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Complete Axioms for Categorical Fixed-Point Operators
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Simpson, A & Plotkin, G 2000, Complete Axioms for Categorical Fixed-Point Operators. in Proceedings of
the 15th Annual IEEE Symposium on Logic in Computer Science. Institute of Electrical and Electronics
Engineers (IEEE), Washington, DC, USA, pp. 30-41. https://doi.org/10.1109/LICS.2000.855753
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Complete Axioms for Categorical Fixed-point Operators
Alex Simpson
∗
and Gordon Plotkin
†
LFCS, Division of Informatics, University of Edinburgh,
JCMB, King’s Buildings, Edinburgh, EH9 3JZ
{als,gdp
}
@dcs.ed.ac.uk
Abstract
We give an axiomatic treatment of fixed-point operators
in categories. A notion of iteration operator is defined, em-
bodying the equational properties of iteration theories. We
prove a general completeness theorem for iteration opera-
tors, relying on a new, purely syntactic characterisation of
the free iteration theory.
We then show how iteration operators arise in axiomatic
domain theory. One result derives them from the existence
of sufficiently many bifree algebras (exploiting the universal
property Freyd introduced in his notion of algebraic com-
pactness). Another result shows that, in the presence of a
parameterized natural numbers object and an equational
lifting monad, any uniform fixed-point operator is neces-
sarily an iteration operator.
1. Introduction
Fixed points play a central r
ˆ
ole in domain theory. Tra-
ditionally, one works with a category such as
Cppo, the
category of
ω-continuous functions between ω
-complete
pointed partial orders. This possesses a least-fixed-point
operator, whose properties are well understood. For exam-
ple, a theorem of Beki
˘
c states that least simultaneous fixed
points can be found in sequence by a form of Gaussian elim-
ination, see e.g. [33]. More generally, the equational theory
between fixed-point terms (
µ
-terms), induced by the least-
fixed-point operator, has been axiomatized as the free
iter-
ation theory of Bloom and
´
Esik [3]. (This theory is known
to be decidable.) Also, Eilenberg [6] and Plotkin [25] gave
an order-free characterisation of the least-fixed-point opera-
tor as the unique fixed-point operator satisfying a condition
known as uniformity, expressed with respect to the subcate-
gory Cppo
⊥
of
strict maps in
Cppo
, see e.g. [15].
Nowadays, one appreciates that
Cppo is one of many
possible categories of “domain-like” structures, each with
∗
Research supported by EPSRC grant GR/K06109.
†
Research supported by EPSRC grant GR/M56333.
an associated fixed-point operator. Not only are there many
familiar order-theoretic variations on the notions of com-
plete partial order and continuous function, but there are
also many categories of “domains” based on somewhat dif-
ferent principles — for example, categories of games and
strategies [21], realizability-based categories [20] and cate-
gories of abstract geometric structures [12]. Thus one needs
generally applicable methods for establishing properties of
the associated fixed-point operators.
In this paper, we analyse the equational properties of
fixed-point operators in arbitrary categories of “domain-
like” structures. In Section 2, we consider the basic notions
of
(parameterized) fixed-point operator,
Conway operator
and iteration operator, developed from analogous notions
in Bloom and
´
Esik’s study of iteration theories [3]. Our defi-
nitions are straightforward adaptations of Bloom and
´
Esik’s
to the general setting of a category with finite products. In
particular, the notion of
iteration operator
is intended to
capture all desirable equational properties of a fixed-point
operator, as exemplified by the many completeness results
for the free iteration theory in [3].
As in the case of the fixed-point operator on
Cppo
, we
also consider a notion of
(parameterized) uniformity
for
(parameterized) fixed-point operators. We define this in
general assuming a suitable functor J : S → D from a
category
S of “strict” maps. In practice, (parameterized)
uniformity serves two purposes. First, it is often satisfied
by a unique (parameterized) fixed-point operator, and so
characterises that operator. Second, any parametrically uni-
form Conway operator is an iteration operator, so parame-
terized uniformity is a convenient tool for establishing that
the equations of an iteration operator are satisfied.
In Section 3, we examine the equational theory of itera-
tion operators. We use a syntax of multisorted fixed-point
terms (
µ
-terms), which can be interpreted in any category
with an iteration operator. In any such category, Bloom
and
´
Esik’s axioms for iteration theories [3] are sound.
Bloom and
´
Esik provide numerous completeness theorems,
demonstrating that the iteration theory axioms are also com-
plete for deriving the valid equations in many familiar cat-
Simpson, A., & Plotkin, G. (2000). Complete Axioms for Categorical Fixed-Point Operators. In
Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science. (pp. 30-41).
