1. Cirquent calculus Episode 15 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

2. About cirquent calculus in general
15.1
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.

3. are general and 0-ary. And the only logical operators are ,  and .
The language of CL5
15.2
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”.

4. Cirquents F G H F Formulas Arcs Groups
15.3
Cirquents
F G H F
Formulas
Arcs
Groups
Every formula should be in (= connected with an arc to) at least one group.

5. 15.4
Formulas as Cirquents F = F

6. 15.4
Cirquents as Circuits
F G H F
Circuit
Sequent

7. 15.4
Cirquents as Circuits
F G H F
Cir
quent

8. 15.4
Cirquents as Circuits
F G H F
Cir
quent

9. Cirquents as Circuits F G H F Cir quent Circuit F G H F   15.4
sequent
Circuit
F G H F  

10. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

11. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

12. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

13. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

14. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

15. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

16. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F

17. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H

18. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

19. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

20. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

21. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

22. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

23. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F G H F

24. Operations on Cirquents
15.5
Operations on Cirquents
F G H F
Merging groups (merging groups #1 and #2):
F G H F
Merging formulas (merging G and H into E): F E F

25. 15.6
Identity I
F F

26. 15.7
Mix
F F
G G
Put one cirquent next to the other

27. 15.7
Mix
F F
G G M
F F
G G
Put one cirquent next to the other

28. 15.7
Mix
F F
G G M
F F
G G
Put one cirquent next to the other

29. 15.7
Mix
F F
G G M
F F
G G
Put one cirquent next to the other

30. 15.7
Mix
F F
G G M
F F
G G
Put one cirquent next to the other

31. 15.7
Mix
F F
G G M
F F
G G
Put one cirquent next to the other

32. 15.8
Exchange
F F
G G
Swap two adjacent formulas or groups

33. Exchange F F G G F F G G Swap two adjacent formulas or groups 15.8

34. Exchange F F G G F F G G Swap two adjacent formulas or groups 15.8

35. Exchange F F G G F F G G Swap two adjacent formulas or groups 15.8

36. Exchange F F G G F F G G Swap two adjacent formulas or groups 15.8

37. Exchange F F G G F F G G F G G F
15.8
Exchange
F F
G G E
F F G G E F G G F
Swap two adjacent formulas or groups

38. Exchange F F G G F F G G F F G G
15.8
Exchange
F F
G G E
F F G G E F F G G
Swap two adjacent formulas or groups

39. Exchange F F G G F F G G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G
Swap two adjacent formulas or groups

40. Exchange F F G G F F G G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G
Swap two adjacent formulas or groups

41. Exchange F F G G F F G G F G F G F G G F
15.8
Exchange
F F
G G E
F F G G E F G F G E F G G F
Swap two adjacent formulas or groups

42. Exchange F F G G F F G G F G F G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G E F G F G
Swap two adjacent formulas or groups

43. Exchange F F G G F F G G F G F G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G E F G F G
Swap two adjacent formulas or groups

44. Exchange F F G G F F G G F G F G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G E F G F G
Swap two adjacent formulas or groups

45. Exchange F F G G F F G G F G F G F G F G
15.8
Exchange
F F
G G E
F F G G E F G F G E F G F G
Swap two adjacent formulas or groups

46. 15.9
Weakening E H F G W E H F G
Delete any arc in any group of the conclusion;
if this leaves some formula “homeless”, delete that formula as well

47. Weakening E H F G E F G H E F G H
15.9
Weakening E H F G 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

48. Weakening E F G H E F G H E F G H
15.9
Weakening 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

49. 15.10
Duplication E H F G
Replace a group with two identical copies

50. Duplication E H F G E H F G Replace a group with two identical copies
15.10
Duplication E H F G D E H F G
Replace a group with two identical copies

51. Duplication E H F G E F G H Replace a group with two identical copies
15.10
Duplication E H F G D E F G H
Replace a group with two identical copies

