arXiv subjects

Martin Rinard

Publications and source records attributed to Martin Rinard.

At least 19 recordsLinked to original sources

arXiv2026

Automated Testing and Repair for Verified Compilers Generated by a Coding Agent

We present an agent based automated testing and repair system for verified compilers that contain four kinds of code: verified code, checked code, unverified code, and specification. We present specialized defect detection techniques that exploit the structure present in such compilers. For each surfaced defect the system invokes a coding agent to repair the defect and validate the repair. We evaluate the system on the Axon compiler, a compiler completely generated by a coding agent operating under developer supervision. The compiler was validated during development on a small benchmark set (the Livermore benchmarks), raising the possibility that its coding agent reward hacked the compiler. We also evaluate the possibility that the repair system reward hacked the repairs and find no evidence of reward hacking in either the Axon compiler or the repairs. We present results that characterize the testing and repair effectiveness and discuss repair characteristics.

cs.SE↗

arXiv2026

Program Analysis with Prophecy and History Variables in the Nexis Compiler

We present prophecy variables for forward formulations of program analysis problems that require information about the future execution of the program. We specify prophecy and history variables via a domain specific language that augments the step rules of the base operational semantics with subset inclusion constraints over the prophecy and history variables. This tight coupling between the prophecy and history variable specification and the operational semantics promotes the construction of correctness and optimality proofs for program transformations, with the proofs structured as forward simulations between the original and transformed versions of the program. In comparison with traditional dataflow approaches, this approach eliminates mechanisms such as explicit control flow graphs, abstraction functions, concretization functions, Galois connections, and separate backward and forward analyses. We present a verified implementation of prophecy and history variables and use the implementation to prove correctness and optimality properties of two classic transformations, partial dead code elimination and lazy code motion, that use both prophecy and history variables. To the best of our knowledge, these proofs are the first machine checked correctness and optimality proofs for these transformations.

cs.PL↗

arXiv2026

Nash Equilibria with Derangement Degree Probabilities

We prove for every $n\ge4$ the existence of an $n$-player game in normal form with integer payoffs that has a unique Nash equilibrium, which is fully mixed. In the equilibrium, each probability weight is an algebraic number of degree $\mathbin{!n}$ (the derangement number), and its minimal polynomial has Galois group $S_{\mathbin{!n}}$ and $\mathbin{!n}+1$ nonzero coefficients.

cs.GT↗

arXiv2026

Quantitative Comparison of Credible Compilation and Verification In Coding Agent Compiler Development

Formal program verification is a longstanding goal in the field. We present the first quantitative comparison of the two primary compiler verification approaches, credible compilation/translation validation and full verification. Working with the first verified compiler developed by a coding agent (operating under human supervision), we present quantitative results from a coding agent implementing several optimizations using these two approaches. The results indicate that 1) verification requires roughly an order of magnitude more development effort than credible compilation, 2) to enhance provability, the coding agent chooses less efficient algorithms and data structures for verified optimizations, and 3) in an attempt to minimize proof effort the coding agent repeatedly implemented optimization scope reductions for verified optimizations, and 4) certificate checking time dominates optimization and certificate generation time for the considered optimizations. Because of the increased proof overhead, verified optimizations required substantially more supervision and coding sessions than credible compilation optimizations. Given the capabilities of a modern coding agent working in this context, implementation efforts for both credible compilation and verified versions remained feasible for the considered optimizations (unreachable code elimination, dead assignment elimination, and constant propagation/folding).

cs.PL↗

arXiv2026

Testing, Credible Compilation, and Verification in the Axon Verified Compiler in Lean and Claude Code

This paper presents the use of testing, credible compilation/translation validation, verification, and audits in the Axon compiler. Axon comes with fully machine checked proofs that guarantee the correctness of the generated code. All code and proofs were written in Lean by Claude Code, with the correctness proofs eliminating any need to audit or examine any verified code. I present a development process for using these validation techniques, evaluate the use of this process during the development of the compiler, and discuss implications for other development efforts.