Washington, DC, USA: IEEE Computer Society. doi: 10.1109/LICS.2000.855753
egories with iteration operators. The first main contribu-
tion of this paper is a precise characterisation of the circum-
stances in which the iteration theory axioms are complete
(Theorem 1). This result accounts for all the examples in
[3]. It shows that, in non-degenerate categories, the sound-
ness of the iteration theory axioms implies their complete-
ness. This explains the ubiquity of completeness results for
the free iteration theory.
Our completeness theorem follows from a new, purely
syntactic characterisation of the free iteration theory as
a maximal theory satisfying two properties:
closed con-
sistency
and typical ambiguity
(Theorem 2). This result,
which is of interest in its own right, was inspired by Stat-
man’s characterisation of βη-equality in the simply-typed
λ-calculus [31].
The remainder of the paper is devoted to providing con-
ditions for establishing the existence (and uniqueness) of
parametrically uniform Conway operators (hence iteration
operators). In one common setting, which arises in ax-
iomatic domain theory [13, 10, 12], one has that the cate-
gory
D
of “domains” is obtained as the co-Kleisli category
of a comonad on the category of strict maps
S. (For exam-
ple,
Cppo
is the co-Kleisli category of the lifting comonad
on Cppo
⊥
.) In axiomatic domain theory, S
satisfies a cu-
rious property, first identified by Freyd [13, 14]: a wide
class of endofunctors on S have initial algebras whose in-
verses are final coalgebras (in Freyd’s terminology, S is
algebraically compact). Following [7], we call such ini-
tial/final algebras/coalgebras
bifree algebras
. (In the exam-
ple of
Cppo
⊥
, every
Cppo
-enriched endofunctor has a
bifree algebra [10].)
In Section 5, we give a quick overview of initial algebras,
final coalgebras and bifree algebras, including a couple of
minor new propositions. Then, in Section 6, we show how
bifree algebras in
S can induce properties of fixed-point op-
erators in
D. This programme was begun by Freyd and oth-
ers [13, 5, 24, 28]. A further step was taken by Moggi, who,
in unpublished work, gave a direct verification of the Beki
˘
c
equality. Here, we give the complete story, showing how the
presence of sufficiently many bifree algebras determines a
unique parametrically uniform Conway operator (hence it-
eration operator).
In Section 7 we show how the Conway operator iden-
tities can be established without assuming the existence of
the bifree algebras used in Section 6. This is possible when
the category S
of “strict” maps arises as the category of al-
gebras for a “lifting monad” on a suitable category of “pre-
domains” C
. (For example, Cpp o
⊥
is the category of alge-
bras for the usual lifting monad on the category Cpo of, not
necessarily pointed, ω-complete partial orders.) Axiomati-
cally, we assume that
C
is a category with finite products, a
monad embodying the equational properties of partial map
classifiers (an
equational lifting monad
[4]), partial func-
tion spaces (
Kleisli exponentials [22, 28]), and a (parameter-
ized) natural numbers object. These conditions are always
satisfied by the categories of predomains that arise in ax-
iomatic and synthetic domain theory [10, 12, 20, 11, 26, 30].
Theorem 4 states that such categories support at most one
uniform recursion operator (a T -fixed-point operator), and
moreover it determines a unique parametrically uniform
Conway operator on the associated category of domains.
Thus, in the presence of a lifting monad and a parameter-
ized natural numbers object, uniformity alone implies all
equational properties of fixed points.
2. Fixed-point operators
In this section we give an overview of the various no-
tions of fixed-point operator we shall be concerned with.
We work with a category,
D
, with distinguished finite prod-
ucts, to be thought of as a category of “domains”. We write
1 for the terminal object.
Definition 2.1 (Fixed-point operator)
A
fixed-point oper-
ator is a family of functions (·
)
∗
:
D(
A, A
) → D
(1
, A
)
such that, for any
f
:
A
-
A, f ◦ f
∗
= f
∗
.
Definition 2.2 (Parameterized fixed-pt. op.) A
parame-
terized fixed-point operator is a family of functions
(
·
)
†
:
D(
X ×
A, A
)
→ D(
X, A
) satisfying:
1. (Naturality.)
For any
g
:
X
-
Y and
f
:
Y
×
A
-
A,
f
†
◦ g = (
f ◦
(g ×
id
A
))
†
:
X
-
A.
2. (Parameterized fixed-point property.)
For any
f
: X
× A
-
A,
f ◦ hid
X
, f
†
i =
f
†
:
X
-
A.
