Infinite Alphabet Research Articles

Amenability of bounded automata groups on infinite alphabets

AbstractWe study the action of groups generated by bounded activity automata with infinite alphabets on their orbital Schreier graphs. We introduce an amenability criterion for such groups based on the recurrence of the first‐level action. This criterion is a natural extension of the result that all groups generated by bounded activity automata with finite alphabets are amenable. Our motivation comes from the investigation of iterated monodromy groups of entire transcendental functions in holomorphic dynamics.

Topological entropy for non-autonomous iterated function systems

In this paper we introduce the topological entropy and the capacity topological entropies of non-autonomous iterated function systems, which extends the notion of the topological entropy of non-autonomous iterated function systems (or NAIFSs for short) on countably infinite alphabets, by using the Carathéodory–Pesin structure (C-P structure). We provide some properties of these notions and give the main results. In fact we give lower and upper estimations of the topological entropy of non-autonomous iterated function systems action by local entropies.

Parikh’s Theorem Made Symbolic

Parikh’s Theorem is a fundamental result in automata theory with numerous applications in computer science. These include software verification (e.g. infinite-state verification, string constraints, and theory of arrays), verification of cryptographic protocols (e.g. using Horn clauses modulo equational theories) and database querying (e.g. evaluating path-queries in graph databases), among others. Parikh’s Theorem states that the letter-counting abstraction of a language recognized by finite automata or context-free grammars is definable in Linear Integer Arithmetic (a.k.a. Presburger Arithmetic). In fact, there is a linear-time algorithm computing existential Presburger formulas capturing such abstractions, which enables an efficient analysis via SMT-solvers. Unfortunately, real-world applications typically require large alphabets (e.g. Unicode, containing a ‍million of characters) — which are well-known to be not amenable to explicit treatment of the alphabets — or even worse infinite alphabets. Symbolic automata have proven in the last decade to be an effective algorithmic framework for handling large finite or even infinite alphabets. A symbolic automaton employs an effective boolean algebra, which offers a symbolic representation of character sets (i.e. in terms of predicates) and often lends itself to an exponentially more succinct representation of a language. Instead of letter-counting, Parikh’s Theorem for symbolic automata amounts to counting the number of times different predicates are satisfied by an input sequence. Unfortunately, naively applying Parikh’s Theorem from classical automata theory to symbolic automata yields existential Presburger formulas of exponential size. In this paper, we provide a new construction for Parikh’s Theorem for symbolic automata and grammars, which avoids this exponential blowup: our algorithm computes an existential formula in polynomial-time over (quantifier-free) Presburger and the base theory. In fact, our algorithm extends to the model of parametric symbolic grammars, which are one of the most expressive models of languages over infinite alphabets. We have implemented our algorithm and show it can be used to solve string constraints that are difficult to solve by existing solvers.

Invariant measures for substitutions on countable alphabets

AbstractIn this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

Saturating automata for game semantics

Saturation is a fundamental game-semantic property satisfied by strategies that interpret higher-order concurrent programs. It states that the strategy must be closed under certain rearrangements of moves, and corresponds to the intuition that program moves (P-moves) may depend only on moves made by the environment (O-moves). We propose an automata model over an infinite alphabet, called saturating automata, for which all accepted languages are guaranteed to satisfy a closure property mimicking saturation. We show how to translate the finitary fragment of Idealized Concurrent Algol (FICA) into saturating automata, confirming their suitability for modelling higher-order concurrency. Moreover, we find that, for terms in normal form, the resultant automaton has linearly many transitions and states with respect to term size, and can be constructed in polynomial time. This is in contrast to earlier attempts at finding automata-theoretic models of FICA, which did not guarantee saturation and involved an exponential blow-up during translation, even for normal forms.Comment: Presented at MFPS 2023

Kernel Words and Gap Sequences of the Tribonacci Word on an Infinite Alphabet

In this paper, we propose a nuanced variation in the kernel words of the tribonacci sequence. Our primary objective is to investigate the intrinsic properties of the kernel words and associated gap sequences when the tribonacci sequence is expanded over an infinite alphabet.

On-the-fly bisimulation equivalence checking for fresh-register automata

Register automata are one of the simplest classes of automata that operate on infinite input alphabets. Each automaton comes equipped with a finite set of registers where it can store data values and compare them with others from the input. Fresh-register automata are additionally able to accept a given data value just if it is fresh in the computation history. One such use for this is representing processes in the π-calculus, where private names need to be fresh with respect to any process context. The bisimilarity problem for fresh-register automata is known to be in NP, when empty registers and duplicate register content are forbidden. In this paper, we investigate on-the-fly algorithms for solving bisimilarity, which attempt to build a bisimulation relation starting from a given input configuration pair. We propose an algorithm that uses concise representations of candidate bisimulation relations based on generating systems. While the algorithm runs in exponential time in the worst case, we demonstrate through a series of benchmarks its efficiency compared to existing algorithms and tools. We moreover define and implement a novel translation from π-calculus processes to fresh-register automata, and use the latter to obtain a (strong early) bisimilarity checking tool for finitary π-calculus processes. Using a series of benchmarks for this, we demonstrate an improvement in run-time compared to another equivalence checker.

