We develop a theory of vector spaces spanned by orbit-finite sets. Using this theory, we give a decision procedure for equivalence of weighted register automata, which are the common generalization of weighted automata and register automata for infinite alphabets. The algorithm runs in exponential time, and in polynomial time for a fixed number of registers. As a special case, we can decide, with the same complexity, language equivalence for unambiguous register automata, which improves previous results in three ways: (a) we allow for order comparisons on atoms, and not just equality; (b) the complexity is exponentially better; and (c) we allow automata with guessing.
Symbolic Finite Automata and Register Automata are two orthogonal extensions of finite automata motivated by real-world problems where data may have unbounded domains. These automata address a demand for a model over large or infinite alphabets, respectively. Both automata models have interesting applications and have been successful in their own right. In this paper, we introduce Symbolic Register Automata, a new model that combines features from both symbolic and register automata, with a view on applications that were previously out of reach. We study their properties and provide algorithms…
Register automata extend classical finite automata with a finite set of registers that can store data from an infinite data domain for later equality comparisons with data from an input data word. While the registers in the original model of register automata, introduced in 1994 by Kaminski and Francez, can only store data occurring in the data word processed so far, we study here the more expressive class of register automata with guessing, where registers can nondeterministically take any value from the infinite data domain, even if this data does not occur in the input data word. It is well…
We investigate the complexity of the containment problem "Does $L(A)\subseteq L(B)$ hold?", where $B$ is an unambiguous register automaton and $A$ is an arbitrary register automaton. We prove that the problem is decidable and give upper bounds on the computational complexity in the general case, and when $B$ is restricted to have a fixed number of registers.
Register automata are finite automata equipped with a finite set of registers ranging over the domain of some relational structure like $(\mathbb N;=)$ or $(\mathbb Q;<)$. Register automata process words over the domain, and along a run of the automaton, the registers can store data from the input word for later comparisons. It is long known that the universality problem, i.e., the problem to decide whether a given register automaton accepts all words over the domain, is undecidable. Recently, we proved the problem to be decidable in 2-ExpSpace if the register automaton under study is over $(\…
We consider one of the weakest variants of cost register automata over a tropical semiring, namely copyless cost register automata over $\mathbb{N}$ with updates using $\min$ and increments. We show that this model can simulate, in some sense, the runs of counter machines with zero-tests. We deduce that a number of problems pertaining to that model are undecidable, in particular equivalence, disproving a conjecture of Alur et al. from 2012. To emphasize how weak these machines are, we also show that they can be expressed as a restricted form of linearly-ambiguous weighted automata.
Cost register automata (CRA) and its subclass, copyless CRA, were recently proposed by Alur et al. as a new model for computing functions over strings. We study some structural properties, expressiveness, and closure properties of copyless CRA. We show that copyless CRA are strictly less expressive than weighted automata and are not closed under reverse operation. To find a better class we impose restrictions on copyless CRA, which ends successfully with a new robust computational model that is closed under reverse and other extensions.
Our starting point are register automata for data words, in the style of Kaminski and Francez. We study the effects of the single-use restriction, which says that a register is emptied immediately after being used. We show that under the single-use restriction, the theory of automata for data words becomes much more robust. The main results are: (a) five different machine models are equivalent as language acceptors, including one-way and two-way single-use register automata; (b) one can recover some of the algebraic theory of languages over finite alphabets, including a version of the Krohn-Rh…
In the past decades, classical results from algebra, including Hilbert's Basis Theorem, had various applications in formal languages, including a proof of the Ehrenfeucht Conjecture, decidability of HDT0L sequence equivalence, and decidability of the equivalence problem for functional tree-to-string transducers. In this paper, we study the scope of the algebraic methods mentioned above, particularily as applied to the equivalence problem for register automata. We provide two results, one positive, one negative. The positive result is that equivalence is decidable for MSO transformations on uno…
We study the universality and inclusion problems for register automata over equality data. We show that the universality and the inclusion problems can be solved with 2-EXPTIME complexity when the input automata are without guessing and unambiguous, improving on the currently best-known 2-EXPSPACE upper bound by Mottet and Quaas. When the number of registers of both automata is fixed, we obtain a lower EXPTIME complexity, also improving the EXPSPACE upper bound from Mottet and Quaas for fixed number of registers. We reduce inclusion to universality, and then we reduce universality to the probl…
A register automaton is a finite automaton with finitely many registers ranging from an infinite alphabet. Since the valuations of registers are infinite, there are infinitely many configurations. We describe a technique to classify infinite register automata configurations into finitely many exact representative configurations. Using the finitary representation, we give an algorithm solving the reachability problem for register automata. We moreover define a computation tree logic for register automata and solve its model checking problem.
