the representation of boolean algebras - units.itthe representation of boolean algebras1 in the...

The representation of Boolean algebrasin the spotlight of a proof checker

Rodica Ceterchi, E. G. Omodeo, Alexandru I. Tomescu

Dip. Matematica e Geoscienze — DMI

Eugenio G. Omodeo Proof-checking Stone’s representation th’m 1/22

The representation of Boolean algebras1

in the spotlight of a proof checker

Rodica Ceterchi, E. G. Omodeo, Alexandru I. Tomescu

Dip. Matematica e Geoscienze — DMI

1Work partially funded by INdAM/GNCS 2013, by Academy of Finland grant250345 (CoECGR), and by FRA-UniTS PUMA


( Ideal ) itinerary of this talk

i Stone’s representation of Boolean algebras

ii The proof assistant Ref

iii Our Stone-related proof-verification experiment


‘Boolean algebras have an almost embarrassingly rich structure.’

[Hal62, p. 10]

A Boolean algebra is. . .

0 a distributive and complemented lattice

1 a ring with 1, meeting the special laws

X + X = 0X · X = X∥∥∥∥∥∥∥∥∥∥∥∥

One often sets:

X =Def X + 1X t Y =Def X · Y + X + YX 6 Y ↔Def X t Y = Y

etc.Eugenio G. Omodeo Proof-checking Stone’s representation th’m 3/22

Digression: The Robbins / McCune axioms

X t Y = Y t X

X t (Y t Z ) = (X t Y ) t Z

X t Y t X t Y = X

Cf. [McC97]


How does one plainly ascertain Boolean laws ?


Fields of sets

One calls field of sets any family B which

is closed under the operations ofintersection ( · ) and symmetric difference ( + ) ;

has⋃B among its members, viz.,

owns a maximum w.r.t. set inclusion ;

differs from {∅} .This clearly is an instance of a Boolean algebra. How general? Arenowned theorem by M. H. Stone [Sto36] gives us the answer:

Every Boolean algebra is isomorphic to a field of sets.


Fields of sets

One calls field of sets any family B which

is closed under the operations ofintersection ( · ) and symmetric difference ( + ) ;

has⋃B among its members, viz.,

owns a maximum w.r.t. set inclusion ;

differs from {∅} .This clearly is an instance of a Boolean algebra. How general? Arenowned theorem by M. H. Stone [Sto36] gives us the answer:

Every Boolean algebra is isomorphic to a field of sets.

( Marshall Stone1903–1989 )

( Alfred Tarski1902–1983 )


‘A cardinal principle of modern mathematical research

may be stated as a maxim: “One must always topologize”.’

One calls Stone space any topological space (X , τ) such that

1 for every p ∈ X and q ∈ X \ {p} there exist P,Q ∈ τ such that

p ∈ P , q ∈ Q , P ∩ Q = ∅ ;

2 every O ⊆ τ s.t.⋃O = X has a finite subfamily F ⊆ O such

that⋃F = X ;

3 there is a β ⊆ τ ∩ {X \ Q : Q ∈ τ } such that

τ = {⋃Y : Y ⊆ β } .

Every Boolean algebra is isomorphic to the fieldτ ∩ {X \ Q : Q ∈ τ }

of all clopen sets of a Stone space (X , τ).Cf. [Sto37], [Sto38, p. 814]


ÆtnaNova aka Ref eree: Cf. [SCO11]

( On-line worksheet )


Interaction with our proof-verifier ( Input )


Interaction with our proof-verifier( Output )


3 basic constituents of a scenario ( examples )

Definition: ( no ∈-recursion here! )

-- After the celebrated paper Sur les ensembles fini ( Tarski, 1924 )

Def Fin: [Finitude] Finite(F) ↔Def

〈∀g ∈ P(P(F))\{∅}, ∃m | g ∩ P(m) = {m}〉


3 basic constituents of a scenario ( examples )

Definition: ( no ∈-recursion here! )

-- After the celebrated paper Sur les ensembles fini ( Tarski, 1924 )

Def Fin: [Finitude] Finite(F) ↔Def

〈∀g ∈ P(P(F))\{∅}, ∃m | g ∩ P(m) = {m}〉

Theorem and Proof: ( Monotonicity of finitude )

Thm fin0. Y ⊇ X & Finite(Y)→ Finite(X). Proof:Suppose_not(y0, x0) =⇒ y0 ⊇ x0 & Finite(y0) & ¬Finite(x0)

