Encoding Sets as Real Numbers Domenico Cantone 1 Alberto Policriti 2 - - PowerPoint PPT Presentation

Encoding Sets as Real Numbers

Domenico Cantone1 Alberto Policriti2

• Dept. of Mathematics and Computer Science, University of Catania, Italy
• Dept. of Mathematics, Computer Science, and Physics, University of Udine, Italy

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 1/18

Sets

Paul Halmos “Naive Set Theory” Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. Sets, as they are usually conceived, have elements or members. An element of a set may be a wolf, a grape, or a pidgeon. It is important to know that a set itself may also be an element of some

• ther set. [...] What may be surprising is not so much that sets

may occur as elements, but that for mathematical purposes no

• ther elements need ever be considered.

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Definition (Hereditarily finite sets) HF :=

n∈N HFn is the collection of all hereditarily finite sets,

where

• HF0 := ∅,

HFn+1 := P(HFn), for n ∈ N.

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Ackermann 1937

Definition NA(x) = Σy∈x2NA(y)

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Ackermann 1937

Definition NA(x) = Σy∈x2NA(y) Example 1 1 0 1 0 is the code (in base 2) of the 26th (non-empty) set, whose elements are the second, the 4th and the 5th sets in Ackermann

• rder (numbering).

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Ackermann 1937

Definition NA(x) = Σy∈x2NA(y) Ackermann Order hi ≺ hj ⇔ NA(hi) = i < j = NA(hj) Ordering hi ≺ hj ⇔ NA(hi) = i < j = NA(hj) ⇔ max(hi \ hj) ≺ max(hj \ hi)

... which is as comparing the binary codes of i and j

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Equality and Extensionality

Uniqueness of (binary) positional notation Two natural numbers are equal if and only if they have the same binary code. Axiom of Extensionality Two sets are equal if and only if they have the same elements.

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What about hypersets?

Definition Ω = {Ω} Questions Extensionality? Can we propose a numerical encoding for hypersets?

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What about hypersets?

Definition Ω = {Ω} Questions Extensionality? Can we propose a numerical encoding for hypersets?

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Extending Ackermann code: A first version

Definition (Dyadic rational numbers) n/2m n, m ∈ N 11010, 101

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Extending Ackermann code: A first version

Definition (Dyadic rational numbers) n/2m n, m ∈ N 11010, 101 Mapping on dyadic rational numbers Find an order of (proper) non well-founded h.f. sets and put them on the right: let ZA the result. Use ZA, that is NA on the left and ZA on the right.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 7/18

Extending Ackermann code: A first version

Definition (Dyadic rational numbers) n/2m n, m ∈ N 11010, 101 Mapping on dyadic rational numbers Find an order of (proper) non well-founded h.f. sets and put them on the right: let ZA the result. Use ZA, that is NA on the left and ZA on the right. Example 1 1 0 1 0 , 1 0 1

is the code of h26 ∪ {1, 3} with, in addition, the first and the third non well-founded sets.

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Extending Ackermann code

Definition RA(x) = Σy∈x2−RA(y)

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Extending Ackermann code

Definition RA(x) = Σy∈x2−RA(y) Ω = {Ω} ξ = 2−ξ . (1)

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Extending Ackermann code

Definition RA(x) = Σy∈x2−RA(y) Ω = {Ω} ξ = 2−ξ . (1)

y x y = x y = 2−x

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Extending Ackermann code

What kind of number is Ω? Ω is irrational. Ω is transcendental: (Gelfond-Schneider theorem: ab is transcendental if a and b are algebraic, 0 = a = 1, and b is irrational) Ω algebraic ⇒ −Ω algebraic ⇒ (G.S.) 2−Ω = Ω would be transcendental. The RA-code 2−1/

√ 2 of {∅}4 is transcendental.

...

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 8/18

Conjectures

Conjecture 1 The function RA is injective on HF0. Conjecture 2 The function RA is injective on HF1/2.

