arXiv subjects

Xiaodong Jia

Publications and source records attributed to Xiaodong Jia.

At least 19 recordsLinked to original sources

arXiv2026

Coherence of Smyth powerspaces

In this paper, we study when the Smyth powerspace $\mathcal{Q}^*_v(X)$ of a topological space $X$ is coherent, and prove that $X$ is coherent and weakly Hausdorff if and only if $\mathcal{Q}^*_v(X)$ is coherent and weakly Hausdorff. We give examples to show that neither coherence nor weak Hausdorffness of $X$ solely implies that $\mathcal{Q}^*_v(X)$ is coherent or weakly Hausdorff. As a byproduct, our work gives an affirmative answer to a question raised by Xu.

math.GN↗

arXiv2026

Forbidden substructures for coherence of domains

A coherent domain, in the sense of Dana Scott's domain theory, is a domain in which the intersection of every two compact saturated subsets is again compact, when the domain is equipped with the Scott topology. Coherence plays key roles in classifying Cartesian closed subcategories of domains and in characterizing Lawson compactness of domains. In this paper, we find two typical domains that fail to be coherent, and prove that a bounded algebraic domain fails to be coherent if and only if it has one of the two typical domains as its Scott-continuous retract. Similar results also generalize to bounded continuous domains, provided that the domains in consideration are hereditarily Lindelöf and weakly Hausdorff in the Scott topology.

math.GN↗

arXiv2026

2026 Roadmap on Artificial Intelligence and Machine Learning for Smart Manufacturing

The evolution of artificial intelligence (AI) and machine learning (ML) is reshaping smart manufacturing by providing new capabilities for efficiency, adaptability, and autonomy across industrial value chains. However, the deployment of AI and ML in industrial settings still faces critical challenges, including the complexity of industrial big data, effective data management, integration with heterogeneous sensing and control systems, and the demand for trustworthy, explainable, and reliable operation in high-stakes industrial environments. In this roadmap, we present a comprehensive perspective on the foundations, applications, and emerging directions of AI and ML in smart manufacturing. It is structured in three parts. The first highlights the foundations and trends that frame the evolution of AI in smart manufacturing. The second focuses on key topics where AI is already enabling advances, including industrial big data analytics, advanced sensing and perception, autonomous systems, additive and laser-based manufacturing, digital twins, robotics, supply chain and logistics optimization, and sustainable manufacturing. The third section explores non-traditional ML approaches that are opening new frontiers, such as physics-informed AI, generative AI, semantic AI, advanced digital twins, explainable AI, RAMS, data-centric metrology, LLMs, and foundation models for highly connected and complex manufacturing systems. By identifying both opportunities and remaining barriers across these areas, this roadmap outlines the advances needed in methods, integration strategies, and industrial adoption. We hope this roadmap will serve as a guide for researchers, engineers, and practitioners to accelerate innovation, align academic and industrial priorities, and ensure that AI-driven smart manufacturing delivers reliable, sustainable, and scalable impact for the future of manufacturing ecosystems.

cs.AI↗

arXiv2026

Notes on countable frames

Matthew de Brecht raised the question of whether countable frames are continuous lattices. We prove that the continuity of a countable frame implies the quasicontinuity of its corresponding spectrum in the dual specialization order. We further show that this question admits a positive answer if the frame's spectrum is a $T_1$ space or a Scott space. In general, we confirm the existence of non-continuous countable frames. This work also partially addresses an open problem proposed by Jimmie Lawson and Michael Mislove in 1990, which concerns the characterization of when the spectrums of spatial frames are Scott spaces.

math.GN↗

arXiv2025

Products of two sober dcpo's need not be sober

We construct two dcpo's whose Scott spaces are sober, but the Scott space of their order product is not sober. This answers an open problem on the sobriety of Scott spaces. Meantime, we show that if $M$ and $N$ are special type of sober complete lattices, then the Scott space of their order product $M\times N$ is sober.

math.GN↗

arXiv2025

Separating domains from algebraic domains

We prove that every domain that fails to be algebraic admits the unit interval $[0, 1]$ as its Scott-continuous retract. As a result, every countable domain is algebraic.

math.GN↗

arXiv2024

A cone-theoretic barycenter existence theorem

We show that every continuous valuation on a locally convex, locally convex-compact, sober topological cone $\mathfrak{C}$ has a barycenter. This barycenter is unique, and the barycenter map $β$ is continuous, hence is the structure map of a $\mathbf V_{\mathrm w}$-algebra, i.e., an Eilenberg-Moore algebra of the extended valuation monad on the category of $T_0$ topological spaces; it is, in fact, the unique $\mathbf V_{\mathrm w}$-algebra that induces the cone structure on $\mathfrak{C}$.

