elements of the winning region for Eve in the knowledge game. We implemented these two algorithms in C++, as well as the one initially proposed, and report on their relative performances on various benchmarks.

cs.LO↗
arXiv2026

Predictive Zonotope Reduction: Precise Runtime Monitoring under Uncertainty

Robots operating in physical environments make control decisions based on uncertain sensor measurements, which can lead to unsafe or suboptimal actions. Runtime monitors that check their behavior against safety specifications must represent this uncertainty soundly. Zonotopes are a widely used representation, but continuously incorporating new measurements grows their order unboundedly, so monitors must periodically apply an over-approximating reduction. The choice of the reduction method substantially affects the zonotope's precision, yet existing approaches typically utilize a fixed method throughout the run, even though the optimal choice depends on the current state. This paper presents a Predictive Zonotope Reduction (PZR) approach, which frames reducer selection as an optimal control problem and solves it using beam-search model predictive control. Policy distillation into a small neural policy further provides substantially higher execution speed than model predictive control while maintaining improved performance, enabling uncertainty-aware runtime monitoring on resource-constrained real-time systems. We implement our approach in the RLola runtime monitoring framework and evaluate it on a 5-degree-of-freedom robotic arm simulated in MuJoCo, with sensor uncertainty modeled according to ISO 5725. Experiments on a Raspberry Pi 5 show that dynamic reduction significantly lowers false-positive rates in monitoring compared with static reduction strategies.

cs.RO↗
arXiv2026

sheval: An RDF data shapes evaluation tool and test-suite for recursive shapes

Two different languages have been developed to validate RDF data based on the concept of a shape: ShEx and SHACL. In each language it is possible to define a shape that refers to itself, which is called a recursive shape. While in the case of ShEx, the semantics of recursive shapes is well defined and is part of the specification, in the case of SHACL, the semantics of recursive shapes is left to the implementation of the different SHACL engines. Consequently, the different SHACL engines show different behaviours when confronted with recursive shapes. In this paper we present sheval: an evaluation framework consisting of a tool and a test suite that can be used to compare the behaviour of different shapes technologies when confronted with recursive definitions. The tool has been used to evaluate and understand the differences in the implementation of recursive shapes in ShEx and SHACL. It provides a framework for testing and comparing the behaviour of different shape engines, helping to identify inconsistencies and potential issues, and providing a basis for further research and development in the field of shape-based validation of RDF data.

cs.DB↗
arXiv2026

Length Generalization for Transformers via Compression

Recent advancements in transformer length generalization theory enable us to reliably predict when a transformer can learn to solve a task. In particular, the C-RASP hypothesis (a formalized version of the so-called RASP-l conjecture) posits that transformers length-generalize on a task if and only if a solution is expressible in the C-RASP language. While this hypothesis has strong empirical validation, theoretical problems arise from the fact that no computable length generalization bounds exist for C-RASP, alongside the discovery of seemingly contradictory experiments. To address these problems, we refine the C-RASP hypothesis utilizing the recently-proposed fragments C-RASP+ and C-RASP1. These fragments have computable length generalization bounds, though in the worst case requiring an extremely large (double exponential) sample size. It is an open question whether these sample size bounds are tight. In this paper, we resolve this open question by providing an exponentially tighter bound. In doing so, we show a polynomial length generalization bound for transformers if we adopt compressed strings, via a novel connection to power words. As an application, we show how this yields a fine-grained analysis of the C-RASP conjecture that resolves contradicting experimental evidence against it.

cs.LG↗
arXiv2026

AirFM-DDA: Air-Interface Foundation Model in the Delay-Doppler-Angle Domain for AI-Native 6G