cs.PL↗

arXiv2026

Backwards Data-Flow Analysis using Prophecy Variables in the BuildIt System

Many program transformations and optimizations require information about the future behavior of the program. A standard way to obtain this information is to build an intermediate program representation, then use a backwards program analysis to propagate relevant information against the flow of control back to the transformation/optimization site. We instead propose to use prophecy variables, which predict information about the future execution of the program, to enable such transformations and optimizations. We implement prophecy variables in BuildIt, a lightweight domain specific language implementation system. BuildIt uses staged compilation to implement high performance domain specific languages embedded within a standard general purpose programming language (C++). The BuildIt first phase uses standard C++ program execution to generate optimized C, C++, and CUDA second phase code. This approach enables BuildIt to eliminate programming language implementation components such as parsers and intermediate representations, delivering a dramatic decrease in the engineering effort required to implement domain specific languages. The combination of prophecy variables and repeated forward program execution enables BuildIt to extend this approach to include transformations and optimizations that require information about the future execution of the program without backwards analyses and without the engineering overhead associated with implementing these analyses. We formalize the use of prophecy variables for this purpose, discuss the implementation of prophecy variables and repeated execution in BuildIt, and present experimental results for BuildIt computations that benefit from optimizations enabled by the information that prophecy variables provide.

cs.PL↗

arXiv2025

Nash Equilibria with Irradical Probabilities

We present for every $n\ge4$ an $n$-player game in normal form with payoffs in $\{0,1,2\}$ that has a unique, fully mixed, Nash equilibrium in which all the probability weights are irradical (i.e., algebraic but not closed form expressible even with $m$-th roots for any integer $m$).

cs.GT↗

arXiv2025

Evaluating the Generalization Capabilities of Large Language Models on Code Reasoning

We assess how the code reasoning abilities of large language models (LLMs) generalize to different kinds of programs. We present techniques for obtaining in- and out-of-distribution programs with different characteristics: code sampled from a domain-specific language, code automatically generated by an LLM, code collected from competitive programming contests, and mutated versions of these programs. We also present an experimental methodology for evaluating LLM generalization by comparing their performance on these programs. We perform an extensive evaluation across 10 state-of-the-art models from the past year, obtaining insights into their generalization capabilities over time and across different classes of programs. Our results highlight that while earlier models exhibit behavior consistent with pattern matching, the latest models exhibit strong generalization abilities on code reasoning.

cs.SE↗

arXiv2024

Latent Causal Probing: A Formal Perspective on Probing with Causal Models of Data

As language models (LMs) deliver increasing performance on a range of NLP tasks, probing classifiers have become an indispensable technique in the effort to better understand their inner workings. A typical setup involves (1) defining an auxiliary task consisting of a dataset of text annotated with labels, then (2) supervising small classifiers to predict the labels from the representations of a pretrained LM as it processed the dataset. A high probing accuracy is interpreted as evidence that the LM has learned to perform the auxiliary task as an unsupervised byproduct of its original pretraining objective. Despite the widespread usage of probes, however, the robust design and analysis of probing experiments remains a challenge. We develop a formal perspective on probing using structural causal models (SCM). Specifically, given an SCM which explains the distribution of tokens observed during training, we frame the central hypothesis as whether the LM has learned to represent the latent variables of the SCM. Empirically, we extend a recent study of LMs in the context of a synthetic grid-world navigation task, where having an exact model of the underlying causal structure allows us to draw strong inferences from the result of probing experiments. Our techniques provide robust empirical evidence for the ability of LMs to induce the latent concepts underlying text.

cs.CL↗

arXiv2024

Edge-dominance games on graphs

We consider zero-sum games in which players move between adjacent states, where in each pair of adjacent states one state dominates the other. The states in our game can represent positional advantages in physical conflict such as high ground or camouflage, or product characteristics that lend an advantage over competing sellers in a duopoly. We study the equilibria of the game as a function of the topological and geometric properties of the underlying graph. Our main result characterizes the expected payoff of both players starting from any initial position, under the assumption that the graph does not contain certain types of small cycles. This characterization leverages the block-cut tree of the graph, a construction that describes the topology of the biconnected components of the graph. We identify three natural types of (on-path) pure equilibria, and characterize when these equilibria exist under the above assumptions. On the geometric side, we show that strongly connected outerplanar graphs with undirected girth at least 4 always support some of these types of on-path pure equilibria. Finally, we show that a data structure describing all pure equilibria can be efficiently computed for these games.

