7.1. Summary
In this thesis we presented a study of competitive optimisation on timed automata. We introduced an important class of strategies, boundary strategies, that suggest to a player a symbolic timed move of the form(b,c,a)— “wait until the value of the clockcis in very close proximity of the integerb, and then execute a transition labelled with the actiona”. The existence of optimal boundary strategies for a competitive optimisation problem allows us to work with the boundary region graph abstraction of the timed automata, which—for a fixed initial state—is a finite graph. Hence, to solve competitive optimisation problems on timed automata, it is sufficient to prove that there exist optimal boundary strategies for both players, and to then solve the corresponding optimisation problems on finite subgraphs of reachable states of the boundary region graph.
We showed that for a number of non-competitive optimisation problems and for competitive optimisation problems with reachability time and average time objectives there exist optimal boundary strategies.
For a noncompetitive optimisation problem on concavely-priced timed automata opti- mal boundary strategies exists if the corresponding cost function is a concave-regular func- tion. Concave-regularity is satisfied by a number of cost functions including reachability- price, average-time, average-price, and average price-per-time-unit. The decision version of optimisation problems for a concave-regular cost function is PSPACE-complete on timed automata with at least three clocks.
Using Bellman’s optimality equations, we showed that reachability-time games are positionally determined, and that in reachability-time games optimal boundary strategies exist for both players. We presented a strategy improvement algorithm to solve these optimality equations. We showed that the complexity of solving reachability-time games is EXPTIME-complete for timed automata with at least two clocks.
Using closed region graph, an abstraction of timed automata, we showed that average- time games are determined, and that in average-time games optimal boundary strategies exist for both players. We showed that the complexity of solving average-time games is EXPTIME-complete for timed automata with at least two clocks.
7.2. Future Work
7.2.1. Implementation
We have implemented the algorithm proposed in Chapter 5 for computing the optimal reachability-time of a timed automaton. The Flex code (lexical grammar) and the Bison code (parser grammar) for the specification of a timed game automaton is given in the Appendix D. A self-explanatory example of our specification language for timed (game) automata is given below.
EXAMPLE 7.2.1(Lightbulb example). The following is a specification of a light bulb from the Example 3.1. The bound of this timed automaton is 2.
system light Bulb begin
automaton light begin
locations_n : {off, bright, dim}; /* locations of Min */
locations_x : {}; /* locations of Max */
clocks : x[2]; /* Declaration of clock variables, the number
* in square brackets denote the bound for * that clock.*/
actions : {press, unpress}; /* set of actions */
invar(off, {x<=2}); /* Invariant for location off */
invar(dim, {x<=2}); /* Invariant for location dim */
invar(bright, {x<=2});/* Invariant for location bright */ trans(off, dim, press, {x >= 0}, {x});
/* specification of a guarded transition. * e.g., trans(l1, l2, a, g, rl) denotes a * transition from locations l1 to l2, * labelled with the action a, the guard of * the transition is g and the set of clocks * to be reset after the transition is rl.*/ trans(dim, bright, press, {x <= 1}, {x});
trans(dim, off, press, {x >= 1}, {x}); trans(bright, off, press, {x >= 0}, {x});
init(dim, {x=0}); /* Set of initial states */
final(bright, {x>0}); /* Set of final states */ end
The computational complexity of reachability-time games (EXPTIME-complete) indi- cates that it is unlikely to have an efficient algorithm for this problem. Our implementation of the strategy improvement algorithm for the reachability-time games is inefficient: it constructs and stores the whole boundary region graph of the timed automaton under consideration. More work is needed to make this implementation of practical interest.
A possible future direction of research work is to investigate symbolic zone-based algorithms [UPP] for competitive optimisation problems studied in this thesis.
7.2.2. Complexity of Competitive Optimisation Problems for Timed Automata with Two Clocks
The exact complexity of the reachability problem on timed automata with two clocks is unknown. By the time-abstract bisimulation property of region equivalence relation, it follows that the problem is in PSPACE. Laroussinie, Markey, and Schnoebelen [LMS04] proved the NP-hardness by giving a reduction from NP-complete subset-sum problem to reachability problem on timed automata with two clocks.
