SLIDE 1 Hyper Natural Deduction for Gödel Logic a natural deduction system for parallel reasoning1
Norbert Preining
Accelia Inc., Tokyo Joint work with Arnold Beckmann, Swansea University
mla 2018 Kanazawa, March 2018
1Partially supported by Royal Society Daiwa Anglo-Japanese Foundation International Exchanges Award
SLIDE 2
Motivation
◮ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248
SLIDE 3
Motivation
◮ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248
◮ The second, deeper objective of this paper is to contribute towards a better understanding of the notion of logical consequence in general, and especially its possible relations with parallel computations
SLIDE 4
Motivation
◮ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248
◮ The second, deeper objective of this paper is to contribute towards a better understanding of the notion of logical consequence in general, and especially its possible relations with parallel computations ◮ We believe that these logics [...] could serve as bases for parallel λ-calculi.
SLIDE 5
Motivation
◮ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248
◮ The second, deeper objective of this paper is to contribute towards a better understanding of the notion of logical consequence in general, and especially its possible relations with parallel computations ◮ We believe that these logics [...] could serve as bases for parallel λ-calculi. ◮ The name “communication rule” hints, of course, at a certain intuitive interpretation that we have of it as corresponding to the idea of exchanging information between two multiprocesses: [...]
SLIDE 6
Motivation
◮ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248
◮ The second, deeper objective of this paper is to contribute towards a better understanding of the notion of logical consequence in general, and especially its possible relations with parallel computations ◮ We believe that these logics [...] could serve as bases for parallel λ-calculi. ◮ The name “communication rule” hints, of course, at a certain intuitive interpretation that we have of it as corresponding to the idea of exchanging information between two multiprocesses: [...]
◮ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics.
SLIDE 7
Setting the stage
SLIDE 8
Setting the stage
IL
SLIDE 9
Setting the stage
IL λ
SLIDE 10
Setting the stage
IL λ ND ⇔
SLIDE 11
Setting the stage
IL λ ND ⇔ Gentzen ’34
SLIDE 12
Setting the stage
IL λ ND ⇔ introduction rule A B A ∧ B elimination rule A ∧ B A
SLIDE 13
Setting the stage
IL λ ND ⇔ introduction rule A B A ∧ B elimination rule A ∧ B A [A] B A → B A A → B B
SLIDE 14
Setting the stage
IL λ ND ⇔ introduction rule A B A ∧ B elimination rule A ∧ B A [A] B A → B A A → B B normalisation
SLIDE 15
Setting the stage
IL λ ND ⇔ LJ ⇔ Gentzen ’34
SLIDE 16
Setting the stage
IL λ ND ⇔ LJ ⇔ Sequent Γ ⇒ A Axiom A ⇒ A Rules Γ ⇒ A ∆ ⇒ B Γ, ∆ ⇒ A ∧ B
SLIDE 17
Setting the stage
IL λ ND ⇔ LJ ⇔ Sequent Γ ⇒ A Axiom A ⇒ A Rules Γ ⇒ A ∆ ⇒ B Γ, ∆ ⇒ A ∧ B Γ ⇒ A Π, A ⇒ B (cut) Γ, Π ⇒ B
SLIDE 18
Setting the stage
IL λ ND ⇔ LJ ⇔ Sequent Γ ⇒ A Axiom A ⇒ A Rules Γ ⇒ A ∆ ⇒ B Γ, ∆ ⇒ A ∧ B Γ ⇒ A Π, A ⇒ B (cut) Γ, Π ⇒ B cut elimination – consistency
SLIDE 19
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard Every proof system hides a model of computation.
SLIDE 20
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL
SLIDE 21
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL Gödel ’32, Dummett ’59
SLIDE 22
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL Gödel ’32, Dummett ’59 IL + LIN (A → B) ∨ (B → A)
SLIDE 23
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL Gödel ’32, Dummett ’59 IL + LIN (A → B) ∨ (B → A) Logic of Linear Kripke Frames
SLIDE 24
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Avron ’91
SLIDE 25
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Hypersequent Γ1 ⇒ ∆1 | . . . | Γn ⇒ ∆n
SLIDE 26
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B Γ ⇒ B | ∆ ⇒ A
SLIDE 27
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A
SLIDE 28
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Avron ’91: Communication between agents
SLIDE 29
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Fermüller ’08: Lorenzen style dialogue games, . . .
