Timed Automata with Polynomial Delay and their Expressiveness

We consider previous models of Timed, Probabilistic and Stochastic Timed Automata, we introduce our model of Timed Automata with Polynomial Delay and we characterize the expressiveness of these models relative to each other. Timed and probabilistic automata may be used for modelling systems that exhibit timed and probabilistic behaviour as diverse as safety-critical navigation, web security, algorithmic trading, search-engine optimization, communication protocol design and hardware failure prediction. It is important to understand the expressiveness power of these models and incorporate them within an expressiveness hierarchy, as properties of given machines may be deduced upwards and downwards the hierarchy. There exists, for example, a trade-off between the expressiveness of a given model of computation and the tractability/decidability of its Model Checking problem. This paper provides a unifying expressiveness framework for the aforementioned models. We define measures on the runs of machines and understand two runs as isomorphic or homomorphic if one can be obtained from another by applying certain transformations. Two machines will then have the same expressiveness power if there exists a bijection between their collection of runs determined by these isomorphisms or homomorphisms. We also introduce our model of Timed Automata with Probabilistic Delay, a restriction of Stochastic Timed Automata to transitions characterized by Taylor Polynomials. Of interest to us is the paper Thin and Thick Timed Regular Languages by Basset and Asarin [2] in which information-theoretic arguments are applied for characterizing trajectories of timed automata. Alur and Dill develop in [1] the theory of timed automata to model the behavior of real-time, safety-critical systems over time, with applications such as navigation, traffic signaling or railway safety. The definition provides a way to include timing information within state-transition machines using real-valued variables called clocks. Stoelinga in [9] provides an introduction to probabilistic automata, describing how distributed systems with discrete probabilities can be modeled and analyzed by means of this model and extending the basic techniques of analysis of non-probabilistic automata to probabilistic systems. Probabilistic timed automata are timed automata extended with discrete probability distributions, and can be used to model timed randomized protocols or fault-tolerant systems. They were introduced by Kwiatowska et al. in [7]. The authors use ar X iv :1 70 3. 08 93 6v 1 [ cs .L O ] 2 7 M ar 2 01 7 probabilistic decision protocols to model real-time models exhibiting probabilistic behavior. Our brief exposition of Stochastic Timed Automata, a relatively new topic in the area, is an adaptation of the model introduced in [3]. A mathematical treatment of concurrent timed and probabilistic systems is given in the monograph [8] Stochastic Timed Automata and related results are considered in Baier et al. [3] and Hartmanns in [6] The theory of Stochastic Processes is given by Grimmett and Stirzaker in [5] In the first section we explain the notation used and introduce Non-deterministic Finite-state Automata. In the second section we introduce adapted versions of Timed Automata, Probabilistic Automata, Probabilistic Timed Automata and Stochastic Timed Automata. In the third section we introduce our model of Timed Automata with Polynomial Delay. In the fourth section we set up the expressiveness framework that will apply to these models and outline our results. The fifth section deals with future work and conclusion. The technical appendices give details of the proofs of our main results. 1 Preliminaries Let N,Q,R denote the natural, rational, real numbers with instances n1, n2, . . . , q1, q2, . . . , x1, x2, . . . Special kinds of natural numbers are members of index sets J denoted by k, t, i, j, m, n, r. In the present work, index sets are always subsets of N. For a function f let dom(f) denote its domain and cod(f) its codomain. A relabelling is any bijective map φ. An embedding is any non-injective map φ. Sequences may be finite or countably infinite. A k-tuple is a sequence of length k. For a k-tuple σ = (s1, s2 . . . sk) let σ(i) = si. For any sequence σ, let σbk be its initial segment of length k. For any sequences σ, σ′ write σ σ′ if σ is an initial segment of σ′. This induces a partial order on the collection of considered sequences. A collection S of sequences is called prefix-free if there exists no σ, σ′ ∈ S such that σ σ′. For any sequence σbk let head(σ) = σ(1) and tail(σ) = σ(2) . . . σ(k) For any sequence σ1, σ2 . . . σk, σk+1 . . . we let #, σ2 . . .#, σk+1 . . . and σ1,# . . . σk,# . . . denote the sub-sequences σ2k and σ2k+1. The padding characters indicate we obtain the information in the sub-sequences by hiding certain components of the initial sequence. For any finite set of positive integers S let μS and ξS denote its minimum and maximum elements respectively. A time sequence τ is a rational-valued sequence τi, the “time values”, that is monotonically increasing and non-convergent. A time-isomorphism is a bijective map ε : τ → τ ′ such that τi τj iff ε(τi) ε(τj) Σ is a finite list of symbols, the alphabet. We shall not impose a limit on the size of Σ, only stipulate that it must be finite. For the present purposes we