cs.GT↗

arXiv2024

Limited-perception games

We study rational agents with different perception capabilities in strategic games. We focus on a class of one-shot limited-perception games. These games extend simultaneous-move normal-form games by presenting each player with an individualized perception of all players' payoff functions. The accuracy of a player's perception is determined by the player's capability level. Capability levels are countable and totally ordered, with a higher level corresponding to a more accurate perception. We study the rational behavior of players in these games and formalize relevant equilibria conditions. In contrast to equilibria in conventional bimatrix games, which can be represented by a pair of mixed strategies, in our limited perception games a higher-order response function captures how the lower-capability player uses their (less accurate) perception of the payoff function to reason about the (more accurate) possible perceptions of the higher-capability opponent. This response function characterizes, for each possible perception of the higher-capability player (from the perspective of the lower-capability player), the best response of the higher capability player for that perception. Since the domain of the response function can be exponentially large or even infinite, finding one equilibrium may be computationally intractable or even undecidable. Nevertheless, we show that for any $\epsilon$, there exists an $\epsilon$-equilibrium with a compact, tractable representation whose size is independent of the size of the response function's domain. We further identify classes of zero-sum limited-perception games in which finding an equilibrium becomes a (typically tractable) nonsmooth convex optimization problem.

cs.GT↗

arXiv2023

Randomness Requirements and Asymmetries in Nash Equilibria

In general, Nash equilibria in normal-form games may require players to play (probabilistically) mixed strategies. We define a measure of the complexity of finite probability distributions and study the complexity required to play Nash equilibria in finite two player $n\times n$ games with rational payoffs. Our central results show that there exist games in which there is an exponential vs. linear gap in the complexity of the mixed distributions that the two players play in the (unique) Nash equilibrium of these games. This gap induces asymmetries in the amounts of space required by the players to represent and sample from the corresponding distributions using known state-of-the-art sampling algorithms. We also establish exponential upper and lower bounds on the complexity of Nash equilibria in normal-form games. These results highlight (i) the nontriviality of the assumption that players can play any mixed strategy and (ii) the disparity in resources that players may require to play Nash equilibria in normal-form games.

cs.GT↗

arXiv2023

TRAFS: A Nonsmooth Convex Optimization Algorithm with $\mathcal{O}\left(\frac{1}{\epsilon}\right)$ Iteration Complexity

We present the Trust Region Adversarial Functional Subdifferential (TRAFS) algorithm for constrained optimization of nonsmooth convex Lipschitz functions. Unlike previous methods that assume a subgradient oracle model, we work with the functional subdifferential defined as a set of subgradients that simultaneously captures sufficient local information for effective minimization while being easy to compute for a wide range of functions. In each iteration, TRAFS finds the best step vector in an $\ell_2$-bounded trust region by considering the worst bound given by the functional subdifferential. TRAFS finds an approximate solution with an absolute error up to $\epsilon$ in $\mathcal{O}\left( \epsilon^{-1}\right)$ or $\mathcal{O}\left(\epsilon^{-0.5} \right)$ iterations depending on whether the objective function is strongly convex, compared to the previously best-known bounds of $\mathcal{O}\left(\epsilon^{-2}\right)$ and $\mathcal{O}\left(\epsilon^{-1}\right)$ in these settings. TRAFS makes faster progress if the functional subdifferential satisfies a locally quadratic property; as a corollary, TRAFS achieves linear convergence (i.e., $\mathcal{O}\left(\log \epsilon^{-1}\right)$) for strongly convex smooth functions. In the numerical experiments, TRAFS is on average 39.1x faster and solves twice as many problems compared to the second-best method.

math.OC↗

arXiv2023

Correlated vs. Uncorrelated Randomness in Adversarial Congestion Team Games

We consider team zero-sum network congestion games with $n$ agents playing against $k$ interceptors over a graph $G$. The agents aim to minimize their collective cost of sending traffic over paths in $G$, which is an aggregation of edge costs, while the interceptors aim to maximize the collective cost by increasing some of these edge costs. To evade the interceptors, the agents will usually use randomized strategies. We consider two cases, the correlated case when agents have access to a shared source of randomness, and the uncorrelated case, when each agent has access to only its own source of randomness. We study the additional cost that uncorrelated agents have to bear, specifically by comparing the costs incurred by agents in cost-minimal Nash equilibria when agents can and cannot share randomness. We consider two natural cost functions on the agents, which measure the invested energy and time, respectively. We prove that for both of these cost functions, the ratio of uncorrelated cost to correlated cost at equilibrium is $O(\min(m_c(G),n))$, where $m_c(G)$ is the mincut size of $G$. This bound is much smaller than the most general case, where a tight, exponential bound of $\Theta((m_c(G))^{n-1})$ on the ratio is known. We also introduce a set of simple agent strategies which are approximately optimal agent strategies. We then establish conditions for when these strategies are optimal agent strategies for each cost function, showing an inherent difference between the two cost functions we study.