SLIDE 30
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A hyper sequent calculi for various logics
SLIDE 31
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A deep inference
SLIDE 32
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ ?
SLIDE 33
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ ? HND ⇔
SLIDE 34
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ HND ⇔ ? ⇐ ⇒?
SLIDE 35
Setting the stage
IL λ ND ⇔ LJ ⇔ ⇐ ⇒ Curry Howard GL HLK ⇔ ? ⇐ ⇒? HND ⇔ today’s topic
SLIDE 36
Previous work
Hirai, FLOPS 2012
A Lambda Calculus for Gödel-Dummett Logics Capturing Waitfreedom ◮ change of both syntax and semantics ◮ different calculus
SLIDE 37
Previous work
Hirai, FLOPS 2012
A Lambda Calculus for Gödel-Dummett Logics Capturing Waitfreedom ◮ change of both syntax and semantics ◮ different calculus
Baaz, Ciabattoni, Fermüller 2000
A Natural Deduction System for Intuitionistic Fuzzy Logic (will be discussed later)
SLIDE 38
Wishlist
Properties we want to have:
(semi) local
◮ construction of deductions: apply ND inspired rules to extend a HND deductions ◮ modularity of deductions: reorder/restructure deductions ◮ analyticity (sub-formula property, . . . )
SLIDE 39
Wishlist
Properties we want to have:
(semi) local
◮ construction of deductions: apply ND inspired rules to extend a HND deductions ◮ modularity of deductions: reorder/restructure deductions ◮ analyticity (sub-formula property, . . . )
normalisation
◮ procedural normalisation via conversion steps
SLIDE 40
Natural Deduction rules
A B ∧-i A ∧ B A ∧ B ∧-e A A ∧ B B A ∨-i A ∨ B B A ∨ B A ∨ B [A] C [B] C ∨-e C [A] B →-i A → B A A → B →-e B ⊥ ⊥I A
SLIDE 41
Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | . . . | Γn ⇒ An
SLIDE 42
Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | . . . | Γn ⇒ An Some rules: Γ ⇒ A | H Γ, B ⇒ C | H ′ →,l Γ, A → B ⇒ C | H | H ′ Γ, A ⇒ B | H →,r Γ ⇒ A → B | H
SLIDE 43
Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | . . . | Γn ⇒ An Some rules: Γ ⇒ A | H Γ, B ⇒ C | H ′ →,l Γ, A → B ⇒ C | H | H ′ Γ, A ⇒ B | H →,r Γ ⇒ A → B | H Γ1 ⇒ A1 | H Γ2 ⇒ A2 | H ′ com Γ1 ⇒ A2 | Γ2 ⇒ A1 | H | H ′ Π, Γ ⇒ A | H split Π ⇒ A | Γ ⇒ A | H
SLIDE 44
Linearity in LJ
⇒ (A → B) ∨ (B → A)
SLIDE 45
Linearity in LJ
⇒ A → B ⇒ (A → B) ∨ (B → A)
SLIDE 46
Linearity in LJ
??? A ⇒ B ⇒ A → B ⇒ (A → B) ∨ (B → A)
SLIDE 47
Linearity in HLK
⇒ (A → B) ∨ (B → A)
SLIDE 48
Linearity in HLK
⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ⇒ (A → B) ∨ (B → A) EC
SLIDE 49
Linearity in HLK
⇒ (A → B) ∨ (B → A) | ⇒ B → A ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ∨ -r ⇒ (A → B) ∨ (B → A) EC
SLIDE 50
Linearity in HLK
⇒ A → B | ⇒ B → A ⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ∨ -r ⇒ (A → B) ∨ (B → A) EC
SLIDE 51
Linearity in HLK
⇒ A → B | B ⇒ A ⇒ A → B | ⇒ B → A → -r ⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ∨ -r ⇒ (A → B) ∨ (B → A) EC
SLIDE 52
Linearity in HLK
A ⇒ B | B ⇒ A ⇒ A → B | B ⇒ A → -r ⇒ A → B | ⇒ B → A → -r ⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ∨ -r ⇒ (A → B) ∨ (B → A) EC
SLIDE 53
Linearity in HLK
A ⇒ A B ⇒ B A ⇒ B | B ⇒ A com ⇒ A → B | B ⇒ A → -r ⇒ A → B | ⇒ B → A → -r ⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) ∨ -r ⇒ (A → B) ∨ (B → A) EC
SLIDE 54
BCF System
Models hyper sequents in natural deduction by combining deductions in ND with a new operator |.
