
Hybrid Logics 5
Note that the above syntax is simply that of ordinary (multi-modal) propositional modal logic
extended with clauses for the state symbols and for Eϕ,@sϕand ↓xj.ϕ. Also, note that, like
propositional variables, nominals and state variables can be used as atomic formulas. The differ-
ence between nominals and state variables is analogous to the difference between constants and
variables in first-order logic: nominals cannot be bound by ↓, and their interpretation is specified
by the model, whereas state variables are interpreted by assignment functions, and they can be
bound by the ↓-binder.
The notions of free and bound state variable are defined as in first-order logic, with ↓as the
only binding operator. Similarly, other syntactic notions (such as substitution, and a state symbol
tbeing substitutable for xin ϕ) are defined as in first-order logic. A sentence is a formula
containing no free state variables. Furthermore, a formula is pure if it contains no propositional
variables, and nominal-free if it contains no nominals.
In the remainder of the chapter we will assume fixed a signature hREL,PROP,NOM,SVARi.
Now for the semantics.
DEFINITION 2. A (hybrid) model Mis a triple M=hM, (RM)R∈REL , V isuch that Mis a
non-empty set, each RMis a binary relation on M, and V:PROP ∪NOM →℘(M)is such
that for all nominals i∈NOM,V(i)is a singleton subset of M. We usually write M(roman
letters) for the domain of a model M, and call the elements of Mstates,worlds or points. Each
RMis an accessibility relation, and Vis the valuation. A frame is defined in the usual way: as a
model without a valuation. If F=hM, (RF)R∈RELiis a frame and Vis a valuation on M, then
M=hF, V iis the model hM, (RF)R∈REL , V i. In this case we, say that Mis based on F, and
that Fis the underlying frame of M.
An assignment gfor Mis a mapping g:SVAR →M. Given an assignment g:SVAR →M,
a state variable x∈SVAR, and a state m∈M, we define gx
m(an x-variant of g) by letting
gx
m(x) = mand gx
m(y) = g(y)for all y6=x.
Let M=hM, (RM)R∈REL, V ibe a model, m∈M, and gan assignment for M. For any
state symbol s∈NOM ∪SVAR, let [s]M,g be the state denoted by s(i.e., for i∈NOM,[i]M,g
is the unique m∈Msuch that V(i) = {m}, and for x∈SVAR,[x]M,g =g(x)). Then the
satisfaction relation is defined as follows:
M, g, m |=⊤
M, g, m |=piff m∈V(p)for p∈PROP
M, g, m |=siff m= [s]M,g for s∈NOM ∪SVAR
M, g, m |=¬ϕiff M, g, m 6|=ϕ
M, g, m |=ϕ1∧ϕ2iff M, g, m |=ϕ1and M, g, m |=ϕ2
M, g, m |=hRiϕiff there is a state m′such that RM(m, m′)and M, g, m′|=ϕ
M, g, m |=Eϕiff there is a state m′∈Msuch that M, g, m′|=ϕ
M, g, m |= @sϕiff M, g, [s]M,g |=ϕfor s∈NOM ∪SVAR
M, g, m |=↓x.ϕ iff M, gx
m, m |=ϕ.
The first six clauses in the definition of the satisfaction relation are similar to the ones for the basic
modal language, except that they are relativized to an additional assignment function. Recall
that nominals and state variables can be used as atomic formulas, in which case they act as
propositional variables that are true at a unique state. The ↓binder binds state variables to the
state where evaluation is being performed (the current world), and @sshifts evaluation to the
state named by s. As in first-order logic, if ϕis a sentence (i.e., a formula with no free state
variables), the truth of ϕat a state in a model does not depend on the assignment. Hence, in this
case we will write M, m |=ϕinstead of M, g, m |=ϕ.