Greybox Learning of Languages Recognizable by Event-Recording Automata

First, we define a greybox learning algorithm to derive a DERA with the minimal number of states that recognizes a target timed language. To achieve the learning of a minimal automaton, we must solve an NP-hard optimization problem: calculating a minimal finite state automaton that separates regular languages of positive and negative examples derived from membership and equivalence queries [11]. For that, we adapt to the timed setting the approach of Chen et al. in [7]. A similar framework has also been studied using SMT-based techniques by Moeller et al. in [16]. Second, we explore methods to circumvent this NP-hard optimization problem when the requirement for minimality is loosened. We introduce a heuristic that computes a candidate automaton in polynomial time using a greedy algorithm. This algorithm strives to maintain a small automaton size but does not guarantee the discovery of the absolute minimal solution, yet guaranteed to terminate with a DERA that accepts the input timed language. We have implemented this heuristic and demonstrate with our prototype that, in practice, it yields compact automata.

Grinchtein et al. in [12] also present an active learning framework for ERA-recognizable languages. However, their learning algorithm generates simple DERAs – these are a canonical form of ERA recognizable languages, closely aligned with the underlying region automaton. In a simple DERA, transitions are labeled with region constraints, and notably, every path leading to an accepting state is satisfiable. This means the sequence of regions is consistent (as in the region automaton), and any two path prefixes reaching the same state are region equivalent. Typically, this results in a prohibitively large number of states in the automaton that maintains explicitly the semantics of regions. In contrast, our approach, while also labeling automaton edges with region constraints, permits the merging of prefixes that are not region equivalent. Enabled by equation 1, this allows for a more compact representation of the ERA languages (we generate DERA with minimal number of states). While our algorithm has been implemented, there is, to the best of our knowledge, no implementation of Grinchtein’s algorithm, certainly due to high complexity of their approach.

Lin et al. in [15] propose another active learning style algorithm for inferring ERA, but with zones (coarser constraints than regions) as guards on the transitions. This allows them to infer smaller automata than [12]. However, it assumes that Teacher is familiar with a specific DERA for the underlying language and is capable of answering syntactic queries about this DERA. In fact, their learning algorithm is recognizing the regular language of words over the alphabet, composed of events and zones of the specific DERA known to Teacher. Unlike this approach, our learning algorithm does not presuppose a known and fixed DERA but only relies on queries that are semantic rather than syntactic. Although the work of [15] may seem superficially similar to ours, we dedicate in Section 3 an entire paragraph to the formal and thorough comparison between our two different approaches.

Recently, Masaki Waga developed an algorithm [24] for learning the class of Deterministic Timed Automata (DTA), which includes ERA recognizable languages. Waga’s method resets a clock on every transition and utilizes region-based guards, often producing complex automata. Consequently, this algorithm does not guarantee the minimality of the resulting automaton, but merely ensures it accepts the specified timed language. In contrast, our approach guarantees learning of automata with the minimal number of states, featuring implicit clock resets that enhance readability. In the experimental section, we detail a performance comparison of our tool against their tool, LearnTA, across various examples. Further, in [20], the authors extend Waga’s approach to reduce the number of clocks in learned automata, though they also do not achieve minimality. Their method also requires a significantly higher number of queries to implement these improvements.

Several other works have proposed learning methods for different models of timed languages, such as, One-clock Timed Automata [25], Real-time Automata [4], Mealy-machines with timers [22, 6]. There have also been approaches other than active learning for learning such models, for example, using Genetic programming [19] and passive learning [23, 8].

