# Project : s4

## Section: New Results

Keywords : control , discrete event system , partial observation , communicating system , logics , mu-calculus , tree automata , winning strategy , parity game .

### Classification and resolution of control problems through the quantified mu-calculus

Participants : Sophie Pinchinat, Stéphane Riedweg.

The theory of control synthesis introduced by Ramadge and Wohnam is a
generic method which can be described as follows: given a program and
some expected behavior, known as *control objective*, the goal
is to produce, by automated methods, a device (*e.g.*, another
program) with two main properties. First, this device must fulfill
some constraints (*e.g.*, it should belong to some particular
class of programs), and second, it should be able to control the
original program (*e.g.*, by synchronous composition) in order to
achieve the required behavior.

We have developed a logical formalism as a general formal language for
the specification of control problems. The proposed framework extends
the Mu-Calculus, a extremely expressive modal logic with fix-points
operators, introduced by Dexter Kozen: we allow for quantifications
over atomic propositions, yielding to a second order logic. We have
established that *checking for the existence of a solution to the
control problem is equivalent to perform verification of
formulas*. Verification of formulas is often called
*Model-Checking*. However, the is logic undecidable, as the
decentralized control problems under partial observation can be
expressed therein. We have explored various fragments of the
logic. The fragments reveal to be expressive enough to specify
interesting control problems, but small enough to remain decidable.
An accurate study of the complexity of the satisfiability and
model-checking problems, in these logical fragments, has been carried
out.

In [19], we consider the fragment
corresponding to the setting where the moves of the systems to be
controlled are fully observable. These are the so called *control problems
under total observation*. The logical setting offers a uniform way to
describe, as parameters, the kind of system (closed or open),
the control objective, the type of interaction between the controller
and the system, optimality criteria (fairness, maximally
permissive), etc. To our knowledge, none of the former approaches can
capture such a wide range of concepts. Moreover, we have established
that model-checking this fragment is decidable and that *the
synthesis of controllers* can be obtained on the basis of the
underlying model checking procedure.

In [26], the case of control requirements for systems under partial observation is studied. We have focused on a fragment expressive enough to specify the unobservable sets of events of (decentralized) controllers, and to allow for the joint unobservability and controllability of an event. We have identified the set of formulas representable by infinite tree automata. Technically, the automata constructions are borrowed from the work of André Arnold et al. [36]. For formulas expressing control requirements, any model of the associated automaton provides an adequate controller. For example, given any Mu-Calculus definable control objective, a maximal permissive controller in some class of controllers under partial observation can be specified by a formula and synthesized in time 3EXP in the size of the formula.

This logical framework brings a new vision of the field, and makes discrete event system control theory much clearer. In particular, it provides a rigorous classification of control problems. Our logical framework is also expected to be relevant to problems related to control theory, such as diagnosis or test generation. This will be the objective of this continuing research work.