Mikołaj Bojańczyk, Bartek Klin, Sławomir Lasota
We study languages over infinite alphabets equipped with some structure that can be tested by recognizing automata. We develop a framework for studying such alphabets and the ensuing automata theory, where the key role is played by an automorphism group of the alphabet. In the process, we generalize nominal sets due to Gabbay and Pitts.
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Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. These automata can be exponentially smaller than classical …
Automata models for data languages (i.e. languages over infinite alphabets) often feature either global or local freshness operators. We show that Bollig et al.'s session automata, which focus on global freshness, are equivalent to regular nondeterministic nominal automata (RNNA), a natural nominal automaton model with explicit name binding that has appeared implicitly in the semantics of nominal Kleene algebra (NKA), an extension of Kleene algebra with name binding. The expected Kleene theorem for NKA is known to fail in one direction, i.e. there are nominal languages that can be accepted by …
The emerging field of Nominal Computation Theory is concerned with the theory of Nominal Sets and its applications to Computer Science. We investigate here the impact of nominal sets on the definition of Cellular Automata and on their computational capabilities, with a special focus on the emergent behavioural properties of this new model and their significance in the context of computation-oriented interpretations of physical phenomena. A preliminary investigation of the relations between Nominal Cellular Automata and Wolfram's Elementary Cellular Automata is also carried out.
Logics and automata models for languages over infinite alphabets, such as Freeze LTL and register automata, serve the verification of processes or documents with data. They relate tightly to formalisms over nominal sets, such as nondetermininistic orbit-finite automata (NOFAs), where names play the role of data. Reasoning problems in such formalisms tend to be computationally hard. Name-binding nominal automata models such as regular nondeterministic nominal automata (RNNAs) have been shown to be computationally more tractable. In the present paper, we introduce a linear-time fixpoint logic Ba…
We define a class of languages of infinite words over infinite alphabets, and the corresponding automata. The automata used for recognition are a generalisation of deterministic Muller automata to the setting of nominal sets. Remarkably, the obtained languages are determined by their ultimately periodic fragments, as in the classical case. Closure under complement, union and intersection, and decidability of emptiness and equivalence are preserved by the generalisation. This is shown by using finite representations of the (otherwise infinite-state) defined class of automata.
This paper provides a coalgebraic approach to the language semantics of two types of non-deterministic automata over nominal sets: non-deterministic orbit-finite automata (NOFAs) and regular nominal non-deterministic automata (RNNAs), which were introduced in previous work. While NOFAs are a straightforward nominal version of non-deterministic automata, RNNAs feature ordinary as well as name binding transitions. Correspondingly, words accepted by RNNAs are strings formed by ordinary letters and name binding letters. Bar languages are sets of such words modulo $α$-equivalence, and to every stat…
We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the level of orbits, which allows for an accurate complexity analysis. The approach is implemented as the library ONS (Ordered Nominal Sets). Our main motivation is nominal automata, which are models for recognising languages over infinite alphabets. We evaluate ONS in two applications: minimisation of automata and active automata learning. In both cases, ONS is com…
Graded Semantics of Nominal Systems
本文提出一种名义设定下的分级单调子框架，用于处理名义自动机的局部新鲜语义及相关行为等价游戏。
Hereditarily finite (HF) set theory provides a standard universe of sets, but with no infinite sets. Its utility is demonstrated through a formalisation of the theory of regular languages and finite automata, including the Myhill-Nerode theorem and Brzozowski's minimisation algorithm. The states of an automaton are HF sets, possibly constructed by product, sum, powerset and similar operations.
This article studies the expressive power of finite automata recognizing sets of real numbers encoded in positional notation. We consider Muller automata as well as the restricted class of weak deterministic automata, used as symbolic set representations in actual applications. In previous work, it has been established that the sets of numbers that are recognizable by weak deterministic automata in two bases that do not share the same set of prime factors are exactly those that are definable in the first order additive theory of real and integer numbers. This result extends Cobham's theorem, w…
We propose a generic categorical framework for learning unknown formal languages of various types (e.g. finite or infinite words, weighted and nominal languages). Our approach is parametric in a monad T that represents the given type of languages and their recognizing algebraic structures. Using the concept of anautomata presentation of T-algebras, we demonstrate that the task of learning a T-recognizable language can be reduced to learning an abstract form of algebraic automaton whose transitions are modeled by a functor. For the important case of adjoint automata, we devise a learning algori…
Logic and Languages of Higher-Dimensional Automata
本文从逻辑角度研究有限高维自动机，证明其语言与MSO可定义的有限有界宽度带接口区间偏序集在序扩张下相互对应，并由此得出该类集合的序扩张仍保持MSO可定义性。
This paper addresses the symbolic representation of non-convex real polyhedra, i.e., sets of real vectors satisfying arbitrary Boolean combinations of linear constraints. We develop an original data structure for representing such sets, based on an implicit and concise encoding of a known structure, the Real Vector Automaton. The resulting formalism provides a canonical representation of polyhedra, is closed under Boolean operators, and admits an efficient decision procedure for testing the membership of a vector.
