Cirquent calculus Episode 15 About cirquent calculus in general

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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 )