Let $\Sigma=\{\sigma_{1},\sigma_{2},\dots,\sigma_{k}\}$ be a finite alphabet. A deterministic finite state automaton (DFA) over $\Sigma$ is a tuple $A=(Q,q_{\sf init},\Sigma,\delta,F)$ where $Q$ is a finite set of states, $q_{\sf init}\in Q$ is the initial state, $\delta:Q\times\Sigma\rightarrow Q$ is a partial transition function, and $F\subseteq Q$ is the set of final (accepting) states. When $\delta$ is a total function, we say that $A$ is total. A DFA $A=(Q,q_{\sf init},\Sigma,\delta,F)$ defines a regular language, noted $L(A)$, which is a subset of $\Sigma^{*}$ and which contains the set of words $w=\sigma_{1}\sigma_{2}\dots\sigma_{n}$ such that there exists an accepting run of $A$ over $w$, i.e., there exists a sequence of states $q_{0}q_{1}\dots q_{n}$ such that $q_{0}=q_{\sf init}$, and for all $i$, $1\leq i\leq n$, $\delta(q_{i-1},\sigma_{i})=q_{i}$, and $q_{n}\in F$. We then say that the word $w$ is accepted by $A$. If such an accepting run does not exist, we say that $A$ rejects the word $w$.

A 3DFA $D$ over $\Sigma$ is a tuple $(Q,q_{\sf init},\Sigma,\delta,{\sf A},{\sf R},{\sf E})$ where $Q$ is a finite set of states, $q_{\sf init}\in Q$ is the initial state, $\delta:Q\times\Sigma\rightarrow Q$ is a total transition function, and ${\sf A}\uplus{\sf R}\uplus{\sf E}$ forms a partition of $Q$ into a set of accepting states ${\sf A}$, a set of rejection states ${\sf R}$, and a set of ‘don’t care’ states ${\sf E}$. The 3DFA $D$ defines the function $D:\Sigma^{*}\rightarrow\{0,1,?\}$ as follows: for $w=\sigma_{0}\sigma_{1}\dots\sigma_{n}$ let $q_{0}q_{1}\dots q_{n+1}$ be the unique run of $D$ on $w$. If $q_{n+1}\in{\sf A}$ (resp. $q_{n+1}\in{\sf R}$), then $D(w)=1$ (resp. $D(w)=0$) and we say that $w$ is strongly accepted (resp. strongly rejected) by $D$, and if $q_{n+1}\in{\sf E}$, then $D(w)={?}$. We interpret “$?$” as either way, i.e., so the word can be either accepted or rejected. We say that a regular language $L$ is consistent with a 3DFA $D$, noted $L\models D$, if for all $w\in\Sigma^{*}$, if $D(w)=0$ then $w\not\in L$, and if $D(w)=1$ then $w\in L$. Accordingly, we define $D^{-}$ as the DFA obtained from $D$ where the partition ${\sf A}\uplus{\sf R}\uplus{\sf E}$ is replaced by $F={\sf R}$, clearly $L(D^{-})=\{w\in\Sigma^{*}\mid D(w)=0\}$, it is thus the set of words that are strongly rejected by $D$. Similarly, we define $D^{+}$ as the DFA obtained from $D$ where the partition ${\sf A}\uplus{\sf R}\uplus{\sf E}$ is replaced by $F={\sf A}$, then $L(D^{+})=\{w\in\Sigma^{*}\mid D(w)=1\}$, it is thus the set of words that are strongly accepted by $D$. Now, it is easy to see that $L\models D$ iff $L(D^{+})\subseteq L$ and $L(D^{-})\subseteq\overline{L}$.

A timed word over an alphabet $\Sigma$ is a finite sequence $(\sigma_{1},t_{1})(\sigma_{2},t_{2})\dots(\sigma_{n},t_{n})$ where each $\sigma_{i}\in\Sigma$ and $t_{i}\in\mathbb{R}_{\geq 0}$, for all $1\leq i\leq n$, and for all $1\leq i<j\leq n$, $t_{i}\leq t_{j}$ (time is monotonically increasing). We use $\mathsf{TW}(\Sigma)$ to denote the set of all timed words over the alphabet $\Sigma$.