The paper shows that there is a deep structure on certain sets of bisimilar Probabilistic Automata (PA). The key prerequisite for these structures is a notion of compactness of PA. It is shown that compact bisimilar PA form lattices. These results are then used in order to establish normal forms not only for finite automata, but also for infinite automata, as long as they are compact.
We introduce nominal string diagrams as, string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application.
In this paper we present our current development on a new formalization of nominal sets in Agda. Our first motivation in having another formalization was to understand better nominal sets and to have a playground for testing type systems based on nominal logic. Not surprisingly, we have independently built up the same hierarchy of types leading to nominal sets. We diverge from other formalizations in how to conceive finite permutations: in our formalization a finite permutation is a permutation (i.e. a bijection) whose domain is finite. Finite permutations have different representations, for i…
A Unified Treatment of Substitution for Presheaves, Nominal Sets, Renaming Sets, and so on
本文通过从幺半范畴作用导出闭幺半结构的方法，统一为预层、名义集合及其变体（如重命名集合）构造替换张量，揭示它们之间的新对应关系。
Permissive-Nominal Logic (PNL) extends first-order predicate logic with term-formers that can bind names in their arguments. It takes a semantics in (permissive-)nominal sets. In PNL, the forall-quantifier or lambda-binder are just term-formers satisfying axioms, and their denotation is functions on nominal atoms-abstraction. Then we have higher-order logic (HOL) and its models in ordinary (i.e. Zermelo-Fraenkel) sets; the denotation of forall or lambda is functions on full or partial function spaces. This raises the following question: how are these two models of binding connected? What trans…
Nominal sets provide a framework to study key notions of syntax and semantics such as fresh names, variable binding and $α$-equivalence on a conveniently abstract categorical level. Coalgebras for endofunctors on nominal sets model, e.g., various forms of automata with names as well as infinite terms with variable binding operators (such as $λ$-abstraction). Here, we first study the behaviour of orbit-finite coalgebras for functors $\bar F$ on nominal sets that lift some finitary set functor $F$. We provide sufficient conditions under which the rational fixpoint of $\bar F$, i.e. the collectio…
Nominal Sets in Rocq
本文在 Rocq 中构造性形式化 nominal sets，利用类型类层级与广义重写机制支持新鲜性、α-等价、名称抽象和有限支撑函数，并为 nominal 框架奠定递归归纳基础。
A popular method for solving reachability in timed automata proceeds by enumerating reachable sets of valuations represented as zones. A naïve enumeration of zones does not terminate. Various termination mechanisms have been studied over the years. Coming up with efficient termination mechanisms has been remarkably more challenging when the automaton has diagonal constraints in guards. In this paper, we propose a new termination mechanism for timed automata with diagonal constraints based on a new simulation relation between zones. Experiments with an implementation of this simulation show sig…
PNL to HOL: from the logic of nominal sets to the logic of higher-order functions
本文提出将许可名义逻辑（PNL）的部分子系统翻译为高阶逻辑（HOL），保留名称与绑定，但不保留等变性质，揭示两者模型间存在丰富的同构子系统。
We investigate a class of nominal algebraic Henkin-style models for the simply typed lambda-calculus in which variables map to names in the denotation and lambda-abstraction maps to a (non-functional) name-abstraction operation. The resulting denotations are smaller and better-behaved, in ways we make precise, than functional valuation-based models. Using these new models, we then develop a generalisation of λ-term syntax enriching them with existential meta-variables, thus yielding a theory of incomplete functions. This incompleteness is orthogonal to the usual notion of incompleteness given …
We investigate the foundations of a theory of algebraic data types with variable binding inside classical universal algebra. In the first part, a category-theoretic study of monads over the nominal sets of Gabbay and Pitts leads us to introduce new notions of finitary based monads and uniform monads. In a second part we spell out these notions in the language of universal algebra, show how to recover the logics of Gabbay-Mathijssen and Clouston-Pitts, and apply classical results from universal algebra.
Tree Regular Model Checking (TRMC) is the name of a family of techniques for analyzing infinite-state systems in which states are represented by terms, and sets of states by Tree Automata (TA). The central problem in TRMC is to decide whether a set of bad states is reachable. The problem of computing a TA representing (an over- approximation of) the set of reachable states is undecidable, but efficient solutions based on completion or iteration of tree transducers exist. Unfortunately, the TRMC framework is unable to efficiently capture both the complex structure of a system and of some of its…
We investigate commutative images of languages recognised by register automata and grammars. Semi-linear and rational sets can be naturally extended to this setting by allowing for orbit-finite unions instead of only finite ones. We prove that commutative images of languages of one-register automata are not always semi-linear, but they are always rational. We also lift the latter result to grammars: commutative images of one-register context-free languages are rational, and in consequence commutatively equivalent to register automata. We conjecture analogous results for automata and grammars w…
Higher-Dimensional Timed Automata for Real-Time Concurrency
本文提出高维时间自动机（HDTAs）以统一建模实时并发系统中的事件并发与时间约束，证明其语言包含不可判定但去时间化语言包含可判定，并探讨了其在可达性检查等实际应用中的潜在优势。
Forest automata (FA) have recently been proposed as a tool for shape analysis of complex heap structures. FA encode sets of tree decompositions of heap graphs in the form of tuples of tree automata. In order to allow for representing complex heap graphs, the notion of FA allowed one to provide user-defined FA (called boxes) that encode repetitive graph patterns of shape graphs to be used as alphabet symbols of other, higher-level FA. In this paper, we propose a novel technique of automatically learning the FA to be used as boxes that avoids the need of providing them manually. Further, we prop…
We give a semantics for the lambda-calculus based on a topological duality theorem in nominal sets. A novel interpretation of lambda is given in terms of adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is necessary).