math.GN↗

arXiv2024

The category of well-filtered dcpos is not $Γ$-faithful

The Ho-Zhao problem asks whether any two dcpo's with isomorphic Scott closed set lattices are themselves isomorphic, that is, whether the category $\mathbf{DCPO}$ of dcpo's and Scott-continuous maps is $Γ$-faithful. In 2018, Ho, Goubault-Larrecq, Jung and Xi answered this question in the negative, and they introduced the category $\mathbf{DOMI}$ of dominated dcpo's and proved that it is {$Γ$-faithful}. Dominated dcpo's subsume many familiar families of dcpo's in domain theory, such as the category of bounded-complete dcpo's and that of sober dcpo's, among others. However, it is unknown whether the category of dominated dcpo's dominates all well-filtered dcpo's, a class strictly larger than that of bounded-complete lattices and that of sober dcpo's. In this paper, we address this very natural question and show that the category $\mathbf{WF}$ of well-filtered dcpo's is not $Γ$-faithful, and as a result of it, well-filtered dcpo's need not be dominated in general. Since not all dcpo's are well-filtered, our work refines the results of Ho, Goubault-Larrecq, Jung and Xi. As a second contribution, we confirm that the Lawson's category of $Ω^{*}$-compact dcpo's is $Γ$-faithful. Moreover, we locate a class of dcpo's which we call weakly dominated dcpo's, and show that this class is $Γ$-faithful and strictly larger than $\mathbf{DOMI}$.

cs.LO↗

arXiv2024

$ω$-well-filtered spaces, revisited

We prove that a $T_0$ topological space is $ω$-well-filtered if and only if it does not admit either the natural numbers with the cofinite topology or with the Scott topology as its closed subsets in the strong topology. Based on this, we offer a refined topological characterization for the $ω$-well-filterification of $T_0$-spaces and solve a problem posed by Xiaoquan Xu. In the setting of second countable spaces, we also characterise sobriety by convergences of certain $Π^0_2$-Cauchy subsets of the spaces.

math.GN↗

arXiv2024

V-Star: Learning Visibly Pushdown Grammars from Program Inputs

Accurate description of program inputs remains a critical challenge in the field of programming languages. Active learning, as a well-established field, achieves exact learning for regular languages. We offer an innovative grammar inference tool, V-Star, based on the active learning of visibly pushdown automata. V-Star deduces nesting structures of program input languages from sample inputs, employing a novel inference mechanism based on nested patterns. This mechanism identifies token boundaries and converts languages such as XML documents into VPLs. We then adapted Angluin's L-Star, an exact learning algorithm, for VPA learning, which improves the precision of our tool. Our evaluation demonstrates that V-Star effectively and efficiently learns a variety of practical grammars, including S-Expressions, JSON, and XML, and outperforms other state-of-the-art tools.

cs.PL↗

arXiv2023

Continuous R-valuations

We introduce continuous $R$-valuations on directed-complete posets (dcpos, for short), as a generalization of continuous valuations in domain theory, by extending values of continuous valuations from reals to so-called Abelian d-rags $R$. Like the valuation monad $\mathbf{V}$ introduced by Jones and Plotkin, we show that the construction of continuous $R$-valuations extends to a strong monad $\mathbf{V}^R$ on the category of dcpos and Scott-continuous maps. Additionally, and as in recent work by the two authors and C. Théron, and by the second author, B. Lindenhovius, M. Mislove and V. Zamdzhiev, we show that we can extract a commutative monad $\mathbf{V}^R_m$ out of it, whose elements we call minimal $R$-valuations. We also show that continuous $R$-valuations have close connections to measures when $R$ is taken to be $\mathbf{I}\mathbb{R}^\star_+$, the interval domain of the extended nonnegative reals: (1) On every coherent topological space, every non-zero, bounded $τ$-smooth measure $μ$ (defined on the Borel $σ$-algebra), canonically determines a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation; and (2) such a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation is the most precise (in a certain sense) continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation that approximates $μ$, when the support of $μ$ is a compact Hausdorff subspace of a second-countable stably compact topological space. This in particular applies to Lebesgue measure on the unit interval. As a result, the Lebesgue measure can be identified as a continuous $\mathbf{I}\mathbb{R}^\star_+$-valuation. Additionally, we show that the latter is minimal.