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The domain of RA

Definition (Set systems) A set system S (x1, . . . , xn) in the set unknowns x1, . . . , xn is a collection of set-theoretic equations of the form:        x1 = {x1,1, . . . , x1,m1} . . . xn = {xn,1, . . . , xn,mn}, (1) with mi ≥ 0 for i ∈ {1, . . . , n}, and where each unknown xi,u, for i ∈ {1, . . . , n} and u ∈ {1, . . . , mi}, occurs among the unknowns x1, . . . , xn. A set system S (x1, . . . , xn) is normal if there exist n pairwise distinct (i.e., non bisimilar) hypersets 1, . . . , n ∈ HF1/2 such that the assignment xi → i satisfies all the set equations of S (x1, . . . , xn).

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Our result

RA is well-given Any finite collection 1, . . . , n of pairwise distinct sets in HF1/2 satisfying S (x1, . . . , xn), univocally determines real numbers RA(1), . . . , RA(n) satisfying:        RA(1) = m1

k=1 2−RA(1,k)

. . . RA(n) = mn

k=1 2−RA(n,k).

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Proof sketch

Definition The multi-set approximating sequence for (the solution of) S (x1, . . . , xn) is

• µj

i | 1 ≤ i ≤ n

• :=

    

• ∅| 1 ≤ i ≤ n
• if j = 0
• µj−1

i,1 , . . . , µj−1 i,mi

• | 1 ≤ i ≤ n
• if j > 0 ,

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 12/18

Proof sketch

Definition Code approximating sequence

• Rµ

A(µj i)| 1 ≤ i ≤ n}

• j∈N by

recursively putting:

• Rµ

A(µ0 i )

= Rµ

A(µj+1 i

) = mi

u=1 2−Rµ

A(µj i,u) .

(2) We also define the corresponding code increment sequence

• δj

i | 1 ≤ i ≤ n

• j∈N by putting, for i ∈ {1, . . . , n} and j ∈ N:

δj

i = Rµ A(µj+1 i

) − Rµ

A(µj i) .

(3)

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Proof sketch

Lemma (i) Rµ

A(µj+1 i

) = δ0

i + · · · + δj i ,

(ii) δ0

i = mi,

(iii) δj+1

i

= mi

u=1 2−Rµ

A(µj i,u) · (2−δj i,u − 1),

(iv) δ2j+1

i

≤ 0 ≤ δ2j

i ,

(v) |δj+1

i

| ≤ |δj

i |, and

(vi) limj→∞ δj

i = 0.

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Theorem For any given normal set system, the corresponding code system admits always a unique solution.

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Theorem For any given normal set system, the corresponding code system admits always a unique solution. Remark While the value of RA

• µ0

i

• is 0 for any i ∈ {1, . . . , n}, the value of

RA

• µ1

i

• —first approximation of RA (µi)—is the cardinality of µi,

and the subsequent approximations oscillate within the interval [0, |µi|].

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Partial results: super-singletons

Definition The elements of the family S of super-singletons S =

• {∅}i | i ∈ N
• ,

are defined recursively as follows: {∅}0 = ∅ and {∅}n+1 = {{∅}n}.

s0 s1 1 s2

1 2

s3

1 √ 2

s4 s5 Ω

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Partial results: super-singletons

Definition The elements of the family S of super-singletons S =

• {∅}i | i ∈ N
• ,

are defined recursively as follows: {∅}0 = ∅ and {∅}n+1 = {{∅}n}. Proposition The following hold: 0 = s0 < · · · s2i < s2i+2 · · · < Ω < · · · s2i+3 < s2i+1 · · · < s1 = 1, and lim

i→∞ s2i = lim i→∞ s2i+1 = Ω.

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Partial results: successors, density, ...

Proposition For all i ∈ N:

1 RA(hi) = RA(hi+1); 2 RA(hi) = RA(hi+2). Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 15/18

Partial results: successors, density, ...

Proposition For all i ∈ N:

1 RA(hi) = RA(hi+1); 2 RA(hi) = RA(hi+2).

Proposition {RA(h) | h ∈ HF} is dense in R.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 15/18

On Conjecture 1

Definition (The Adjunctive Hierarchy) Let A =

n∈N An, where

A0 = {∅}, An+1 = {(x with y) | x, y ∈ An} ∪ {∅}.

