In this section we show how to associate a generalised operator with a family of quantifiers so as to enable the translation from fixed-point logics to infinitary logics. We recall that generalised operators are defined in terms of many-sorted structures. We now generalise the notion of a Lindström quantifier and introduce the notion of a many-sorted quantifier.
Definition 3.17. Let τ := (R, S, ζ) be a many-sorted relational vocabulary. Let G be a
class of τ-structures and let ar : S → N0. We associate the pair (G, ar) with a many-sorted
quantifier QG,ar. We call G the class of structures and ar the arity of the quantifier QG,ar.
For a logic L the extension L(QG,ar) is defined by extending the formula formation rules for
L as follows:
For each s ∈ S let ⃗xs, ⃗xs
1, and ⃗xs2 be ar(s)-tuples of element variables and let ϕDs
and ϕ≈
s be L(QG,ar)-formulas. For each R ∈ R and i ∈ [rR] let ⃗xRi be an ar(ζ(R)(i))-
length tuple of element variables. For each R ∈ R let ϕR be an L(QG,ar)-formula.
Then
ϕ ≡ QG,ar[(ϕDs )s∈S,(ϕ≈s)s∈S][(⃗xR1, . . . , ⃗xRrR)ϕR]R∈R
is a formula of L(QG,ar). We have
free(ϕ) = [ s∈S [(free(ϕD s) \ ⃗xs) ∪ (free(ϕ≈s) \ (⃗x1s∪ ⃗x2s))] ∪ [ R∈R (free(ϕR) \ ( [ i∈[rR] ⃗ xRi )). Let I := ⟨(ϕD
s)s∈S,(ϕ≈s)s∈S,(ϕR)R∈R⟩. The semantics of the formula ϕ is defined for a
structure A ∈ fin[ρ] and assignment α to the free variables in ψ as follows A |= ψ[α] if, and only if, I(A, α) is defined and I(A, α) ∈ G
We note that many-sorted quantifiers can equivalently be thought of as Boolean-valued relational generalised operators that quantify over the universe. We also note that many- sorted quantifiers generalise Lindström quantifiers in the sense that we can identify each Lindström quantifier with a many-sorted quantifier with a single sort.
As for generalised operators, for a family of many-sorted quantifiers Q we let L(Qe)
be defined from L(Q) by restricting ourselves to formulas in which each application of a many-sorted quantifier in Q has only trivial domain and equality formulas. In this case we omit the domain and equality formulas when applying the quantifier.
We aim to define a general method for associating a family of generalised operators with a family of many-sorted quantifiers. We do this in such a way as to enable a translation from each extension of fixed-point logic by a family of generalised operators to infinite families of first-order formulas extended by the corresponding family of many-sorted quantifiers. This translation generalises the translation from FPC to infinite families of FO+C-formulas (or just single Cω-formulas) given by Grädel and Otto [21]. We now review this translation and
then discuss what would be needed to generalise it.
Let θ(⃗x) be an FPC-formula. We begin by unrolling the fixed-point operators in θ(⃗x) in order to define a sequence of FOC-formulas (θn(⃗x))n∈N, each of which capture the meaning
of θ on structures of size n and such that there is a constant bound on the width of these formulas. For more details on this ‘unrolling’ argument please see [35]. We then aim to define an equivalent family of FO+C-formulas by defining a translation from FOC-formulas to FO+C-formulas. This translation involves recursively removing any reference to the number-sort and replacing instances of universal and existential quantification over the number domain with large disjunctions and conjunctions. To be precise, we recursively define for each n ∈ N and each subformula ϕn(⃗y, ⃗ν) of θn a formula ϕn, ⃗m(⃗y) for each ⃗m ∈ N|⃗ν| such
that for any A ∈ fin[τ, n] and any assignment α to ⃗y in A we have A |= ϕn, ⃗m[α] if, and only if,
A |= ϕn[αm⃗
⃗ν]. In other words, we recursively define for each subformula a family of formulas,
one for each assignment to the free number variables, each of which has the same meaning as the original subformula for the given assignment to the number variables.
We can translate a subformula with an existential or universal quantifier at its head that binds a number variable to a large disjunction or conjunction indexed by the pos- sible assignments to that variable. For example, we translate a formula of the form
ψn(⃗y, ⃗µ) ≡ ∃ν ϕn(⃗y, ν, ⃗µ) to the family of formulas (ψn, ⃗m)n∈N, ⃗m∈[n]|⃗µ|
0 where each ψn, ⃗m(⃗x) ≡
W
a∈[n]0ϕn,a, ⃗m(⃗x). It is crucial at this point to note that we may assume without a loss of
generality that every counting operator binds only a single element variable [21]. As such, we do not need to consider the problem of how to translate counting operators that bind number variables.
In contrast, an arbitrary generalised operator may bind element and number variables (e.g. the rank operator) and there is no known normal form that would allow us to restrict our attention to generalised operators that only bind element variables. As such, we cannot assume that all number variables are bound by universal or existential quantifiers and so we cannot handle the binding of number variables using disjunctions and conjunctions, and must consider how to translate applications of generalised operators that bind number variables. In order to address this we introduce for each operator a family of quantifiers each of which is applied to the entire family of formulas generated by the translation. Each of these quantifiers is defined over an extended vocabulary defined by ‘copying’ each relation symbol for each possible assignment to the bound number variables. We now define these quantifiers formally.
