The word problem for -categories

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```					  The word problem for ΣΠ-categories,
i.e. properties of free ΣΠ-categories (II)

Robin Cockett
Department of Computer Science
University of Calgary

* Luigi Santocanale *
LIF, Marseille
e
Universit´ de Provence
FRANCE

Cape Town, Category Theory 2009

1/26
Outline

Introductory remarks
ΣΠ-categories and their theory
Context, some motivations
The free ΣΠ-category

“En route” towards a decision procedure
The CT2006 results

Softness, cardinals, and (some hints on) a decision procedure
Understanding softness
Pushouts-pullbacks, bouncing

2/26
Outline

Introductory remarks
ΣΠ-categories and their theory
Context, some motivations
The free ΣΠ-category

“En route” towards a decision procedure
The CT2006 results

Softness, cardinals, and (some hints on) a decision procedure
Understanding softness
Pushouts-pullbacks, bouncing

3/26
ΣΠ-categories and their theory

A ΣΠ-category :
a category with (chosen) ﬁnite products and ﬁnite coproducts.

Binary products:
,
[X,A] × [X,B ] −−  −→ [X,A × B ]
πi
−
[ Xi , A ] −→ [ X0 × X1 , A ]

such that

πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
. . . and dual equations.

4/26
ΣΠ-categories and their theory

A ΣΠ-category :
a category with (chosen) ﬁnite products and ﬁnite coproducts.

Binary products:
,
[X,A] × [X,B ] −−  −→ [X,A × B ]
πi
−
[ Xi , A ] −→ [ X0 × X1 , A ]

such that

πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
. . . and dual equations.

4/26
ΣΠ-categories and their theory

A ΣΠ-category :
a category with (chosen) ﬁnite products and ﬁnite coproducts.

Binary products:
,
[X,A] × [X,B ] −−  −→ [X,A × B ]
πi
−
[ Xi , A ] −→ [ X0 × X1 , A ]

such that

πi ( f0 , f1 ) = fi ,       π0 (f ), π1 (f ) = f .

Binary coproducts: dual nats { , } and σj , j = 0, 1, . . .
. . . and dual equations.

4/26
Units, empty products and coproducts,
terminal and initial objects

The units:

[ X , 1 ] = { !X }      [ 0, A ] = { ?A }

Some derived equations:

{!X , !Y } = !X +Y         →
:X +Y − 1
!0 = ? 1       →
:0− 1

5/26
Units, empty products and coproducts,
terminal and initial objects

The units:

[ X , 1 ] = { !X }      [ 0, A ] = { ?A }

Some derived equations:

{!X , !Y } = !X +Y         →
:X +Y − 1
!0 = ? 1       →
:0− 1

5/26
The word problem for ΣΠ-categories?
Elementary but important mathematical theory,

A challenge for category theorists and logicians:
Joyal & Hu 96, Hu 98,
Dosen 99, Dosen & Petric 2009,
Hughes 2000, Hughes & van Glabbek 03,
Joyal 95,
Cockett & Seely 2001,

From Whitman’s condition to softess:
mimicking the theory of free lattices,

Joyal’s theory of communication:
free ΣΠ-categories are semantics for communication protocols,
we tackle equivalence of these protocols.

6/26
The word problem for ΣΠ-categories?
Elementary but important mathematical theory,

A challenge for category theorists and logicians:
Joyal & Hu 96, Hu 98,
Dosen 99, Dosen & Petric 2009,
Hughes 2000, Hughes & van Glabbek 03,
Joyal 95,
Cockett & Seely 2001,

From Whitman’s condition to softess:
mimicking the theory of free lattices,

Joyal’s theory of communication:
free ΣΠ-categories are semantics for communication protocols,
we tackle equivalence of these protocols.

6/26
The word problem for ΣΠ-categories?
Elementary but important mathematical theory,

A challenge for category theorists and logicians:
Joyal & Hu 96, Hu 98,
Dosen 99, Dosen & Petric 2009,
Hughes 2000, Hughes & van Glabbek 03,
Joyal 95,
Cockett & Seely 2001,

From Whitman’s condition to softess:
mimicking the theory of free lattices,

Joyal’s theory of communication:
free ΣΠ-categories are semantics for communication protocols,
we tackle equivalence of these protocols.

6/26
The word problem for ΣΠ-categories?
Elementary but important mathematical theory,

A challenge for category theorists and logicians:
Joyal & Hu 96, Hu 98,
Dosen 99, Dosen & Petric 2009,
Hughes 2000, Hughes & van Glabbek 03,
Joyal 95,
Cockett & Seely 2001,

From Whitman’s condition to softess:
mimicking the theory of free lattices,

Joyal’s theory of communication:
free ΣΠ-categories are semantics for communication protocols,
we tackle equivalence of these protocols.