SLIDE 55
BCF System
Models hyper sequents in natural deduction by combining deductions in ND with a new operator |. Example linearity: From From {A} A and {B} B
SLIDE 56 BCF System
Models hyper sequents in natural deduction by combining deductions in ND with a new operator |. Example linearity: From From {A} A and {B} B
{A} A (com) B {B} B (com) A
SLIDE 57 BCF System
Models hyper sequents in natural deduction by combining deductions in ND with a new operator |. Example linearity: From From {A} A and {B} B
{A} A (com) B {B} B (com) A then {A} A (com) B A → B {B} B (com) A B → A etc
SLIDE 58
Discussion of the BCF system
◮ direct translation from HLK ◮ inductive definition ◮ easy to translate proofs back and forth ◮ normalisation only via translation to HLK
SLIDE 59
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!
SLIDE 60
Our approach to Hyper Natural Deduction
SLIDE 61
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A
SLIDE 62
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A ∆ B
SLIDE 63
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A com B ∆ B
SLIDE 64
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A com B ∆ B com A
SLIDE 65
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A com B ∆ B com A
SLIDE 66
Our approach to Hyper Natural Deduction
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A Γ A com B ∆ B com 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
SLIDE 67
Rules of HNGL
Rules for NJ plus
k[Γ], ∆
A r:k SptΓ,∆ A
SLIDE 68
Rules of HNGL
Rules for NJ plus
k[Γ], ∆
A r:k SptΓ,∆ A Γ A r: ComA,B B
SLIDE 69
Rules of HNGL
Rules for NJ plus
k[Γ], ∆
A r:k SptΓ,∆ A Γ A r: ComA,B B Γ A ∆ A r: Ctr A
SLIDE 70
Rules of HNGL
Rules for NJ plus
k[Γ], ∆
A r:k SptΓ,∆ A Γ A r: ComA,B B Γ A ∆ A r: Ctr A Γ A r: Rep A
SLIDE 71
Rules of HNGL
Rules for NJ plus
k[Γ], ∆
A r:k SptΓ,∆ A Γ A r: ComA,B B Γ A ∆ A r: Ctr A Γ A r: Rep A A prederivation is a well-formed derivation tree based on the rules of HNGL.
SLIDE 72
Hyper rules
Applies to k prehyper deductions and produces another prehyper deduction: R1 · · · Rk h-r R
SLIDE 73
Hyper rules
Applies to k prehyper deductions and produces another prehyper deduction: R1 · · · Rk h-r R
Hyper rule h-r for NJ rule r
R1 Γ · · · · · A → B R2 ∆ · · · · · A h-→-e R1 R2 Γ · · · · · A → B ∆ · · · · · A →-e B
SLIDE 74
Hyper communication rule
R1 Γ · · · · · A R2 ∆ · · · · · B h-Com R1 R2 Γ · · · · · A x: ComA,B B ∆ · · · · · B ¯ x: ComB,A A
SLIDE 75
Hyper splitting rule
R Γ, ∆ · · · · · A h-Spt R
k[Γ], ∆
· · · · · A x:k SptΓ,∆ A Γ, l[∆] · · · · · A ¯ x:l Spt∆,Γ A
SLIDE 76
Hyper contraction and repetition rules
R Γ · · · · · A ∆ · · · · · A h-Ctr R Γ · · · · · A ∆ · · · · · A x: Ctr A R Γ · · · · · A h-Rep R Γ · · · · · A x: Rep A
SLIDE 77 Why this verbosity?
Natural deduction, as well as Sequent calculus, define a partial
- rder of rule instances, and any linearisation that agrees with
the partial order gives a valid derivation.
SLIDE 78 Why this verbosity?
Natural deduction, as well as Sequent calculus, define a partial
- rder 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.
SLIDE 79
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
SLIDE 80
Proof of linearity - HNGL version
1[A]
x: ComA,B B
1→-i A → B
∨-i C
2[B]
x: ComA,B A
2→-i B → A
∨-i C y: Ctr C
SLIDE 81 HNGL deduction
A B h-Com A x: ComA,B B B ¯ x: ComB,A A h-→-i
1[A]
x: ComA,B B
1 →:i
A → B B ¯ x: ComB,A A h-→-i
1[A]
x: ComA,B B
1 →:i
A → B
2[B]
¯ x: ComB,A A
2 →:i
B → A h-∨-i
1[A]
x: ComA,B B
1 →:i
A → B ∨:i C
2[B]
¯ x: ComB,A A
2 →:i
B → A h-∨-i
1[A]
x: ComA,B B
1 →:i
A → B ∨:i
2[B]
¯ x: ComB,A A
2 →:i
B → A ∨:i
SLIDE 82
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.
