Linear time-dependent constraints programming with MSVL

Abstract

This paper investigates specifying and solving linear time-dependent constraints with an interval temporal logic programming language MSVL. To this end, linear constraint statements involving linear equality and non-strict inequality are first defined. Further, the time-dependent relations in the constraints are specified by temporal operators, such as . Thus, linear time-dependent constraints can be smoothly incorporated into MSVL. Moreover, to solve the linear constraints within MSVL by means of reduction, the operational semantics for linear constraints is given. In particular, semantic equivalence rules and transition rules within a state are presented, which enable us to reduce linear equations, inequalities and optimization problems in a convenient way. Besides, the operational semantics is proved to be sound. Finally, a production scheduling application is provided to illustrate how our approach works in practice.

Appendix

1.1 The proof of Lemma 2

Proof

In Yang and Duan (2008), it has been proved that by using semantic equivalence rules in Figs. 6 and 7, any MSVL program q without linear constraints can be rewritten into an equivalent program q″ in the form of

$$q'' \equiv\bigvee\limits^{k}_{i=1} (q_{ei}\wedge \varepsilon ) \vee\bigvee\limits^{h}_{j=1} (q_{cj} \wedge\bigcirc q_{fj}) $$

such that q≡q″, where k+h≥1, q _fj is a general program in MSVL whereas q _ei and q _cj are true or all state formulas of the form: $(x_{1} =c_{1}) \wedge\cdots\wedge(x_{l} =c_{l}) \wedge \dot{p}_{x_{1}} \wedge\cdots\wedge\dot{p}_{x_{l}}$ with c _r ∈D (1≤r≤l) and $\dot{p}_{x}$ denoting p _x or ¬p _x . Obviously, this form is a special instance of the normal form defined in Definition 8 with q _mj ≡true. Let q′ be q″. Thus, the conclusion holds. □

1.2 The proof of Theorem 5

Proof

If q is an MSVL program without linear constraints, we can rewrite it into its normal form according to Lemma 2 by semantic equivalence rules in Figures 6 and 7. Then we only need to prove that for any linear constraint statement, it can be reduced into its normal form. The proof proceeds on the structure of the statements:

(a) If q is linear equality statement le ₁=le ₂, we have

$$\begin{array}{rcl@{\hspace{2.7cm}}r} \mathit {le}_1=\mathit {le}_2&\equiv& \mathit {le}_1=\mathit {le}_2 \wedge \mathsf {true}\\ &\equiv& \mathit {le}_1=\mathit {le}_2 \wedge(\varepsilon \vee\bigcirc \mathsf {true}) & (\mathsf {true}\equiv \varepsilon \vee \bigcirc \mathsf {true})\\ & \equiv& \mathit {le}_1=\mathit {le}_2 \wedge \varepsilon \vee \mathit {le}_1=\mathit {le}_2 \wedge \bigcirc \mathsf {true}&\mbox {(a1)}\\ \end{array} $$

In light of the definition of linear expressions, we discuss it in two cases:

(i) If either of le ₁ and le ₂ contains the temporal operator ◯, le ₁=le ₂∧ε≡false. Note that at the current state, the linear expression with can be evaluated into a constant by evaluation rules, so we treat such linear expression as a constant. By rules (Equ1, 2, 4) in Table 2 and (AEqu13) in Table 1, we can transpose all the temporal linear expressions to the left side with the next operator outmost and all the state linear expressions to the right side. After that, le ₁=le ₂ is rewritten into $\bigcirc \mathit {le}_{1}'=\mathit {le}_{2}'$, where $\mathit {le}_{1}'$ is a general linear expression while $\mathit {le}_{2}'$ is a state linear expression. Thus, its normal form is

$$\begin{array}{l@{\hspace{1.8cm}}r} \mbox{(a1)}\equiv\bigcirc \mathit {le}_1'=\mathit {le}_2' \wedge\bigcirc \mathsf {true}& \mbox{(AEqu13 in Table 1, Equ1, 2, 4 in Table 2)} \end{array} $$

where the present component q _cj is true.

