arXiv subjects

C. -H. Luke Ong

Publications and source records attributed to C. -H. Luke Ong.

At least 19 recordsLinked 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

Converse Barrier Certificates for Set-Based Stochastic Reach-Avoid Verification

Recent work established sufficient and necessary barrier-like conditions for infinite-horizon reach-avoid verification of stochastic discrete-time systems from a single initial state. Whether such a converse characterization extends to a set of initial states, however, remains open. In this paper, we answer this question affirmatively for compact initial sets. We consider a uniform reach-avoid specification requiring the reach-avoid probability to exceed a prescribed threshold for every initial state in a compact set. Under appropriate assumptions, including continuous system transitions, together with a strict uniform probability margin, we extend the pointwise converse characterization to the uniform setting.

eess.SY↗

arXiv2026

LazyHMC: Hamiltonian Monte Carlo Simulation for Lazy, Infinite Dimensional Probabilistic Programs

Hamiltonian Monte Carlo (HMC) is a successful generic inference method in probabilistic programming, but in its ordinary formulation it needs gradients and finite-dimensional parameter spaces. In Haskell, lazy evaluation lets probabilistic programs express stochastic processes and other non-parametric Bayesian models over implicit infinite-dimensional spaces. This paper develops new formulations of gradient-based HMC for this infinite-dimensional setting, via lazy evaluation. For automatic differentiation, we provide an analysis based on a new notion of "piecewise analytic under cylindrical analytic partition" (PACAP), to show that even if a program is infinite-dimensional and defined lazily, the gradient of the likelihood function is finitely supported. For the Monte Carlo method itself, we develop several HMC variants and a No-U-Turn Sampler that operate over the infinite-dimensional parameter space but are still productive because of lazy evaluation. Experiments cover Gaussian mixture clustering, random walks, and piecewise-constant regression with Poisson-process changepoints.

stat.ML↗

arXiv2026

Contextual MetaML: Syntax and Full Abstraction

MetaML-style metaprogramming languages allow programmers to construct, manipulate and run code. In the presence of higher-order references for code, ensuring type safety is challenging, as free variables can escape their binders. In this paper, we present Contextual MetaML, \textit{the first metaprogramming language that supports storing and running open code under a strong type safety guarantee}. The type system utilises contextual modal types to track and reason about free variables in code explicitly. A crucial concern in metaprogramming-based program optimisations is whether the optimised program preserves the meaning of the original program. Addressing this question requires a notion of program equivalence and techniques to reason about it. In this paper, we provide a semantic model that captures contextual equivalence for Contextual MetaML, establishing \textit{the first full abstraction result for an imperative MetaML-style language}. Our model is based on traces derived via operational game semantics, where the meaning of a program is modelled by its possible interactions with the environment. We also establish a novel closed instances of use theorem that accounts for both call-by-value and call-by-name closing substitutions.

cs.PL↗

arXiv2026

Quantitative Verification of Finite-Time Constrained Occupation Measures for Continuous-time Stochastic Systems

This paper addresses the quantitative verification of finite-time constrained occupation time for stochastic continuous-time systems governed by stochastic differential equations (SDEs). Unlike classical reachability analysis, which focuses on single-event properties such as entering a target set, many autonomous tasks-including surveillance, wireless charging, and chemical mixing-require a system to accumulate a prescribed duration within a target region while strictly maintaining safety constraints. We propose a barrier-certificate framework to compute rigorous upper and lower bounds on the probability that such cumulative specifications are satisfied over a finite time horizon. By introducing a stopped process that freezes the system once it reaches the boundary of the safe set, we derive three classes of certificates: one for upper bounds and two for lower bounds. The proposed approaches are validated through numerical examples implemented using semidefinite programming.

eess.SY↗

arXiv2026

Quantitative Verification of Constrained Occupation Time for Stochastic Discrete-time Systems

This paper addresses the quantitative verification of constrained occupation time in stochastic discrete-time systems, focusing on the probability of visiting a target set at least $k$ times while maintaining safety. Such cumulative properties are essential for certifying repeated behaviors like surveillance and periodic charging. To address this, we present the first barrier certificate framework capable of certifying these behaviors. We introduce multiplicative stochastic barrier functions that encode visitation counts implicitly within the algebraic structure of a scalar barrier. By adopting a switched-system reformulation to handle safety, we derive rigorous probabilistic bounds for both finite and infinite horizons. Specifically, we show that dissipative barriers establish upper bounds ensuring the exponential decay of frequent visits, while attractive barriers provide lower bounds via submartingale analysis. The efficacy of the proposed framework is demonstrated through numerical examples.

eess.SY↗

arXiv2026

PAC Finite-Time Safety Guarantees for Stochastic Systems with Unknown Disturbance Distributions