Observe that a parameterized fixed-point operator corre-
sponds to a family of fixed-point operators (·
)
∗
X
in the co-
Kleisli categories
D
X
×
(
−
)
of X
×
(
−) comonads. In this
formulation, naturality states that, for any g : X
-
Y ,
the induced functor H
g
: D
Y
×
(
−)
→ D
X×(−
)
preserves
the fixed-point operators in the sense that, for any endomor-
phism in D
Y
×(
−
)
, given by a morphism
f : Y
× A
-
A
in D, it holds that H
g
(
f
∗
Y
) = (
H
g
f)
∗
X
.
In practice, well-behaved fixed-point operators satisfy
many other equations that do not follow from the fixed-point
property alone.
Definition 2.3 (Dinaturality) A fixed-point operator is
said to be
dinatural
if, for every f
:
A
-
B
and
g
:
B
-
A
, it holds that
(f ◦ g)
∗
= f ◦
(g
◦
f
)
∗
.
Definition 2.4 (Conway operator) A
Conway operator is
a parameterized fixed-point operator that, in addition, satis-
fies:
3. (Parameterized dinaturality.)
For any f
: X
× B
-
A and g : X
×
A
-
B,
f
◦ hid
X
, (
g
◦ hπ
1
, f
i
)
†
i
= (
f ◦ hπ
1
, gi)
†
:
X
-
A.
4. (Diagonal property.)
For any f
: X ×
A
× A
-
A
,
(
f
◦ (id
X
× ∆))
†
=
(
f
†
)
†
:
X
-
A
(where
∆ : A
-
A
×
A is the
diagonal map).
It is easily seen that (parameterized) dinaturality implies the
(parameterized) fixed-point property, so 2 of Definition 2.2
is redundant in the axiomatization of Conway operators.
The reason for singling out dinaturality is that it is a con-
cept that makes sense for unparameterized fixed-point op-
erators. It is also a powerful property. In special circum-
stances, it alone characterises a unique well-behaved fixed-
point operator [29].
Mainly, however, we shall be interested in well behaved
parameterized fixed-point operators, and the notion of Con-
way operator is appropriate. Conway operators are so
named because their axioms correspond to those of the
Con-
way theories
of Bloom and
´
Esik [3]. They have also arisen
independently in work of M. Hasegawa [16] and Hyland,
who established a connection with Joyal, Street and Verity’s
notion of trace [18]. The definition of trace makes sense
in any braided monoidal category. Hasegawa and Hyland
showed that, in the special case that the monoidal product
is cartesian, traces are in one-to-one correspondence with
Conway operators.
There are many alternative axiomatizations for Conway
operators. The axioms for a trace provide one possibility.
Other options are discussed in [3, 16]. The following im-
portant property often appears in variant axiomatizations.
Proposition 2.5 (Beki
˘
c property) For any Conway opera-
tor, given
h
f, g
i
: X ×
A
×
B
-
A ×
B, it holds that
hf, g
i
†
=
hh, (g
◦
(hid
X
, h
i ×
id
B
))
†
i
:
X
-
A ×
B,
where
h
= (
f ◦ h
id
X×A
, g
†
i
)
†
: X
-
A.
In spite of such consequences, there are basic equalities
that Conway operators do not necessarily satisfy; for exam-
ple, it is not true in general that f
∗
= (
f ◦
f
)
∗
for an en-
domorphism f
. The
commutative identities
of Bloom and
´
Esik [3] ensure that such “missing” equalities do hold.
Definition 2.6 (Iteration operator)
An
iteration operator
is a Conway operator that, in addition, satisfies:
5. (Commutative identities.)
Given
f
: X
× A
m
-
A and morphisms
ρ
1
, . . . , ρ
m
: A
m
-
A
m
such that each ρ
i
=
h
p
i1
, . . . , p
im
i is a tuple of projections (i.e. each p
ij
is
one of the
m projections
π
1
, . . . , π
m
:
A
m
-
A),
it holds that
hf
◦ (id
X
×
ρ
1
)
, . . . , f ◦ (
id
X
×
ρ
m
)i
†
=
∆
m
◦ (f ◦ (id
X
×
∆
m
))
†
: X
-
A
m
(Here
A
m
is the
m-fold power
A
×
. . . × A
, and
∆
m
: A
-
A
m
is the diagonal.)
1
The complex formulation of the commutative identities
means that they can be hard to establish in practice. One
way of reducing the complexity is to look for simpler
equational axiomatizations. For example,
´
Esik [9] has re-
cently proved that it suffices to consider certain instances
of the commutative identities generated, in an appropriate
sense, by finite groups. However, in many situations, it is
more convenient to derive the commutative identities from
(more easily established) non-equational properties that im-
ply them. Many examples of such properties are given by
Bloom and
´
Esik [3]. In this paper we shall be concerned
with one such property:
(parameterized) uniformity
.