52. Duplication E H F G E F G H Replace a group with two identical copies
15.10
Duplication E H F G D E F G H
Replace a group with two identical copies

53. Duplication E H F G E F G H Replace a group with two identical copies
15.10
Duplication E H F G D E F G H
Replace a group with two identical copies

54. -Introduction E H F G Merge two adjacent formulas F and G into FG
15.11 E H F G
Merge two adjacent formulas F and G into FG

55. -Introduction E H F G E H F G
15.11 E H F G  E H F G
Merge two adjacent formulas F and G into FG

56. -Introduction E H F G E F G H
15.11 E H F G  E F G H
Merge two adjacent formulas F and G into FG

57. -Introduction E H F G E F G H
15.11 E H F G  E F G H
Merge two adjacent formulas F and G into FG

58. -Introduction E H F G E F G H
15.11 E H F G  E F G H
Merge two adjacent formulas F and G into FG

59. -Introduction E H F G E H F G In the premise:
15.12 E H F G  E H F G
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.

60. -Introduction E H F G E F G H In the premise:
15.12 E H F G  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.

61. -Introduction E H F G E F G H In the premise:
15.12 E H F G  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.

62. -Introduction E H F G E F G H In the premise:
15.12 E H F G  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.

63. -Introduction E H F G E F  G H In the premise:
15.12 E H F G  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.

64. -Introduction E H F G  E F  G H In the premise:
15.12 E H F G   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.

65. -Introduction E F G H E F  G H In the premise:
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.

66. -Introduction E F G H E F  G H In the premise:
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.

67. -Introduction E F G H E F  G H In the premise:
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.

68. -Introduction E H F G E F  G H In the premise:
15.12 E H F G  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.

69. -Introduction E H F G  E F  G H In the premise:
15.12 E H F G   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.

70. -Introduction E F G H E F  G H In the premise:
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.

71. -Introduction E F G H E F  G H In the premise:
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.

72. -Introduction E F G H E F  G H In the premise:
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.

73. -Introduction E F G H E F  G H In the premise:
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.

74. -Introduction E H F G  E F  G H In the premise:
15.12 E H F G   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.

75. -Introduction E F G H E F  G H In the premise:
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.

76. -Introduction E F G H E F  G H In the premise:
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.

77. -Introduction E F G H E F  G H In the premise:
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.

78. -Introduction E F G H E F  G H In the premise:
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.

79. -Introduction E H F G K E H F G K In the premise:
15.12 E H F G K  E H F G 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.

80. -Introduction E H F G K E F G H K In the premise:
15.12 E H F G 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.

81. -Introduction E H F G K E F G H K In the premise:
15.12 E H F G 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.

82. -Introduction E H F G K E F G H K In the premise:
15.12 E H F G 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.

83. -Introduction E H F G K E F G H K In the premise:
15.12 E H F G 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.

84. -Introduction E H F G K E F G H K In the premise:
15.12 E H F G 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.

85. -Introduction E H F G K E F  G H K In the premise:
15.12 E H F G 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.

86. -Introduction E H F G K  E F  G H K In the premise:
15.12 E H F G 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.

87. -Introduction E F G H K E F  G H K In the premise:
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.

88. -Introduction E F G H K E F  G H K In the premise:
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.

89. -Introduction E F G H K E F  G H K In the premise:
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.

90. -Introduction E F G H K E F  G H K In the premise:
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.

91. -Introduction E F G H K E F  G H K In the premise:
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.

92. -Introduction E H F G K E F  G H K In the premise:
15.12 E H F G 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.