math.GN↗

arXiv2022

Addition Machines, Automatic Functions and Open Problems of Floyd and Knuth

Floyd and Knuth investigated in 1990 register machines which can add, subtract and compare integers as primitive operations. They asked whether their current bound on the number of registers for multiplying and dividing fast (running in time linear in the size of the input) can be improved and whether one can output fast the powers of two summing up to a positive integer in subquadratic time. Both questions are answered positively. Furthermore, it is shown that every function computed by only one register is automatic and that automatic functions with one input can be computed with four registers in linear time; automatic functions with a larger number of inputs can be computed with 5 registers in linear time. There is a nonautomatic function with one input which can be computed with two registers in linear time.

cs.CC↗

arXiv2021

The Central Valuations Monad

We give a commutative valuations monad Z on the category DCPO of dcpo's and Scott-continuous functions. Compared to the commutative valuations monads given in [Jia et al., 2021], our new monad Z is larger and it contains all push-forward images of valuations on the unit interval [0,1] along lower semi-continuous maps. We believe that this new monad will be useful in giving domain-theoretic denotational semantics for statistical programming languages with continuous probabilistic choice.

cs.LO↗

arXiv2021

A Domain-Theoretic Approach to Statistical Programming Languages

We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin' s monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations. (Implementations by themselves are not addressed here.) Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: all measures on the real line can be modeled by minimal valuations on the domain $\mathbf{I}\mathbb{R}_\bot$ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples in order to demonstrate that our semantics computes exactly as one would expect, and in order to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance. Our examples also include some useful, non-trivial cases of distributions on higher-order objects.

cs.LO↗

arXiv2021

A Derivative-based Parser Generator for Visibly Pushdown Grammars

In this paper, we present a derivative-based, functional recognizer and parser generator for visibly pushdown grammars. The generated parser accepts ambiguous grammars and produces a parse forest containing all valid parse trees for an input string in linear time. Each parse tree in the forest can then be extracted also in linear time. Besides the parser generator, to allow more flexible forms of the visibly pushdown grammars, we also present a translator that converts a tagged CFG to a visibly pushdown grammar in a sound way, and the parse trees of the tagged CFG are further produced by running the semantic actions embedded in the parse trees of the translated visibly pushdown grammar. The performance of the parser is compared with a popular parsing tool ANTLR and other popular hand-crafted parsers. The correctness of the core parsing algorithm is formally verified in the proof assistant Coq.

cs.PL↗

arXiv2021

Separating minimal valuations, point-continuous valuations and continuous valuations

We give two concrete examples of continuous valuations on dcpo's to separate minimal valuations, point-continuous valuations and continuous valuations: (1) Let $\mathcal J$ be the Johnstone's non-sober dcpo, and $μ$ be the continuous valuation on $\mathcal J$ with $μ(U) =1$ for nonempty Scott opens $U$ and $μ(U) = 0$ for $U=\emptyset$. Then $μ$ is a point-continuous valuation on $\mathcal J$ that is not minimal. (2) Lebesgue measure extends to a measure on the Sorgenfrey line $\mathbb R_{l}$. Its restriction to the open subsets of $\mathbb R_{l}$ is a continuous valuation $λ$. Then its image valuation $\overlineλ$ through the embedding of $\mathbb R_{l}$ into its Smyth powerdomain $\mathcal Q\mathbb R_{l}$ in the Scott topology is a continuous valuation that is not point-continuous. We believe that our construction $\overlineλ$ might be useful in giving counterexamples displaying the failure of the general Fubini-type equations on dcpo's.

cs.LO↗

arXiv2021

Semantics for Variational Quantum Programming

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide the first denotational analysis that relates models of classical probabilistic programming to models of quantum programming.

cs.LO↗

arXiv2021

Commutative Monads for Probabilistic Programming Languages

A long-standing open problem in the semantics of programming languages supporting probabilistic choice is to find a commutative monad for probability on the category DCPO. In this paper we present three such monads and a general construction for finding even more. We show how to use these monads to provide a sound and adequate denotational semantics for the Probabilistic FixPoint Calculus (PFPC) -- a call-by-value simply-typed lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. We also show that in the special case where we consider continuous dcpo's, then all three monads coincide with the valuations monad of Jones and we fully characterise the induced Eilenberg-Moore categories by showing that they are all isomorphic to the category of continuous Kegelspitzen of Keimel and Plotkin.

cs.LO↗