SLIDE 83
Discussion
◮ Hyper rules – derivations are completely in ND style ◮ Hyper rules mimic HLK/BCF system ◮ natural style of deduction
SLIDE 84
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)
SLIDE 85
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!
SLIDE 86
Towards an explicit definition
SLIDE 87
Proof criteria
What about the following proof part: B x: ComB,A A E A ¯ x: ComA,B B F E ∧ F
SLIDE 88
Equivalence classes
c1 ∧ c2 ¯ c2 c3 . . . cn ¯ c1
SLIDE 89
Equivalence classes
c1 ∧ c2 ¯ c2 c3 . . . cn ¯ c1 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.
SLIDE 90
Another criteria
What about this: B x: ComB,A A E x: ComE,F F F ¯ x: ComF,E E A ¯ x: ComA,B B
SLIDE 91
Another criteria
What about this: B x: ComB,A A E x: ComE,F F F ¯ x: ComF,E E A ¯ x: ComA,B B Criterion 2: There is a total order on communication and split labels that is compatible with the order on the branches.
SLIDE 92
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
SLIDE 93
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 Ec ⊆ 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, Ec) is a disjoint union of trees, we call G a C-graph or canopy graph.
SLIDE 94
Motivation of these concepts
Consider the following hyper-sequent derivation:
B ⇒ B C, B ⇒ B A ⇒ A com1 C, B ⇒ A | A ⇒ B C ⇒ C C, B ⇒ C A ⇒ A com2 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
SLIDE 95
Motivation of these concepts
Consider the following hyper-sequent derivation:
B ⇒ B C, B ⇒ B A ⇒ A com1 C, B ⇒ A | A ⇒ B C ⇒ C C, B ⇒ C A ⇒ A com2 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] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
SLIDE 96
Motivation of these concepts II
[C] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
SLIDE 97
Motivation of these concepts II
[C] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
and the associated graph x1 x2 ¯ x2 ¯ x1 y z w u v
SLIDE 98
Motivation of these concepts II
[C] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
connectivity condition does not hold for u x1 x2 ¯ x2 ¯ x1 y z w
SLIDE 99
Motivation of these concepts II
[C] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
cut at the contraction, conn. comp. fall apart x1 x2 ¯ x2 ¯ x1 y z w
SLIDE 100
Motivation of these concepts II
[C] x1: ComC,A A [B] x2: ComB,A A y: Ctr A z B → A w C → (B → A) [A] ¯ x1: ComA,C C [A] ¯ x2: ComA,B B u:∧-i B ∧ C v A → (B ∧ C)
cut at the contraction, conn. comp. fall apart x1 x2 ¯ x2 ¯ x1 y z w Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).
SLIDE 101
Explicit definition of HND for Gödel logics
A finite set of pre-derivations R (together with a total order on labels) forms a hyper natural deduction iff ◮ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . .
SLIDE 102
Explicit definition of HND for Gödel logics
A finite set of pre-derivations R (together with a total order on labels) forms a hyper natural deduction iff ◮ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . . ◮ Independence of premises for non-unary logical rules r and communication: The connected components in Cut(Drop(G(R), r)) of premises of r are disjoint.
SLIDE 103
Explicit definition of HND for Gödel logics
A finite set of pre-derivations R (together with a total order on labels) forms a hyper natural deduction iff ◮ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . . ◮ Independence of premises for non-unary logical rules r and communication: The connected components in Cut(Drop(G(R), r)) of premises of r are disjoint. ◮ Local dependence of contraction premises r: The connected components in Cut(Drop(G(R), r)) of premises of r are equal.
SLIDE 104
Core lemma
Chain lemma – in a GLHD the following figure cannot appear. x1 u1 y1 u+
1 ∗ t ∗ t ∗ t
x2 u2 y2 u+
2 ∗ t ∗ t ∗ t
. . . xℓ uℓ yℓ u+
ℓ ∗ t ∗ t ∗ t
SLIDE 105
Normalisation
SLIDE 106
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule:
SLIDE 107
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule: [A] B →-i A → B Γ A →-e B
SLIDE 108
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule: [A] B →-i A → B Γ A →-e B converts to Γ A B
SLIDE 109
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule: [A] B →-i A → B Γ A →-e B converts to Γ A B Effect of normalisation: hourglass form of derivation, eliminations followed by introductions.
