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
1 Partially supported by Royal Society Daiwa Anglo-Japanese Foundation International Exchanges Award ñ 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 ñ We believe that these logics [...] could serve as bases for parallel λ-calculi. ñ 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: [...] ñ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics.
Motivation
ñ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248 ñ We believe that these logics [...] could serve as bases for parallel λ-calculi. ñ 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: [...] ñ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics.
Motivation
ñ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248 ñ 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 ñ 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: [...] ñ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics.
Motivation
ñ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248 ñ 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 ñ We believe that these logics [...] could serve as bases for parallel λ-calculi. ñ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics.
Motivation
ñ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248 ñ 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 ñ We believe that these logics [...] could serve as bases for parallel λ-calculi. ñ 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: [...] Motivation
ñ Arnon Avron: Hypersequents, Logical Consequence and Intermediate Logics for Concurrency Ann.Math.Art.Int. 4 (1991) 225-248 ñ 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 ñ We believe that these logics [...] could serve as bases for parallel λ-calculi. ñ 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: [...] ñ General aim: provide Curry-Howard style correspondences for parallel computation, starting from logical systems with good intuitive algebraic / relational semantics. IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard IL a LJ a ND ⇐⇒ λ
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ a ND ⇐⇒ λ
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ a ND ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL λ IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a ND λ IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a ND λ
Gentzen ’34 IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a ND λ
introduction rule elimination rule AB A ∧ B A ∧ B A IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a ND λ
introduction rule elimination rule AB A ∧ B A ∧ B A [A] B AA → B A → B B IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard a LJ ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a ND λ
introduction rule elimination rule AB A ∧ B A ∧ B A [A] B AA → B A → B B normalisation IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a LJ a ND λ
Gentzen ’34 IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a LJ a ND λ
Sequent Axiom Rules Γ ⇒ A A ⇒ A Γ ⇒ A ∆ ⇒ B Γ , ∆ ⇒ A ∧ B IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a LJ a ND λ
Sequent Axiom Rules Γ ⇒ A A ⇒ A Γ ⇒ A ∆ ⇒ B Γ , ∆ ⇒ A ∧ B
Γ ⇒ A Π,A ⇒ B (cut) Γ , Π ⇒ B IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Curry Howard ⇐⇒
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
IL a LJ a ND λ
Sequent Axiom Rules Γ ⇒ A A ⇒ A Γ ⇒ A ∆ ⇒ B Γ , ∆ ⇒ A ∧ B
Γ ⇒ A Π,A ⇒ B (cut) Γ , Π ⇒ B
cut elimination – consistency IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Gödel ’32, Dummett ’59
GL a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Every proof system hides a model of computation. IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL Every proof system hides a model of computation.
ILa+ LINHND(A → B) ∨⇐⇒(B? → A) ?
Logictoday’s of Linear topic Kripke Frames
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Gödel ’32, Dummett ’59
GL Every proof system hides a model of computation.
a HND ⇐⇒? ?
Logictoday’s of Linear topic Kripke Frames
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Gödel ’32, Dummett ’59
GL IL + LIN (A → B) ∨ (B → A) Every proof system hides a model of computation.
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Gödel ’32, Dummett ’59
GL IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL a HLK
Avron ’91 IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Hypersequent GL a HLK Γ1 ⇒ ∆1 | ... | Γn ⇒ ∆n IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND(com) ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK Γ ⇒ B | ∆ ⇒ A IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK (com) Γ ⇒ B | ∆ ⇒ A IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK (com) Γ ⇒ B | ∆ ⇒ A
Avron ’91: Communication between agents IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK (com) Γ ⇒ B | ∆ ⇒ A
Fermüller ’08: Lorenzen style dialogue games, . . . IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK (com) Γ ⇒ B | ∆ ⇒ A
hyper sequent calculi for various logics IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
Γ ⇒ A ∆ ⇒ B GL a HLK (com) Γ ⇒ B | ∆ ⇒ A
deep inference IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL a HLK ? IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND ⇐⇒? ?