6/26
ΣΠs: a theory of communication
Objects of ΣΠ(A) are channels,
arrows of ΣΠ(A) are mediating protocols.

G
M        GH

mediating protocol

1st channel                                                                2nd channel

!"# !"#
#  /// '&%\$ '&%\$            !"#
'&%\$        !"#
'&%\$
/                                                /
                     M                              /× //
:  G //:        y                      r          :  H //:
!"#
'&%\$          !"#
'&%\$                                                                        !"#
'&%\$          !"#
'&%\$
                                                   
q                                                                                      p 
hooks
left user                                                                                                     right user

7/26
ΣΠ(A), the free ΣΠ category on A

free ΣΠ-category generated by A:
η
A GG               G ΣΠ(A)
GG                    
GG
GG
GGF
GG
GG            ∃!F o
GG              ˜     ΣΠ-functor
GG
GG 
5      o
C          ΣΠ-category

ΣΠ-functor:
preserves ﬁnite products and ﬁnite coproducts,

8/26
ΣΠ(A), the free ΣΠ category on A

free ΣΠ-category generated by A:
η
A GG               G ΣΠ(A)
GG                    
GG
GG
GGF
GG
GG            ∃!F o
GG              ˜     ΣΠ-functor
GG
GG 
5      o
C          ΣΠ-category

ΣΠ-functor:
preserves ﬁnite products and ﬁnite coproducts,

8/26
CS01 : a Lambek-style construction of ΣΠ(A)
f
→
x− y
η(f )
−→
η(x) − − η(x)
−
!
R1
→
X − 1
f                            f                       g
→
Xi − A                          →
X − A                      →
X − B
Li ×                                   R×
πi (f )                        f ,g
−−
X0 × X1 − − → A                       −−
X −−→ A × B

−
?
L0
→
0− A
f                    g                           f
X − A
→         Y − A
→                            →
X − Aj
L+                                   Rj +
{f ,g }                      σj (f )
X +Y −−→ A
−−                             −−
X − − → A0 + A1

9/26
CS01: conﬂuence modulo equations

Proposition
The cut-elimination procedure is conﬂuent modulo the equations:

πi ( f , g ) =    πi (f ), πi (g )   σj ({f , g }) = {σj (f ), σj (g )}

πi (σj (f )) = σj (πi (f ))

{ f11 , f12 , f21 , f22 } = {f11 , f21 }, {f12 , f22 }

πi (!) = !                            σj (?) = ?

{!, !} = !                              ?, ?     = ?

!0 =?1

10/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
→
1. The functor η : A − ΣΠ(A) is full and faithful.
2. Generators are atomic:
→
[ η(a), Yj ] − [ η(a),             Yj ]
j                                     j

→
[ Xi , η(b) ] − [       Xi , η(b) ]
i                           i

are isomorphisms.
3. ΣΠ(A) is soft:

i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]

                                                        
i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

is a pushout.
11/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
→
1. The functor η : A − ΣΠ(A) is full and faithful.
2. Generators are atomic:
→
[ η(a), Yj ] − [ η(a),             Yj ]
j                                     j

→
[ Xi , η(b) ] − [       Xi , η(b) ]
i                           i

are isomorphisms.
3. ΣΠ(A) is soft:

i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]

                                                        
i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

is a pushout.
11/26
Abstract characterization of free ΣΠ-cats [CS01, J95]
→
1. The functor η : A − ΣΠ(A) is full and faithful.
2. Generators are atomic:
→
[ η(a), Yj ] − [ η(a),             Yj ]
j                                     j

→
[ Xi , η(b) ] − [       Xi , η(b) ]
i                           i

are isomorphisms.
3. ΣΠ(A) is soft:

i,j [ Xi , Yj          ]                       G        j[     i       Xi , Yj ]

                                                        
i [ Xi ,           j   Yj ]                     G[        i   Xi ,         j   Yj ]

is a pushout.
11/26
Characterization of free ΣΠ-categories (cont.)

Theorem
The pair (η, ΣΠ(A)) satisﬁes 1,2,3.

→
If C is a ΣΠ-category “generated” by A, F : A − C,
and (F , C) satisﬁes 1,2,3,
ˆ      →
then the extension F : ΣΠ(A) − C is an equivalence.

