Supervisory control synthesis for deterministic context free specification languages

Abstract

This paper describes two steps in the generalization of supervisory control theory to situations where the specification is modeled by a deterministic context free language (DCFL). First, it summarizes a conceptual iterative algorithm from Schneider et al. (2014) solving the supervisory control problem for language models. This algorithm involves two basic iterative functions. Second, the main part of this paper presents an implementable algorithm realizing one of these functions, namely the calculation of the largest controllable marked sublanguage of a given DCFL. This algorithm least restrictively removes controllability problems in a deterministic pushdown automaton realizing this DCFL.

Appendix A: Counterexample

In this appendix, following Schmuck et al. (2014) (Appendix A) the algorithm presented by Griffin (2008, p.827) and Griffin (2007, p.64) is applied to an example. It will be shown that this represents a counterexample to Theorem 3.5 by Griffin (2008) and Theorem 5.2.5 by Griffin (2007) since the final DPDA does not realize the supremal controllable sublanguage of the given prefix closed deterministic context free specification language (which is required to be a subset of the given prefix closed regular plant language). Therefore we claim that the problem of automatically calculating a supremal controllable sublanguage of a DCFL was not solved by Griffin in (2007, 2008).

The algorithm is initialized with a DFA G and a DPDA M realizing the plant and the specification, respectively, s.t. $\mathcal {L}_{m}(G)=\mathcal {L}_{um}(G), \mathcal {L}_{m}(M)=\mathcal {L}_{um}(G)$, and $\mathcal {L}_{m}(M)\subseteq \mathcal {L}_{m}(G)$. Observe that the DFA G and the DPDA M depicted in Fig. 8 satisfy these requirements since their languages are given by

$$\begin{array}{@{}rcl@{}} &&\mathcal{L}_{m}(G)=\mathcal{L}_{um}(G)=\{a^{n},a^{n}b,a^{n}bu|n\in \mathbb{N}\}\quad\text{and}\\ &&\mathcal{L}_{m}(M)=\mathcal{L}_{um}(M)=\{a^{n},a^{m}b,a^{k}bu\big|n,m,k\in \mathbb{N},m>0,k>1\}, \end{array} $$

Fig. 8

DFA G (left) and DPDA M (right) realizing plant and specification, respectively

where Σ_c = {a, b} and Σ _{u c} = {u}. Using these automata, the construction follows seven steps.

• 1. Construct M ^′, depicted in Fig. 9, by making M scan its entire input, using the algorithm in Hopcroft and Ullman (1979, Lemma10.3).

• 2. Construct a DPDA M ^″ that accepts the complement of L _M using the algorithm by Hopcroft and Ullman (1979, Thm.10.1). For the considered example M ^″ is identical to M ^′ in Fig. 9 but with exchanged non-marking and marking states, i.e., F ^″ = {q _d }.

• 3. Construct a DPDA M ^″′ that accepts $\mathrm {L}^{\mathrm {M}}_{\mathrm {c}}(M)\cap \mathrm {L}_{m}(G)$, i.e., calculate the cross product of G and M ^″ using the algorithm by Hopcroft and Ullman (1979, Thm.6.5).

• 4. Construct M ₁, depicted in Fig. 10, as the accessible part of M ^″′, using the algorithm by Griffin (2006, Thm.4.1).

• 5. Construct a predicting machine to observe so called μ-reverse paths using the algorithm by Hopcroft and Ullman (1979, p.240). For the considered example, the construction simply defines an additional stack symbol μ _γ ,∀γ∈Γ s.t.

  $$\mu_{\gamma}:=\left\{q\in Q_{1}\setminus F_{1}\left|\exists q^{\prime}\in F_{1}, v\in{\Sigma}_{\mathsf{uc}}^{*}\cdot (\bot,(q,w,\gamma\cdot s))\vdash^{\ast}_{M_{1}}(\bot,(q^{\prime},w\cdot v,s^{\prime}))\right.\right\} $$

  which denotes the set of unmarked states in M ₁ from which a derivation starting with stack-top γ, generating a sequence of uncontrollable symbols $v\in {\Sigma }_{\text {uc}}^{*}$ and reaching a marking state q ^′ (i.e., a so called μ-reverse path), exists. For M ₁, depicted in Fig. 10, this gives μ _□ = {(p ₁, q ₁)} and μ _∙ = ∅. The predicting machine $M_{1}^{\mu }$, depicted in Fig. 11, is then identical to M ₁ but uses pairs [γ, μ _γ ] as stack.

