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Episode 15. Cirquent calculus. About cirquent calculus in general The language of CL5 Cirquents Cirquents as circuits Formulas as cirquents Operations on cirquents The rules of inference of CL5 The soundness and completeness of CL5

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cirquent calculus

Episode 15

Cirquent calculus
  • About cirquent calculus in general
  • The language of CL5
  • Cirquents
  • Cirquents as circuits
  • Formulas as cirquents
  • Operations on cirquents
  • The rules of inference of CL5
  • The soundness and completeness of CL5
  • A cirquent calculus system for classical logic
  • CL5 versus affine logic

0

about cirquent calculus in general

15.1

About cirquent calculus in general

Cirquent calculus is a new proof-theoretic approach, introduced

recently in “Introduction to cirquent calculus and abstract resource

semantics”. Its invention was motivated by the needs of computability

logic, which had stubbornly resisted any axiomatization attempts within

the framework of the traditional proof-theoretic approaches such as

sequent calculus or Hilbert-style systems.

The main distinguishing feature of cirquent calculus from the known

approaches is sharing: it allows us to account for the possibility of shared

resources (say, formulas) between different parts of a proof tree.

The version of cirquent calculus presented here can be called shallow

as it limits cirquents to depth 2. Deep versions of cirquent calculus, with

no such limits, are being currently developed.

the language of cl5

15.2

The language of CL5

The cirquent calculus system that we consider here is called CL5.

CL5 axiomatizes the fragment of computability logic where all letters

are general and 0-ary. And the only logical operators are ,  and .

Furthermore, as in systems G1, G2 and G3 (Episodes 4 and 5),  is

only allowed on atoms (if this condition is not satisfied, the formula

should be rewritten into an equivalent one using the double negation and

DeMorgan’s principles). And FG is understood as an abbreviation of

EF.

We agree that, throughout this episode, “formula” exclusively means

a formula of the above fragment of the language of computability logic.

CL5 has 7 rules of inference: Identity, Mix, Exchange, Weakening,

Duplication, -Introduction and -Introduction. We present those rules,

as well as the concept of a cirquent, very informally through examples

and illustrations. More formal definitions, if needed, can be found in

“Introduction to cirquent calculus and abstract resource semantics”.

slide4

Cirquents

15.3

F G H F

Formulas

Arcs

Groups

Every formula should be in (= connected with an arc to) at least one group.

slide6

Cirquents as Circuits

15.4

F G H F

Circuit

Sequent

slide7

Cirquents as Circuits

15.4

F G H F

Cir

quent

slide8

Cirquents as Circuits

15.4

F G H F

Cir

quent

slide9

Cirquents as Circuits

sequent

sequent

sequent

F G H F

Circuit

15.4

F G H F

Cir

quent

slide10

Operations on Cirquents

F G H F

15.5

F G H F

Merging groups (merginggroups #1 and #2):

slide11

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide12

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide13

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide14

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide15

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide16

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

slide17

Operations on Cirquents

F

G

H

F

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

slide18

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide19

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide20

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide21

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide22

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide23

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

G

H

F

slide24

Operations on Cirquents

15.5

F G H F

Merging groups (merginggroups #1 and #2):

F G H F

Merging formulas (merging G and H into E):

F

E

F

slide25

Identity

I

F F

15.6

slide26

Mix

F F

G G

15.7

Put one cirquent next to the other

slide27

Mix

F F

G G

F F

G G

15.7

M

Put one cirquent next to the other

slide28

Mix

F F

G G

15.7

M

F F

G G

Put one cirquent next to the other

slide29

Mix

F F

G G

15.7

M

F F

G G

Put one cirquent next to the other

slide30

Mix

F F

G G

15.7

M

F F

G G

Put one cirquent next to the other

slide31

Mix

F F

G G

15.7

M

F F

G G

Put one cirquent next to the other

slide32

Exchange

F F

G G

15.8

Swap two adjacent formulas or groups

slide33

Exchange

F F

G G

F F

G G

15.8

E

Swap two adjacent formulas or groups

slide34

Exchange

F F

G G

15.8

E

F F

G

G

Swap two adjacent formulas or groups

slide35

Exchange

F F

G G

15.8

E

F F

G

G

Swap two adjacent formulas or groups

slide36

Exchange

F F

G G

15.8

E

F F

G

G

Swap two adjacent formulas or groups

slide37

Exchange

F F

G G

E

F

F

G

G

15.8

E

F F

G

G

Swap two adjacent formulas or groups

slide38

Exchange

F F

G G

E

15.8

E

F F

G

G

F

F

G

G

Swap two adjacent formulas or groups

slide39

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

Swap two adjacent formulas or groups

slide40

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

Swap two adjacent formulas or groups

slide41

Exchange

F F

G G

E

E

F

G

F

G

15.8

E

F F

G

G

F

G

F

G

Swap two adjacent formulas or groups

slide42

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

E

F

G

F

G

Swap two adjacent formulas or groups

slide43

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

E

F

G

F

G

Swap two adjacent formulas or groups

slide44

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

E

F

G

F

G

Swap two adjacent formulas or groups

slide45

Exchange

F F

G G

E

15.8

E

F F

G

G

F

G

F

G

E

F

G

F

G

Swap two adjacent formulas or groups

slide46

Weakening

E

F

G

H

W

E

F

G

H

15.9

Delete any arc in any group of the conclusion;