Register automata (RAs) are finite automata extended with a finite set of registers to store and compare data from an infinite domain. We study the concept of synchronizing data words in RAs: does there exist a data word that sends all states of the RA to a single state? For deterministic RAs with k registers (k-DRAs), we prove that inputting data words with 2k+1 distinct data from the infinite data domain is sufficient to synchronize. We show that the synchronization problem for DRAs is in general PSPACE-complete, and it is NLOGSPACE-complete for 1-DRAs. For nondeterministic RAs (NRAs), we sh…
Timed automata and register automata are well-known models of computation over timed and data words respectively. The former has clocks that allow to test the lapse of time between two events, whilst the latter includes registers that can store data values for later comparison. Although these two models behave in appearance differently, several decision problems have the same (un)decidability and complexity results for both models. As a prominent example, emptiness is decidable for alternating automata with one clock or register, both with non-primitive recursive complexity. This is not by cha…
We study synthesis of reactive systems interacting with environments using an infinite data domain. A popular formalism for specifying and modelling such systems is register automata and transducers. They extend finite-state automata by adding registers to store data values and to compare the incoming data values against stored ones. Synthesis from nondeterministic or universal register automata is undecidable in general. However, its register-bounded variant, where additionally a bound on the number of registers in a sought transducer is given, is known to be decidable for universal register …
Register pushdown automata (RPDA) is an extension of classical pushdown automata to handle data values in a restricted way. RPDA attracts attention as a model of a query language for structured documents with data values. The membership and emptiness problems for RPDA are known to be EXPTIME-complete. This paper shows the membership problem becomes PSPACE-complete and NP-complete for nondecreasing and growing RPDA, respectively, while the emptiness problem remains EXPTIME-complete for these subclasses.
We study synthesis of reactive systems interacting with environments using an infinite data domain. A popular formalism for specifying and modelling such systems is register automata and transducers. They extend finite-state automata by adding registers to store data values and to compare the incoming data values against stored ones. Synthesis from nondeterministic or universal register automata is undecidable in general. However, its register-bounded variant, where additionally a bound on the number of registers in a sought transducer is given, is known to be decidable for universal register …
We propose a novel automata model over the alphabet of rational numbers, which we call register automata over the rationals (RA-Q). It reads a sequence of rational numbers and outputs another rational number. RA-Q is an extension of the well-known register automata (RA) over infinite alphabets, which are finite automata equipped with a finite number of registers/variables for storing values. Like in the standard RA, the RA-Q model allows both equality and ordering tests between values. It, moreover, allows to perform linear arithmetic between certain variables. The model is quite expressive: i…
We develop a general framework for the specification and implementation of systems whose executions are words, or partial orders, over an infinite alphabet. As a model of an implementation, we introduce class register automata, a one-way automata model over words with multiple data values. Our model combines register automata and class memory automata. It has natural interpretations. In particular, it captures communicating automata with an unbounded number of processes, whose semantics can be described as a set of (dynamic) message sequence charts. On the specification side, we provide a loca…
We propose TOPL automata as a new method for runtime verification of systems with unbounded resource generation. Paradigmatic such systems are object-oriented programs which can dynamically generate an unbounded number of fresh object identities during their execution. Our formalism is based on register automata, a particularly successful approach in automata over infinite alphabets which administers a finite-state machine with boundedly many input-storing registers. We show that TOPL automata are equally expressive to register automata and yet suitable to express properties of programs. Compa…