〈y0, x0〉↪→Tpow1 =⇒ Py0 ⊇ Px0Use_def(Finite) =⇒ Stat1 :¬〈∀g ∈ P(Px0)\{∅},∃m |

g ∩ Pm = {m}〉 & 〈∀g ′ ∈ P(Py0)\{∅},∃m | g ′ ∩ Pm = {m}〉〈Py0,Px0〉↪→Tpow1 =⇒ P(Py0)⊇ P(Px0)

〈g0, g0〉↪→Stat1(Stat1?) =⇒ ¬〈∃m | g0 ∩ Pm = {m}〉 &〈∃m | g0 ∩ Pm = {m}〉

Discharge =⇒ QedEugenio G. Omodeo Proof-checking Stone’s representation th’m 11/22

4th, major constituent of a scenario (example)

A construct for proof reuseTheory finite_image (s0 , g(X ))

Finite(s0)

End finite_image

Enter_theory finite_image...

......

...

Enter_theory Set_theory

Within a scenario, the discourse can momentarily digress into a‘Theory’ that enforces certain local assumptions.At the end of the digression, the upper theory will be re-entered.




Finite(s0)

=⇒ (fΘ)

Finite({ g(x) : x ∈ s0 }

)fΘ ⊆ s0 & 〈 ∀ t ⊆ fΘ | g(t) = g(s0) ↔ t = fΘ 〉

End finite_image

As an outcome of the digression, the Theory will be able toinstantiate new theorems




Finite(s0)

=⇒ (fΘ)

Finite({ g(x) : x ∈ s0 }

)fΘ ⊆ s0 & 〈 ∀ t ⊆ fΘ | g(t) = g(s0) ↔ t = fΘ 〉

End finite_image

As an outcome of the digression, the Theory will be able toinstantiate new theorems: possibly involving new symbols,whose definition it encapsulates.


Our experiment, in digits

The script-file containing our verified formal derivation of Stone’sresults ( with many asides ) from ‘first principles’:

comprises 42 definitions;

proves 210 theorems,

organized in 19 Theory s

Its processing takes ca. 25 seconds.

http://www2.units.it/eomodeo/StoneReprScenario.html


http://www2.units.it/eomodeo/StoneReprScenario.html

Conclusions and future work

Proof-verification can highly benefit from representation theoremsof the kind illustrated by Stone’s results on Boolean algebras.

On the human side, such results disclose new insights byshedding light on a discipline from unusual angles

on the technological side, they enable the transfer of proofmethods from one realm of mathematics to another.

An envisaged continuation of the present work will be in thedirection of MV-algebras (cf. [Mun86])


Thank you for your attention!


Bibliografic references


Paul R. Halmos. Algebraic Logic.AMS Chelsea Publishing, Providence, Rhode Island, 1962.

W. W. McCune. Solution of the Robbins problem.J. of Automated Reasoning, 19(3):263–276, 1997.

D. Mundici. Interpretation of AF C ∗-algebras in Łukasiewiczsentential calculus.J. Funct. Anal., 65:15–63, 1986.

J.T. Schwartz, D. Cantone, and E.G. Omodeo.Computational Logic and Set Theory - Applying FormalizedLogic to Analysis.Springer, 2011.

Marshall H. Stone.The theory of representations for Boolean algebras.Transactions of the American Mathematical Society,40:37–111, 1936.


Marshall H. Stone. Applicationsof the theory of Boolean rings to general topology.Transactions of the American Mathematical Society,41:375–481, 1937.

Marshall H. Stone.The representation of Boolean algebras.Bulletin of the American Mathematical Society, 44(Part1):807–816, 1938.


Excerpt 1 from scenario on Stone’s theorems

Theory pord(dd, Le(U,V)

)〈∀x, y | {x, y}⊆ dd → (

Le(x, y) & Le(y, x) ↔ x = y)〉

〈∀x, y, z | {x, y, z}⊆ dd → Le(x, y) & Le(y, z) → Le(x, z)〉=⇒ (poIsoΘ)

poIsoΘ = {[x, {v ∈ dd | Le(v, x)}] : x ∈ dd}

〈∀x | x ∈ dd → Le(x, x) & poIsoΘ�x = {v ∈ dd | Le(v, x)}〉〈∀x, y | {x, y}⊆ dd → (

Le(x, y) ↔ poIsoΘ�x ⊆ poIsoΘ�y)〉

1–1(poIsoΘ) & dom(poIsoΘ) = dd

End pord

Figure: Interface of a representation Theory about partial orderings


Excerpt 2 from scenario on Stone’s theoremsTheory booleanRing(bb, ·,÷)