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL a HLK a HND ? IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND
today’s topic
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL a HLK a HND ⇐⇒? ? IL + LIN (A → B) ∨ (B → A)
Logic of Linear Kripke Frames
Every proof systemGödel hides a’32 model, Dummett of computation. ’59
a HND
Setting the stage
Curry Howard IL a LJ a ND ⇐⇒ λ
GL a HLK a HND ⇐⇒? ?
today’s topic Baaz, Ciabattoni, Fermüller 2000 A Natural Deduction System for Intuitionistic Fuzzy Logic (will be discussed later)
Previous work
Hirai, FLOPS 2012 A Lambda Calculus for Gödel-Dummett Logics Capturing Waitfreedom ñ change of both syntax and semantics ñ different calculus Previous work
Hirai, FLOPS 2012 A Lambda Calculus for Gödel-Dummett Logics Capturing Waitfreedom ñ change of both syntax and semantics ñ different calculus
Baaz, Ciabattoni, Fermüller 2000 A Natural Deduction System for Intuitionistic Fuzzy Logic (will be discussed later) normalisation ñ procedural normalisation via conversion steps
Wishlist
Properties we want to have: (semi) local
ñ construction of deductions: apply ND inspired rules to extend a HND deductions ñ modularity of deductions: reorder/restructure deductions ñ analyticity (sub-formula property, . . . ) Wishlist
Properties we want to have: (semi) local
ñ construction of deductions: apply ND inspired rules to extend a HND deductions ñ modularity of deductions: reorder/restructure deductions ñ analyticity (sub-formula property, . . . ) normalisation ñ procedural normalisation via conversion steps Natural Deduction rules
AB A ∧ B A ∧ B ∧-i ∧-e A ∧ B A B
[A] [B] A B A ∨ B C C ∨-i ∨-e A ∨ B A ∨ B C
[A] B AA → B →-i →-e A → B B
⊥ ⊥I A Some rules:
Γ ⇒ A | H Γ ,B ⇒ C | H 0 →,l Γ ,A → B ⇒ C | H | H 0
Γ ,A ⇒ B | H →,r Γ ⇒ A → B | H
0 Γ1 ⇒ A1 | H Γ2 ⇒ A2 | H com 0 Γ1 ⇒ A2 | Γ2 ⇒ A1 | H | H
Π, Γ ⇒ A | H split Π ⇒ A | Γ ⇒ A | H
Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | ... | Γn ⇒ An 0 Γ1 ⇒ A1 | H Γ2 ⇒ A2 | H com 0 Γ1 ⇒ A2 | Γ2 ⇒ A1 | H | H
Π, Γ ⇒ A | H split Π ⇒ A | Γ ⇒ A | H
Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | ... | Γn ⇒ An Some rules:
Γ ⇒ A | H Γ ,B ⇒ C | H 0 →,l Γ ,A → B ⇒ C | H | H 0
Γ ,A ⇒ B | H →,r Γ ⇒ A → B | H Hypersequent Calculus
Hypersequent: Γ1 ⇒ A1 | ... | Γn ⇒ An Some rules:
Γ ⇒ A | H Γ ,B ⇒ C | H 0 →,l Γ ,A → B ⇒ C | H | H 0
Γ ,A ⇒ B | H →,r Γ ⇒ A → B | H
0 Γ1 ⇒ A1 | H Γ2 ⇒ A2 | H com 0 Γ1 ⇒ A2 | Γ2 ⇒ A1 | H | H
Π, Γ ⇒ A | H split Π ⇒ A | Γ ⇒ A | H ??? A ⇒ B
⇒ A → B
Linearity in LJ
⇒ (A → B) ∨ (B → A) ??? A ⇒ B
Linearity in LJ
⇒ A → B ⇒ (A → B) ∨ (B → A) Linearity in LJ
??? A ⇒ B ⇒ A → B ⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
A ⇒ B | B ⇒ A → -r
⇒ 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) EC
Linearity in HLK
⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
A ⇒ B | B ⇒ A → -r
⇒ A → B | B ⇒ A → -r
⇒ A → B | ⇒ B → A ∨ -r
⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r
Linearity in HLK
⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) EC ⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
A ⇒ B | B ⇒ A → -r