12/26
Outline

Introductory remarks
ΣΠ-categories and their theory
Context, some motivations
The free ΣΠ-category

“En route” towards a decision procedure
The CT2006 results

Softness, cardinals, and (some hints on) a decision procedure
Understanding softness
Pushouts-pullbacks, bouncing

13/26
A result presented at CT06

Theorem
In ΣΠ(A) coproducts are weakly disjoint:
g
X                    G Y1

f                           σ1

       σ0           
Y0                  G Y0 + Y1

14/26
A result presented at CT06

Theorem
In ΣΠ(A) coproducts are weakly disjoint:
g
X                                  jGR Y1
∃h                  j jjjj
'           jj?Y1
j
jjjj
0
f                                          σ1

? Y0

 ×                  σ0             
Y0                                   G Y0 + Y1

14/26
. . . and some Corollaries

decide in linear time whether
an object of ΣΠ(A) is isomorphic to 0 or 1,
an arrow of ΣΠ(A) factors through 0 and or 1.

a simple characterization of monic coproduct injections:

→
σ0 : A − A + B

is monic iﬀ either B is not pointed or A is pointed.

15/26
. . . and some Corollaries

decide in linear time whether
an object of ΣΠ(A) is isomorphic to 0 or 1,
an arrow of ΣΠ(A) factors through 0 and or 1.

a simple characterization of monic coproduct injections:

→
σ0 : A − A + B

is monic iﬀ either B is not pointed or A is pointed.

15/26
. . . and some Corollaries

decide in linear time whether
an object of ΣΠ(A) is isomorphic to 0 or 1,
an arrow of ΣΠ(A) factors through 0 and or 1.

a simple characterization of monic coproduct injections:

→
σ0 : A − A + B

is monic iﬀ either B is not pointed or A is pointed.

15/26
. . . and some Corollaries

decide in linear time whether
an object of ΣΠ(A) is isomorphic to 0 or 1,
an arrow of ΣΠ(A) factors through 0 and or 1.

a simple characterization of monic coproduct injections:

→
σ0 : A − A + B

is monic iﬀ either B is not pointed or A is pointed.

15/26
Outline

Introductory remarks
ΣΠ-categories and their theory
Context, some motivations
The free ΣΠ-category

“En route” towards a decision procedure
The CT2006 results

Softness, cardinals, and (some hints on) a decision procedure
Understanding softness
Pushouts-pullbacks, bouncing

16/26
Softness: the focus of a decision procedure

A decision procedure focuses on the homset [ X × Y , A + B ].

For example:
→
l e t equal f g : X − A × B =
let
f = f1 , f2
and
g = g1 , g2
in
e q u a l f1 g1 && e q u a l f2 g2

17/26
Softness: the focus of a decision procedure

A decision procedure focuses on the homset [ X × Y , A + B ].

For example:
→
l e t equal f g : X − A × B =
let
f = f1 , f2
and
g = g1 , g2
in
e q u a l f1 g1 && e q u a l f2 g2

17/26
Understanding softness
The homset [ X × Y , A + B ] is . . .

18/26
Understanding softness
The homset [ X × Y , A + B ] is . . .

the pushout

[X,A] + [X,B ] + [Y,A] + [Y,B ]         G [X × Y,A] + [X × Y,B ]

                                
[X,A + B ] + [Y,A + B ]               G [X × Y,A + B ]

18/26
Understanding softness
The homset [ X × Y , A + B ] is . . .

the colimit of the “diagram of cardinals”:
π0                     σ0
[X × Y,A] o                 [X,A]              G [X,A + B ]
y                                        y

π1                                             σ1

[Y,A]                                          [X,B ]

σ0                                             π0

          σ1                     π1

[Y,A + B ] o                [Y,B ]             G [X × Y,B ]

18/26
Understanding softness
The homset [ X × Y , A + B ] is . . .

the quotient of

[X,A + B ] + [Y,A + B ] + [X × Y,A] + [X × Y,B ]

under the equivalence relation generated by elementary pairs (f , g ):

h ∈ [X,A]
mm         QQQ
π0 mmm              QQQσ0
mmmm                    QQQ
QQQ
vmmm                              @
f ∈ [X × Y,A]                             g ∈ [X,A + B ]

f = π0 (h)                                σ0 (h) = g

18/26
(In)deﬁnite arrows

Deﬁnition
An arrow is indeﬁnite if it factors through 0 or through 1.
Otherwise, it is deﬁnite.

A simple decision procedure for indeﬁnite maps:
l e t equal f g = (∗f g f a c t o r through 0∗)
let
f = f ;?
and
g = g ;?
in
i f cod f i s p o i n t e d t h e n t r u e
else
f =g

19/26
(In)deﬁnite arrows

Deﬁnition
An arrow is indeﬁnite if it factors through 0 or through 1.
Otherwise, it is deﬁnite.

