A Restricted Class of 1-register Machines Presburger Arithmetic with Divisibility Veriication Example: Timing-based Mutual Exclusion Symbolic Computation Computing Consistent Parameter Valuations 3.1 a Decidability Result

New decidability results concerning two-way counter machines and applications. 15 machine instruction that adds (or subtracts) the input variable a to the register corresponds, then, to a transition labeled with (y = a; y := 0) (or (x = a; x := 0), respectively). Transition labels of the form (x y), for 2 f=; <; >g, which correspond to test instructions, can be eliminated by duplicating each state so that all states have at most one incoming transition. In certain cases, given a parametric timed automaton A, we can construct a restricted 1-register machine M A that accepts ?(A), and thus reduce the emptiness problem for parametric timed automata with two clocks to the emptiness problem for restricted 1-register machines. A recent result in IJTW93] shows that the emptiness problem is decidable for deterministic restricted 1-register machines. The problem is still open for nondeterministic restricted 1-register machines. 14 Theorem Existential Presburger arithmetic with divisibility]. Let be a quantiier-free formula over the primitives of addition, the integer divisibility relation, comparisons, and integer constants, and let T = N. We can construct a parametric timed automaton A with two clocks such that the formula is satissable over the natural numbers ii ?(A) is nonempty. Proof. First we transform the formula into a formula 0 such that is satissable ii 0 is satissable, and 0 is a positive boolean combination of atoms of the form (= a), < a, ajb, and :(ajb), where is a linear term (with positive coeecients), and a, b are variables with b > 0. This can be achieved in a straightforward way by introducing extra variables. We now construct a parametric timed automaton A 0 with two clocks x and y such that the parameters of A 0 are the variables of 0. For an atom = (k 1 a 1 + + k m a m + k m+1 = a), the automaton A consists of a single path whose transition labels form the following sequence: Atoms of the form (< a) are handled similarly. For an atom = (ajb), the automaton A is (x; y := 0); (x = a; x := 0) ; (x = a; y = b); resembling the automaton of Figure 1. The atom = :(ajb) is equivalent to (a > b) _ 0 , where 0 = :(ajb) ^ a b. The automaton A 0 is (x; y := 0); (x …