(ii) Otherwise, neither of le ₁ and le ₂ contains the temporal operator. By (Equ1, 2) in Table 2 and laws in Table 1, le ₁=le ₂ can be rewritten into x=le′ by transposing all the expressions except variable x into the right side, so we obtain the normal form:

$$\begin{array}{l@{\hspace{2.5cm}}r} \mbox{(a1)}\equiv x=\mathit {le}' \wedge \varepsilon \vee x=\mathit {le}' \wedge\bigcirc \mathsf {true}& \hfill \mbox{(Table 1, Equ1, 2 in Table 2)}\\ \end{array} $$

(b) If q is c∗x⇐le, it can be reduced similarly as case (a).

(c) If q is LTET inequality statement le ₁≤le ₂, we have the following:

(i) If neither le ₁ nor le ₂ involves any temporal operators

$$\begin{array}{lll} \mathit {le}_1\leq \mathit {le}_2&\equiv& \exists x: x=\mathit {le}_2-\mathit {le}_1 \wedge x\geq 0\hfill \mbox{(InEqu5 in Table 3)}\\ & \stackrel {x}{\equiv }& x=\mathit {le}_2-le_1 \wedge x\geq0 \hfill \mbox{(InEqu5 in Table 3)}\\ &\equiv& x=\mathit {le}_2-\mathit {le}_1 \wedge0\leq x \wedge(\varepsilon \vee\bigcirc \mathsf {true})\qquad (\mbox{InEqu3 in Table 3, Fig.~7})\\ & \equiv& x=\mathit {le}_2-\mathit {le}_1 \wedge0\leq x \wedge \varepsilon \vee x=\mathit {le}_2-\mathit {le}_1 \wedge0\leq x \wedge\bigcirc \mathsf {true}\end{array} $$

Therefore, le ₁≤le ₂ has already in its normal form.

(ii) Otherwise, by using (InEqu1, 6, AEqu13), we transpose all the temporal linear expressions to the left side with the next operator outmost and all the state linear expressions to the right side. Then le ₁≤le ₂ is reduced into $\bigcirc \mathit {le}_{1}'\leq \mathit {le}_{2}'$, where $\mathit {le}_{1}'$ is a general linear expression while $\mathit {le}_{2}'$ is a state linear expression. Further,

$$\begin{array}{l@{\hspace{1.4cm}}l} \mathit {le}_1\leq \mathit {le}_2 \equiv\bigcirc \mathit {le}_1'\leq \mathit {le}_2' \wedge\bigcirc \mathsf {true}&\mbox{(Figure 7, Table 1, InEqu1, 6 in Table 3) } \end{array} $$

Other composite statements can be transformed into their normal forms by first expanding their definitions and then their reductions come down to the basic cases above. For example, GTET statement le ₁≥le ₂ is rewritten as −le ₁≤−le ₂, which can be reduced into its normal form in case (c). □

1.3 Basic operational semantics of MSVL

The basic operational rules of MSVL in Yang and Duan (2008) are given below. In Fig. 5, evaluation rules of arithmetic expressions are presented by rules (A1–A5) while evaluation rules of boolean expressions are shown by rules (B1–B6). These rules enable us to obtain the overall value of arithmetic and boolean expressions respectively after solving variables in them. Figure 6 concerns semantic equivalence rules to normalize a program without linear constraints at a state while Fig. 7 lists semantic equivalence rules regarding true and false. In Fig. 8, rules (TR1) and (TR2) deal with interval transition between different states whereas (Nor-Or) and (CONG II) handle transition within a state. Rule (TR1) claims that the current state s _i is appended to the model σ _i−1, p requests to be executed at the next state s _i+1, and the number of states increases by one. Rule (TR2) declares that s _i is attached to σ _i−1 and the final configuration is reached. (Non-Or) tells us that we can arbitrarily select one subprogram to execute when p ₁∨p ₂ is encountered in programs. Rule (CONG II) characterizes the replacement of prog with prog′ in configurations as long as prog≡prog′.

Fig. 5

Evaluation rules of arithmetic and boolean expressions

Fig. 6

Semantic equivalence rules

Fig. 7

Semantic equivalence rules of truth values

Fig. 8

Interval transition rules and transition rules within a state

1.4 Pseudo-code of Simplex algorithm

The pseudo-code of Simplex algorithm is shown in Table 16, which is directly taken from Chap. 29 in Cormen et al. (2005).

Table 16 Simplex algorithm