if this leaves some formula “homeless”, delete that formula as well

slide47

Weakening

E

F

G

H

15.9

W

E

F

G

H

W

E

F

G

H

Delete any arc in any group of the conclusion;

if this leaves some formula “homeless”, delete that formula as well

slide48

Weakening

15.9

E

F

G

H

W

E

F

G

H

W

E

F

G

H

Delete any arc in any group of the conclusion;

if this leaves some formula “homeless”, delete that formula as well

slide49

Duplication

E

F

G

H

15.10

Replace a group with two identical copies

slide50

Duplication

E

F

G

H

D

E

F

G

H

15.10

Replace a group with two identical copies

slide51

Duplication

E

F

G

H

D

15.10

E

F

G

H

Replace a group with two identical copies

slide52

Duplication

E

F

G

H

D

15.10

E

F

G

H

Replace a group with two identical copies

slide53

Duplication

E

F

G

H

D

15.10

E

F

G

H

Replace a group with two identical copies

slide54

-Introduction

E

F

G

H

15.11

Merge two adjacent formulas F and G into FG

slide55

-Introduction

E

F

G

H

E

F

G

H

15.11

Merge two adjacent formulas F and G into FG

slide56

-Introduction

E

F

G

H

15.11

E

F

G

H

Merge two adjacent formulas F and G into FG

slide57

-Introduction

E

F

G

H

15.11

E

F

G

H

Merge two adjacent formulas F and G into FG

slide58

-Introduction

E

F

G

H

15.11

E

F

G

H

Merge two adjacent formulas F and G into FG

slide59

-Introduction

E

F

G

H

E

F

G

H

15.12

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide60

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide61

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide62

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide63

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide64

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide65

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide66

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide67

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide68

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide69

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide70

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide71

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide72

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide73

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide74

-Introduction

E

F

G

H

15.12

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide75

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide76

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide77

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide78

-Introduction

15.12

E

F

G

H

E

F

G

H

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide79

-Introduction

E

F

G

H

K

E

F

G

H

K

15.12

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide80

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide81

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide82

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide83

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide84

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide85

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide86

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide87

-Introduction

15.12

E

F

G

H

K

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide88

-Introduction

15.12

E

F

G

H

K

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide89

-Introduction

15.12

E

F

G

H

K

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide90

-Introduction

15.12

E

F

G

H

K

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide91

-Introduction

15.12

E

F

G

H

K

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide92

-Introduction

E

F

G

H

K

15.12

E

F

G

H

K

In the premise:

  • F and G are adjacent formulas, and no group contains both of them together;
  • Every group containing F is immediately followed by a group containing G, and vice

versa: every group containing G is immediately preceded by a group containing F.

To obtain the conclusion:

  • Merge each group that contains F with its right neighbor (that contains G);
  • Then merge F and G into FG.
slide93

Proof of Blass’s Principle

15.13

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide94

Proof of Blass’s Principle

15.13

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide95

Proof of Blass’s Principle

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

15.13

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide96

Proof of Blass’s Principle

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

15.13

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide97

Proof of Blass’s Principle

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

15.13

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide98

Proof of Blass’s Principle

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

15.13

P

Q

R

S

(

R

)

(

Q

S

)

P

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide99

Proof of Blass’s Principle

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

15.13

P

Q

R

S

(

R

)

(

Q

S

)

P

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide100

Proof of Blass’s Principle

15.13

P

Q

R

S

(

R

)

(

Q

S

)

P

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide101

Proof of Blass’s Principle

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

15.13

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide102

Proof of Blass’s Principle

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

15.13

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide103

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide104

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

P

P

Q

Q

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide105

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide106

Proof of Blass’s Principle

P

Q

R

S

R

Q

S

P

15.13

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide107

Proof of Blass’s Principle

P

Q

R

S

R

Q

S

P

15.13

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide108

Proof of Blass’s Principle

P

Q

R

S

R

Q

S

P

15.13

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide109

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide110

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide111

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide112

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide113

Proof of Blass’s Principle

15.13

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide114

Proof of Blass’s Principle

P

P

P

P

Q

Q

Q

Q

R

R

R

R

S

S

S

S

15.13

I I I I

M…M

E…E

P

Q

R

S

R

Q

S

P

P

Q

R

S

R

Q

S

P

P

P

Q

Q

R

R

S

S

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

(

P

P

Q

Q

)

)

(

(

R

R

S

S

)

)

(

(

R

R

)

)

(

(

Q

Q

S

S

)

)

P

P

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P

slide115

Soundness and completeness

15.14

Theorem 15.1. For any formula F, the following statements are

equivalent:

(i) F is provable in CL5.

