[

# [

[ Dipartimento di Matematica “T. Levi Civita”. Univ. di Padova. Via Trieste, 63. Padova. Italy
###### Abstract

We give here a constructive account of the frequentist approach to probability, by means of natural density. Using this notion of natural density, we introduce some probabilistic versions of the Limited Principle of Omniscience. Finally we give an attempt general definition of probability structure which is pointfree and takes into account abstractely the process of probability assignment.

Natural density and probability, constructively]Natural density and probability, constructively.

S. Maschio]Samuele Maschio

## 1 Introduction

Natural density ([niven1951]) provides a notion of size for subsets of natural numbers. Classically the density of is defined as

 δ(A)=limn→∞|A∩{1,...,n}|n

provided this limit exists. Since classically subsets of are in bijective correspondence with sequences of s and s, this notion of density still works for such sequences and can provide a constructive account of frequentist probability. Here we study such a notion of frequentist probability in a constructive framework and we introduce some weak probabilistic forms of the limited principles of omniscience. Our treatment is informal à la Bishop, but it can be formalized in the extensional level of the Minimalist Foundation (see [mtt], [m09]) with the addition of the axiom of unique choice. Finally we will propose a more general pointfree notion of probabilistic structure inspired by the properties which hold in our constructive treatment of frequentist probability and in other classical situations related to Kolmogorov’s axiomatic probability and to fuzzy sets.

## 2 A constructive account of natural density

### 2.1 Potential events

A potential event is a sequence of s and s.

 e(n)∈{0,1}[n∈N+]

Potential events form a set with extensional equality, that is two potential events and are equal if . This set, which we denote with , can be endowed with a structure of boolean algebra as follows:

1. the bottom is ;

2. the top is ;

3. the conjunction of and in is ;

4. the disjunction of of and in is ;

5. the negation of is ;

6. for every , if and only if .

Potential events can be understood as sequences of outcomes (“success” or “failure”) in a sequence of iterated trials.

For every potential event , one can define a sequence of rational numbers

 Φ(e)(n)∈Q[n∈N+]

taking to be . This sequence is called the sequence of rates of success (or frequencies) of the potential event .

### 2.2 Actual events

Actual events are those potential events for which the sequence of rates of success can be shown constructively to be convergent. {defi} An actual event is a pair where is a potential event and is a strictly increasing sequence of natural numbers such that

We denote with the set of actual events and two actual events and are equal, , if .

Let us now recall the definition of Bishop real numbers from [BB85].

{defi}

A Bishop real is a sequence of rational numbers such that

 |x(n)−x(m)|≤1n+1m[n∈N+, m∈N+]

Two Bishop reals and are equal if . The set of Bishop reals is denoted with .

We now show that one can well define a notion of probability on actual events. {prop} If is an actual event, then is a Bishop real number. {proof} Suppose . Then and hence

 |Φ(e∘γ)(n)−Φ(e∘γ)(m)|=|Φ(e)(γ(n))−Φ(e)(γ(m))|≤1m<1n+1m.

This implies that is a Bishop real.

{prop}

If and are equal actual events, then and are equal Bishop reals.

{proof}

Suppose and are equal actual events. Since for every , or and , then

 ∣∣(Φ(e)∘γ)(n)−(Φ(e′)∘γ′)(n)∣∣=∣∣Φ(e)(γ(n))−Φ(e)(γ′(n))∣∣≤1n<2n[n∈N+].

From this it follows that and are equal Bishop reals.

As a consequence of the previous two propositions we can give the following {defi} The function is defined as follows: if is an actual event, then

 P(e,α):=Φ(e)∘γ.

### 2.3 Properties of actual events

We show here some basic properties of . However, before proceeding, we need a trivial but useful result: {lem} If is an actual event and is an increasing sequence of positive natural numbers such that for every , then is an actual event. {proof}

This is an immediate consequence of the definition of actual event.

We first show that impossible events are actual null events, i.e. the so called strictness.

{prop}

is an actual event and . {proof} for every and so in particular for every . This means that is an actual event and .

Then we show that actual events and satisfy involution.

{prop}

If is an actual event, then is an actual event and

 P(¬e,γ)=R1−P(e,γ)\lx@notefootnoteWefollowherethedefinitionsoftheoperationsandrelationsbetweenrealsgiveninBishop′sbook\@@cite[cite][\@@bibrefBB85].
{proof}

Let be an actual event. It is immediate to see that

for every . From this it follows that

 ∣∣Φ(¬e)(γ(n)+m)−Φ(¬e)(γ(n)+m′)∣∣=∣∣1−Φ(e)(γ(n)+m)−1+Φ(e)(γ(n)+m′)∣∣=
 =∣∣Φ(e)(γ(n)+m′)−Φ(e)(γ(n)+m)∣∣≤1n[n∈N+, m∈N, m′∈N].

Hence is an actual event. Moreover for every

 |P(¬e,γ)(n)−(1−P(e,γ))(n)|=|1−P(e,γ)(n)−(1−P(e,γ)(2n))|=
 =|P(e,γ)(2n)−P(e,γ)(n)|≤1n+12n<2n

so .