Entropy of Local Homeomorphisms With Applications to Infinite Alphabet Shift Spaces

Abstract In this paper, we introduce topological entropy for dynamical systems generated by a single local homeomorphism (Deaconu–Renault systems). More precisely, we generalize Adler, Konheim, and McAndrew’s definition of entropy via covers and Bowen’s definition of entropy via separated sets. We propose a definition of factor map between Deaconu–Renault systems and show that entropy (via separated sets) always decreases under uniformly continuous factor maps. Since the variational principle does not hold in the full generality of our setting, we show that the proposed entropy via covers is a lower bound to the proposed entropy via separated sets. Finally, we compute entropy for infinite graphs (and ultragraphs) and compare it with the entropy of infinite graphs defined by Gurevich.

Catalan numbers as discrepancies for a family of substitutions on infinite alphabets

In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function is exponential (and guided by the inflation factor), the behaviour of the second-order term is strikingly different. For the finite setting, it is known that the second term is also exponential or exponential times a polynomial. We exhibit a large family of examples where the second term is at least exponential in n divided by half-integer powers of n, where n is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

We prove a variety of improved uniform continuity bounds for entropies of both classical random variables on an infinite state space and of quantum states of infinite-dimensional systems. We obtain the first tight continuity estimate on the Shannon entropy of random variables with a countably infinite alphabet. The proof relies on a new mean-constrained Fano-type inequality. We then employ this classical result to derive a tight energy-constrained continuity bound for the von Neumann entropy. To deal with more general entropies in infinite dimensions, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e.g</i> . α-Rényi and α-Tsallis entropies, we develop a novel approximation scheme based on operator Hölder continuity estimates. Finally, we settle an open problem raised by Shirokov [1], [2] regarding the characterisation of states with finite entropy.

Spectral properties of substitutions on compact alphabets

AbstractWe consider substitutions on compact alphabets and provide sufficient conditions for the diffraction to be pure point, absolutely continuous and singular continuous. This allows one to construct examples for which the Koopman operator on the associated function space has specific spectral components. For abelian bijective substitutions, we provide a dichotomy result regarding the spectral type of the diffraction. We also provide the first example of a substitution that has countably infinite Lebesgue spectral components and countably infinite singular continuous components. Lastly, we give a non‐constant length substitution on a countably infinite alphabet that gives rise to substitutive Delone sets of infinite type. This extends the spectral theory of substitutions on finite alphabets and Delone sets of finite type with inflation symmetry.

Dimension-Free Empirical Entropy Estimation

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} -- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest.

Tail Redundancy and its Characterization of Compression of Memoryless Sources

We formalize the tail redundancy of a collection of distributions over a countably infinite alphabet, and show that this fundamental quantity characterizes the asymptotic per-symbol minimax redundancy of universally compressing sequences generated i.i.d. from a collection P of distributions over a countably infinite alphabet. Contrary to the worst case formulations of universal compression, finite single letter minimax (average case) redundancy of P does not automatically imply that the expected minimax redundancy of describing length-n strings sampled i.i.d. from P grows sublinearly with n. Instead, we prove that universal compression of length-n i.i.d. sequences from P is characterized by how well the tails of distributions in P can be universally described, showing that the asymptotic per-symbol redundancy of i.i.d. strings is equal to the tail redundancy.

Fast computations on ordered nominal sets

Nominal automata are models for recognising languages over infinite alphabets, based on the algebraic notion of nominal set. Motivated by their use in automata theory, we show how to compute efficiently with nominal sets over the so-called total order symmetry, a variant which allows to compare alphabet letters for equality as well as their respective order. We develop an explicit finite representation of such nominal sets and basic constructions thereon. The approach is implemented as the library Ons (Ordered Nominal Sets), enabling programming with infinite sets. Returning to our motivation of nominal automata, we evaluate Ons in two applications: minimisation of automata and active automata learning. In both cases, Ons is competitive compared to existing implementations and outperforms them for certain classes of inputs.

Computability of Data-Word Transductions over Different Data Domains

In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $\omega$-words). The notion of computability is defined through Turing machines with infinite inputs which can produce the corresponding infinite outputs in the limit. We use non-deterministic transducers equipped with registers, an extension of register automata with outputs, to describe specifications. Being non-deterministic, such transducers may not define functions but more generally relations of data $\omega$-words. In order to increase the expressive power of these machines, we even allow guessing of arbitrary data values when updating their registers. For functions over data $\omega$-words, we identify a sufficient condition (the possibility of determining the next letter to be outputted, which we call next letter problem) under which computability (resp. uniform computability) and continuity (resp. uniform continuity) coincide. We focus on two kinds of data domains: first, the general setting of oligomorphic data, which encompasses any data domain with equality, as well as the setting of rational numbers with linear order; and second, the set of natural numbers equipped with linear order. For both settings, we prove that functionality, i.e. determining whether the relation recognized by the transducer is actually a function, is decidable. We also show that the so-called next letter problem is decidable, yielding equivalence between (uniform) continuity and (uniform) computability. Last, we provide characterizations of (uniform) continuity, which allow us to prove that these notions, and thus also (uniform) computability, are decidable. We even show that all these decision problems are PSpace-complete for $(\mathbb{N},&lt;)$ and for a large class of oligomorphic data domains, including for instance $(\mathbb{Q},&lt;)$.