We investigate the problem of establishing finite-time probabilistic safety guarantees for discrete-time stochastic dynamical systems subject to unknown disturbance distributions, using barrier certificate methods. Our approach develops a data-driven safety certification framework that relies only on a finite collection of independent and identically distributed (i.i.d.) disturbance samples. Within this framework, we propose a certification procedure such that, with confidence at least $1-δ$ over the sampled disturbances, if the output of the certification procedure is accepted, the probability that the system remains within a prescribed safe set over a finite horizon is at least $1-ε$. A key challenge lies in formally characterizing the probably approximately correct (PAC) generalization behavior induced by finite samples. To address this, we derive PAC generalization bounds using tools from VC dimension, scenario optimization, and Rademacher complexity. These results illuminate the fundamental trade-offs between sample size, model complexity, and safety tolerance, providing both theoretical insight and practical guidance for designing reliable, data-driven safety certificates in discrete-time stochastic systems.

eess.SY↗

arXiv2025

Quantitative Verification of Omega-regular Properties in Probabilistic Programming

Probabilistic programming provides a high-level framework for specifying statistical models as executable programs with built-in randomness and conditioning. Existing inference techniques, however, typically compute posterior distributions over program states at fixed time points, most often at termination, thereby failing to capture the temporal evolution of probabilistic behaviors. We introduce temporal posterior inference (TPI), a new framework that unifies probabilistic programming with temporal logic by computing posterior distributions over execution traces that satisfy omega-regular specifications, conditioned on possibly temporal observations. To obtain rigorous quantitative guarantees, we develop a new method for computing upper and lower bounds on the satisfaction probabilities of omega-regular properties. Our approach decomposes Rabin acceptance conditions into persistence and recurrence components and constructs stochastic barrier certificates that soundly bound each component. We implement our approach in a prototype tool, TPInfer, and evaluate it on a suite of benchmarks, demonstrating effective and efficient inference over rich temporal properties in probabilistic models.

cs.PL↗

arXiv2024

Guaranteed Bounds on Posterior Distributions of Discrete Probabilistic Programs with Loops

We study the problem of bounding the posterior distribution of discrete probabilistic programs with unbounded support, loops, and conditioning. Loops pose the main difficulty in this setting: even if exact Bayesian inference is possible, the state of the art requires user-provided loop invariant templates. By contrast, we aim to find guaranteed bounds, which sandwich the true distribution. They are fully automated, applicable to more programs and provide more provable guarantees than approximate sampling-based inference. Since lower bounds can be obtained by unrolling loops, the main challenge is upper bounds, and we attack it in two ways. The first is called residual mass semantics, which is a flat bound based on the residual probability mass of a loop. The approach is simple, efficient, and has provable guarantees. The main novelty of our work is the second approach, called geometric bound semantics. It operates on a novel family of distributions, called eventually geometric distributions (EGDs), and can bound the distribution of loops with a new form of loop invariants called contraction invariants. The invariant synthesis problem reduces to a system of polynomial inequality constraints, which is a decidable problem with automated solvers. If a solution exists, it yields an exponentially decreasing bound on the whole distribution, and can therefore bound moments and tail asymptotics as well, not just probabilities as in the first approach. Both semantics enjoy desirable theoretical properties. In particular, we prove soundness and convergence, i.e. the bounds converge to the exact posterior as loops are unrolled further. On the practical side, we describe Diabolo, a fully-automated implementation of both semantics, and evaluate them on a variety of benchmarks from the literature, demonstrating their general applicability and the utility of the resulting bounds.

cs.PL↗

arXiv2024

Static Posterior Inference of Bayesian Probabilistic Programming via Polynomial Solving

In Bayesian probabilistic programming, a central problem is to estimate the normalised posterior distribution (NPD) of a probabilistic program with conditioning via score (a.k.a. observe) statements. Most previous approaches address this problem by Markov Chain Monte Carlo and variational inference, and therefore could not generate guaranteed outcomes within a finite time limit. Moreover, existing methods for exact inference either impose syntactic restrictions or cannot guarantee successful inference in general. In this work, we propose a novel automated approach to derive guaranteed bounds for NPD via polynomial solving. We first establish a fixed-point theorem for the wide class of score-at-end Bayesian probabilistic programs that terminate almost-surely and have a single bounded score statement at program termination. Then, we propose a multiplicative variant of Optional Stopping Theorem (OST) to address score-recursive Bayesian programs where score statements with weights greater than one could appear inside a loop. Finally, we use polynomial solving to implement our fixed-point theorem and OST variant. To improve the accuracy of the polynomial solving, we further propose a truncation operation and the synthesis of multiple bounds over various program inputs. Our approach can handle Bayesian probabilistic programs with unbounded while loops and continuous distributions with infinite supports. Experiments over a wide range of benchmarks show that compared with the most relevant approach (Beutner et al., PLDI 2022) for guaranteed NPD analysis via recursion unrolling, our approach is more time efficient and derives comparable or even tighter NPD bounds. Furthermore, our approach can handle score-recursive programs which previous approaches could not.

cs.PL↗

arXiv2024

Diagonalisation SGD: Fast & Convergent SGD for Non-Differentiable Models via Reparameterisation and Smoothing

