Lecture Notes in Computer Science 3472

544 Séverine Colin and Leonardo Mariani
The operators ¬, ∨ , ∧ and ⇒ are standard operators (respectively not, or,
and, and implies), but they are defined on the domain {true, false, ⊥}; define(c)
allows to know if c is defined, [c1, c2) is true if and only if c2 was never true
since the last time c1 was observed to be true, including the state when c1 was
true; ↑ c occurs when condition c changes from false to true; ↓ c occurs when
condition c changes from true to false; and “e when c” allows to know if event
e occurs when condition c is true.
A model for this logic is a tuple (S,τ,LC , LE ), where S = {s0,s1,...} is a set
of states, τ is a mapping from S to the time domain (which could be integer,
rationale, or real), LC is a total function from S ×C to {true, false, ⊥} where C
is a countable set of primitive conditions, and LE is a partial function from S ×E
to De where E is a countable set of primitive events and De is the domain of the
event. Intuitively, LC assigns to each state the truth value of all the primitive
conditions; since conditions are interpreted over a 3-valued logic, the truth value
of primitive conditions can be true, false or ⊥ (undefined). Similarly, in each
state s, LE (s, e) is defined for each event e that occurs at s and gives the value
of the primitive event e. The mapping τ defines the time associated to each
state, and it satisfies the requirement that τ(si) i ⇒ τ(sj ) > t
defined D t M (defined(c)) = true if D t M (c) �= ⊥
false otherwise
pair D t M ([e1, e2)) = true if ∃ t0.t0 ≤ t : M , t0 |= e1 and
∀ t ′ .t0 ≤ t ′ ≤ t ⇒ M , t ′ � e2
= false otherwise
negation D t M (¬c) = true if D t M (c) =false
= ⊥ if D t M (c) =⊥
= false if D t M (c) =true
disjunction D t M (c1 ∧ c2) = true if D t M (c1) orD t M (c2) istrue
= false if D t M (c1) =D t M (c2) =false
= ⊥ otherwise
conjunction D t M (c1 ∨ c2) = D t M (¬(¬c1 ∧¬c2))
implication D t M (c1 ⇒ c2) = D t M (¬c1 ∧ c2)
Table 18.1. Denotation of conditions
Before defining the semantics of a condition c holding in a given model M
(c) that
(c) is
at time t, i.e. M ,t |= c, we define the semantics of the denotation Dt M
associates to each condition c its truth value for model M and time t. Dt M
defined in a recursive way in the Table 18.1.
The value of Dt M for a primitive condition ck is given by the truth value of
the condition ck evaluated on state si that is the last discrete state recognized
before time t; inthecaseofdefined, theresultofDt M is true if the condition c
is defined at state t, otherwise the result is false; inthecaseofpair, thevalueof