Moreover actual events are closed under disjunction of incompatible actual events:

{prop}

If and are actual events with , then if

 η:=λn.(γ(2n)+γ′(2n))

the pair is an actual event. {proof} Since , we have that . Moreover is clearly stricly increasing, as and are so. Moreover for every , it is immediate to see that .

In particular for every ,

 ∣∣Φ(e+e′)(η(n)+i)−Φ(e+e′)(η(n)+j)∣∣≤
 ≤∣∣Φ(e)(γ(2n)+γ′(2n)+i)−Φ(e)(γ(2n)+γ′(2n)+j)∣∣+

Hence is an actual event.

We now show that the set of actual events with null probability is downward closed: {prop} Let be an actual event with and let be a potential event such that , then is an actual event. {proof} For every and

 ∣∣Φ(e′)(γ(6n)+i)−Φ(e′)(γ(6n)+j)∣∣≤∣∣Φ(e′)(γ(6n)+i)∣∣+∣∣Φ(e′)(γ(6n)+j)∣∣≤
 ≤|Φ(e)(γ(6n)+i)|+|Φ(e)(γ(6n)+j)|≤
 ≤|Φ(e)(γ(6n)+i)−Φ(e)(γ(6n))|+|Φ(e)(γ(6n)+j)−Φ(e)(γ(6n))|+2|Φ(e)(γ(6n))|≤
 ≤16n+16n+226n=1n

by using the fact that is an actual event and . Hence is an actual event. Moreover satisfies monotonicity: {prop} If and are actual events and , then

 P(e,γ)≤P(e′,γ′).
{proof}

First of all if we take to be the sequence , then and are actual events equal to and respectively as a consequence of proposition 2.3.

In order to show that , we must prove that for every

 (P(e,η)−P(e′,η))(n)≤1n

But . Hence we must prove that

 Φ(e)(η(2n))≤Φ(e′)(η(2n))+1n[n∈N+]

But this is trivially true, because from we can deduce that

 Φ(e)(n)≤Φ(e′)(n)[n∈N+].

Finally we prove that satisfies a form of modularity: {prop} If , , , are actual events, then

 P(e∨e′,δ)+P(e∧e′,γ)=RP(e,α)+P(e′,β).
{proof}

Using proposition 2.3, if , then

So we must prove that . However this is immediate as for every

 e(n)+e′(n)=(e∨e′)(n)+(e∧e′)(n)

from which it immediately follows that for every

 Φ(e,n)+Φ(e′,n)=Φ(e∧e′,n)+Φ(e∨e′,n).

### 2.4 Regular events

Among potential events, there are some events which can be considered sort of “deterministic”: their sequence of outcomes of trials have a periodic behaviour, up to a possible finite number of accidental errors in the recording of the results: these events are here called regular: {defi} Let and be two finite lists of elements of with length and , respectively. We define the potential event as follows

 {∥α,π∥(i):=αi if i∈N+,i≤ℓ(α)∥α,π∥(i):=πrm(i−ℓ(α)−1,ℓ(π))+1 if i>ℓ(α) (1)

where and denote the th component of and , respectively and where for all and , is the remainder of divided by . The potential events of this form are called regular.

The goal of the following propositions is to show that regular events are actual. We first show that regular events without errors are actual events and their probability is equal to the frequency of their period.

{prop}

For every finite non-empty list , is an actual event and

 P(∥[],π∥,λn.4nm)=R∑mk=1πkm.
{proof}

For every and every

 |Φ(∥[],π∥)(4nm+i)−Φ(∥[],π∥)(4nm+j)|=
 =∣∣ ∣∣∑mk=1πkm((4n+qt(i,m))m4nm+i)+∑rm(i,m)k=1πk4nm+i−∑mk=1πkm((4n+qt(j,m))m4nm+j)−∑rm(j,m)k=1πk4nm+j∣∣ ∣∣≤

Hence is an actual event. Moreover for every

 P(∥[],π∥,λn.4nm)(i)=∑mk=1πkm

Hence

The next step consists in proving that regular events definitely equal to are actual events and their probability is .

{prop}

If is a finite list of s and s with length , then is an actual event and . {proof} Suppose and with . Then

 |Φ(∥α,[0]∥)(2nm+i)−Φ(∥α,[0]∥)(2nm+j)|≤
 ≤|Φ(∥α,[0]∥)(2nm+i)|+|Φ(∥α,[0]∥)(2nm+j)|=
 =α1+...+αm2nm+i+α1+...+αm2nm+j≤2m2nm=1n

Hence is an actual event. Moreover for every

 (P(∥α,[0]∥,λn.2nm)(n)=α1+...+αm2nm≤12n<2n

Hence .

Finally we prove that actual events are closed under shift to the right of terms and that probability is preserved by these shifts.