Using countdown games [JLS07], we proved that reachability games on timed au- tomata with two clocks are EXPTIME-hard. Our initial guess was to use a similar approach—first prove that one-player countdown games are PSPACE-complete and then give a reduction from one-player countdown games to reachability problem on timed automata with two clocks—to prove PSPACE-hardness of this problem. However, one player countdown games turned out to be NP-complete.
The question of the exact complexity of the reachability problem, and hence of all the other one-player optimisation problems (reachability price, discounted price, average price, price-per-reward average, etc.) on concavely-priced timed automata with two clocks remains open.
However, for timed automata with one clock we can generalise the construction of Laroussinie, Markey, and Schnoebelen [LMS04]—which they used to show NLOGSPACE- membership of the reachability problem for one clock timed automata—to obtain an abstraction similar to the boundary region graph whose size is polynomial in the size of timed automata. Using such abstraction and the techniques presented in this thesis, it can be shown that for one clock timed automata noncompetitive optimisation problems for concave-regular cost functions are in NLOGSPACE. Similarly it is easy to show that for one-clock timed automata reachability-time games are in PTIME, and average-time games are in NP∩co-NP.
7.2.3. Maximisation Problem on Concave-Priced Timed Automata
In Chapter 4 we showed that minimisation problems on concavely-priced timed automata for concave-regular cost functions are PSPACE-complete. Using similar arguments it can be shown that maximisation problems on convexly-priced timed automata for convex-regular cost functions are PSPACE-complete.
We do not know much about maximisation problem on concavely-priced timed automata. However, for the following class of cost functions, it can easily be shown that
both maximisation and minimisation problems on concavely-priced timed automata are PSPACE-complete, as a monotonic function is both quasiconcave and quasiconvex [BV04]. DEFINITION7.2.2(Monotonic-Regular Cost Function). A cost functionCost:PreRuns→R
ismonotonic-regularif it satisfies the following properties.
(1) (Monotonicity). For every regionR∈ Rand for every run typeΛ∈Types(R), there isN∈ N, such that for every states ∈ Rand for everyn≥ N, the functionCostΛn,s
is monotonic on∆Λn,s.
(2) (Regular Lipschitz-continuity). There is a constantκ ≥ 0, such that for every region R∈ Rand for everypositional run typeΛ∈Types(R), there isN∈N, such that for every states∈ Rand for everyn ≥N, the functionCostnΛ,sisκ-continuous on∆Λn,s.
(3) (Positional optimality).There is a positional optimal strategy forCostinTb. (4) (Uniform convergence).For everys ∈Swe have that
CostTb
∗(s,[s]) =nlim→∞CostnTb,∗(s,[s]).
7.2.4. Optimisation on Probabilistic Timed Automata
In this thesis, we considered competitive optimisation problem for the systems having time- critical behaviours modelled using timed automata. One possible direction for the future work in this line of research is to extend these techniques to solve competitive optimisation problems for systems having both time-critical and probabilistic behaviours.
Markov decision processes (MDPs) extend finite automata by providing a probability distribution on the successor states for every transition. Timed automata, as we discussed in this thesis, extend finite automata by providing a mechanism to constrain the transitions with continuous time. Probabilistic timed automata—a hybrid between MDPs and timed automata—were suggested by Kwiatkowska et al. [KNSS99] as a modelling formalism for system exhibiting both timed and probabilistic characteristics. We propose the study of optimisation problems (expected reachability-price, expected discounted-price, and expected average-price) onconcavely-priced probabilistic timed automata which, arguably, can be used to model a larger class of scheduling problems than MDPs, concavely-priced timed automata and probabilistic timed automata.
We conjecture that optimisation problems for these three objectives are decidable. We claim that an abstraction—let us call it boundary region MDP—similar to boundary region graph extended with probability distribution on the edges may be suitable to solve these optimisation problems on probabilistic timed automata.