A simple decision procedure for indeﬁnite maps:
l e t equal f g = (∗f g f a c t o r through 0∗)
let
f = f ;?
and
g = g ;?
in
i f cod f i s p o i n t e d t h e n t r u e
else
f =g

19/26
A useful Lemma

If
π : f = f0 g1 f2 . . . gn = g
is a path of elementary pairs crossing a corner,
then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite.

Consider

. . . fi−1 = π0 (hi ) σ0 (hi ) = gi   gi = σ1 (hi+1 ) π0 (hi+1 )   = fi+1 . . .

then hi and hi+1 are copointed and [g ] = [f ] as well.

Lemma
∗
→
If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of
elementary pairs, then π “bounces” along one side of this diagram.

20/26
A useful Lemma

If
π : f = f0 g1 f2 . . . gn = g
is a path of elementary pairs crossing a corner,
then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite.

Consider

. . . fi−1 = π0 (hi ) σ0 (hi ) = gi   gi = σ1 (hi+1 ) π0 (hi+1 )   = fi+1 . . .

then hi and hi+1 are copointed and [g ] = [f ] as well.

Lemma
∗
→
If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of
elementary pairs, then π “bounces” along one side of this diagram.

20/26
A useful Lemma

If
π : f = f0 g1 f2 . . . gn = g
is a path of elementary pairs crossing a corner,
then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite.

Consider

...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

then hi and hi+1 are copointed and [g ] = [f ] as well.

Lemma
∗
→
If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of
elementary pairs, then π “bounces” along one side of this diagram.

20/26
A useful Lemma

If
π : f = f0 g1 f2 . . . gn = g
is a path of elementary pairs crossing a corner,
then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite.

Consider

...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

then hi and hi+1 are copointed and [g ] = [f ] as well.

Lemma
∗
→
If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of
elementary pairs, then π “bounces” along one side of this diagram.

20/26
A useful Lemma

If
π : f = f0 g1 f2 . . . gn = g
is a path of elementary pairs crossing a corner,
then [f ] = [g ] ∈ [ X × Y , A + B ] is indeﬁnite.

Consider

...                σ0 (hi ) = gi   gi = σ1 (hi+1 )                    ...

then hi and hi+1 are copointed and [g ] = [f ] as well.

Lemma
∗
→
If [f ] ∈ [ X × Y , A + B ] is deﬁnite and π : f − g is a path of
elementary pairs, then π “bounces” along one side of this diagram.

20/26
Softness for deﬁnite maps
The previous Lemma transforms – for deﬁnite maps – the cardinal
diagram from

π0                   σ0
[X × Y,A] o               [X,A]                G [X,A + B ]
y                                        y

π1                                             σ1

[Y,A]                                          [X,B ]

σ0                                             π0

         σ1                   π1

[Y,A + B ] o              [Y,B ]               G [X × Y,B ]

21/26
Softness for deﬁnite maps
The previous Lemma transforms – for deﬁnite maps – the cardinal
diagram to

π0                 σ0
[X × Y,A] o             [X,A]              G [X,A + B ]

[X × Y,A]                                  [X,A + B ]
y                                         y
π1                                        σ1

[Y,A]                                     [X,B ]
σ0                                        π0
                                     
[Y,A + B ]                                 [X × Y,B ]

σ1                 π1
[Y,A + B ] o            [Y,B ]             G [X × Y,B ]

21/26
A new main result
Theorem
A pushout diagram

[X,A]
q                MMM
π0 qqq                    MMσ0
qqq                         MMM
xqqq                               MM8
[X × Y,A]                                   [X,A + B ]

8        x
Po

is also a pullback.
∗
→
That is: given π : f − g a path “bouncing” on the upper row,
there exists a unique h such that

f = π0 (h) ,                          σ0 (h) = g .

22/26
A new main result
Theorem
A pushout diagram

[X,A]
q                MMM
π0 qqq                    MMσ0
qqq                         MMM
xqqq                               MM8
[X × Y,A]                                   [X,A + B ]

8        x
Po

is also a pullback.
∗
→
That is: given π : f − g a path “bouncing” on the upper row,
there exists a unique h such that

f = π0 (h) ,                          σ0 (h) = g .

22/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

let equivalent f g =
(∗ f g a r e both d e f i n i t e
→           →
f : X × Y − A , g : X − A + B ∗)
find h s . t .
f = σ0 (h) && π0 (h) = g

Uniqueness of such h “makes it is easy” to ﬁnd it.