cs.GT↗

arXiv2023

Depth-bounded Epistemic Logic

Epistemic logics model how agents reason about their beliefs and the beliefs of other agents. Existing logics typically assume the ability of agents to reason perfectly about propositions of unbounded modal depth. We present DBEL, an extension of S5 that models agents that can reason about epistemic formulas only up to a specific modal depth. To support explicit reasoning about agent depths, DBEL includes depth atoms Ead (agent a has depth exactly d) and Pad (agent a has depth at least d). We provide a sound and complete axiomatization of DBEL. We extend DBEL to support public announcements for bounded depth agents and show how the resulting DPAL logic generalizes standard axioms from public announcement logic. We present two alternate extensions and identify two undesirable properties, amnesia and knowledge leakage, that these extensions have but DPAL does not. We provide axiomatizations of these logics as well as complexity results for satisfiability and model checking. Finally, we use these logics to illustrate how agents with bounded modal depth reason in the classical muddy children problem, including upper and lower bounds on the depth knowledge necessary for agents to successfully solve the problem.

cs.AI↗

arXiv2023

Sound Explanation for Trustworthy Machine Learning

We take a formal approach to the explainability problem of machine learning systems. We argue against the practice of interpreting black-box models via attributing scores to input components due to inherently conflicting goals of attribution-based interpretation. We prove that no attribution algorithm satisfies specificity, additivity, completeness, and baseline invariance. We then formalize the concept, sound explanation, that has been informally adopted in prior work. A sound explanation entails providing sufficient information to causally explain the predictions made by a system. Finally, we present the application of feature selection as a sound explanation for cancer prediction models to cultivate trust among clinicians.

cs.LG↗

arXiv2023

Emergent Representations of Program Semantics in Language Models Trained on Programs

We present evidence that language models (LMs) of code can learn to represent the formal semantics of programs, despite being trained only to perform next-token prediction. Specifically, we train a Transformer model on a synthetic corpus of programs written in a domain-specific language for navigating 2D grid world environments. Each program in the corpus is preceded by a (partial) specification in the form of several input-output grid world states. Despite providing no further inductive biases, we find that a probing classifier is able to extract increasingly accurate representations of the unobserved, intermediate grid world states from the LM hidden states over the course of training, suggesting the LM acquires an emergent ability to interpret programs in the formal sense. We also develop a novel interventional baseline that enables us to disambiguate what is represented by the LM as opposed to learned by the probe. We anticipate that this technique may be generally applicable to a broad range of semantic probing experiments. In summary, this paper does not propose any new techniques for training LMs of code, but develops an experimental framework for and provides insights into the acquisition and representation of formal semantics in statistical models of code. Our code is available at https://github.com/charlesjin/emergent-semantics.

cs.LG↗

arXiv2023

Depth-bounded epistemic logic

Epistemic logics model how agents reason about their beliefs and the beliefs of other agents. Existing logics typically assume the ability of agents to reason perfectly about propositions of unbounded modal depth. We present DBEL, an extension of S5 that models agents that can reason about epistemic formulas only up to a specific modal depth. To support explicit reasoning about agent depths, DBEL includes depth atoms $E_a^d$ (agent $a$ has depth exactly $d$) and $P_a^d$ (agent $a$ has depth at least $d$). We provide a sound and complete axiomatization of DBEL. We extend DBEL to support public announcements for bounded depth agents and show how the resulting DPAL logic generalizes standard axioms from public announcement logic. We present two alternate extensions and identify two undesirable properties, amnesia and knowledge leakage, that these extensions have but DPAL does not. We provide axiomatizations of these logics as well as complexity results for satisfiability and model checking. Finally, we use these logics to illustrate how agents with bounded modal depth reason in the classical muddy children problem, including upper and lower bounds on the depth knowledge necessary for agents to successfully solve the problem.

cs.LO↗