Analysing oscillatory trends of discrete-state stochastic processes through HASL statistical model checking

Abstract

The application of formal methods to the analysis of stochastic oscillators has been at the focus of several research works in recent times. In this paper, we provide insights on the application of an expressive temporal logic formalism, namely the hybrid automata stochastic logic (HASL), to that issue. We show how one can take advantage of the expressive power of the HASL logic to define and assess relevant characteristics of (stochastic) oscillators.

Appendix

Theorem 3

If a trace $\sigma _A$ is noisy periodic w.r.t. amplitude levels $L,H\!\in \!\mathbb {N}$ then it is accepted by automaton $\mathcal{A}_{per}$ with parameters $L$, $H$, $initT\!\in \!\mathbb {R^+}$ and $N\!\in \!\mathbb {N}$

Proof

By hypothesis $\sigma _A$ (the projection, w.r.t an observed species $A$, of a trace $\sigma$ an $n$-dimensional DESP $\mathcal{D}$) is noisy periodic w.r.t. the partition of species $A$’s domain into regions $low=(-\infty ,L)$, $mid=[L,H)$ and $high=[H,\infty )$.

The initial state of the synchronised process $\mathcal{D}\times \mathcal{A}_{per}$ will be $(\sigma _A[0],l_0,Val_0)$ with $Val_0$ being the initial valuation with $Val_0(x)\!=\!0$ for all variables $\forall x\!\in \!X$ of $A_{per}$.

To demonstrate that $\sigma _A$ is accepted by $\mathcal{A}_{per}$ we need to show that starting from the initials state $(\sigma _A[0],l_0,Val_0)$ a final state of $\mathcal{D}\times \mathcal{A}_{per}$, i.e. a state such that the current location is the accepting location end of $\mathcal{A}_{per}$, is reached. For this we proceed by induction w.r.t. the number of subsequent sojourns in the $low$ and $high$ regions. In the remainder, we use the notation $(s,l,Val)\xrightarrow {*} (s^{\prime },l^{\prime }, Val^{\prime })$ to indicate that state $(s^{\prime },l^{\prime },Val^{\prime })$ of process $\mathcal{D}\times \mathcal{A}_{per}$ is reachable from $(s,l,Val)$. We split the demonstration in parts corresponding to the traversal of $\mathcal{A}_{per}$ locations resulting from synchronisation with trace $\sigma _A$:

init :

Let $l_0\!\in \!\mathbb {N}$ be the index of the first state of $\sigma _A$ that belongs to $low$ and that follows $\sigma _A@initT$ (where $initT$ is the parameter of $\mathcal{A}_{per}$), that is: $l_0\!=\!min\{i\!\in \!\mathbb {N}\mid i> i_{initT}\wedge \sigma [i]\!\in \!low\}$, with $i_{initT}$ being the index of the state $\sigma _A$ is in at time $initT$ (observe that since $\sigma _A$ is assumed noisy periodic then $min\{i\!\in \!\mathbb {N}\mid i> i_{initT}\wedge \sigma [i]\!\in \!low\}$ is guaranteed to exist). Thus because of the structure of $\mathcal{A}_{per}$, $(\sigma _A[0],l_0,Val_0)\xrightarrow {*} (\sigma [l_0],\mathbf{low}, Val_{l_0})$, with $Val_{l_0}(x)\!=\!0$, for $x\!\ne \! n$ and $Val_{l_0}(n)\!=\!-1$.

low $\rightarrow$ mid $\rightarrow$ high :

Since $\sigma _A$ is noisy periodic then $\exists m_1,h_1\!\in \!\mathbb {N}:h_1\!>\!m_1\!>\!l_0$ such that $\sigma _A[ m_1]\!\in \!mid$, $\sigma _A[ h_1]\!\in \!high$ hence, because of the structure of $\mathcal{A}_{per}$, it follows that $(\sigma [l_0],\mathbf{low}, Val_{l_0})\xrightarrow {*} (\sigma [m_1],\mathbf{mid}, Val_{m_1}) \xrightarrow {*} (\sigma [h_1],\mathbf{high}, Val_{h_1})$ with $Val_{h_1}(top)=1$, because of the update $\{top:{=}1\}$ of the arc leading to location high.

