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

encoding sets as real numbers
SMART_READER_LITE
LIVE PREVIEW

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

Encoding Sets as Real Numbers Domenico Cantone 1 Alberto Policriti 2 Dept. of Mathematics and Computer Science, University of Catania, Italy Dept. of Mathematics, Computer Science, and Physics, University of Udine, Italy Domenico Cantone, Alberto


slide-1
SLIDE 1

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

slide-2
SLIDE 2

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.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 2/18

slide-3
SLIDE 3

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.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 3/18

slide-4
SLIDE 4

Ackermann 1937

Definition NA(x) = Σy∈x2NA(y)

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18

slide-5
SLIDE 5

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).

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18

slide-6
SLIDE 6

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 4/18

slide-7
SLIDE 7

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.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 5/18

slide-8
SLIDE 8

What about hypersets?

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 6/18

slide-9
SLIDE 9

What about hypersets?

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 6/18

slide-10
SLIDE 10

Extending Ackermann code: A first version

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

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

slide-11
SLIDE 11

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

slide-12
SLIDE 12

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.

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

slide-13
SLIDE 13

Extending Ackermann code

Definition RA(x) = Σy∈x2−RA(y)

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

slide-14
SLIDE 14

Extending Ackermann code

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

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

slide-15
SLIDE 15

Extending Ackermann code

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

y x y = x y = 2−x

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

slide-16
SLIDE 16

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

slide-17
SLIDE 17

Conjectures

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 9/18

slide-18
SLIDE 18

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).

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 10/18

slide-19
SLIDE 19

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).

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 11/18

slide-20
SLIDE 20

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

slide-21
SLIDE 21

Proof sketch

Definition Code approximating sequence

A(µj i)| 1 ≤ i ≤ n}

  • j∈N by

recursively putting:

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)

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

slide-22
SLIDE 22

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.

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

slide-23
SLIDE 23

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 13/18

slide-24
SLIDE 24

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|].

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 13/18

slide-25
SLIDE 25

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

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 14/18

slide-26
SLIDE 26

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.

Domenico Cantone, Alberto Policriti Encoding Sets as Real Numbers 14/18

slide-27
SLIDE 27

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

slide-28
SLIDE 28

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

slide-29
SLIDE 29

On Conjecture 1

Definition (The Adjunctive Hierarchy) Let A =

n∈N An, where

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

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

slide-30
SLIDE 30

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}})
  • .

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

slide-31
SLIDE 31

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

slide-32
SLIDE 32

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

slide-33
SLIDE 33

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

slide-34
SLIDE 34

On Conjecture 1

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

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

slide-35
SLIDE 35

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

slide-36
SLIDE 36

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

slide-37
SLIDE 37

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

slide-38
SLIDE 38

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.

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

slide-39
SLIDE 39

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

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