Theorem
There exists an algorithm to decide whether two parallel arrow-terms f , g
of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in

23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

let equivalent f g =
(∗ f g a r e both d e f i n i t e
→           →
f : X × Y − A , g : X − A + B ∗)
find h s . t .
f = σ0 (h) && π0 (h) = g

Uniqueness of such h “makes it is easy” to ﬁnd it.

Theorem
There exists an algorithm to decide whether two parallel arrow-terms f , g
of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in

23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

let equivalent f g =
(∗ f g a r e both d e f i n i t e
→           →
f : X × Y − A , g : X − A + B ∗)
find h s . t .
f = σ0 (h) && π0 (h) = g

Uniqueness of such h “makes it is easy” to ﬁnd it.

Theorem
There exists an algorithm to decide whether two parallel arrow-terms f , g
of ΣΠ(A) are equal in [ X , A ]. The procedure runs in time polynomial in

size(X ) · size(A) .

23/26
Deciding whether [f ] = [g ] in [ X0 × X1 , A0 + A1 ]

let equivalent f g =
(∗ f g a r e both d e f i n i t e
→           →
f : X × Y − A , g : X − A + B ∗)
find h s . t .
f = σ0 (h) && π0 (h) = g

Uniqueness of such h “makes it is easy” to ﬁnd it.

Theorem
There exists an algorithm to decide whether two parallel arrow-terms f , g
of ΣΠ(∅) are equal in [ X , A ]. The procedure runs in time linear in

(hgt(X ) + hgt(A)) · size(X ) · size(A) .

23/26
Background on bouncers
Lemma
Consider a pushout in Set

A AA
}            AA g
}}               AA
}}
f
}                     AA
~}}                        2
B                                  C

2        ~
Po

TFAE:
the diagram is a weak pullback,
Ker (f ) and Ker (g ) commute.
A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that
(a0 , a1 ) ∈ Ker (f ) ,                      (a1 , a2 ) ∈ Ker (g ) .
24/26
Background on bouncers
Lemma
Consider a pushout in Set

A AA
}            AA g
}}               AA
}}
f
}                     AA
~}}                        2
B                                  C

2        ~
Po

TFAE:
the diagram is a weak pullback,
Ker (f ) and Ker (g ) commute.
A bouncer for (a0 , a2 ) ∈ A is a1 ∈ A such that
(a0 , a1 ) ∈ Ker (f ) ,                      (a1 , a2 ) ∈ Ker (g ) .
24/26
Bouncers within ΣΠ(A)

A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that

f     GA
wY X              FF
π0 www                    FF σ
w                         FF 0
ww                            FF
ww                                 F4
X ×Y                    h             A` + B
GG                                 x
GG                            xx
GG                        xx
π0 GGG                     xσ
5              # xxx 0
X          GA
g

25/26
Bouncers within ΣΠ(A)

A bouncer for (f , g ) ∈ [ X , A ] is an h ∈ [ X , A ] such that

f     GA
x`
X              EE
π0 xxx                    EE σ
x                         EE 0
xx                            EE
xx                                 E4
X ×0                    h             A+1
FF                                 y`
FF                            yy
FF                        yy
π0 FFF                     yσ
4              # yyy 0
X          GA
g

25/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

z    π0 (?)
f =
σ0 (!) idx

: (0 × 0) + x              G (1 + 1) × x

z    π1 (?)
g=
σ1 (!) idx

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

z    π0 (?)
f =
σ0 (!) idx
z    π1 (?)                              G (1 + 1) × x
h=                     : (0 × 0) + x
σ0 (!) idx
z    π1 (?)
g=
σ1 (!) idx

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

z    π0 (?)
f =
σ0 (!) idx
z    π0 (?)                               G (1 + 1) × x
h =                    : (0 × 0) + x
σ1 (!) idx
z    π1 (?)
g=
σ1 (!) idx

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

z    π0 (?)
f =
σ0 (!) idx
z    π0 (?)                               G (1 + 1) × x
h =                    : (0 × 0) + x
σ1 (!) idx
z    π1 (?)
g=
σ1 (!) idx

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26
A minimal nontrivial bouncer
In ΣΠ(∅) all bouncer are trivial. This is not the case in general:

z    π0 (?)
f =
σ0 (!) idx
z    π0 (?)                               G (1 + 1) × x
h =                    : (0 × 0) + x
σ1 (!) idx
z    π1 (?)
g=
σ1 (!) idx

Proposition
In ΣΠ(A) (f , g ) have a bouncer if and only if (g , f ) have a
bouncer.

Proposition
In ΣΠ(A) there exists at most one bouncer for (f , g ).
26/26

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