high $\rightarrow$ mid $\rightarrow$ low :

similarly since $\sigma _A$ is noisy periodic then $\exists l_1,m_{1b}\!\in \!\mathbb {N}:l_1\!>\!m_{1b}\!>\!h_1$ such that $\sigma _A[ m_{1b}]\!\in \!mid$, $\sigma _A[ l_1]\!\in \!low$. hence $(\sigma [h_1],\mathbf{high}, Val_{h_1})\xrightarrow {*} (\sigma [m_{1b}],\mathbf{mid}, Val_{m_{1b}}) \xrightarrow {*} (\sigma [l_1],\mathbf{low}, Val_{l_1})$ with $Val_{m_{1b}}(top)=1$, $Val_{m_{1b}}(n)=-1$ hence $Val_{l_1}(top)=0$, $Val_{l_1}(n)=0$, $Val_{l_1}(t)=0$, since, because of $Val_{m_{1b}}$, location low is entered through edge mid $\xrightarrow {E,(n\!=\! -1 \wedge top=1),\{n+\!+,top:{=}\!0,t:{=}\!0\}}$ low

induction :

since $\sigma _A$ is noisy periodic then the $high$, $low$ regions are entered infinitely often, hence the [low $\rightarrow$ mid $\rightarrow$ high] and [high $\rightarrow$ mid $\rightarrow$ low] steps of the proof hold for each successive iteration. This means that if $l_i$ is the index corresponding to the $i$th time that $\sigma _A$ enters the $low$ region after having sojourned in the $high$ region then because of the periodicity of $\sigma _A$ $\exists m_{i+1},h_{i+1},m{(i+1)b}\!\in \!\mathbb {N}:l_{i+1}\!>\!m_{(i+1)b}\!>\!h_1\!>\!m_1\!>\!l_i$ such that $\sigma _A[ m_{ib}],\sigma _A[ m_{(i+1)b}]\!\in \!mid$, $\sigma _A[ l_i], \sigma _A[ l_{i+1}]\!\in \!low$, $\sigma _A[ h_{(i+1)}]\!\in \!high$, hence $(\sigma [l_i],\mathbf{low}, Val_{l_i})\xrightarrow {*} (\sigma [l_{i+1}],\mathbf{low}, Val_{l_{i+1}})$, with $Val_{l_{i+1}}(n) \!=\! Val_{l_{i}}(n) +1$.

termination, [low $\rightarrow$ end :

] By induction we have seen that $\forall i\!\in \!\mathbb {N}$, $(\sigma [l_i],\mathbf{low}, Val_{l_i})\xrightarrow {*} (\sigma [l_{i+1}],\mathbf{low}, Val_{l_{i+1}})$. Thus on the $(N-1)$th iteration $(\sigma [l_{N\!-\!1}],\mathbf{low}, Val_{l_{N\!-\!1}}){\xrightarrow {*}}$ $(\sigma [l_{N}],\mathbf{low}, Val_{l_{N}})$ with $Val_{l_N}(n)=N$ which enables low $\xrightarrow {\sharp ,(n\!=\! N ),\emptyset }$ end, hence $(\sigma [l_{N}],\mathbf{low}, Val_{l_{N}})\xrightarrow {*} (\sigma [l_{N}],$ $\mathbf{end}, Val_{l_{N}})$ and $\sigma _A$ is accepted. $\square$

Theorem 4

If $\sigma _A$ is a $\delta$-separated (with $\delta \in \mathbb {R}^+$) alternating trace then it is accepted by automaton $\mathcal{A}_{peaks}$ with parameters $\delta$, $initT\!\in \!\mathbb {R^+}$ and $N\!\in \!\mathbb {N}$.

Proof

By hypothesis $\sigma _A$ is $\delta$-separated alternating, which means it contains an infinity of $\delta$-separated local maxima interleaved with $\delta$-separated minima.