Some Notes on the Classification of Shift Spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

The aim of this article is to find appropriate definitions for shifts of finite type and sofic shifts in a general context of symbolic dynamics. We start showing that the classical definitions of shifts of finite type and sofic shifts, as they are given in the context of finite-alphabet shift spaces on the one-dimensional monoid $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, do not fit for shift spaces over infinite alphabet or on other monoids. Therefore, by examining the core features in the classical definitions of shifts of finite type and sofic shifts, we propose general definitions that can be used in any context. The alternative definition given for shifts of finite type inspires the definition of a new class of shift spaces which intersects with the class of sofic shifts and includes shifts of finite type. This new class is named finitely defined shifts, and the non-finite-type shifts in it are named shifts of variable length. For the specific case of infinite-alphabet shifts on the lattice $\mathbb{N}$ or $\mathbb{Z}$ with the usual sum, shifts of variable length can be interpreted as the topological version of variable length Markov chains.

On Universal D-Semifaithful Coding for Memoryless Sources With Infinite Alphabets

The problem of variable length and fixed-distortion universal source coding (or D-semifaithful source coding) for stationary and memoryless sources on countably infinite alphabets ($\infty$-alphabets) is addressed in this paper. The main results of this work offer a set of sufficient conditions (from weaker to stronger) to obtain weak minimax universality, strong minimax universality, and corresponding achievable rates of convergences for the worse-case redundancy for the family of stationary memoryless sources whose densities are dominated by an envelope function (or the envelope family) on $\infty$-alphabets. An important implication of these results is that universal D-semifaithful source coding is not feasible for the complete family of stationary and memoryless sources on $\infty$-alphabets. To demonstrate this infeasibility, a sufficient condition for the impossibility is presented for the envelope family. Interestingly, it matches the well-known impossibility condition in the context of lossless (variable-length) universal source coding. More generally, this work offers a simple description of what is needed to achieve universal D-semifaithful coding for a family of distributions $\Lambda$. This reduces to finding a collection of quantizations of the product space at different block-lengths -- reflecting the fixed distortion restriction -- that satisfy two asymptotic requirements: the first is a universal quantization condition with respect to $\Lambda$, and the second is a vanishing information radius (I-radius) condition for $\Lambda$ reminiscent of the condition known for lossless universal source coding.

The universal rewrite system coded

The universal rewrite system (URS), first formulated by Rowlands and Diaz [2002], which may well be the meta-pattern driving the systems of the physical universe, has been realized, in the companion paper, in the deterministic Turing machine and other devices foundational to computational theory. Here, we show that we can extend the results to actual coding of a given finite section of the fundamentally infinite system. We also apply the principle of the URS to the universal Turing machine and to variable finite automata over an infinite alphabet, and explain its use in cryptography and the decision problem.

Simultaneous wireless information and power transfer with fixed and adaptive modulation

Activating Wireless Power Transfer (WPT) in Radio-Frequency (RF) to provide on-demand energy supply to widely deployed Internet of Everything devices is a key to the next-generation energy self-sustainable 6G network. However, Simultaneous Wireless Information and Power Transfer (SWIPT) in the same RF bands is challenging. The majority of previous studies compared SWIPT performance to Gaussian signaling with an infinite alphabet, which is impossible to implement in any realistic communication system. In contrast, we study the SWIPT system in a well-known Nakagami-m wireless fading channel using practical modulation techniques with finite alphabet. The attainable rate-energy-reliability tradeoff and the corresponding rationale are revealed for fixed modulation schemes. Furthermore, an adaptive modulation-based transceiver is provided for further expanding the attainable rate-energy-reliability region based on various SWIPT performances of different modulation schemes. The modulation switching thresholds and transmit power allocation at the SWIPT transmitter and the power splitting ratios at the SWIPT receiver are jointly optimized to maximize the attainable spectrum efficiency of wireless information transfer while satisfying the WPT requirement and the instantaneous and average BER constraints. Numerical results demonstrate the SWIPT performance of various fixed modulation schemes in different fading conditions. The advantage of the adaptive modulation-based SWIPT transceiver is validated.

Topological Ramsey spaces of equivalence relations and a dual Ramsey theorem for countable ordinals

We define a collection of topological Ramsey spaces consisting of equivalence relations on ω with the property that the minimal representatives of the equivalence classes alternate according to a fixed partition of ω. To prove the associated pigeonhole principles, we make use of the left-variable Hales-Jewett theorem and its extension to an infinite alphabet. We also show how to transfer the corresponding infinite-dimensional Ramsey results to equivalence relations on countable limit ordinals (up to a necessary restriction on the set of minimal representatives of the equivalence classes) in order to obtain a dual Ramsey theorem for such ordinals.