Model learning (a.k.a. active automata learning) is a highly effective technique for obtaining black-box finite state models of software components. Thus far, generalisation to infinite state systems with inputs/outputs that carry data parameters has been challenging. Existing model learning tools for infinite state systems face scalability problems and can only be applied to restricted classes of systems (register automata with equality/inequality). In this article, we show how we can boost the performance of model learning techniques by extracting the constraints on input and output paramete…
Programs with dynamic allocation are able to create and use an unbounded number of fresh resources, such as references, objects, files, etc. We propose History-Register Automata (HRA), a new automata-theoretic formalism for modelling such programs. HRAs extend the expressiveness of previous approaches and bring us to the limits of decidability for reachability checks. The distinctive feature of our machines is their use of unbounded memory sets (histories) where input symbols can be selectively stored and compared with symbols to follow. In addition, stored symbols can be consumed or deleted b…
We study alternating register automata on data words and data trees in relation to logics. A data word (resp. data tree) is a word (resp. tree) whose every position carries a label from a finite alphabet and a data value from an infinite domain. We investigate one-way automata with alternating control over data words or trees, with one register for storing data and comparing them for equality. This is a continuation of the study started by Demri, Lazic and Jurdzinski. From the standpoint of register automata models, this work aims at two objectives: (1) simplifying the existent decidability pr…
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…
National and Kapodistrian University of Athens（雅典国立与卡波季斯特里亚大学）；National Center for Scientific Research “Demokritos”（德谟克利特国家科学研究中心）；University of Piraeus（比雷埃夫斯大学）
We propose an automaton model which is a combination of symbolic and register automata, i.e., we enrich symbolic automata with memory. We call such automata Symbolic Register Automata (SRA). SRA extend the expressive power of symbolic automata, by allowing Boolean formulas to be applied not only to the last element read from the input string, but to multiple elements, stored in their registers. SRA also extend register automata, by allowing arbitrary Boolean formulas, besides equality predicates. We study the closure properties of SRA under union, intersection, concatenation, Kleene closure, c…
Recently there has been a significant effort to handle quantitative properties in formal verification and synthesis. While weighted automata over finite and infinite words provide a natural and flexible framework to express quantitative properties, perhaps surprisingly, some basic system properties such as average response time cannot be expressed using weighted automata, nor in any other know decidable formalism. In this work, we introduce nested weighted automata as a natural extension of weighted automata which makes it possible to express important quantitative properties such as average r…
Multiset automata are a class of automata for which the symbols can be read in any order and obtain the same result. We investigate weighted multiset automata and show how to construct them from weighted regular expressions. We present training methods to learn the weights for weighted regular expressions and for general multiset automata from data. Finally, we examine situations in which inside weights can be computed more efficiently.
We propose an automaton model which is a combination of symbolic and register automata, i.e., we enrich symbolic automata with memory. We call such automata Register Match Automata (RMA). RMA extend the expressive power of symbolic automata, by allowing formulas to be applied not only to the last element read from the input string, but to multiple elements, stored in their registers. RMA also extend register automata, by allowing arbitrary formulas, besides equality predicates. We study the closure properties of RMA under union, concatenation, Kleene+, complement and determinization and show t…
The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all these differently flavoured categories of nominal sets are monadic over supported sets. Thus, supported sets provide a canonical finite way to represent nominal sets and the automata therein, e.g. register automata. Name binding in supported sets is modelled by a functor following the idea of de Bruijn indices. This functor lifts to the well-known abstraction…
In this paper, we first study the conversion of weighted two-way automata to one-way automata. We show that this conversion preserves the unambiguity but does not preserve the determinism. Yet, we prove that the conversion of an unambiguous weighted one-way automaton into a two-way automaton leads to a deterministic two-way automaton. As a consequence, we prove that unambiguous weighted two-way automata are equivalent to deterministic weighted two-way automata in commutative semirings.