We recall the basic notation: if $\sigma _A$ is alternating then

$$\begin{aligned} \sigma _A=*,m[1],*,M[1],*,m[2],*,\ldots \end{aligned}$$

where $m[i]$ (res[. $M[i]$) are the local minima (resp. maxima) and $*$ denotes an arbitrary long sequence of states. $\sigma ^M_A\!=\! M[1],M[2],\ldots$, respectively $\sigma ^m_A\!=\! m[1],m[2],\ldots$, is the projection of $\sigma _A$ consisting of the local maxima, respectively minima, of $\sigma _A$. If $\sigma _A$ is $\delta$-separated alternating then:

$$\begin{aligned} \sigma _A=\sigma ^1_A, m_\delta [1],\sigma ^2_A, M_\delta [1],\ldots \end{aligned}$$

where each $\sigma ^i_A$ ($i\in \mathbb {N}$) is a finite alternating sequence of non-$\delta$separated local minima/maxima and $m_\delta [i]$ (resp. $M_\delta [i]$) are the $\delta$-separated local minima (resp. maxima) For $\delta \in \mathbb {R}^+$ we denote $\sigma ^{M_\delta }_A\!=\! M_\delta [1],M_\delta [2],\ldots$, respectively $\sigma ^{m_\delta }_A\!=\! m_\delta [1],m_\delta [2],\ldots$, the projection of $\sigma _A$ consisting of the $\delta$-separated local maxima, respectively minima, of $\sigma _A$.

Let $(\sigma _A[0],l_0,Val_0)$ be the initial state of the process corresponding to the synchronisation of $\sigma _A$ with $\mathcal{A}_{peaks}$, with $Val_0$ being the initial valuation with $Val_0(x)\!=\!0$ for all variables $\forall x\!\in \!X$ of $A_{peaks}$. We need to show that a final state $(\sigma _A[n],\mathbf{end},Val_n)$ is reached when $\sigma _A$ is synchronised with $A_{peaks}$. For this we proceed by induction w.r.t. the number of subsequent $\delta$-separated max and min encountered during the synchronisation of $\delta _A$. We split the demonstration in parts corresponding to the traversal of $\mathcal{A}_{peaks}$ locations resulting from synchronisation with trace $\sigma _A$.

${\varvec{l}}_{0}\rightarrow$ start :

Because of the structure of $\mathcal{A}_{peaks}$, we have that $(\sigma _A[0],l_0,Val_0)\xrightarrow {*} (\sigma _A@initT,\mathbf{start}, Val_{\text {start}})$, with $Val_{\text {start}}(n)\!=\!-1$, $Val_{\text {start}}(x)\!=\! A$ and $Val_{\text {start}}(t)\!=\!initT$.

start $\rightarrow$ Max or start $\rightarrow$ Min :

Let $i_0\!\in \!\mathbb {N}$ be the index of the first state of $\sigma _A$ that follows $\sigma _A@initT$ and that is distanced more than $\delta$ from $\sigma _A@initT$, i.e. $i_0\!=\!min\{i\!\in \!\mathbb {N}\mid \sigma _A@initT - \sigma _A[i_0]>\delta \}$, (observe that since $\sigma _A$ is assumed to be $\delta$-separated alternating then $min\{i\!\in \!\mathbb {N}\mid \sigma _A@initT - \sigma _A[i_0]>\delta \}$ is guaranteed to exist). Furthermore, let $i_{initT}$ be the index of state $\sigma _A@initT$, i.e. $\sigma _A[i_{initT}]\!=\!\sigma _A@initT$, and $\Delta T=\sum _{i=0}^{n \mid i_{initT}+n<i_0} time(\sigma _A,i_{initT}+i)$, be the time distance from $initT$ and the instant when state $\sigma _A[i_0]$ is entered. Thus because of the structure of $\mathcal{A}_{peaks}$, either $(\sigma _A@initT,\mathbf{start}, Val_{\text {start}})\xrightarrow {*} (\sigma [i_0],\mathbf{Max}, Val_{i_0})$, if $(\sigma _A@initT \!-\! \sigma _A[i_0])\!<\!0$, or $(\sigma _A@initT,\mathbf{start}, Val_{\text {start}})\xrightarrow {*} (\sigma [i_0],\mathbf{Min}, Val_{i_0})$, if $(\sigma _A@initT \!-\! \sigma _A[i_0])\!>\!0$, with $Val_{i_0}(x)\!=\!A$, $Val_{i_0}(t)\!=\!Val_{\text {start}}(t)+\Delta T$.