To define (parameterized) uniformity, we suppose given
a category
S
with finite products and the same objects as D,
together with a functor J
:
S → D
that strictly preserves
finite products and is the identity on objects. We use the
symbol
◦
for maps in
S
, and we call morphisms in D
in the image of
J
strict
. (We really just need the subcategory
of D
consisting of strict maps, but the functorial formulation
will be helpful in Section 6. Observe that morphisms given
purely by the finite product structure on
D are strict.)
Definition 2.7 (Uniformity)
A fixed-point operator is said
to be uniform (with respect to J
) if, for any f :
A
-
A
,
g
: B
-
B
and h : A
◦ B
, Jh
◦
f =
g ◦ Jh implies
g
∗
= Jh ◦ f
∗
.
Definition 2.8 (Parameterized uniformity)
A parameter-
ized fixed-point operator is said to be parametrically uni-
form if, for any f
:
X
×
A
-
A
,
g : X × B
-
B
and h : A ◦ B
,
Jh
◦ f
= g
◦
(
id
X
×
Jh
) implies
g
†
= Jh ◦ f
†
.
Observe that parameterized uniformity is just the state-
ment that the fixed-point operator
(
·)
∗
X
in each co-Kleisli
category
D
X
×(−)
is uniform with respect to the composite
functor
S → D → D
X×(−
)
. Hasegawa gives an interesting
reformulation of parameterized uniformity directly in terms
of a trace [16]. If S is defined to be the subcategory of
morphisms given purely by the finite product structure on
D, then parameterized uniformity is exactly the functorial
dagger implication for base morphisms of [3].
Proposition 2.9 Any parametrically uniform Conway op-
erator is an iteration operator.
The proof is an easy application of the strictness of all diag-
onals ∆
m
: A
-
A
m
.
The converse to proposition 2.9 does not hold in general,
see [8].
1
Strictly speaking, we consider only instances of the commutative iden-
tities of [3] in which their “surjective base morphism”
ρ is a diagonal ∆
m
.
The general commutative identities of [3] follow from such instances, us-
ing properties of Conway operators.
3. Completeness
In this section we introduce Bloom and
´
Esik’s
iteration
theories [3], using a syntax of multisorted fixed-point terms
(µ-terms). We prove a very general completeness theorem
(Theorem 1) for the free iteration theory relative to interpre-
tations in categories with iteration operators. The complete-
ness theorem follows from a new syntactic characterisation
of the free iteration theory (Theorem 2).
We assume given a nonempty collection of base types
(or sorts), over which α, β, . . .
range. Types
σ, τ, . . . are
either base types or product types
σ
1
×
. . . × σ
n
(for
n ≥
0
). We use
σ
n
as an abbreviation for the n
-fold
power σ × . . .
×
σ. We assume also a signature given
by a set Σ
of function symbols, each with an associated
typing information of the form
(
α
1
, . . . , α
n
;
β) (there is no
loss of generality in considering only base types here). We
loosely refer to both (
α
1
, . . . , α
n
;
β) and
n
as the arity
of
the function symbol. Constants are considered as func-
tion symbols with arity 0. We assume a countably infi-
nite set of variable symbols
x, y, . . .
A variable is a pair,
written x
σ
, consisting of a variable symbol and a type (we
omit the type superscript when convenient). Terms and
their types are given by: each variable
x
σ
is a term of
type σ; if
t
1
, . . . , t
n
are terms of (base) types
α
1
, . . . , α
n
and
f
is a function symbol of arity (α
1
, . . . , α
n
; β
) then
f(t
1
, . . . , t
n
) is a term of (base) type β
; if
t
1
, . . . , t
n
are
terms of types σ
1
, . . . , σ
n
then ht
1
, . . . , t
n
i is a term of type
σ
1
× . . . × σ
n
; if t
is a term of type
σ
1
× . . . ×
σ
n
then
π
i
t, where
1
≤
i
≤ n, is a term of type
σ
i
; if t is a term
of type σ
then µx
σ
. t
is a term of type σ. As usual, the
variable
x is bound by µ
in µx. t. We identify terms up to
α-equivalence, writing t
≡
t
0
for the identity of terms. We
write
t(x
σ
1
1
, . . . , x
σ
k
k
) : τ for a term of type
τ
all of whose
free variables are contained in x
σ
1
1
, . . . , x
σ
k
k
. We call a term
with no free variables closed
. We write the substitution of
n
terms
t
1
, . . . , t
n
for n distinct free variables x
1
, . . . , x
n
(of
the correct types) in a term t as
t[
t
1
, . . . , t
n
/ x
1
, . . . , x
n
]
.