93. Proof of Blass’s Principle
15.13 ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

94. Proof of Blass’s Principle
15.13  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

95. Proof of Blass’s Principle
15.13  P ( Q  ) R S  P R Q S ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

96. Proof of Blass’s Principle
15.13 P ( Q  ) R S  P R Q S ( P  Q )  ( R  S ) ( P  R )  ( Q  S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

97. Proof of Blass’s Principle
15.13  P ( Q  ) R S  P R Q S ( P  Q )  ( R  S ) ( P  R )  ( Q  S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

98. Proof of Blass’s Principle
15.13 P  Q R  S ( P  R )  ( Q  S )  P ( Q  ) R S  P R Q S ( P  Q )  ( R  S ) ( P  R )  ( Q  S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

99. Proof of Blass’s Principle
15.13 P  Q R  S ( P  R )  ( Q  S )  P ( Q  ) R S  P R Q S ( P  Q )  ( R  S ) ( P  R )  ( Q  S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

100. Proof of Blass’s Principle
15.13 P  Q R  S ( P  R )  ( Q  S )  ( P  Q )  ( R  S ) ( P  R )  ( Q  S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

101. Proof of Blass’s Principle
15.13 P Q  R S P ( R ) Q S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

102. Proof of Blass’s Principle
15.13  P Q  R S P ( R ) Q S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

103. Proof of Blass’s Principle
15.13 P  Q R  S P  R Q  S  P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

104. Proof of Blass’s Principle
15.13 P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( P P   Q Q )  ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

105. Proof of Blass’s Principle
15.13 P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

106. Proof of Blass’s Principle
15.13 P Q  R S P R Q S P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

107. Proof of Blass’s Principle
15.13  P P Q  R S P R Q S  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

108. Proof of Blass’s Principle
15.13  P P Q  R S P R Q S  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

109. Proof of Blass’s Principle
15.13 P Q R  S P  R Q  S  P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

110. Proof of Blass’s Principle
15.13 P Q R  S P  R Q  S   P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

111. Proof of Blass’s Principle
15.13 P Q R S P  R Q  S    P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

112. Proof of Blass’s Principle
15.13 P Q R S P R Q  S     P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

113. Proof of Blass’s Principle
15.13 P Q R S P R Q S     P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

114. Proof of Blass’s Principle
15.13
I I I I P P S S Q Q R R
M…M P P S S Q Q R R
E…E P Q R S P R Q S     P  Q R  S P  R Q  S  P P   Q Q R R   S S ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ( P P   Q Q ) )   ( ( R R   S S ) ) ( ( P P   R R ) )   ( ( Q Q   S S ) )  ( ) ( ) ( P  Q )  ( R  S )  ( P  R )  ( Q  S )

115. Soundness and completeness
15.14
Soundness and completeness
Theorem 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.

116. CL5 versus classical logic
15.15
CL5 versus classical logic
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 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.

117. Contraction A cirquent-calculus system for classical logic
15.16
A cirquent-calculus system for classical logic
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 H F
Merge two adjacent and identical formulas F and F into F

118. Contraction A cirquent-calculus system for classical logic
15.16
A cirquent-calculus system for classical logic
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 H F C E H F
Merge two adjacent and identical formulas F and F into F

119. Contraction A cirquent-calculus system for classical logic
15.16
A cirquent-calculus system for classical logic
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 H F C E F F H
Merge two adjacent and identical formulas F and F into F

120. Contraction A cirquent-calculus system for classical logic
15.16
A cirquent-calculus system for classical logic
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 H F C E F F H
Merge two adjacent and identical formulas F and F into F

121. Contraction A cirquent-calculus system for classical logic
15.16
A cirquent-calculus system for classical logic
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 H F C E F H
Merge two adjacent and identical formulas F and F into F

122. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I

123. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P

124. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P

125. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P

126. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P

127. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P

128. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P

129. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P

130. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P

131. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P P

132. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P

133. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P

134. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P P

135. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P P

136. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P P

137. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P  P

138. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P  P  P P 

139. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P  P  P P  P

140. 15.17
Example
Now P(PP) becomes provable, and so do any other classical tautologies: P P I P P I M P P P P E P P P P C P P P  P P  P  P
 ( ) P  P

141. ( ) ( ) CL5 versus multiplicative affine logic
15.18
CL5 versus multiplicative affine logic
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 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 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 )  ( P  R )  ( Q  S )