Max $\rightarrow$ NoisyDec $\rightarrow$ Min :

Let us assume that from state $(\sigma _A@initT,\mathbf{start}, Val_{\text {start}})$ we reached $(\sigma [i_0],\mathbf{Max},$ $Val_{i_0})$, which means that from $\sigma _A@initT$ the trace $\sigma _A$ reaches a state $\sigma _A[i_0]>\sigma _A@initT+\delta$ without ever descending below $\sigma @initT-\delta$, i.e. $\forall j< i_0, \sigma _A[j]\ge \sigma _A@initT - \delta$. Since $\sigma _A$ is $\delta$-separated alternating and since we assumed to have reached configuration $(\sigma [i_0],\mathbf{Max}, Val_{i_0})$ then the first $\delta$-separated point is a maximum and in particular is a state $\sigma _A[H_0]=M_\delta [1]$, $i_0\!<\!H_0$, such that $\sigma _A[H_0]$ is the largest state reachable from $\sigma _A[i_0]$ (i.e. $\sigma _A[j^{\prime }]\!\le \sigma _A[H_0], \forall j^{\prime }, i_0\!<\!j^{\prime }\!<\! H_0$) and from which a $\delta$-distanced smaller state $\sigma _A[k_0]$, $i_0\!<\! H_0\!<\! k_0$ is reachable with $k_0\!=\! inf\{k\mid \sigma _A[k]< \sigma _A[H_0]-\delta \wedge \sigma _A[k^{\prime }]\!\le \sigma _A[H_0], \forall k^{\prime }, H_0\!<\!k^{\prime }\!<\! k\}$. Thus, because of the structure of $\mathcal{A}_{peaks}$, it follows that $(\sigma [i_0],\mathbf{Max}, Val_{i_0})\xrightarrow {*} (\sigma [k_0],\mathbf{Min}, Val_{k_0})$, where $Val_{k_0}(Lmax[\sigma [H_0]])=Val_{i_0}(Lmax[\sigma [H_0]])+1$ $Val_{k_0}(n_M)=Val_{i_0}(n_M)+1$, $Val_{k_0}(x)=\sigma _A[k_0]$, $Val_{k_0}(Smax)=Val_{i_0}(Smax)+1$, $Val_{k_0}(t)=Val_{i_0}(t)$ $+\Delta T$, with $\Delta T=\sum _{j=i_0}^{k_0} time(\sigma _A,j)$.

Min $\rightarrow$ NoisyInc $\rightarrow$ Max :

On the contrary if we assume that from state $(\sigma _A@initT,\mathbf{start}, Val_{\text {start}})$ we reached $(\sigma [i_0],\mathbf{Min}, Val_{i_0})$, which means that from $\sigma _A@initT$ the trace $\sigma _A$ reaches a state $\sigma _A[i_0]<\sigma _A@initT-\delta$ without ever entering the $\sigma @initT+\delta$ region, i.e. $\forall j< i_0, \sigma _A[j]\le \sigma _A@initT + \delta$. Since $\sigma _A$ is $\delta$-separated alternating and since we assumed to have reached configuration $(\sigma [i_0],\mathbf{Min}, Val_{i_0})$ then the first $\delta$-separated point is a minimum and in particular is a state $\sigma _A[h_0]=m_\delta [1]$, $i_0\!<\!h_0$, such that $\sigma _A[h_0]$ is the smallest state reachable from $\sigma _A[i_0]$ (i.e. $\sigma _A[j^{\prime }]\!\ge \sigma _A[h_0], \forall j^{\prime }, i_0\!<\!j^{\prime }\!<\! h_0$) and from which a $\delta$-distanced larger state $\sigma _A[k_0]$, $i_0\!<\! h_0\!<\! k_0$ is reachable with $k_0\!=\! inf\{k\mid \sigma _A[k]>\sigma _A[h_0]-\delta \wedge \sigma _A[k^{\prime }]\!\ge \sigma _A[h_0], \forall k^{\prime }, h_0\!<\!k^{\prime }\!<\! k\}$. Thus, because of the structure of $\mathcal{A}_{peaks}$, it follows that $(\sigma [i_0],\mathbf{Min}, Val_{i_0})\xrightarrow {*} (\sigma [k_0],\mathbf{Max}, Val_{k_0})$, where $Val_{k_0}(Lmin[\sigma [h_0]])=Val_{i_0}(Lmin[\sigma [h_0]])+1$, $Val_{k_0}(n_m)=Val_{i_0}(n_m)+1$, $Val_{k_0}(x)=\sigma _A[k_0]$, $Val_{k_0}(Smin)=Val_{i_0}(Smin)+1$, $Val_{k_0}(t)=Val_{i_0}(t)$ $+\Delta T$, with $\Delta T=\sum _{j=i_0}^{k_0} time(\sigma _A,j)$.