bb 6=∅〈∀x, y | {x, y}⊆ bb → x · y ∈ bb〉〈∀x, y | {x, y}⊆ bb → x ÷ y ∈ bb〉〈∀x, y, z | {x, y, z}⊆ bb → x · (y · z) = (x · y) · z〉〈∀x, y, z | {x, y, z}⊆ bb → x ÷ (y ÷ z) = (x ÷ y)÷ z〉〈∀x, y, z | {x, y, z}⊆ bb → (x ÷ y) · z = z · y ÷ z · x〉〈∀x, y | {x, y}⊆ bb → x ÷ x = y ÷ y〉〈∀x, y | {x, y}⊆ bb → x ÷ (y ÷ x) = y〉〈∀x | x ∈ bb → x · x = x〉

=⇒ (zzΘ)

zzΘ = arb (bb)÷ arb (bb)

〈∀x | (x ∈ bb → x ÷ x = zzΘ & x ÷ zzΘ = x & zzΘ ÷ x = x) & zzΘ ∈ bb〉〈∀x, y | x, y ∈ bb → x ÷ y = y ÷ x〉〈∀x, y | x, y ∈ bb → x · y = y · x〉〈∀x | x ∈ bb → zzΘ · x = zzΘ〉〈∀u, v | {u, v}⊆ bb & u · v = u & v · u = v → u = v〉

End booleanRing


Excerpt 3 from scenario on Stone’s theoremsOur next Theory, presupposing the definition of symmetricdifference, shows that rings of sets match the assumptions of theTheory booleanRing:

Theory protoBoolean(dd)

∅6=⋃

dd

〈∀x, y | {x, y}⊆ dd → x ∩ y ∈ dd〉〈∀x, y | {x, y}⊆ dd → x4y ∈ dd〉

=⇒dd 6=∅〈∀x ∈ dd, y ∈ dd, z ∈ dd | x ∩ (y ∩ z) = (x ∩ y) ∩ z〉〈∀x ∈ dd, y ∈ dd, z ∈ dd | x4(y4z) = (x4y)4z〉〈∀x ∈ dd, y ∈ dd, z ∈ dd | (x4y) ∩ z = z ∩ y4z ∩ x〉〈∀x ∈ dd, y ∈ dd | x4x = y4y〉〈∀x ∈ dd, y ∈ dd | x4(y4x) = y〉〈∀x ∈ dd | x ∩ x = x〉

End protoBoolean



Two claims, proved inside the background Theory, namelySet_theory, and presupposing the definition of P, show that thefamily of all subsets, and the one of all finite and cofinite subsets,of a non-void set constitute instances of protoBoolean:

Thm . W 6=∅ & {X,Y}⊆ PW → {X ∩ Y,X4Y}⊆ PW &⋃(PW)6=∅.

Thm . W 6=∅ & D = {s ⊆W | Finite(s) ∨ Finite(W\s)} & {X,Y}⊆ D →{X ∩ Y,X4Y}⊆ D &

⋃D 6=∅.



Another Theory, akin to the preceding one, introduces a slightlymore specific algebraic variety than the one treated byprotoBoolean:

Theory archeoBoolean(dd)

∅6=⋃

dd

〈∀x, y, z | {x, y}⊆ dd & z ⊆ x ∪ y → z ∈ dd〉=⇒〈∀x, y | {x, y}⊆ dd → x ∩ y ∈ dd〉〈∀x, y | {x, y}⊆ dd → x4y ∈ dd〉dd 6=∅

End archeoBoolean



After switching back to the background Set_theory level, oneproves that there are fields of sets which are instances ofprotoBoolean but are not instances of archeoBoolean. Indeed, thecollection of all finite and cofinite subsets of an infinite set is notclosed with respect to inclusion.

Thm . ¬Finite(W) & D = {s ⊆W | Finite(s) ∨ Finite(W\s)}→W ∈ D & 〈∃z ⊆W | z /∈ D〉.

Surprisingly enough, it is unnecessary to resort to a theory ofcardinals of any sophistication in order to get the result just cited:the distinction between finite sets ( see above ) and sets which arenot finite more than suffices for that purpose.


the representation of boolean algebras - units.itthe representation of boolean algebras1 in the...

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