SLIDE 110
Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ σ0 A → B x: ComA→B,C C ∆ σ1 C ¯ x: ComC,A→B A → B Π σ2 A →-e B
SLIDE 111
Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ σ0 A → B x: ComA→B,C C ∆ σ1 C ¯ x: ComC,A→B A → B Π σ2 A →-e B first try: use dual labels as channels to communicate sub-derivations
Γ σ0 A → B Π σ2 A →-e B x: ComB,C C ∆ σ1 C ¯ x: ComC,B B
SLIDE 112
Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ σ0 A → B x: ComA→B,C C ∆ σ1 C ¯ x: ComC,A→B A → B Π σ2 A →-e B first try: use dual labels as channels to communicate sub-derivations
Γ σ0 A → B Π σ2 A →-e B x: ComB,C C ∆ σ1 C ¯ x: ComC,B B
SLIDE 113
Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ σ0 A → B x: ComA→B,C C ∆ σ1 C ¯ x: ComC,A→B A → B Π σ2 A →-e B converts to (similar to cut-elimination in HLK)
Γ σ0 A → B
1[Π]
σ2 A →-e B
1Sy Γ,Π
B x: ComB,C C
2[Γ]
σ0 A → B Π σ2 A →-e B
2S ¯ y Π,Γ
B ∆ σ1 C ¯ x: ComC,B B contr B
SLIDE 114
Conversions
◮ proof follows Troelstra/Schwichtenberg proof ◮ detour conversions, simplification conversion and permutation conversions as there, with cases for cut and split added ◮ branches and tracks ◮ double induction on cut-rank and ordinal sum of critical label sequences
SLIDE 115
Conversions
◮ proof follows Troelstra/Schwichtenberg proof ◮ detour conversions, simplification conversion and permutation conversions as there, with cases for cut and split added ◮ branches and tracks ◮ double induction on cut-rank and ordinal sum of critical label sequences
Theorem
Contraction, communication and splitting permutation conversions convert hyper natural deductions into hyper natural deductions.
SLIDE 116 Results
Theorem (Normalisation)
Hyper Natural Deduction for Gödel Logic admits (weak)
- normalisation. That is, there is a way to move all elimination
rules above introduction rules by applying the above conversions.
SLIDE 117 Results
Theorem (Normalisation)
Hyper Natural Deduction for Gödel Logic admits (weak)
- normalisation. That is, there is a way to move all elimination
rules above introduction rules by applying the above conversions.
Theorem (Sub-formula property)
Let R be a normal hyper natural deduction with derived hypersequent H . Then each formula in R is a subformula of a formula in H .
SLIDE 118
Returning to our wishlist
(semi) local construction of deductions:
apply ND inspired rules to extend a HND deductions
SLIDE 119
Returning to our wishlist
(semi) local construction of deductions:
apply ND inspired rules to extend a HND deductions
modularity of deductions:
reorder/restructure deductions
SLIDE 120
Returning to our wishlist
(semi) local construction of deductions:
apply ND inspired rules to extend a HND deductions
modularity of deductions:
reorder/restructure deductions
analyticity (sub-formula property)
SLIDE 121
Returning to our wishlist
(semi) local construction of deductions:
apply ND inspired rules to extend a HND deductions
modularity of deductions:
reorder/restructure deductions
analyticity (sub-formula property) normalisation procedural normalisation via conversion steps
SLIDE 122
Further steps
◮ Extend hyper natural deduction to first order ◮ Reconsidering BCF system in the light of our procedural definition ◮ Develop term systems (“parallel λ”) and establish Curry-Howard correspondences ◮ Investigate confluence of normalisation ◮ Connections to process algebra or other systems ◮ Extension to other hyper sequent systems
SLIDE 123
Further steps
◮ Extend hyper natural deduction to first order ◮ Reconsidering BCF system in the light of our procedural definition ◮ Develop term systems (“parallel λ”) and establish Curry-Howard correspondences ◮ Investigate confluence of normalisation ◮ Connections to process algebra or other systems ◮ Extension to other hyper sequent systems Thanks for your attention! Ref: Beckmann, A. and P., N. Hyper Natural Deductions, to appear in Journal of Logic and Computation.
