The increasing compute demands of AI systems has led to the emergence of services that train models on behalf of clients lacking necessary resources. However, ensuring correctness of training and guarding against potential training-time attacks, such as data poisoning and backdoors, poses challenges. Existing works on verifiable training largely fall into two classes: proof-based systems, which can be difficult to scale, and ``optimistic'' methods that consider a trusted third-party auditor who replicates the training process. A key challenge with the latter is that hardware nondeterminism between GPU types during training prevents an auditor from replicating the training process exactly, and such schemes are therefore non-robust. We propose a method that combines training in a higher precision than the target model, rounding after intermediate computation steps, and storing rounding decisions based on an adaptive thresholding procedure, to successfully control for nondeterminism. Across three different NVIDIA GPUs (A40, Titan XP, RTX 2080 Ti), we achieve exact training replication at FP32 precision for both full-training and fine-tuning of ResNet-50 (23M) and GPT-2 (117M) models. Our verifiable training scheme significantly decreases the storage and time costs compared to proof-based systems.

As demand for LLM inference grows, it is becoming increasingly important that providers and their customers can verify that inference processes are performed correctly, without errors or tampering. However, re-running the same inference process twice often leads to different results due to benign numerical noise, making it difficult to distinguish legitimate variation from actual problems. To address this problem, we introduce Token-DiFR (Token-Divergence-From-Reference), a method for verifying inference outputs by comparing generated tokens against predictions made by a trusted reference implementation conditioned on the same random seed. Sampling seed synchronization tightly constrains valid outputs, leaving providers minimal room to deviate from correct inference, which allows output tokens themselves to serve as auditable evidence of correctness at zero additional cost to the provider. Token-DiFR reliably identifies sampling errors, simulated bugs, and model quantization, detecting 4-bit quantization with AUC $>$ 0.999 within 300 output tokens. For applications requiring sample-efficient forward-pass verification, we additionally introduce Activation-DiFR, a scheme that uses random orthogonal projections to compress activations into compact fingerprints for subsequent verification. Activation-DiFR detects 4-bit quantization with AUC $>$ 0.999 using just 2 output tokens, while reducing communication overhead by 25-75% relative to existing methods. We release an open-source integration with vLLM to accelerate practical deployment of verifiable inference.

We study stochastic planning problems in Markov Decision Processes (MDPs) with goals specified in Linear Temporal Logic (LTL). The state-of-the-art approach transforms LTL formulas into good-for-MDP (GFM) automata, which feature a restricted form of nondeterminism. These automata are then composed with the MDP, allowing the agent to resolve the nondeterminism during policy synthesis. A major factor affecting the scalability of this approach is the size of the generated automata. In this paper, we propose a novel GFM state-space reduction technique that significantly reduces the number of automata states. Our method employs a sophisticated chain of transformations, leveraging recent advances in good-for-games minimisation developed for adversarial settings. In addition to our theoretical contributions, we present empirical results demonstrating the practical effectiveness of our state-reduction technique. Furthermore, we introduce a direct construction method for formulas of the form $\mathsf{G}\mathsf{F}φ$, where $φ$ is a co-safety formula. This construction is provably single-exponential in the worst case, in contrast to the general doubly-exponential complexity. Our experiments confirm the scalability advantages of this specialised construction.

Financial institutions deploy Large Language Models (LLMs) for reconciliations, regulatory reporting, and client communications, but nondeterministic outputs (output drift) undermine auditability and trust. We quantify drift across five model architectures (7B-120B parameters) on regulated financial tasks, revealing a stark inverse relationship: smaller models (Granite-3-8B, Qwen2.5-7B) achieve 100% output consistency at T=0.0, while GPT-OSS-120B exhibits only 12.5% consistency (95% CI: 3.5-36.0%) regardless of configuration (p<0.0001, Fisher's exact test). This finding challenges conventional assumptions that larger models are universally superior for production deployment. Our contributions include: (i) a finance-calibrated deterministic test harness combining greedy decoding (T=0.0), fixed seeds, and SEC 10-K structure-aware retrieval ordering; (ii) task-specific invariant checking for RAG, JSON, and SQL outputs using finance-calibrated materiality thresholds (plus or minus 5%) and SEC citation validation; (iii) a three-tier model classification system enabling risk-appropriate deployment decisions; and (iv) an audit-ready attestation system with dual-provider validation. We evaluated five models (Qwen2.5-7B via Ollama, Granite-3-8B via IBM watsonx.ai, Llama-3.3-70B, Mistral-Medium-2505, and GPT-OSS-120B) across three regulated financial tasks. Across 480 runs (n=16 per condition), structured tasks (SQL) remain stable even at T=0.2, while RAG tasks show drift (25-75%), revealing task-dependent sensitivity. Cross-provider validation confirms deterministic behavior transfers between local and cloud deployments. We map our framework to Financial Stability Board (FSB), Bank for International Settlements (BIS), and Commodity Futures Trading Commission (CFTC) requirements, demonstrating practical pathways for compliance-ready AI deployments.

