SLIDE 1
Proof nets for sumproduct logic Willem Heijltjes LFCS School of - - PowerPoint PPT Presentation
Proof nets for sumproduct logic Willem Heijltjes LFCS School of - - PowerPoint PPT Presentation
Proof nets for sumproduct logic Willem Heijltjes LFCS School of Informatics University of Edinburgh Kananaskis, 11-12 June 2011 In part 2 Recap The soundness proof The full net calculus a A B 0 1 f f 0 + + g 1 f f
SLIDE 2
SLIDE 3
The full net calculus
A a B 1 f + ι0 ι1 f + g f × g f × π0 π1
SLIDE 4
Equivalence and saturation
+
⇔
+ +
SLIDE 5
Equivalence and saturation
×
⇔
× ×
SLIDE 6
Equivalence and saturation
+ 1
⇔
+ 1 + 1
SLIDE 7
Equivalence and saturation
× 1
⇔
× 1 × 1
SLIDE 8
The soundness proof
To show: σR = σS ⇒ R ⇔ S
SLIDE 9
The soundness proof: first intuition
Saturation allows induction on paths in (
∗)
R R′ R′′ . . . σR = σS . . . S′′ S′ S For each step in ( ) there is a corresponding one in (⇔)
SLIDE 10
The soundness proof: first intuition
Saturation allows induction on paths in (
∗)
R R′ R′′ . . . σR = σS . . . S′′ S′ S For each step in ( ) there is a corresponding one in (⇔) R ⇔ R0 ⇔ R1 ⇔ . . . ⇔ Rm ?? Sn ⇔ . . . ⇔ S1 ⇔ S0 ⇔ S But this only shifts the problem: how to show that σR = σS gives Rm ⇔ Sn ?
SLIDE 11
’Desaturation’ is not trivial
× × + 1 × × + 1 × × + 1
S ⊆ σR does not necessarily mean σS = σR
SLIDE 12
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 13
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 14
Atoms and units
A
R,S
− → Y X
R,S
− → A 1
R,S
− → Y X
R,S
− → 0
A + 1 ×
In these cases no rewrite rules apply, and R = σR = σS = S
SLIDE 15
Atoms and units
Nets corresponding to initial and terminal maps.
+ ⇒ + × ⇒ ×
If X = 0 then X
R
− → Y ⇒∗
∗⇐
X
S
− → Y Similar for Y = 1
SLIDE 16
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 17
Coproduct source or product target
+ 1 ⇒ + 1 + R
⇒∗
R′ + R′′ + S
⇒∗
S′ + S′′
SLIDE 18
Coproduct source or product target
+ 1 + 1 σR′ + σR′′ ∗ σR′ + σR′′
= σR
σS′ + σS′′ ∗ σS′ + σS′′
= σS σR = σS means that σR′ = σS′ and σR′′ = σS′′
SLIDE 19
Coproduct source or product target
σR = σS means that σR′ = σS′ and σR′′ = σS′′
+ R
⇒∗
R′ + R′′
- +
S
⇒∗
S′ + S′′
SLIDE 20
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 21
Nets from a product into a coproduct
× + × + A A + × × 1 × + 1
SLIDE 22
Nets from a product into a coproduct
× + × + A A + × × 1 × + 1
SLIDE 23
Nets from a product into a coproduct
× + × + A A + × × 1 × + 1
SLIDE 24
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 25
Initial and terminal nets
Call nets 0
R
− → Y initial and X
S
− → 1 terminal
A + × 1 ∗ A + × 1
σR and σS are full: they have all possible unit links (but no atomic links)
SLIDE 26
Points and copoints
Points and copoints are maps out of 1 and into 0 respectively X
!
− → 1
p
− → P Q
q
− → 0
?
− → Y (co)pointed maps are those that factor through a (co)point (co)pointed objects are those that admit (co)points P := 1 | P×P | P+Y | Y +P Q := 0 | Q+Q | Q×X | X ×Q
SLIDE 27
Points and copoints
Points and copoints are maps out of 1 and into 0 respectively X
!
− → 1
p
− → P Q
q
− → 0
?
− → Y (co)pointed maps are those that factor through a (co)point (co)pointed objects are those that admit (co)points P := 1 | P×P | P+Y | Y +P Q := 0 | Q+Q | Q×X | X ×Q (co)pointed nets will be those consisting of rooted unit links
1 p p′ × p +
SLIDE 28
Points and copoints
Pointed nets may rewrite by moving their links in parallel
× 1 ⇔ × 1 + 1 ⇔ + 1 × p
⇔
p′ × × p
⇔
p′ + p′
(all links in p and p′ connect to the left root)
SLIDE 29
Points and copoints
A A + + B 1 × × 1 1
SLIDE 30
Points and copoints
A A + + B 1 × × 1 1
SLIDE 31
Points and copoints
A A + + B 1 × × 1 1
SLIDE 32
Points and copoints
A A + + B 1 × × 1 1
SLIDE 33
Points and copoints
A A + + B 1 × × 1 1
SLIDE 34
Points and copoints
A A + + B 1 × × 1 1
SLIDE 35
Points and copoints
A A + + B 1 × × 1 1
SLIDE 36
Points and copoints
A A + + B 1 × × 1 1
SLIDE 37
Points and copoints
A A + + B 1 × × 1 1
SLIDE 38
Bipointed nets
Bipointed maps (or disconnects) are both pointed and copointed. There is exactly one b : Q → P for copointed Q and pointed P, and none for other X, Y . Q
q ! ! ? ?
