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Predicative semantics of modal logics 4
Law 2
(A : (y,x) B) : (x,z) C = A : (y,x) (B : (x,z) C).
Generic composition and its inverse can be used to define modalities. These properties make
the composition a useful technical tool for linking temporal logics. Generic composition has also
been applied to define a variety of healthiness conditions and parallel compositions. The above
laws and a series of other laws can be found in [2].
Interpreting modalities
Under Kripke semantics [1], modal logics are logical systems of relations (called “accessibility
relations”). Here, we represent a specification as a predicate on a modal variable (e.g. x) and
an auxiliary variable (e.g. y). The modal variable records the observable aspect related to the
accessibility of the modalities, while the auxiliary variable records the unrelated observable
aspect. For now, the variables are left untyped. These logical variables will later be typed
in temporal logics. A logical variable may split into several ones, and its type becomes the
product of several types. The semantic space is the set of all such specifications (e.g. denoted
by A ). An accessibility relation R = R(x, x) is denoted by a predicate on two variables: the
modal variable x and the overlined modal variable x . Overlined variables only appear in the
accessibility relations. Each accessibility relation determines a pair of modalities.
Def 3 ♦ A P ̂= P : x ˜R and A P ̂= P / x R .
The operator ♦ A P informally means that “the predicate P may be true” and is defined as a
generic composition of the specification P and the converse relation ˜R ; its dual modality A P
informally means that “the predicate P must be true” is defined with an inverse operator.
If we replace the accessibility relation with its converse, we will obtain a pair of converse modalities.
Def 4 ˜♦A P ̂= P : x R and ˜ A P ̂= P / x ˜R .
Generic composition and its inverse can be regarded as parameterised modal operators. They
have a designated interface and are more convenient than traditional relational composition in
this context for two reasons. Firstly, the abservable aspects (described by the auxiliary variable)
unrelated to the accessibility relation can be excluded from the interface of the relational composition.
Secondly, the predicate on the left-hand side of a generic composition (or its inverse) can
be either a specification (without overlined variables) or an accessibility relation (with overlined
variables). Thus the operators can be directly used to represent the composition of accessibility
relations (i.e. the composition of modalities).
Report No. 301,
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