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[MATH] is negative, then [MATH] , and if a point of [MATH] is positive, then [MATH] . If we do this then by checking the sign of an element of [MATH] we determine an element of [MATH] |
The construction of [MATH] proceeds as follows. We immediately remove from [MATH] the intervals [MATH] [MATH] and [MATH] . Then we activate the following inductive procedure. Suppose that at stage [MATH] it holds |
[MATH] for all [MATH] . We add then to [MATH] the points [MATH] and [MATH] . At the same time we remove from [MATH] the intervals [MATH] and [MATH] . Otherwise, let [MATH] be the first stage in which a digit different from 0 appears in |
[MATH] , say, [MATH] . We want then the closest point of [MATH] to 0 to be positive. To this aim, we add to [MATH] the point [MATH] alone. Moreover, we remove from [MATH] the intervals |
[MATH] [MATH] (notice that 0 was removed from [MATH] already before the start of the inductive procedure), and [MATH] . In this case the description of [MATH] is complete after stage [MATH] . The case |
[MATH] is analogous. Corollary 4.21 [MATH] Proof. By Propositions 4.19 and 4.20 For [MATH] we see instead that a precise characterisation is given by |
[MATH] Theorem 4.22 [MATH] for [MATH] Proof. We prove the statement for [MATH] by replacing [MATH] through its well known strongly Weihrauch equivalent version [MATH] (see BdBP12 , Corollary 4.6] ). The cases |
[MATH] are then as usual proved by transitivity of [MATH] Let then [MATH] be given, which means that we are provided with a sequence of rational open intervals [MATH] such that [MATH] . We now construct the new closed set [MATH] as the set of all points (with polar coordinates, as in the proofs of Theorem 4.8 and Propo... |
satisfying the following three conditions: (1) [MATH] (2) [MATH] (3) [MATH] Intuitively, we draw in [MATH] part of the circular crown between the circles of radius [MATH] and [MATH] centered at the origin. We then remove around a point [MATH] a little open portion of the crown as soon as we know that |
[MATH] (see Figure ). It is immediate to see that if [MATH] then [MATH] and [MATH] . Hence it remains to prove that we can compute a name of [MATH] |
To see that (a name for) [MATH] as an element of [MATH] is computable from (the given name for) [MATH] , observe that all the conditions (1)–(3) are |
[MATH] in [MATH] To see that also (a name for) [MATH] as an element of [MATH] is computable from (the given name for) [MATH] , observe that [MATH] is the closure of the set |
[EQUATION] We claim that we can enumerate, and even decide, this subset of [MATH] effectively from [MATH] . The conditions [MATH] and |
[MATH] are immediately decidable for rational numbers. Hence, to determine whether [MATH] it remains only to analyze the condition (3) in the definition of [MATH] . To this aim, for [MATH] , let then |
[MATH] be minimal such that [MATH] . Then, for [MATH] condition (3) is equivalent to [MATH] . Since [MATH] and [MATH] are rational numbers, we can find effectively such [MATH] and then decide whether [MATH] |
Therefore the set [MATH] is decidable and we can enumerate its members (as pairs of real numbers) for the positive information on [MATH] |
Corollary 4.23 [MATH] for [MATH] Proof. By Theorem 4.7 .(2) and Theorem 4.22 5. Approximate projections Since, as we have seen, the (exact) projection operators are computationally quite hard, it might be reasonable to consider some approximate versions of them. In many practical circumstances, we may indeed be content... |
Definition 5.1 Given a metric space [MATH] [MATH] , a point [MATH] and a nonempty set [MATH] we say that [MATH] is a [MATH] -projection point of [MATH] |
onto [MATH] if [MATH] . In other words, the [MATH] -projection points of [MATH] onto [MATH] are the points of [MATH] which are at minimal distance from [MATH] up to an error of [MATH] times the distance itself. |
Notice that if [MATH] then for any [MATH] [MATH] is the unique [MATH] -projection point of [MATH] onto [MATH] . In general, for any [MATH] |
[MATH] -projection points of [MATH] onto [MATH] exist unless [MATH] . As when dealing with exact projections, we will be interested in the case where [MATH] is closed; in this situation [MATH] -projection points of any [MATH] onto [MATH] do exist for any [MATH] . If [MATH] is a computable metric space, the multi-valued... |