A timed language is a (possibly infinite) set of timed words. In existing literature, a timed language is called timed regular when there exists a timed automaton ‘recognizing’ the language. Timed Automata (TA) [2] extend deterministic finite-state automata with clocks. In what follows, we will use a subclass of TA, where clocks have a pre-assigned semantics and are not reset arbitrarily. This class is known as the class of Event-recording Automata (ERA) [3]. We now introduce the necessary vocabulary and notations for their definition.

A clock is a non-negative real valued variable, that is, a variable ranging over $\mathbb{R}_{\geq 0}$. Let $K$ be a positive integer. An atomic $K$-constraint over a clock $x$, is an expression of the form $x=c$, $x\in(c,d)$ or $x>K$ where $c,d\in\mathbb{N}\cap[0,K]$. A $K$-constraint over a set of clocks $X$ is a conjunction of atomic $K$-constraints over clocks in $X$. Note that, $K$-constraints are closely related to the notion of zones [10] as in the literature of TA. However, zones also allow difference between two clocks, that are not allowed in $K$-constraints.

An elementary $K$-constraint over a clock $x$, is an atomic $K$-constraint where the intervals are restricted to unit intervals; more formally, it is an expression of the form $x=c$, $x\in(d,d+1)$ or $x>K$ where $c,d,d+1\in\mathbb{N}\cap[0,K]$. A simple $K$-constraint over $X$ is a conjunction of elementary $K$-constraints over clocks in $X$, where each variable $x\in X$ appears exactly in one conjunct. The definition of simple constraints also appear in earlier works, for example [12]. One can again note that, simple constraints closely relate to the classical notion of regions [2]. However, again, regions consider difference between two clocks as well, which simple constraints do not. Interestingly, we do not need the granularity that regions offer (by comparing clocks) for our purpose, because a sequence of simple constraints induce a unique region in the classical sense (see Lemma 2.6).

A valuation for a set of clocks $X$ is a function ${v\colon X\rightarrow\mathbb{R}_{\geq 0}}$. A valuation $v$ for $X$ satisfies an atomic $K$-constraint $\psi$, written as $v\models\psi$, if: $v(x)=c$ when $\psi:=x=c$, $v(x)\in(c,d)$ when $\psi:=x\in(c,d)$ and $v(x)>K$ when $\psi:=x>K$.

A valuation $v$ satisfies a $K$-constraint over $X$ if $v$ satisfies all its conjuncts. Given a $K$-constraint $\psi$, we denote by $[\![\psi]\!]$ the set of all valuations $v$ such that $v\models\psi$. We say that a simple constraint $r$ satisfies an elementary constraint $\psi$, noted $r\models\psi$, if $[\![r]\!]\subseteq[\![\psi]\!]$. It is easy to verify that for any simple constraint $r$ and any elementary constraint $\psi$, either $[\![r]\!]\cap[\![\psi]\!]=\emptyset$ or $[\![r]\!]\subseteq[\![\psi]\!]$.

Let $\Sigma=\{\sigma_{1},\sigma_{2},\dots,\sigma_{k}\}$ be a finite alphabet. The set of event-recording clocks associated to $\Sigma$ is denoted by $X_{\Sigma}=\{x_{\sigma}\mid\sigma\in\Sigma\}$. We denote by $\mathsf{C}(\Sigma,K)$ the set of all $K$-constraints over the set of clocks $X_{\Sigma}$ and we use $\mathsf{Reg}(\Sigma,K)$ to denote the set of all simple $K$-constraints over the clocks in $X_{\Sigma}$. For simple $K$-constraints we adopt the notation $\mathsf{Reg}$, since, as remarked earlier, these kinds of constraints relate closely to the well-known notion of regions. Since simple $K$-constraints are also $K$-constraints, we have that $\mathsf{Reg}(\Sigma,K)\subset\mathsf{C}(\Sigma,K)$.

For the semantics, initially, all the clocks start with the value $0$ and then they all elapse at the same rate. For every transition on a letter $\sigma$, once the transition is taken, its corresponding recording clock $x_{\sigma}$ gets reset to the value $0$.