1
p
P A bipointed net is one Q
q
− → P (copointed) or Q
p
− → P (pointed)
q p
SLIDE 39
Bipointed nets
Two parallel bipointed nets are always equivalent
1 + + 1 × × 1
SLIDE 40
Bipointed nets
1 + + 1 × × 1
SLIDE 41
Bipointed nets
1 + + 1 × × 1
SLIDE 42
Bipointed nets
1 + + 1 × × 1
SLIDE 43
Bipointed nets
1 + + 1 × × 1
SLIDE 44
Bipointed nets
1 + + 1 × × 1
SLIDE 45
Bipointed nets
1 + + 1 × × 1
SLIDE 46
Bipointed nets
1 + + 1 × × 1
SLIDE 47
Bipointed nets
1 + + 1 × × 1
SLIDE 48
Bipointed nets
1 + + 1 × × 1
SLIDE 49
Bipointed nets
The saturation of a bipointed net is full
1 + + 1 × × 1
SLIDE 50
Bipointed nets
1 + + 1 × × 1
SLIDE 51
Bipointed nets
1 + + 1 × × 1
SLIDE 52
Bipointed nets
1 + + 1 × × 1
SLIDE 53
Bipointed nets
1 + + 1 × × 1
SLIDE 54
Bipointed nets
1 + + 1 × × 1
SLIDE 55
Bipointed nets
1 + + 1 × × 1
SLIDE 56
Bipointed nets
1 + + 1 × × 1
SLIDE 57
Bipointed nets
1 + + 1 × × 1
SLIDE 58
Bipointed nets
1 + + 1 × × 1
SLIDE 59
Bipointed nets
1 + + 1 × × 1
SLIDE 60
Bipointed nets
1 + + 1 × × 1
SLIDE 61
Bipointed nets
1 + + 1 × × 1
SLIDE 62
Bipointed nets
1 + + 1 × × 1
SLIDE 63
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 64
Inductive saturation
Saturated nets need not factor through injections/projections
A A + × × 1 × + 1
SLIDE 65
Inductive saturation
Saturation of X
R′
− → Y
ι0
− → Y + Z = X
R
− → Y + Z
× σR′ + ∗ × σR +
SLIDE 66
Inductive saturation
Saturation of X
R′
− → Y
ι0
− → Y + Z = X
R
− → Y + Z
× σR′ + ∗ × σR + q + ∗ q q′ q′′ +
SLIDE 67
Inductive saturation
Saturation of X
R′
− → Y
ι0
− → Y + Z = X
R
− → Y + Z
× σR′ + ∗ × σR + q + ∗ q q′ q′′ + q′′ ∗ full
SLIDE 68
Inductive saturation
A A + × × 1 × + 1
SLIDE 69
Inductive saturation
A A + × × 1 × + 1
SLIDE 70
Inductive saturation
A A + × × 1 × + 1
SLIDE 71
Inductive saturation
A A + × × 1 × + 1
SLIDE 72
Inductive saturation
A A + × × 1 × + 1
SLIDE 73
Inductive saturation
A A + × × 1 × + 1
SLIDE 74
Inductive saturation
A A + × × 1 × + 1
SLIDE 75
Inductive saturation
A A + × × 1 × + 1
SLIDE 76
Inductive saturation
A A + × × 1 × + 1
SLIDE 77
Inductive saturation
A A + × × 1 × + 1
SLIDE 78
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 79
Matching injections and projections
Equivalent nets may factor through different injections or projections, but to allow induction nets must at least have the same domain and codomain.
× R + × + S T × +
Idea: ‘highest’ links, and in particular rooted links, are most significant (downward movement in saturation is unrestricted)
SLIDE 80
If σR contains a rooted link, so does some S ⇔ R
A A + × × 1 × + 1
SLIDE 81
If σR contains a rooted link, so does some S ⇔ R
A A + × × 1 × + 1
SLIDE 82
If σR contains a rooted link, so does some S ⇔ R
A A + × × 1 × + 1
SLIDE 83
If σR contains a rooted link, so does some S ⇔ R
A A + × × 1 × + 1
SLIDE 84
If σR contains a rooted link, so does some S ⇔ R
A A + × × 1 × + 1
SLIDE 85
A A + × × 1 × + 1
SLIDE 86
A A + × × 1 × + 1
SLIDE 87
A A + × × 1 × + 1
SLIDE 88
A A + × × 1 × + 1
SLIDE 89
A A + × × 1 × + 1
SLIDE 90
A A + × × 1 × + 1
SLIDE 91
A A + × × 1 × + 1
SLIDE 92
A A + × × 1 × + 1
SLIDE 93
A A + × × 1 × + 1
SLIDE 94
A A + × × 1 × + 1
SLIDE 95
A A + × × 1 × + 1
SLIDE 96
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 97
Injections into pointed objects
× R′ + × S′ + × σR=σS + q + q′ + full +
SLIDE 98
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 99
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 100
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 101
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 102
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 103
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 104
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 105
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 106
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 107
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 108
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 109
Injections into pointed objects
A A + × × 1 × + 1
SLIDE 110
Injections into pointed objects
A A + × × 1 × + 1
Now the induction hypothesis can be applied
SLIDE 111
The soundness proof
To prove: X
R
− → Y ⇔ X
S
− → Y given σR = σS
◮ One of X and Y is an atom or unit ◮ X is a coproduct or Y a product ◮ X is a product and Y a coproduct × +
◮ Some dynamics of rewriting and saturation ◮ Saturated nets need not factor through injections/projections ◮ R and S may factor through different injections/projections ◮ σR = σS after, but not before, adding an injection/projection
SLIDE 112