induction :

since $\sigma _A$ is $\delta$-separated alternating then it contains and infinity of $\delta$-separated maxima and minima. hence the [Max $\rightarrow$ NoisyDec $\rightarrow$ Min] and [Min $\rightarrow$ NoisyInc $\rightarrow$ Max] steps of the proof hold for each successive iteration. This means that if $k_i,K_i\in \mathbb {N}$ are the index corresponding to the state of $\sigma _A$ determining the detection of the $i$th $\delta$-separated maximum, resp. minimum, then $\forall i\ge 0$ either $(\sigma [K_i],\mathbf{Min}, Val_{K_i})\xrightarrow {*} (\sigma [k_i],\mathbf{Max}, Val_{k_i})$ $\xrightarrow {*} (\sigma [K_{i+1}],\mathbf{Min}, Val_{K_{i+1}})$, if $(\sigma _A@initT,\mathbf{start},$ $Val_{\text {start}})\xrightarrow {*} (\sigma [i_0],\mathbf{Max}, Val_{i_0})$, or $(\sigma [k_i],\mathbf{Max}, Val_{k_i})$ $\xrightarrow {*} (\sigma [K_i],\mathbf{Min}, Val_{K_i}) \xrightarrow {*} (\sigma [k_{i+1}],\mathbf{Max}, Val_{k_{i+1}})$, if $(\sigma _A@initT,\mathbf{start}, Val_{\text {start}})\xrightarrow {*} (\sigma [i_0],\mathbf{Min}, Val_{i_0})$, given that $Val_{K_i}(n_m)\!<\!N_m-1$ and $Val_{k_i}(n_M)\!<\!N_M-1$.

termination, [Max $\rightarrow$ NoisyDec $\rightarrow$ end :

] By induction we have seen that $\forall i\!\in \!\mathbb {N}$, given that $Val_{k_i}(n_M)\!<\!N_M-1$, we have $(\sigma [k_i],\mathbf{Max}, Val_{k_i})\xrightarrow {*} (\sigma [K_i],\mathbf{Min}, Val_{K_i})$ where $Val_{K_i}(n_M)=Val_{k_i}(n_M+1)$. Thus on he $(N_M-1)$th iteration we have $Val_{k_{n_M-1}}=N_M-1$, hence on detection of the $N_M$th $\delta$-separated maximum the LHA edge NoisyDec $\xrightarrow [\{Lmax[x]++,Smax+=x,n_M++\}]{\sharp ,(x\!>\! A\!+\!\delta )\wedge (n_M\!=\! N_M-1)}$ end is enabled hence $(\sigma [k_{n_M-1}],\mathbf{Max}, Val_{k_{n_M-1}})\xrightarrow {*}$ $(\sigma [K_{n_M}],\mathbf{end}, Val_{K_{n_M}})$, and $\sigma _A$ is accepted.

termination, [Min $\rightarrow$ NoisyInc $\rightarrow$ end :

] specular to [termination, [Max $\rightarrow$ NoisyDec $\rightarrow$ end]].$\square$