Given a timed word ${\sf tw}=(\sigma_{1},t_{1})(\sigma_{2},t_{2})\dots(\sigma_{n},t_{n})$, we associate with it a clocked word $\mathsf{cw}(\mathsf{tw})=(\sigma_{1},v_{1})(\sigma_{2},v_{2})\dots(\sigma_{n},v_{n})$ where each $v_{i}:X_{\Sigma}\rightarrow\mathbb{R}_{\geq 0}$ maps each clock of $X_{\Sigma}$ to a real-value as follows: $v_{i}(x_{\sigma})=t_{i}-t_{j}$ where $j=\max\{l<i\mid\sigma_{l}=\sigma\}$, with the convention that $\max(\emptyset)=0$. In words, the value $v_{i}(x_{\sigma})$ is the amount of time elapsed since the last occurrence of $\sigma$ in tw; which is why we call the clocks $x_{\sigma}$ ‘recording’ clocks. Although not explicitly, clocks are implicitly reset after every occurrence of their associated events.

A timed word $\mathsf{tw}=(\sigma_{1},t_{1})(\sigma_{2},t_{2})\dots(\sigma_{n},t_{n})$ with its clocked word ${\sf cw}({\sf tw})=(\sigma_{1},v_{1})(\sigma_{2},v_{2})\dots(\sigma_{n},v_{n})$, is accepted by $A$ if there exists a sequence of states $q_{0}q_{1}\dots q_{n}$ of $A$ such that $q_{0}=q_{\sf init}$, $q_{n}\in F$, and for all $1\leq i\leq n$, there exists $e=(q_{i-1},\sigma,\psi,q_{i})\in E$ such that $\sigma_{i}=\sigma$, and $v_{i}\models\psi$. The set of all timed words accepted by $A$ is called the timed language of $A$, and will be denoted by $L^{\mathsf{tw}}(A)$. We now ask a curious reader: is it possible to construct an ERA with two (or less) states that accepts the timed language represented in Figure 1? We will answer this question in Section 5.

A timed language $L$ is $K$-ERA (resp. $K$-DERA) definable if there exists a $K$-ERA (resp. $K$-DERA) $A$ such that $L^{\mathsf{tw}}(A)=L$. Now, due to Lemma 2.2, a timed language $L$ is $K$-ERA definable iff it is $K$-DERA definable.

over $(\Sigma,K)$ are finite sequences $(\sigma_{1},g_{1})(\sigma_{2},g_{2})\dots(\sigma_{n},g_{n})$ where each $\sigma_{i}\in\Sigma$ and $g_{i}\in\mathsf{C}(\Sigma,K)$, for all $1\leq i\leq n$. Similarly, a region word over $(\Sigma,K)$ is a finite sequence $(\sigma_{1},r_{1})(\sigma_{2},r_{2})\dots(\sigma_{n},r_{n})$ where each $\sigma_{i}\in\Sigma$ and $r_{i}\in{\sf Reg}(\Sigma,K)$ is a simple $K$-constraint, for all $1\leq i\leq n$. Lemma 2.6 will justify why we refer to symbolic words over simple constraints as ‘region’ words.

We are now equipped to define when a timed word ${\sf tw}$ is compatible with a symbolic word $\mathsf{sw}$. Let $\mathsf{tw}=(\sigma_{1},t_{1})(\sigma_{2},t_{2})\dots(\sigma_{n},t_{n})$ be a timed word with its clocked word being $(\sigma_{1},v_{1})(\sigma_{2},v_{2})\dots(\sigma_{n},t_{n})$, and let $\mathsf{sw}=(\sigma^{\prime}_{1},g_{1})(\sigma^{\prime}_{2},g_{2})\dots(\sigma^{\prime}_{m},g_{m})$ be a symbolic word. We say that $\mathsf{tw}$ is compatible with $\mathsf{sw}$, noted $\mathsf{tw}\models\mathsf{sw}$, if we have that: $(i)$ $n=m$, i.e., both words have the same length, $(ii)$ $\sigma_{i}=\sigma^{\prime}_{i}$ and $(iii)$ $v_{i}\models g_{i}$, for all $i$, $1\leq i\leq n$. We denote by $[\![\mathsf{sw}]\!]=\{\mathsf{tw}\in{\sf TW}(\Sigma)\mid\mathsf{tw}\models\mathsf{sw}\}$. We say that a symbolic word $\mathsf{sw}$ is consistent if $[\![\mathsf{sw}]\!]\neq\emptyset$, and inconsistent otherwise. We will use ${\sf RegL}(\Sigma,K)$ to denote the set of all consistent region words over the set of all $K$-simple constraints $\mathsf{Reg}(\Sigma,K)$.