Let τ = (R, S, ζ) be a many-sorted relational vocabulary. Let G be a class of τ-structures, let n ∈ N, and let ar : S × [2] → N. We associate (G, n, ar) with the many-sorted quantifier
QG,n,ar defined as follows. Let
Rn,ar= {R⃗b1,...,⃗b rR : R ∈ R, ⃗b1 ∈[n] ar(ζ(R)(1),2) 0 , . . . ,⃗brR ∈[n] ar(ζ(R)(rR),2) 0 }, where each R⃗b
1,...,⃗brR is a fresh relation symbol. Let ζn,arbe defined such that ζn,ar(R⃗b1,...,⃗brR) =
ζ(R) for all R⃗b
1,...,⃗brR ∈ Rn,ar. Let τn,ar:= (Rn,ar, S, ζn,ar). For each τn,ar-structure A of size
nlet ⊎s∈SAs be the universe of A and let A∗ be the τ-structure defined as follows. For each s ∈ S let A∗s = Aar(s,1)s ×[n]ar(s,2)0 . For each R ∈ R let RA
∗
be defined such that for all i ∈ [rR]
and all ⃗ai⃗bi ∈ A∗ζ(R)(i), (⃗a1⃗b1, . . . , ⃗arR⃗brR) ∈ R
A∗
if, and only if, (⃗a1, . . . , ⃗arR) ∈ R
A
⃗b1,...,⃗brR. Let
A∗ = (⊎s∈SA∗,(RA∗)R∈R). Let Gn,ar := {A ∈ fin[τn,ar] : A∗ ∈ G}. Let ar1 : S → N0 be
defined such that ar1(s) = ar(s, 1) for all s ∈ S. Let Q
G,n,ar := QGn,ar,ar1.
Notice that if ar : S × [2] → N0 is such that ar(s, 2) = 0 for all s ∈ S then for all n ∈ N
we have τn,ar= τ and so Gn,ar= G and QG,n,ar= QG,ar.
Definition 3.18. Let τ = (R, S, ζ) be a many-sorted relational vocabulary. Let G be a class
of τ-structures. We call the set of quantifiers QG= {QG,ar,n : n ∈ N, ar : S × [2] → N0}the
vectorised family of many-sorted quantifiers (or just the vectorised many-sorted quantifier) generated by G.
We now define for each almost relational Boolean-valued generalised operator Ω a cor- repsonding family of many-sorted quantifiers QΩ. Let Ω := ΩE,ar be an almost relational
Boolean-valued generalised operator and let τ := (R, F , S, ζ) be the vocabulary of Ω. We note that since Ω is almost relational it follows that each function symbol F ∈∈ F is a constant symbol. Let τrel := (R, S, ζ
R). For each α : F → N0 and A ∈ fin[τrel] let (A|α) be the τ-structure (U(A), (RA)R∈R,(α(F ))F ∈F). We can think of each α : F → N0as an assignment
to the constant symbols and each τ-structure (A|α) as an expansion of A with constant sym- bols F assigned according to α. For each α : F → N0let GE,α:= {A ∈ fin[τrel] : E(A|α) = 1}.
In order to avoid excessive subscripts we write QE,α,n,ar to denote the quantifier QGE,α,n,ar
for each n ∈ N. Let
QΩ := {QE,α,n,ar: n ∈ N, α : F → N0}.
and let
QE,α:= QGE,α = {QE,α,n,ar: n ∈ N, ar : S × [2] → N0}.
We call QΩ the set of quantifiers correpsonding to Ω. We note that when Ω is relational
α: F → N0 is the empty function and A = (A|α). In this case we omit α in the subscript.
the set of quantifiers correpsonding to Ω. Let Ω be a set of almost relational generalised
operators. Let
QΩ= [
Ω∈Ω
QΩ.
We call QΩthe set of quantifiers corresponding to Ω. In order to illustrate the idea behind this
definition we consider the counting operator as an example and construct the corresponding set of quantifiers. We see that the set of counting quantifiers corresponds to the counting operator.
Example 3.19. We recall that the counting operator ΩEcnt has the vocabulary τset = ({U}, ∅, {s}, ζ). The corresponding set of Boolean-valued generalised operators ΩEB
cnt has the vocabulary τB
set= ({U}, {t}, {s}, ζB), where t is some constant function symbol. From [21] it follows that FPC ≡ FPN(Ω
Ecnt) ≡ FPN(ΩEcnt,ar) where ar : {s} × [2] → N0 is defined such that ar(s, 1) = 1 and ar(s, 2) = 0. We construct the set of quantifiers corresponding to ΩEcnt,ar. Let α : {t} → N0 and let G := GEB
cnt,α. Since ar(s, 2) = 0 it follows that for all
n ∈ N we have Gn,ar= G and QEcntB ,α,n,ar= QG,n,ar= QG,ar1, where ar1 : {s} → N0 is defined
such that ar1(s) = 1. Recall that G is defined such that for any A ∈ fin[τset], A ∈ G if, and
only if, |UA| ≥ α(t). The Lindström quantifier ∃≥α(t) is also defined by G and so we can
identify QG,ar1 with ∃≥α(t). It follows that for all n ∈ N and α : {t} → N0 we can identify
QEB
cnt,α,n,ar with the counting quantifier ∃
≥α(t) and so QΩEcnt,ar= QΩEB cnt ,ar= {QEB cnt,α,n,ar: n ∈ N, α : {t} → N0} = {∃≥k : k ∈ N 0}.