Hyper Natural Deduction for Gdel Logic a natural deduction system - PowerPoint PPT Presentation

BCF System Models hyper sequents in natural deduction by combining deductions in ND with a new operator | .

BCF System Models hyper sequents in natural deduction by combining deductions in ND with a new operator | . Example linearity: From { A } { B } From and A B

BCF System Models hyper sequents in natural deduction by combining deductions in ND with a new operator | . Example linearity: From { A } { B } From and A B { A } { B } one derives A B (com) B (com) A

BCF System Models hyper sequents in natural deduction by combining deductions in ND with a new operator | . Example linearity: From { A } { B } From and A B { A } { B } one derives A B (com) B (com) A { A } { B } B A then (com) B (com) A etc A → B B → A

Discussion of the BCF system ◮ direct translation from HLK ◮ inductive definition ◮ easy to translate proofs back and forth ◮ normalisation only via translation to HLK

Discussion of the BCF system ◮ direct translation from HLK ◮ inductive definition ◮ easy to translate proofs back and forth ◮ normalisation only via translation to HLK Not a solution to our problem!

Our approach to Hyper Natural Deduction

Our approach to Hyper Natural Deduction Γ Γ ⇒ A ∆ ⇒ B (com) A Γ ⇒ B | ∆ ⇒ A

Our approach to Hyper Natural Deduction Γ ∆ Γ ⇒ A ∆ ⇒ B (com) A B Γ ⇒ B | ∆ ⇒ A

Our approach to Hyper Natural Deduction Γ ∆ Γ ⇒ A ∆ ⇒ B (com) A B Γ ⇒ B | ∆ ⇒ A com B

Our approach to Hyper Natural Deduction Γ ∆ Γ ⇒ A ∆ ⇒ B (com) A B Γ ⇒ B | ∆ ⇒ A com com B A

Our approach to Hyper Natural Deduction Γ ∆ Γ ⇒ A ∆ ⇒ B (com) A B Γ ⇒ B | ∆ ⇒ A com com B A

Our approach to Hyper Natural Deduction Γ ∆ Γ ⇒ A ∆ ⇒ B (com) A B Γ ⇒ B | ∆ ⇒ A com com B A ◮ consider sets of derivation trees ◮ divide communication (and split) into two dual parts ◮ search for minimal set of conditions that provides sound and complete deduction system

Rules of HNGL Rules for NJ plus k [ Γ ], ∆ A r : k Spt Γ , ∆ A

Rules of HNGL Rules for NJ plus k [ Γ ], ∆ Γ A A r : k Spt Γ , ∆ A r : Com A,B B

Rules of HNGL Rules for NJ plus k [ Γ ], ∆ Γ A A r : k Spt Γ , ∆ A r : Com A,B B Γ ∆ A A r : Ctr A

Rules of HNGL Rules for NJ plus k [ Γ ], ∆ Γ A A r : k Spt Γ , ∆ A r : Com A,B B Γ ∆ Γ A A A r : Rep A r : Ctr A

Rules of HNGL Rules for NJ plus k [ Γ ], ∆ Γ A A r : k Spt Γ , ∆ A r : Com A,B B Γ ∆ Γ A A A r : Rep A r : Ctr A A prederivation is a well-formed derivation tree based on the rules of HNGL .

Hyper rules Applies to k prehyper deductions and produces another prehyper deduction: R 1 · · · R k h - r R

Hyper rules Applies to k prehyper deductions and produces another prehyper deduction: R 1 · · · R k h - r R Hyper rule h - r for NJ rule r Γ ∆ · · · · · · R 1 R 2 · · · · A → B A h - → -e Γ ∆ · · · · · · · · · · R 1 R 2 A → B A → -e B

Hyper communication rule Γ ∆ · · · · · · R 1 R 2 · · · · A B h -Com Γ ∆ · · · · · · · · · · R 1 R 2 A B x : Com A,B B x : Com B,A A ¯

Hyper splitting rule Γ , ∆ · · · R · · A h -Spt k [ Γ ], ∆ Γ , l [ ∆ ] · · · · · · · · · · R A A x : k Spt Γ , ∆ A x : l Spt ∆ , Γ A ¯

Hyper contraction and repetition rules Γ ∆ Γ · · · · · · · · · R R · · · · · · A A A h -Ctr h -Rep Γ ∆ Γ · · · · · · · · · · · · · · · R R A A A x : Ctr x : Rep A A

Why this verbosity? Natural deduction, as well as Sequent calculus, define a partial order of rule instances, and any linearisation that agrees with the partial order gives a valid derivation.

Why this verbosity? Natural deduction, as well as Sequent calculus, define a partial order of rule instances, and any linearisation that agrees with the partial order gives a valid derivation. In the case of Hyper Natural Deductions we have multiple trees with multiple partial orders, but due to the connections between prederivations via communication rules, the final HNGL does not define a unique derivation order.

