IIST and UNU - UNU-IIST - United Nations University

Linking Duration Calculus and TLA 14
in an interval of length l is identified as follows:
segnum(l, a, c) ̂= max {m + n | ma + nc < l, |m−n| 1}.
Since a slower automaton can only produce less segments in an interval, the number
segnum(l, inf U, inf V ) has provided the maximum number of segments for a general automaton
〈 U ↑ q ↓ V 〉 . Note that we assume (inf U + inf V ) > 0 and consider only non-Zeno automata.
Besides full segments, an interval may include partial segments on both ends. Their lengths are
bounded by sup U or sup V , depending on the state. This is why according to the definition,
we have segnum(a + c, a, c) = 1 so that the last segment on either end can be regarded as a
partial segment. The purpose is to enumerate all possibilities for each interval and ensure that
in every case, the right-hand side of (5) refines the left-hand side. In the following laws, we use
a ⊳ b ⊲ c to denote the value a if b = 1 , or the value c if b = 0 .
Def 9 A characteristic function f can be decomposed as two functions g and h under the restrictions
of U and V , if (inf U + inf V ) > 0 and for any t 0 and any n segnum(t, inf U, inf V )
and any t 0 , t 1 , · · · , t n+1 and t ′ 0 , t′ 1 , · · · , t′ n+1 such that t 0, t n+1 sup U, t ′ 0 , t′ n+1 sup V , t 1, · · · , t n ∈ U
and t ′ 1 , · · · , t′ n ∈ V , we have: if ∑ n+1
k=0 (t k ⊳ 2 | k ⊲ t ′ k ) = t then ∑ n+1
k=0 (g(t k) ⊳ 2 | k ⊲ h(t ′ k ))
f(t); and if ∑ n+1
k=0 (t′ k ⊳ 2 | k ⊲ t k) = t then ∑ n+1
k=0 (h(t′ k ) ⊳ 2 | k ⊲ g(t k)) f(t) .
Law 9 If the function f can be decomposed as g and h under the restrictions of U and V ,
the law (5) holds.
The following theorem reveals that if g and h are maximally minimised (see Law 8), then Law 9
is complete.
Theorem 2 (Completeness) If (5) holds but the function f cannot be decomposed as g and
h under the restrictions of U and V , then there exist g ′ and h ′ ∫
∫
such that D p g(l) =
D p g ′ ∫ ∫
(l) and D p h(l) = D p h ′ ∫ ∫
∫
(l) and D p f(l) ⊇ (D p g ′ (l) ⊳ q ⊲
D p h ′ (l)) ∧ 〈 U ↑ q ↓ V 〉 , and f can be decomposed as g ′ and h ′ under the restrictions of
U and V .
Unfortunately the precondition of the complete law is too complicated to check in practice. We
must consider its useful special cases. If the abstract specification is related to only intervals
no longer than the sum of the minimum lengths of the two phases, then the precondition of the
decomposition can be reduced to a constraint on at most three segments.
Law 10 If a + c > 0 , U = [a, b] , V = [c, d] , t f(t) for any t > a + c , and for any t, we have
g(t 0 ) + g(t 1 ) + h(t 2 ) f(t) for any t 0 , t 1 ∈ [a, b] and t 2 ∈ [c, d] such that t 0 + t 1 + t 2 = t , and
h(t 0 ) + h(t 1 ) + g(t 2 ) f(t) for any t 0 , t 1 ∈ [c, d] and t 2 ∈ [a, b] such that t 0 + t 1 + t 2 = t , then
the law (5) holds.
Report No. 301,
UNU-IIST, P.O. Box 3058, Macau