• 6. Construct M ₂, depicted in Fig. 12, by deleting all transitions

  $$\delta_{cp}:=\left\{(q,\sigma,\gamma,\gamma^{\prime}\cdot s,q^{\prime})\in\delta_{M}\left| \left( \begin{array}{l} (q,\sigma,\gamma,[\gamma^{\prime},\mu_{\gamma^{\prime}}]\cdot s,q^{\prime})\in \delta_{M_{1}^{\mu}}\\ \wedge\sigma\in {\Sigma}_{\textsf{c}} \wedge q^{\prime}\in\mu_{\gamma^{\prime}} \end{array}\right)\right.\right\} $$

  in M which produce a stack-top in q ^′ which enables a μ-reverse path starting in q ^′. For M and $M_{1}^{\mu }$, depicted in Figs. 8 and 11, respectively, observe that $e=((p_{0},q_{0}),b,[\bullet ,\mu _{\bullet }],\lambda ,(p_{1},q_{1}))\in \delta _{M_{1}^{\mu }}$ is the only ingoing transition to (p ₁, q ₁) (where the only μ-reverse path starts for stack-top □, since μ _□ = {(p ₁, q ₁)}) and, since M ₁ is trim, eventually leads to the stack-top □ in (p ₁, q ₁). Using the corresponding transition to e in M, this gives δ _{c p} = {(q ₀, b,∙, λ, q ₁)}. By deleting δ _{c p} in M, we obtain M ₂, depicted in Fig. 12.

• 7. Construct M ₃, depicted in Fig. 13, as the accessible part of M ₂, using the algorithm by Griffin (2006, Thm.4.1). If δ _{c p} in step 6 is empty, the algorithm terminates. Otherwise, the algorithm is restarted with M = M ₃.

Fig. 9

DPDA M ^′, with Ψ=Φ∖u,∙,∙ and Φ={a,□,□;a,∙,∙;b,□,□;b,∙,∙;u,□,□;u,∙,∙}

Fig. 10

DPDA M ₁, with (i j):=(p _i , q _j )

Fig. 11

DPDA $M_{1}^{\mu }$ with μ _□ = {(p ₁, q ₁)} and μ _∙ = ∅. The set of μ-reverse paths is depicted in red (dashed) while the set of edges in δ _{c p} is depicted in blue (dotted)

Fig. 12

DPDA M ₂

Fig. 13

DPDA M ₃

Obviously, M ₃ does not have further controllability problems. Therefore, the algorithm would redo steps 1-6 and then return M ₃.

Now observe that the specification language $\mathcal {L}_{m}(M)$ restricts the plant language $\mathcal {L}_{m}(G)$ such that u cannot occur after exactly one a. This generates a controllability problem for the word ab only. Using (3) in Section 3, the supremal controllable sublanguage of $\mathcal {L}_{m}(M)$ for this example is given by

$$\begin{array}{@{}rcl@{}} \mathcal{K}&=&\{w\in \mathcal{L}_{m}(M)| \forall w^{\prime}\sqsubseteq w\cdot w^{\prime}\neq ab\} =\mathcal{L}_{m}(M)\setminus\{ab\} \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} &=&\{a^{n},a^{m}b,a^{k}bu\big|n,m,k\in \mathbb{N},m,k>1\} \end{array} $$

implying that $\mathcal {L}_{m}(M_{3})=\{a^{n}\,|\,n\in \mathbb {N}\}$ is a strict subset of $\mathcal {K}$ which is an obvious contradiction to Theorem 3.5 by Griffin (2008). Furthermore, $\mathcal {K}$ in (20) cannot be realized using the state and transition structure of M and only deleting existing transitions.

The automatic synthesis of a DPDA realizing $\mathcal {K}$ for this example is provided as an example within our pushdown-plug-in for libFAUDES (2013).