It is well-known that the reparameterisation gradient estimator, which exhibits low variance in practice, is biased for non-differentiable models. This may compromise correctness of gradient-based optimisation methods such as stochastic gradient descent (SGD). We introduce a simple syntactic framework to define non-differentiable functions piecewisely and present a systematic approach to obtain smoothings for which the reparameterisation gradient estimator is unbiased. Our main contribution is a novel variant of SGD, Diagonalisation Stochastic Gradient Descent, which progressively enhances the accuracy of the smoothed approximation during optimisation, and we prove convergence to stationary points of the unsmoothed (original) objective. Our empirical evaluation reveals benefits over the state of the art: our approach is simple, fast, stable and attains orders of magnitude reduction in work-normalised variance.

cs.LG↗

arXiv2023

Fast and Correct Gradient-Based Optimisation for Probabilistic Programming via Smoothing

We study the foundations of variational inference, which frames posterior inference as an optimisation problem, for probabilistic programming. The dominant approach for optimisation in practice is stochastic gradient descent. In particular, a variant using the so-called reparameterisation gradient estimator exhibits fast convergence in a traditional statistics setting. Unfortunately, discontinuities, which are readily expressible in programming languages, can compromise the correctness of this approach. We consider a simple (higher-order, probabilistic) programming language with conditionals, and we endow our language with both a measurable and a smoothed (approximate) value semantics. We present type systems which establish technical pre-conditions. Thus we can prove stochastic gradient descent with the reparameterisation gradient estimator to be correct when applied to the smoothed problem. Besides, we can solve the original problem up to any error tolerance by choosing an accuracy coefficient suitably. Empirically we demonstrate that our approach has a similar convergence as a key competitor, but is simpler, faster, and attains orders of magnitude reduction in work-normalised variance.

cs.PL↗

arXiv2021

Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost Everywhere

We study the differential properties of higher-order statistical probabilistic programs with recursion and conditioning. Our starting point is an open problem posed by Hongseok Yang: what class of statistical probabilistic programs have densities that are differentiable almost everywhere? To formalise the problem, we consider Statistical PCF (SPCF), an extension of call-by-value PCF with real numbers, and constructs for sampling and conditioning. We give SPCF a sampling-style operational semantics a la Borgstrom et al., and study the associated weight (commonly referred to as the density) function and value function on the set of possible execution traces. Our main result is that almost-surely terminating SPCF programs, generated from a set of primitive functions (e.g. the set of analytic functions) satisfying mild closure properties, have weight and value functions that are almost-everywhere differentiable. We use a stochastic form of symbolic execution to reason about almost-everywhere differentiability. A by-product of this work is that almost-surely terminating deterministic (S)PCF programs with real parameters denote functions that are almost-everywhere differentiable. Our result is of practical interest, as almost-everywhere differentiability of the density function is required to hold for the correctness of major gradient-based inference algorithms.

cs.LO↗

arXiv2021

Higher-Order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This paper studies the logical properties of a very general class of infinite ranked trees, namely those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal mu-calculus, three main problems: model-checking, logical reflection (aka global model-checking, that asks for a finite description of the set of elements for which a formula holds) and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems we provide an effective solution. This is obtained thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

cs.LO↗

arXiv2020

The Difference Lambda-Calculus: A Language for Difference Categories

Cartesian difference categories are a recent generalisation of Cartesian differential categories which introduce a notion of "infinitesimal" arrows satisfying an analogue of the Kock-Lawvere axiom, with the axioms of a Cartesian differential category being satisfied only "up to an infinitesimal perturbation". In this work, we construct a simply-typed calculus in the spirit of the differential lambda-calculus equipped with syntactic infinitesimals and show how its models correspond to difference lambda-categories, a family of Cartesian difference categories equipped with suitably well-behaved exponentials.

cs.LO↗

arXiv2020

Collapsible Pushdown Parity Games

This paper studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

cs.FL↗

arXiv2020

The Extended Theory of Trees and Algebraic (Co)datatypes

The first-order theory of finite and infinite trees has been studied since the eighties, especially by the logic programming community. Following Djelloul, Dao and Frühwirth, we consider an extension of this theory with an additional predicate for finiteness of trees, which is useful for expressing properties about (not just datatypes but also) codatatypes. Based on their work, we present a simplification procedure that determines whether any given (not necessarily closed) formula is satisfiable, returning a simplified formula which enables one to read off all possible models. Our extension makes the algorithm usable for algebraic (co)datatypes, which was impossible in their original work due to restrictive assumptions. We also provide a prototype implementation of our simplification procedure and evaluate it on instances from the SMT-LIB.

cs.LO↗

arXiv2020

Change Actions: Models of Generalised Differentiation

Cai et al. have recently proposed change structures as a semantic framework for incremental computation. We generalise change structures to arbitrary cartesian categories and propose the notion of change action model as a categorical model for (higher-order) generalised differentiation. Change action models naturally arise from many geometric and computational settings, such as (generalised) cartesian differential categories, group models of discrete calculus, and Kleene algebra of regular expressions. We show how to build canonical change action models on arbitrary cartesian categories, reminiscent of the Fàa di Bruno construction.

cs.LO↗