We study explorability, a measure of nondeterminism in pushdown automata, which generalises history-determinism. An automaton is k-explorable if, while reading the input, it suffices to follow k concurrent runs, built step-by-step based only on the input seen so far, to construct an accepting one, if it exists. We show that the class of explorable PDAs lies strictly between history-deterministic and fully nondeterministic PDAs in terms of both expressiveness and succinctness. In fact increasing explorability induces an infinite hierarchy: each level k defines a strictly more expressive class than level k-1, yet the entire class remains less expressive than general nondeterministic PDAs. We then introduce a parameterized notion of explorability, where the number of runs may depend on input length, and show that exponential explorability precisely captures the context-free languages. Finally, we prove that explorable PDAs can be doubly exponentially more succinct than history-deterministic ones, and that the succinctness gap between deterministic and 2-explorable PDAs is not recursively enumerable. These results position explorability as a robust and operationally meaningful measure of nondeterminism for pushdown systems.

We are currently in the "large-scale era" of machine learning (ML), where the exciting capabilities of

A central question in computer science and statistics is whether efficient algorithms can achieve the information-theoretic limits of statistical problems. Many computational-statistical tradeoffs have been shown under average-case assumptions, but since statistical problems are average-case in nature, it has been a challenge to base them on standard worst-case assumptions. In PAC learning where such tradeoffs were first studied, the question is whether computational efficiency can come at the cost of using more samples than information-theoretically necessary. We base such tradeoffs on $\mathsf{NP}$-hardness and obtain: $\circ$ Sharp computational-statistical tradeoffs assuming $\mathsf{NP}$ requires exponential time: For every polynomial $p(n)$, there is an $n$-variate class $C$ with VC dimension $1$ such that the sample complexity of time-efficiently learning $C$ is $Θ(p(n))$. $\circ$ A characterization of $\mathsf{RP}$ vs. $\mathsf{NP}$ in terms of learning: $\mathsf{RP} = \mathsf{NP}$ iff every $\mathsf{NP}$-enumerable class is learnable with $O(\mathrm{VCdim}(C))$ samples in polynomial time. The forward implication has been known since (Pitt and Valiant, 1988); we prove the reverse implication. Notably, all our lower bounds hold against improper learners. These are the first $\mathsf{NP}$-hardness results for improperly learning a subclass of polynomial-size circuits, circumventing formal barriers of Applebaum, Barak, and Xiao (2008).

Consider an agent acting to achieve its temporal goal, but with a "trembling hand". In this case, the agent may mistakenly instruct, with a certain (typically small) probability, actions that are not intended due to faults or imprecision in its action selection mechanism, thereby leading to possible goal failure. We study the trembling-hand problem in the context of reasoning about actions and planning for temporally extended goals expressed in Linear Temporal Logic on finite traces (LTLf), where we want to synthesize a strategy (aka plan) that maximizes the probability of satisfying the LTLf goal in spite of the trembling hand. We consider both deterministic and nondeterministic (adversarial) domains. We propose solution techniques for both cases by relying respectively on Markov Decision Processes and on Markov Decision Processes with Set-valued Transitions with LTLf objectives, where the set-valued probabilistic transitions capture both the nondeterminism from the environment and the possible action instruction errors from the agent. We formally show the correctness of our solution techniques and demonstrate their effectiveness experimentally through a proof-of-concept implementation.