Definition 5.2 Given a computable metric space [MATH] and [MATH] the [MATH] -approximate negative, positive and total closed projection operators on [MATH] are the partial multi-valued functions [MATH] [MATH] and |
[MATH] which associate to every [MATH] (with Cauchy representation) and every closed [MATH] (with negative, positive and total representation, respectively) the set of the [MATH] -projection points of [MATH] onto [MATH] |
Thus [MATH] [MATH] , and [MATH] The [MATH] -approximate negative, positive and total projections operators for compact sets are defined by replacing [MATH] [MATH] and [MATH] with [MATH] [MATH] , and [MATH] respectively. These are denoted [MATH] [MATH] , and [MATH] |
The first observations about the approximated operators partly mimic the ones we made for the exact operators. Fact 5.3 Let [MATH] be a computable metric space and [MATH] |
(4) If [MATH] is computably compact [MATH] [MATH] [MATH] (5) [MATH] [MATH] for [MATH] Proof. (1) and (2) are obvious. (3) and (4) can be proved exactly as Facts 4.4 and 4.5 |
respectively: indeed those proofs consist of transformations of the input and do not use any specific feature of the functions involved. |
(5) follows from the proofs of the analogous results in Theorem 4.6 , since the [MATH] -projection points of [MATH] onto the compact sets [MATH] and [MATH] constructed there are also [MATH] -projection points of [MATH] |
onto the original closed set [MATH] Notice that in (5) above [MATH] is missing – we show below in Proposition 5.5 that this does not hold. The proof of the analogous result in Theorem 4.6 |
cannot be translated to the approximate setting. Indeed if we repeat that construction then to obtain the [MATH] -projection points of [MATH] onto [MATH] we need to have a [MATH] -projection point of [MATH] onto [MATH] for |
[EQUATION] Hence no specific [MATH] will work for all [MATH] and [MATH] . Even viewing [MATH] as part of the input we do not solve the problem: from the negative information on [MATH] we obtain only lower bounds for [MATH] |
5.1. Approximated negative projection operators The following results characterizes the computational complexity of negative approximated projection operators on [MATH] for all [MATH] |
Theorem 5.4 For every [MATH] and [MATH] [MATH] Proof. For the right-to-left direction, observe that [MATH] by Fact 5.3 .(2). For the other direction, consider an input [MATH] . Since we can compute [MATH] , we denote by [MATH] the strict lower bound for [MATH] computed at stage |
[MATH] , so that [MATH] . We set [MATH] We now define the negative closed set [EQUATION] To see that we can compute a [MATH] -name of [MATH] observe that [MATH] is defined by a [MATH] -formula with [MATH] as a parameter. |
Intuitively, [MATH] is constituted by “copies” of different subsets of [MATH] translated onto different levels of the space [MATH] , so that (i) each copy lies at distance [MATH] from the adjacent copies, (ii) on the |
[MATH] -th level we remove the points of [MATH] that are “too far” from [MATH] according to the approximation of [MATH] that we know at that stage. Notice that the [MATH] -th level of [MATH] is nonempty if and only if [MATH] , and this happens for some [MATH] (for all [MATH] |
when [MATH] ) because [MATH] . Therefore [MATH] is a valid input for [MATH] If [MATH] , then [MATH] because [MATH] . We have shown |
[MATH] . Finally [MATH] by BdBP12 , Corollary 4.9] For [MATH] we only state the bounds given in the following proposition. We recall the use of the finite-parallelization operator that maps any given multi-valued [MATH] to [MATH] |
defined as [MATH] for all [MATH] Proposition 5.5 For every [MATH] and [MATH] [EQUATION] Proof. The first inequality follows from Fact 5.3 .(2) and the fact that [MATH] by BG11b , Theorem 8.5] |
We proceed to show that [MATH] Given an input [MATH] , we can use [MATH] to decide whether or not [MATH] . If yes, we can output [MATH] . If no, we can compute a lower bound [MATH] . Since we know [MATH] as a compact set, we can subsequently compute some [MATH] such that [MATH] . Consider the slices [MATH] , which we c... |