Given t (~y, x
σ
1
1
, . . . , x
σ
n
n
) :
σ
1
× . . . ×
σ
n
, we use the con-
venient notation µh
x
σ
1
1
, . . . , x
σ
n
n
i. t to represent the term
µx
σ
1
×
...×σ
n
. t
[
π
1
x, . . . , π
n
x / x
1
, . . . , x
n
]
.
A theory
,
T , is a typed congruence relation on terms that:
contains the product equations, i.e. T ` π
i
ht
1
. . . , t
k
i
=
t
i
and T ` t = hπ
1
t, . . . , π
k
t
i (for
t
:
σ
1
×
. . .
×
σ
k
); and is
closed under substitution (i.e. if T `
t = t
0
and
s
: σ
then
T ` t
[s/x
σ
] = t
0
[
s/x
σ
]
). For any theory, T ` t
= t
0
if and
only if T ` (t = t
0
)[h
x
1
σ
1
, . . . , x
n
σ
n
i/x
σ
1
×
...×
σ
n
]
where
x
1
, . . . , x
n
are fresh variables. Thus a theory is determined
by its equations between terms whose only free variables
are of base type. We say that T
is
consistent if there are two
terms
t, t
0
of the same type such that
T 6`
t = t
0
. We say
that T is closed-consistent
if there are two such terms that
are closed.
We now axiomatize Conway theories, in which µ corre-
sponds to a Conway operator, and iteration theories
, identi-
fying the equational properties of an iteration operator.
Definition 3.1 (Conway theory) A theory
T is said to be a
Conway theory if it satisfies two axioms:
1. (Dinaturality.)
For any t(~z , y
τ
) :
σ
and t
0
(~z, x
σ
) :
τ,
T `
µx. t
[
t
0
/y
] =
t
[
µy. t
0
[
t/x]
/y
] : σ.
2. (Diagonal property.)
For t(~z, x
σ
, y
σ
) :
σ
, T ` µx. t[x/y
] =
µx. µy. t
: σ.
These axioms are just the multisorted version of the axiom-
atization given by Corollary 6.2.5 of [3], where dinaturality
and the diagonal property are called the
composition iden-
tity and the
double dagger identity respectively.
Definition 3.2 (Iteration theory)
We say that a Conway
theory T is an iteration theory if it satisfies the following
axiom schema.
3. (Amalgamation.)
For any terms
t
1
, . . . , t
n
(
~z, x
σ
1
, . . . , x
σ
n
) : σ
and
s(
~z, y
σ
) : σ, suppose t
i
[y, . . . , y / x
1
, . . . , x
n
]
≡ s
,
for all i
with
1 ≤ i ≤
n
, then it follows that
T `
µ
h
x
1
, . . . , x
n
i.
ht
1
, . . . , t
n
i
=
h
µy. s, . . . , µy. si
: σ
n
.
Amalgamation is very close to the commutative identities
of [3] (as in Definition 2.6). An alternative formulation is
employed in [17], whose alphabetic identification identity
is equivalent to amalgamation.
We write F for the smallest Conway theory (generated
by the given base types and signature), and I for the small-
est iteration theory. As is shown in [3], I
completely cap-
tures the valid identities in a wide class of models, includ-
ing
Cppo. We write I
∗
for the smallest iteration theory
in which all closed terms (with identical types) are equated.
Although I
∗
is not a closed-consistent theory, it is nonethe-
less consistent. In fact,
I
∗
exactly captures the valid identi-
ties in
Cppo
⊥
. Our aim in this section is to prove a general
completeness theorem, accounting for all such complete-
ness results for I and
I
∗
.
First, we consider the interpretation of µ-terms in any
category D with finite products and a family of functions
(·)
†
: D(
X ×
A, A)
→ D
(
X, A
) satisfying the naturality
property of Definition 2.2. An interpretation is determined
by a function
[[
·]]
from base types to objects of D, together
with a function mapping each function symbol f
, of arity
(α
1
, . . . , α
k
; β)
, to a map
[[f
]] : [[
α
1
]]×. . .
×
[[
α
k
]]
-
[[β
]]
.
The first function extends (using products) to one from ar-
bitrary types to objects of
D
, and the second extends to
one mapping any term t(
x
σ
1
1
, . . . , x
σ
k
k
) : τ
to a morphism