Proof of linearity - GLC version C = (A → B) ∨ (B → A) , A ⇒ A B ⇒ B com A ⇒ B | B ⇒ A → ,r ⇒ A → B | B ⇒ A → ,r ⇒ A → B | ⇒ B → A ∨ 1 ,r ⇒ C | ⇒ B → A ∨ 2 ,r ⇒ C | ⇒ C EC ⇒ C

Proof of linearity - HNGL version 1 [A] 2 [B] x : Com A,B x : Com A,B B A 1 → -i A → B 2 → -i B → A ∨ -i ∨ -i C C y : Ctr C

HNGL deduction A B h -Com A B x : Com A,B ¯ x : Com B,A B A h - → -i 1 [A] x : Com A,B B x : Com B,A B ¯ 1 → :i A A → B h - → -i 1 [A] 2 [B] x : Com A,B ¯ x : Com B,A B A 1 → :i 2 → :i A → B B → A h - ∨ -i 1 [A] 2 [B] x : Com A,B B x : Com B,A ¯ 1 → :i A 2 → :i A → B ∨ :i B → A C h - ∨ -i 1 [A] 2 [B] x : Com A,B x : Com B,A ¯ B A 1 → :i 2 → :i A → B B → A ∨ :i ∨ :i

Results on HNGL Theorem If A is GLC derivable, then A is also HNGL derivable. Theorem If A is HNGL derivable, then A is also GLC derivable. Theorem The system HNGL is sound and complete for infinitary propositional Gödel logic.

Discussion ◮ Hyper rules – derivations are completely in ND style ◮ Hyper rules mimic HLK/BCF system ◮ natural style of deduction

Discussion ◮ Hyper rules – derivations are completely in ND style ◮ Hyper rules mimic HLK/BCF system ◮ natural style of deduction ◮ but: procedural definition (like BCF system): ◮ difficult to check whether a given figure forms a proof ◮ difficult to reason on normalisation (needs reshuffling of proof trees)

Discussion ◮ Hyper rules – derivations are completely in ND style ◮ Hyper rules mimic HLK/BCF system ◮ natural style of deduction ◮ but: procedural definition (like BCF system): ◮ difficult to check whether a given figure forms a proof ◮ difficult to reason on normalisation (needs reshuffling of proof trees) We need criteria to check whether a set of trees forms a proof!

Towards an explicit definition

Proof criteria What about the following proof part: B A x : Com B,A A x : Com A,B ¯ B E F E ∧ F

Equivalence classes c 1 c n c 2 ¯ c 2 c 3 c 1 ¯ . . . ∧

Equivalence classes c 1 c n c 2 ¯ c 2 c 3 c 1 ¯ . . . ∧ Criterion 1 : The sets of trees connected to the sub-trees routed in the predecessors of any non-unary logical rule need to be disjoint.

Another criteria What about this: B F x : Com B,A A x : Com F,E ¯ E E A x : Com E,F x : Com A,B ¯ F B

Another criteria What about this: B F x : Com B,A A x : Com F,E ¯ E E A x : Com E,F x : Com A,B ¯ F B Criterion 2 : There is a total order on communication and split labels that is compatible with the order on the branches.

Canopy graphs Two operations on labeled directed graphs: Cut ( G , E) drops a set of edges from the graph Drop ( G , N) drops a set of nodes and related edges that are reachable from all nodes labeled with a name in N

Canopy graphs Two operations on labeled directed graphs: Cut ( G , E) drops a set of edges from the graph Drop ( G , N) drops a set of nodes and related edges that are reachable from all nodes labeled with a name in N Definition Let G = (V, E, N, f ) be a labeled graph, and let E c ⊆ E be the set of symmetric edges, that is the set of all edges (r, s) ∈ E where also (s, r) ∈ E . If Cut ( G , E c ) is a disjoint union of trees, we call G a C-graph or canopy graph .

Motivation of these concepts Consider the following hyper-sequent derivation: B ⇒ B C ⇒ C C, B ⇒ B A ⇒ A C, B ⇒ C A ⇒ A com 1 com 2 C, B ⇒ A | A ⇒ B C, B ⇒ A | A ⇒ C ∧ -r C, B ⇒ A | C, B ⇒ A | A ⇒ B ∧ C contr C, B ⇒ A | A ⇒ B ∧ C ⇒ C → (B → A) | ⇒ A → B ∧ C

Motivation of these concepts Consider the following hyper-sequent derivation: B ⇒ B C ⇒ C C, B ⇒ B A ⇒ A C, B ⇒ C A ⇒ A com 1 com 2 C, B ⇒ A | A ⇒ B C, B ⇒ A | A ⇒ C ∧ -r C, B ⇒ A | C, B ⇒ A | A ⇒ B ∧ C contr C, B ⇒ A | A ⇒ B ∧ C ⇒ C → (B → A) | ⇒ A → B ∧ C And the following intended HND proof: [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C)

Motivation of these concepts II [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C)

Motivation of these concepts II [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C) and the associated graph x 1 x 2 x 2 ¯ x 1 ¯ y z w u v

Motivation of these concepts II [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C) connectivity condition does not hold for u x 1 x 2 x 2 ¯ x 1 ¯ y z w

Motivation of these concepts II [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C) cut at the contraction, conn. comp. fall apart x 1 x 2 x 2 ¯ x 1 ¯ y z w

Motivation of these concepts II [C] [B] x 1 : Com C,A x 2 : Com B,A A A y : Ctr A z B → A w C → (B → A) [A] [A] x 1 : Com A,C ¯ x 2 : Com A,B ¯ C B u : ∧ -i B ∧ C v A → (B ∧ C) cut at the contraction, conn. comp. fall apart x 1 x 2 x 2 ¯ x 1 ¯ y z w Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).