[MATH] That [MATH] is straightforward, e.g. via the independent choice theorem implying the closure under compositional product of the class of non deterministic functions with finitely many mind changes BdBP12 , Theorem 7.6] ). To see that this is strict, we observe that |
[MATH] . Since the degree of [MATH] admits a single-valued representative (for example unique choice BdBP12 ), the closed choice elimination theorem (as stated in LRP15 , Theorem 2.1] implies that if [MATH] , then [MATH] . That this is impossible can be seen using Hertling’s level |
Her96 , which is an ordinal invariant of Weihrauch degrees defined as follows: Given a function [MATH] , let [MATH] , let [MATH] be the closure of the set of discontinuity points of [MATH] , and for limit ordinal [MATH] , let [MATH] The level of [MATH] is the least [MATH] such that [MATH] (if this ever happens). The le... |
Notice that our proof shows in fact that [EQUATION] and hence [MATH] , for every computable metric space [MATH] 5.2. Approximated positive projection operators |
Theorem 5.6 For every [MATH] and [MATH] [MATH] Proof. By Proposition 3.1 it suffices to show that [MATH] and by Fact 5.3 .1 we can assume that |
[MATH] is computable. Given [MATH] let [EQUATION] This is a [MATH] -condition, hence [MATH] is computable from [MATH] as a nonempty member of [MATH] |
Let now [MATH] and notice that: (1) if [MATH] then by the definition of [MATH] we have [MATH] (2) if [MATH] then [MATH] and inductively we can prove that for every |
[MATH] there exists [MATH] such that [MATH] which implies [MATH] ; hence [MATH] We need to show that from [MATH] and the original input |
[MATH] we can compute an effective Cauchy sequence [MATH] converging to a point [MATH] . To compute [MATH] set [MATH] and start a recursive procedure which will stop after finitely many steps. Given [MATH] |
use (the name of) [MATH] to check whether there exists [MATH] such that [MATH] ; in this case we stop the recursion. If instead [MATH] |
for every [MATH] it follows that [MATH] and hence there exists [MATH] such that [MATH] Since this is a [MATH] property, we can search for such a [MATH] |
until we find one. The recursion will stop when either we find [MATH] such that [MATH] or we see that [MATH] (if the first alternative never occurs, such a [MATH] |
exists since [MATH] because [MATH] ). In the first case let [MATH] while in the second case let [MATH] It is clear from the construction that if [MATH] and [MATH] |
we have [MATH] for every [MATH] . This implies that if for some [MATH] we have [MATH] with [MATH] then the sequence [MATH] converges to [MATH] , which belongs to [MATH] by (1) above. If instead for every [MATH] the first possibility never occurs it means that [MATH] , so that by (2) |
[MATH] , and indeed [MATH] converges to [MATH] However, the convergence of [MATH] does not suffice, and we need to check that we actually defined an effective Cauchy sequence. For this it suffices to show that [MATH] for every [MATH] . This is obvious if |
[MATH] [MATH] and [MATH] . Now assume that neither [MATH] nor [MATH] have been defined using [MATH] . In other words, [MATH] and [MATH] where [MATH] and [MATH] . Then |
[EQUATION] The last possibility (by the observation above) is that [MATH] with [MATH] and [MATH] with [MATH] . In this case notice that, since [MATH] , we have |
[MATH] . Then [EQUATION] Theorem 5.7 For every [MATH] and [MATH] [MATH] Proof. By Proposition 3.1 it suffices to show that [MATH] |
As usual, it suffices to show the reduction for [MATH] . Fix [MATH] such that [MATH] and notice that [MATH] . Given [MATH] closed and nonempty in [MATH] , with |
[MATH] rational open balls in [MATH] , we compute a sequence [MATH] in [MATH] by setting [MATH] for the least [MATH] such that [MATH] if such a [MATH] exists, and otherwise setting [MATH] |
If [MATH] , then [MATH] for every [MATH] , which implies [MATH] . Hence [MATH] If instead [MATH] then [MATH] is the least natural number such that [MATH] and [MATH] is a discrete subset of the closed interval [MATH] . In fact, for all [MATH] |
[MATH] for some [MATH] and, for [MATH] sufficiently large, [MATH] . This implies that [MATH] Moreover, if [MATH] then [MATH] for some [MATH] and hence |