The success of large foundation models is catalyzing a new paradigm for AI-native 6G network design: wireless foundation models for physical-layer design. However, existing models often operate on channel state information (CSI) in the spatial-temporal-frequency (STF) domain, where multipath components are superimposed and structurally entangled. This hinders the learning of a universal channel representation. Their reliance on global attention also incurs prohibitive overhead. In this paper, we propose AirFM-DDA, an Air-interface Foundation Model in the Delay-Doppler-Angle (DDA) domain. AirFM-DDA reparameterizes CSI into the DDA domain to resolve multipath components along physically meaningful axes and employs window-based attention with frame-structure-aware positional encoding. Extensive experiments demonstrate transferability across scenarios, tasks, datasets, and antenna configurations. For channel prediction and estimation, AirFM-DDA generalizes zero-shot to unseen cities, achieving average normalized mean-square error (NMSE) gains of 4.9-8.5 dB over the strongest baselines. With only 10% labeled data, it achieves average gains of 12.0 percentage points in Top-1 accuracy for beam prediction and 3.4 percentage points in F1 score for line-of-sight (LoS) identification. It further transfers across simulated datasets and adapts to measured data and different antenna arrays. Compared with global attention, window-based attention reduces training and inference costs by nearly an order of magnitude.

cs.LG↗
arXiv2026

AxQM: A Textbook-Scale Benchmark for Formal Proof Synthesis in a Library of Finite-Dimensional Quantum Mechanics

Formalizing mathematics in a proof assistant, where a machine checks every definition, statement and proof, has set a new standard of rigor. Large language models are now capable of formalizing autonomously, even at the scale of whole textbooks. We bring this standard of rigor to physics, where theoretical arguments carry idealizations that are rarely stated fully, and any logical gaps could have a cascading effect on interdependent results. Recognizing the need to evaluate autoformalization systems for physics, we release AxQM, 1,019 kernel-checkable proof-synthesis tasks over 479 items drawn from the textbook Quantum Computation and Quantum Information by Nielsen and Chuang. The tasks are stated in a custom Lean library of finite-dimensional quantum mechanics. By task count, it is the largest proof-synthesis benchmark in physics by a factor of four. AxQM is derived from a near-complete formalization of the formal portions of the textbook, so every task is guaranteed a solution, which we keep private. Grading of the benchmark is done deterministically by the Lean kernel, which checks that the proof compiles, that no sorry appears in it or in any declaration it depends on, and that it introduces no new axioms.

quant-ph↗
arXiv2026

Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph $F \in \mathcal{F}$. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems $\textrm{HomInd}(\mathcal{F})$ of deciding homomorphism indistinguishability over $\mathcal{F}$ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of $\textrm{HomInd}(\mathcal{F})$, Seppelt (MFCS 2024) showed that $\textrm{HomInd}(\mathcal{F})$ is in randomised polynomial time for every graph class $\mathcal{F}$ of bounded treewidth that can be defined in counting monadic second-order logic $\mathsf{CMSO}_2$. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in $\mathsf{PTIME}$. For $\mathsf{CMSO}_2$-definable graph classes $\mathcal{F}$ of bounded pathwidth, we improve the previous complexity upper bound for $\textrm{HomInd}(\mathcal{F})$ from $\mathsf{PTIME}$ to $\mathsf{C}_=\mathsf{L}$ and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as $\mathsf{C}_=\mathsf{L}$-complete.

cs.CC↗
arXiv2026

Self-extensional logics of formal inconsistency: Decidability and limits for paraconsistency

RmbC is a self-extensional paraconsistent logic in the family of Logics of Formal Inconsistency (LFIs). This system is obtained from mbC (the basic LFI) by adding the replacement property via two global inference rules. RmbC is characterized by a non-explosive negation $\neg$ and a consistency operator $\circ$, which recovers the principle of explosion in a controlled way. Together with its principal axiomatic extensions, RmbC admits a standard Lindenbaum--Tarski algebraization, with Boolean algebras with LFI operators (BALFIs) as its algebraic semantics. In this paper, we study how far this self-extensional paraconsistent behavior can be extended axiomatically, starting from RmbC. We classify pairs of very natural consistency axioms according to whether they preserve paraconsistency or force classical collapse; identify six algebraically equivalent explosive cores; and isolate a separate structural obstruction for the combination of excluded middle for $\neg$ with an involutive negation. We also investigate, for the first time, the decidability of this family of self-extensional LFIs. As a first result, we prove the finite model property for RmbC with respect to BALFI semantics via an algebraic filtration, which yields decidability, and transfer this result to several paraconsistent axiomatic extensions of RmbC. Finally, we establish a 2-EXPTIME upper bound for the validity problem of RmbC and a coNP-hardness lower bound.

cs.LO↗