(ii) F is valid.

(iii) F is uniformly valid.

Furthermore, there is an effective procedure that takes a CL5-proof of

any formula F and constructs a uniform solution for F. We call this

property of CL5 (and the same property of any other deductive system)

uniform-constructive soundness.

slide116

CL5 versus classical logic

15.15

Remember that we see the atoms of classical logic as 0-ary elementary letters, while,

on the other hand, the atoms of CL5 are 0-ary general letters.

Let us for now disregard this difference and see no distinction between the two sorts

of atoms. That is, let us see the formulas of CL5 as formulas of propositional classical

logic. An interesting question to ask then is how CL5 compares with classical logic.

Here is an answer:

Fact 15.2. Every formula provable in CL5 is a tautology of classical logic, but not

vice versa: some tautologies are not provable in CL5 (and hence not valid in

computability logic when their atoms are seen as general atoms).

The simplest example of a tautology not provable in CL5 is P(PP).

Indeed, this formula (cirquent) could only be derived by -Introduction from the

premise

 P PP

With a little thought one can see that the above cirquent, in turn, cannot be derived.

slide117

A cirquent-calculus system for classical logic

E

F

F

H

15.16

The next question to ask is how to strengthen CL5 so that it proves all tautologies

(and only tautologies). The answer turns out to be very simple. All it takes to extend

CL5 to a sound and complete system for classical logic is to add to it the contraction

rule:

Contraction

Merge two adjacent and identical formulas F and F into F

slide118

A cirquent-calculus system for classical logic

E

F

F

H

C

E

F

F

H

15.16

The next question to ask is how to strengthen CL5 so that it proves all tautologies

(and only tautologies). The answer turns out to be very simple. All it takes to extend

CL5 to a sound and complete system for classical logic is to add to it the contraction

rule:

Contraction

Merge two adjacent and identical formulas F and F into F

slide119

A cirquent-calculus system for classical logic

E

F

F

H

C

15.16

The next question to ask is how to strengthen CL5 so that it proves all tautologies

(and only tautologies). The answer turns out to be very simple. All it takes to extend

CL5 to a sound and complete system for classical logic is to add to it the contraction

rule:

Contraction

E

F

F

H

Merge two adjacent and identical formulas F and F into F

slide120

A cirquent-calculus system for classical logic

E

F

F

H

C

15.16

The next question to ask is how to strengthen CL5 so that it proves all tautologies

(and only tautologies). The answer turns out to be very simple. All it takes to extend

CL5 to a sound and complete system for classical logic is to add to it the contraction

rule:

Contraction

E

F

F

H

Merge two adjacent and identical formulas F and F into F

slide121

A cirquent-calculus system for classical logic

E

F

F

H

C

15.16

The next question to ask is how to strengthen CL5 so that it proves all tautologies

(and only tautologies). The answer turns out to be very simple. All it takes to extend

CL5 to a sound and complete system for classical logic is to add to it the contraction

rule:

Contraction

E

F

H

Merge two adjacent and identical formulas F and F into F

slide122

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

slide123

Example

I

I

P

P

P

P

M

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

slide124

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

slide125

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

slide126

Example

I

I

P

P

P

P

E

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

slide127

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

slide128

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

slide129

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

slide130

Example

I

I

P

P

P

P

C

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

slide131

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

slide132

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

slide133

Example

I

I

P

P

P

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

slide134

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

slide135

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

slide136

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

slide137

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

slide138

Example

I

I

P

P

P

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

slide139

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

P

P

P

slide140

Example

I

I

P

P

P

P

15.17

Now P(PP) becomes provable, and so do any other classical tautologies:

M

P

P

P

P

E

P

P

P

P

C

P

P

P

P

P

P

P

 ( )

P

P

slide141

CL5 versus multiplicative affine logic

15.18

Affine logic is a variation of the famous linear logic. Multiplicative affine logic is

obtained from system G1 (see Episode 4) by deleting Contraction (as for linear logic,

it further deletes Weakening as well).

Our CL5 is also obtained by deleting Contraction from a deductive system for

classical logic, and it is natural to ask how the two compare. Here is the answer:

Fact 15.3. Every formula provable in multiplicative affine logic is also provable in

CL5, but not vice versa: some formulas provable in CL5 are not provable in

affine logic.

Blass’s principle

proven on slide 15.13 is an example of a formula provable in CL5 but not in affine

logic. In fact, one can show that any proof of Blass’s principle in G1 would

require using not only Contraction, but also Weakening. On the other hand, our

CL5-proof of it used neither Weakening nor Contraction (nor Duplication).

(

)

(

)

(

P

Q

)

(

R

S

)

(

R

)

(

Q

S

)

P