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On Conjecture 1

Definition (The Adjunctive Hierarchy) Let A =

n∈N An, where

A0 = {∅}, An+1 = {(x with y) | x, y ∈ An} ∪ {∅}. Definition (with-point) Let h =

• hℓ with hr

be such that hr / ∈ hℓ and NA (hr) = max{NA(x) | x ∈ h}. We denote by Ph, and call it the with-point of h, the element of R2

• 2−RA(hℓ), 2−2−RA(hr)

=

• RA({hℓ}), RA ({{hr}})
• .

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On Conjecture 1

y x P{∅} = P(∅ with ∅) 1/2 1 1 Ph = P(hℓ with hr) RA

• hℓ

= 2−RA(hℓ) RA ({{hr}}) = 2−2−RA(hr) Ph = P(hℓ with hr) E ≡ y = 2−x

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 16/18

On Conjecture 1

y x P{∅} = P(∅ with ∅) 1/2 1 1 Ph = P(hℓ with hr) RA

• hℓ

= 2−RA(hℓ) RA ({{hr}}) = 2−2−RA(hr) Ph = P(hℓ with hr) E ≡ y = 2−x

Remark Given h ∈ HF∗, the area of the rectangle whose diagonal is the segment [(0, 0), Ph] is the code of {h}: 2−RA(hℓ) · 2−2−RA(hr) = 2−

• RA(hℓ)+2−RA(hr)

= 2−RA(h) = RA({h}).

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 16/18

On Conjecture 1

y x

(0, 0) YP2k = Y2k = Y2−ρ = YRA({{hk}}) XPj = Xj = Xλ = XRA({hj}) RA({hj}) = 2−λ RA({{hk}}) = 2−2−ρ P(hj with hk) = Phi = Pi HPi = Hi = Hα = Hλ+2−ρ 2−λ2−2−ρ = RA ({(hj with hk)})

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 16/18

On Conjecture 1

Conjecture For all i, j ∈ N∗, i = j ⇒ Hi = Hj.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 16/18

On Conjecture 2

E x = y

{∅} {∅, {∅}} {{∅}} {{∅}, {{∅}}} {∅}3

• {∅}3 with {∅}3

{∅}4 Ω {Ω} = Ω

Figure: A geometric construction of Ω, obtained as the limit curve of the one starting from {∅} and passing through {∅}n, for n ∈ N∗.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 17/18

On Conjecture 2

... generalizing Consider x = {a, x}, for some fixed a ∈ HF1/2. The above x has code RA(x) satisfying RA(x) = 2−RA(a) + 2−RA(x), whose value we want to determine as the limit of the sequence of codes of sets x0 = {a}, xi+1 = {a, xi}, for i ∈ N.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 17/18

On Conjecture 2

E x = y

{a} = x0 {a, x0} = x1

• x1 with x1

{a, x1} = x2

• x2 with x2

{a, x2} = x3

• x3 with x3

{a, x3} = x4 {a, x} = x

Figure: The geometric construction of the solution of x = {a, x},

• btained as the limit curve of the one starting from x0 = {a} and passing

through xn+1 = {a, xn}, for n ∈ N.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 17/18

On Conjecture 2

E x = y

{a} = x0

1

{b} = x0

2

{a, x0

2} = x1 1

{b, x0

1} = x1 2

{a, x1

2} = x2 1

{b, x1

1} = x2 2

{a, x2

2} = x3 1

{b, x2

1} = x3 2

x1 x2

Figure: The geometric construction of the solution of

x1 = {a, x2}, x2 = {b, x1} obtained as the limit curve of the one starting from x0

1 = {a}, x0 2 = {b} and passing through xn+1 1

= {a, xn

2}, xn+1 2

= {b, xn

1}, for

n ∈ N.

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Conclusions

Hypersets as limits of Sets Alternative approach to bisimulation computation Graphs (labelled on both nodes and edges) can be encoded/compressed by reals Elegant encoding

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