In this paper, we define a new kind of weighted tree automata where the weights are only supported by final states. We show that these automata are sequentializable and we study their closures under classical regular and algebraic operations. We then use these automata to compute the subtree kernel of two finite tree languages in an efficient way. Finally, we present some perspectives involving the root-weighted tree automata.
In this paper we prove the equivalence of sequential, Mealy-type and Moore-type weighted finite automata with output, with respect to various semantics which are defined here.
We show that weighted automata over the field of two elements can be exponentially more compact than non-deterministic finite state automata. To show this, we combine ideas from automata theory and communication complexity. However, weighted automata are also efficiently learnable in Angluin's minimal adequate teacher model in a number of queries that is polynomial in the size of the minimal weighted automaton.. We include an algorithm for learning WAs over any field based on a linear algebraic generalization of the Angluin-Schapire algorithm. Together, this produces a surprising result: weigh…
Motivated by the successful application of the theory of regular languages to formal verification of finite-state systems, there is a renewed interest in developing a theory of analyzable functions from strings to numerical values that can provide a foundation for analyzing {\em quantitative} properties of finite-state systems. In this paper, we propose a deterministic model for associating costs with strings that is parameterized by operations of interest (such as addition, scaling, and $\min$), a notion of {\em regularity} that provides a yardstick to measure expressiveness, and study decisi…
We propose a new symbolic trace semantics for register automata (extended finite state machines) which records both the sequence of input symbols that occur during a run as well as the constraints on input parameters that are imposed by this run. Our main result is a generalization of the classical Myhill-Nerode theorem to this symbolic setting. Our generalization requires the use of three relations to capture the additional structure of register automata. Location equivalence $\equiv_l$ captures that symbolic traces end in the same location, transition equivalence $\equiv_t$ captures that the…
In this paper, we study active learning algorithms for weighted automata over a semiring. We show that a variant of Angluin's seminal \LStar\ algorithm works when the semiring is a principal ideal domain, but not for general semirings such as the natural numbers.
We introduce session automata, an automata model to process data words, i.e., words over an infinite alphabet. Session automata support the notion of fresh data values, which are well suited for modeling protocols in which sessions using fresh values are of major interest, like in security protocols or ad-hoc networks. Session automata have an expressiveness partly extending, partly reducing that of classical register automata. We show that, unlike register automata and their various extensions, session automata are robust: They (i) are closed under intersection, union, and (resource-sensitive…
We use a non-deterministic variant of storage types to develop a framework for the approximation of automata with storage. This framework is used to provide automata-theoretic views on the approximation of multiple context-free languages and on coarse-to-fine parsing.
We present an algorithm to build an automaton from a rational expression. This approach introduces support for extended weighted expressions. Inspired by derived-term based algorithms, its core relies on a different construct, rational expansions. We introduce an inductive algorithm to compute the expansion of an expression from which the automaton follows. This algorithm is independent of the size of the alphabet, and actually even supports infinite alphabets. It can easily be accommodated to generate deterministic (weighted) automata. These constructs are implemented in Vcsn, a free-software…
We develop a new bisimulation (pseudo)metric for weighted finite automata (WFA) that generalizes Boreale's linear bisimulation relation. Our metrics are induced by seminorms on the state space of WFA. Our development is based on spectral properties of sets of linear operators. In particular, the joint spectral radius of the transition matrices of WFA plays a central role. We also study continuity properties of the bisimulation pseudometric, establish an undecidability result for computing the metric, and give a preliminary account of applications to spectral learning of weighted automata.
Hankel matrices (aka connection matrices) of word functions and graph parameters have wide applications in automata theory, graph theory, and machine learning. We give a characterization of real-valued functions on nested words recognized by weighted visibly pushdown automata in terms of Hankel matrices on nested words. This complements C. Mathissen's characterization in terms of weighted monadic second order logic.
The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalisin…
In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of $ω$-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted context-free languages, or $ω$-algebraic series, can be represented as solutions of mixed $ω$-algebraic systems of equations and by weighted $ω$-pushdown automata. In our first main result, …
