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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

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Cirquent calculus

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  1. 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

  2. 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.

  3. 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”.

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

  5. Formulas as Cirquents 15.4 F = F

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

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

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

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

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

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

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

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

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

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

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

  17. 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):

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. Identity I F F 15.6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  41. 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

  42. 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

  43. 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

  44. 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

  45. 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

  46. 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

  47. 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

  48. 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

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

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

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