The above lemma states that, two timed words that are both compatible with one region word, cannot be ‘distinguished’ using a DERA. This can be proved using Lemma $18$ of [12].

Note that, every $K$-DERA $A$ can be transformed into another $K$-DERA $A^{\prime}$ where every guard present in $A^{\prime}$ is a simple $K$-constraint. This can be constructed as follows: for every transition $(q,\sigma,g,q^{\prime})$ in $A$ where $g$ is a $K$-constraint (and not a simple $K$-constraint), $A^{\prime}$ contains the transitions $(q,\sigma,r,q^{\prime})$ such that $r\models g$. Since, for every $K$-constraint $g$, there are only finitely-many (possibly, exponentially many) simple $K$-constraints $r$ that satisfy $g$, this construction indeed yields a finite automaton. As for an example, the transition from $q_{1}$ to $q_{2}$ in the automaton in Figure 1 contains a non-simple $K$-constraint as guard. This transition can be replaced with the following set of transitions without altering its timed language: $(q_{1},b,x_{a}=1\wedge x_{b}=0,q_{2}),(q_{1},b,x_{a}=1\wedge 0<x_{b}<1,q_{2}),(q_{1},b,x_{a}=1\wedge x_{b}=1,q_{2}),(q_{1},b,x_{a}=1\wedge x_{b}>1,q_{2})$. Note that, $A^{\prime}$ in the above construction is different from the “simple DERA” for $A$, as defined in [12]. A key difference is that, in $A^{\prime}$ there can be paths leading to an accepting state that are not satisfiable by any timed words, which is not the case in the simple DERA of $A$. This relaxation ensures, $A^{\prime}$ here has the same number of states as $A$, in contrast, in a simple DERA of [12], the number of states can be exponentially larger than the original DERA.

With the above construction in mind, given a $K$-DERA $A$, we associate two regular languages to it, called the syntactic region language of $A$ and the region language of $A$, both over the finite alphabet $\Sigma\times{\sf Reg}(\Sigma,K)$, as follows:

• •

  the syntactic region language of $A$, denoted $L^{\sf s\cdot rw}(A)$, is the set of region words ${\sf rw}=(\sigma_{1},r_{1})(\sigma_{2},r_{2})\dots(\sigma_{m},r_{m})$ such that there exists a sequence of states $q_{0}q_{1}\dots q_{n}$ in $A$ and $(i)$ $q_{0}=q_{\sf init}$, $q_{n}\in F$, and for all $1\leq i\leq n$, there exists $e=(q_{i-1},\sigma,\psi,q_{i})\in E$ such that $\sigma_{i}=\sigma$, and $r_{i}\models\psi$. It is easy to see that $A$ plays the role of a DFA for this language and that this language is thus a regular language over $\Sigma\times{\sf Reg}(\Sigma,K)$.

• •

  the (semantic) region language of $A$, denoted $L^{\sf rw}(A)$, is the subset of $L^{\sf s\cdot rw}(A)$ that is restricted to the set of region words that are consistent. This language is thus the intersection of $L^{\sf s\cdot rw}(A)$ with ${\sf RegL}(\Sigma,K)$. And so, it is also a regular language over $\Sigma\times{\sf Reg}(\Sigma,K)$. (This is precisely the language accepted by a “simple DERA” corresponding to $A$, as defined in [12]).