[EQUATION] and thus [MATH] . We thus showed that [MATH] We have proved that [MATH] is a singleton and we now show that its unique element [MATH] can be used to compute |
[MATH] . Given then such [MATH] , we produce the [MATH] -name [MATH] of [MATH] in the following way. For each [MATH] we check whether [MATH] or not. Notice that this test is decidable because [MATH] belongs to [MATH] and [MATH] is not an accumulation point of this set. If the answer is positive, we put [MATH] , otherwi... |
Corollary 5.8 For every [MATH] and [MATH] [MATH] Proof. By Theorem 5.6 and Theorem 5.7 Corollary 5.9 For every [MATH] and [MATH] [MATH] |
Proof. By Proposition 3.5 , Theorem 5.4 and Corollary 5.8 5.3. Total approximated projection operators Our classification of projection operators ends finally with a computable version of projection, that can be therefore used in concrete applications. |
Theorem 5.10 For every [MATH] and [MATH] [MATH] is computable. Proof. By Fact 5.3 .1 we can assume that [MATH] is computable. We give an algorithm to determine some [MATH] for every |
[MATH] with [MATH] We know already that total information on [MATH] allows us to compute [MATH] and then [MATH] . We construct by induction an approximate projection point of [MATH] onto [MATH] as follows. |
At stage [MATH] we check whether [EQUATION] Notice that at least one of these two conditions holds, and we stop when we verify one of them. If ( ) is verified before ( ii ), we let |
[MATH] , and move to step [MATH] . If instead ( ii ) is verified before ( ), we inspect the dense sequence in [MATH] searching for some |
[MATH] such that [MATH] (a suitable [MATH] always exists in this case) and then let [MATH] for all [MATH] We now show that the algorithm works. First we need to check that [MATH] |
is an effective Cauchy sequence converging to some [MATH] , and to this end it suffices to check that [MATH] for all [MATH] . This is trivial if at stage [MATH] and as well at stage [MATH] the condition ( is satisfied first, or alternatively if ( ii ) has been verified at some stage [MATH] . The interesting case is the... |
[EQUATION] We then need to check that [MATH] . If ( ) has always been verified, then [MATH] , since [MATH] . If at stage [MATH] ii ) is verified, then [MATH] where [MATH] was picked so that |
[MATH] 6. An application: the Whitney Extension Theorem Projection points are often used in mathematics. An example is the Whitney Extension Theorem, originally proved in Whi34 , and dealing with differentiable functions in [MATH] . This theorem considers a real-valued continuous function [MATH] defined on a closed [MA... |
is closed, we cannot even attempt to compute the partial derivatives of [MATH] at many boundary points of [MATH] . However we can have also a set of continuous functions (the pseudo-derivatives of [MATH] ) defined also on [MATH] which satisfy Taylor’s formulas and hence behave like the partial derivatives of degree |
[MATH] of [MATH] [MATH] and this set of functions are collectively called a jet ). The Whitney Extension Theorem asserts that under these hypotheses |
[MATH] can be extended to some [MATH] , so that [MATH] and its partial derivatives extend the elements of the jet. A classical proof of the Whitney Extension Theorem is contained in Ste70 , Chapter VI] , and we follow Stein’s proof to provide a computable version. Starting with the closed set [MATH] , Stein defines a f... |
extension of [MATH] by [EQUATION] (Notice that, for a given [MATH] and after we fix [MATH] [MATH] and [MATH] , in fact we obtain a linear operator from the space of jets to [MATH] .) |
If [MATH] is given with total information, variations of [MATH] and [MATH] can be computed. Thus at first sight the only essentially non-computable step (by Proposition 4.20 and Theorem |
4.22 ) in Stein’s proof is the choice of [MATH] . To overcome this obstacle [MATH] can be replaced with some other point of [MATH] |
which is close enough to [MATH] , and “close enough” depends only from the size of [MATH] . This suggests that the multi-valued functions naturally associated to the Whitney Extension Theorem are actually computable without resorting to any projections. However there is another, subtler, point that needs to be taken in... |