The depth-bounded fragment of the pi-calculus is an expressive class of systems enjoying decidability of some important verification problems. Unfortunately membership of the fragment is undecidable. We propose a novel type system, parameterised over a finite forest, that formalises name usage by pi-terms in a manner that respects the forest. Type checking is decidable and type inference is computable; furthermore typable pi-terms are guaranteed to be depth bounded. The second contribution of the paper is a proof of equivalence between the semantics of typable terms and nested data class memor…
The category of nominal sets is locally monoidal closed
本文证明名义集合范畴Nom上的每个幺半群闭结构都能诱导所有切片范畴上的相应结构，从而Nom是局部分离闭的。
We investigate final coalgebras in nominal sets. This allows us to define types of infinite data with binding for which all constructions automatically respect alpha equivalence. We give applications to the infinitary lambda calculus.
Probabilistic programming interfaces for random graphs: Markov categories, graphons, and nominal sets
本文研究图上概率编程语言的语义模型，建立其与图子的联系，证明良态等式理论与图子一一对应，并通过Markov范畴、测度论概率和名义集合三种构造证明所有图子均源于等式理论，从而构建了图概率编程新模型。
Staton has shown that there is an equivalence between the category of presheaves on (the opposite of) finite sets and partial bijections and the category of nominal restriction sets: see [2, Exercise 9.7]. The aim here is to see that this extends to an equivalence between the category of cubical sets introduced in [1] and a category of nominal sets equipped with a "01-substitution" operation. It seems to me that presenting the topos in question equivalently as 01-substitution sets rather than cubical sets will make it easier (and more elegant) to carry out the constructions and calculations ne…
A Complexity Dichotomy for Semilinear Target Sets in Automata with One Counter
该研究针对三类单计数器系统，通过引入Presburger公式的通用框架，证明了半线性目标集问题的复杂性二分定理，且可判定公式所属的复杂度侧。
Game semantics has been used with considerable success in formulating fully abstract semantics for languages with higher-order procedures and a wide range of computational effects. Recently, nominal games have been proposed for modelling functional languages with names. These are ordinary, stateful games cast in the theory of nominal sets developed by Pitts and Gabbay. Here we take nominal games one step further, by developing a fully abstract semantics for a language with nominal general references.
Petri Automata
本文提出 Petri 自动机模型，用于识别扩展正则表达式（含交集、逆和全关系）所定义的图语言，并证明相应的 Kleene 定理，进而获得无恒等式的 Kleene 格等式理论的 EXPSPACE 完全判定性。
Learning Quantitative Automata Modulo Theories
本文提出从输入序列赋值上的约束集合中学习确定性定量自动机的问题，并提出了主动学习算法 QUINTIC，通过演绎推理在自动机空间上执行完整搜索，保证最小化和正确终止，评估表明其能有效学习多种定量自动机。
We prove that the determinacy of Gale-Stewart games whose winning sets are infinitary rational relations accepted by 2-tape Büchi automata is equivalent to the determinacy of (effective) analytic Gale-Stewart games which is known to be a large cardinal assumption. Then we prove that winning strategies, when they exist, can be very complex, i.e. highly non-effective, in these games. We prove the same results for Gale-Stewart games with winning sets accepted by real-time 1-counter Büchi automata, then extending previous results obtained about these games. Then we consider the strenghs of determi…
First-Order Quantification over Automata
针对无限字模型上全称量化依赖补全自动机的难题，本文提出直接计算一元编码自然数公式中全称一阶量词对自动机影响的算法，以支持多种算术理论的可实现判定过程。
This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of "$k$-automatic" sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting…
I introduce renaming-enriched sets (rensets for short), which are algebraic structures axiomatizing fundamental properties of renaming (also known as variable-for-variable substitution) on syntax with bindings. Rensets compare favorably in some respects with the well-known foundation based on nominal sets. In particular, renaming is a more fundamental operator than the nominal swapping operator and enjoys a simpler, equationally expressed relationship with the variable freshness predicate. Together with some natural axioms matching properties of the syntactic constructors, rensets yield a trul…
This letter proposes a novel reinforcement learning method for the synthesis of a control policy satisfying a control specification described by a linear temporal logic formula. We assume that the controlled system is modeled by a Markov decision process (MDP). We convert the specification to a limit-deterministic generalized Büchi automaton (LDGBA) with several accepting sets that accepts all infinite sequences satisfying the formula. The LDGBA is augmented so that it explicitly records the previous visits to accepting sets. We take a product of the augmented LDGBA and the MDP, based on which…
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