{defi}

If is a potential event, then is the potential event defined by

 {e+(1):=0e+(n+1):=e(n) for n∈N+ (2)
{prop}

If is an actual event, then is an actual event and . {proof} If and , then is less or equal than the sum of , and . However this sum is equal to

 Φ(e)(γ(3n)+i)γ(3n)+i+1+|Φ(e)(γ(3n)+i)−Φ(e)(γ(3n)+j)|+Φ(e)(γ(3n)+j)γ(3n)+j+1
 ≤1γ(3n)+j+1+13n+1γ(3n)+j+1≤313n=1n

Hence is an actual event. Moreover

 |Φ(e)(γ(3n))−Φ(e)(γ(n))|+∣∣∣Φ(e)(γ(3n))1γ(3n)+1∣∣∣≤1n+13n+1<2n

Hence .

Putting these results together we obtain the following: {thm} For every regular event , there exists a strictly increasing sequence of natural numbers such that is an actual event and . {proof} First of all, notice that where is applied times. By proposition 2.4, we know that there exists such that is an actual event and, as a consequence of propositions 2.4 and 2.4, there exists such that is an actual event.

Moreover it is clear that , thus, using proposition 2.3, we obtain that there exists such that is an actual event.

Finally, by propositions 2.3 and 2.3 and, by propositions 2.4, 2.4, 2.4 and 2.3, this is equal to .

We conclude this section with the following result:

{thm}

Regular events form a boolean algebra with the operations inherited by the algebra of potential events. {proof} First of all and . Suppose now that and are regular events. Then where and are obtained by changing each term of the finite lists in . Without loss of generality, we can suppose that . It is immediate to check that

 ∥α,π∥∧∥β,ψ∥=∥γ,ρ∥
 ∥α,π∥∨∥β,ψ∥=∥∥γ′,ρ′∥∥

where , ,

 {γi:=αi∧βi if i≤ℓ(α)γi:=πrm(i−ℓ(α)−1,ℓ(π))+1∧βi if ℓ(α)
 {γ′i:=αi∨βi if i≤ℓ(α)γ′i:=πrm(i−ℓ(α−1,ℓ(π))+1∨βi if ℓ(α)
 ⎧⎪⎨⎪⎩ρi:=πrm(ℓ(β)−ℓ(α)+i−1,ℓ(π))+1∧ψrm(i−1,ℓ(ψ))+1ρ′i:=πrm(ℓ(β)−ℓ(α)+i−1,ℓ(π))+1∨ψrm(i−1,ℓ(ψ))+1 for every 1≤i≤ℓ(π)ℓ(ψ). (5)

## 3 Probabilistic limited principles of omniscience

The limited principle of omniscience, for short , is a non-constructive principle which is weaker than the law of excluded middle (see [bishfound], [bridges1987varieties] ) which is very important for constructive reverse mathematics (see [Ishihara2006]). It can be formulated in our framework as follows

 LPO:(∀e∈P)((∀n∈N+)(e(n)=0)∨(∃n∈N+)(e(n)=1))

Having introduced a notion of probability on binary sequences, we can weaken this axiom by restricting it to some subset of potential events or by weakening the disjunction.

Let us first introduce some abbreviations: we will write

1. for the set of non-empty lists of elements of a set ;

2. for the set of increasing sequences of positive natural numbers.

3. for ;

4. for if depends on a real number ;

5. for ;

6. for .

In such a way we can define subsets , and of consisting of actual, regular and null events, respectively.

Before proceeding, let us prove a simple, but very important fact. {prop} Let ;

1. if , then ;

2. if , then .

{proof}

The second point is immediate since means exactly .

Suppose now that and . By definition of between real numbers, there exists such that , i.e. ,

 γ(m)∑i=1me(i)>γ(m)>0

Hence there exists such that and . Thus .

We can now introduce a new family of “probabilistic” versions of the limited principle of omniscience.

{defi}

For every subset of we define the following principles:

1. LPO

2. -LPO

3. -LPO

First of all, one can notice that is exactly . Let us now study the relation between these probabilistic versions of . As a direct consequence of proposition 3 we have the following {prop}If is a subset of , then

1. --

2. -

{proof}

Points (1) and (2) follow from points (1) and (2) in proposition 3, respectively. Moreover we have {prop}If are subsets of , then

1. --

2. --

{proof}

This is an obvious consequence of the definitions.

Some of these principles can be proven constructively. {prop}The following hold:

1. -

2. -

3. -

4. -

5. -

{proof}

(1) holds as it reduces to control, for each in , a finite number of entries (those in and ). (2) is a consequence of and proposition 3. For point (3), if in , then . Hence we can decide whether or . Point (4) is obvious, since means, by definition, that . (5) is a consequence of and proposition 3. For point (6), there exist potential events for which there is no such that . Consider for example the sequence

 10110011110000...

which alternates a group of ones and a group of zeros increasing at each step.

Moreover we have the following result. {prop} For every subset of

 (P-LPO[E]∧LPO[N])→LPO[E]

In particular

1. -

2. -

3. -

{proof}

Suppose holds. Then which is equivalent to . If holds too, then which is equivalent to which is . In particular, since and propositions 3 and 3 hold, , and follow.

Hence we can summarize the situation through the following diagram.

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