given above the case distinction is not computable. Thus we need to provide an effective way, given [MATH] and [MATH] , to compute [MATH] without knowing whether [MATH] . Here the projection operators, which are defined over |
[MATH] , come back into the picture and appear to be essential: when we do not know positively that [MATH] they are used to compute [MATH] in a way that is compatible with both cases. Only by showing that approximate projections are indeed sufficient it is possible to find a computable version of the Whitney Extension ... |
Summing up, assuming [MATH] is represented with total information and using Theorem 5.10 , we show that the multi-valued function associated to the Whitney Extension Theorem is computable. As mentioned in the introduction, full details of this result will be included in a forthcoming paper ( GM ). |
# Source: arxiv 1805.12079 # Title: Higher-order CPM Constructions # Sections: all # Downloaded: 2026-03-03T02:41:17.064860+00:00 |
Higher-order CPM Constructions Abstract We define a higher-order generalisation of the CPM construction based on arbitrary finite abelian group symmetries of symmetric monoidal categories. We show that our new construction is functorial, and that its closure under iteration can be characterised by seeing the constructi... |
Introduction The CPM Construction is of cardinal importance to the categorical study of quantum theory , where it provides the canonical model of mixed-state quantum behaviour. It has been extensively studied, both axiomatically |
and concretely . Recently, applications of the CPM construction in the context of compositional distributional models of meaning |
have prompted renewed interest on iterated CPM constructions , with the discovery of new features due to their additional degrees of freedom |
In this work, we define a theory of higher-order CPM constructions, which we characterise as Eilenberg-Moore algebras for a certain monad. We connect to the recent work on iterated CPM constructions |
. We provide a very broad family of examples obtained from categories of free finite-dimensional modules over commutative semirings |
, showing that they can all be understood within the framework of categorical probabilistic theories The traditional CPM construction |
In the traditional formulation of and subsequent work, the CPM construction can be understood in terms of two separate steps: doubling and discarding . By doubling , we mean the passage from a dagger-compact category [MATH] to the corresponding doubled category [MATH] . By discarding , we mean the introduction of an en... |
2.1 Doubling The doubling step of the traditional CPM construction can be understood as the passage from [MATH] to the sub-category [MATH] of [MATH] obtained as the image of the following doubling functor: |
[EQUATION] This work uses a conjugation functor [MATH] which is both strict monoidal—i.e. [MATH] —and involutive—i.e. [MATH] . The [MATH] functor above respects the dagger, but if [MATH] is equipped with the tensor product inherited from [MATH] then the functor is not strict monoidal: |
[EQUATION] The functor [MATH] becomes strict monoidal if we equip the image with a different tensor product—analogous to the one of |
on morphisms—which respects the structure of objects/morphisms: [EQUATION] From this moment forward, when saying “monoidal” we will always mean “strict monoidal”. Under this new tensor product, [MATH] is a dagger SMC, and the doubling functor is dagger monoidal . |
2.2 Discarding The discarding step of the traditional CPM construction can be understood as the introduction of a family of effects [MATH] in [MATH] (the discarding maps ) which respect the tensor product of [MATH] in the following sense: |
[EQUATION] where [MATH] are the symmetry isomorphisms of [MATH] . Traditionally, the chosen family is [MATH] , where by [MATH] and [MATH] we will denote the cups and caps for the compact closed structure of [MATH] |
Given the above, the CPM category [MATH] can be defined as the smallest sub-category of [MATH] containing the doubled category [MATH] and all the effects [MATH] |
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