⇒ A → B | B ⇒ A → -r
⇒ A → B | ⇒ B → A ∨ -r
Linearity in HLK
⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) EC ⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
A ⇒ B | B ⇒ A → -r
⇒ A → B | B ⇒ A → -r
Linearity in HLK
⇒ A → B | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ B → A ∨ -r ⇒ (A → B) ∨ (B → A) | ⇒ (A → B) ∨ (B → A) EC ⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
A ⇒ B | B ⇒ A → -r
Linearity in HLK
⇒ 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) EC ⇒ (A → B) ∨ (B → A) A ⇒ A B ⇒ B com
Linearity in HLK
A ⇒ B | B ⇒ A → -r ⇒ 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) EC ⇒ (A → B) ∨ (B → A) Linearity in HLK
A ⇒ A B ⇒ B com A ⇒ B | B ⇒ A → -r ⇒ 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) EC ⇒ (A → B) ∨ (B → A) Example linearity: From
{A} {B} From and A B
{A} {B} one derives A B (com) (com) B A
{A} {B} A B then (com) (com) etc B A A → B B → A
BCF System
Models hyper sequents in natural deduction by combining deductions in ND with a new operator |. {A} {B} one derives A B (com) (com) B A
{A} {B} A B then (com) (com) etc B A A → B B → 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} A B then (com) (com) etc B A A → B B → 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) (com) B 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) (com) B A
{A} {B} A B then (com) (com) etc B A A → B B → A Not a solution to our problem!
Discussion of the BCF system
ñ direct translation from HLK
ñ inductive definition
ñ easy to translate proofs back and forth
ñ normalisation only via translation to HLK Discussion of the BCF system
ñ direct translation from HLK
ñ inductive definition
ñ easy to translate proofs back and forth
ñ normalisation only via translation to HLK
Not a solution to our problem! ΓΓ ∆
AA B com com B A
Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction Γ ∆
A B com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction
Γ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A A ΓΓ ∆
AA B com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction
Γ ∆ Γ ⇒ A ∆ ⇒ B (com) Γ ⇒ B | ∆ ⇒ A A B ΓΓ ∆
AA B com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction
Γ ∆ Γ ⇒ A ∆ ⇒ B (com) B Γ ⇒ B | ∆ ⇒ A com A B ΓΓ ∆
AA B com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction
Γ ∆ Γ ⇒ A ∆ ⇒ B (com) ⇒ | ⇒ A B Γ B ∆ A com com B A ΓΓ ∆
AA B com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system
Our approach to Hyper Natural Deduction
Γ ∆ Γ ⇒ A ∆ ⇒ B (com) ⇒ | ⇒ A B Γ B ∆ A com com B A ΓΓ ∆
AA B com com B A
Our approach to Hyper Natural Deduction
Γ ∆ Γ ⇒ A ∆ ⇒ B (com) ⇒ | ⇒ A B Γ B ∆ A com com B A
ñ consider sets of derivation trees ñ divide communication (and split) into two dual parts ñ search for minimal set of conditions that provides sound and complete deduction system Γ
r : Com A A,B B
Γ ∆ Γ
A A A r : Ctr r : Rep A A
A prederivation is a well-formed derivation tree based on the rules of HNGL.
Rules of HNGL
Rules for NJ plus
k[Γ ], ∆
A r :k Spt Γ ,∆ A Γ ∆ Γ
A A A r : Ctr r : Rep A A
A prederivation is a well-formed derivation tree based on the rules of HNGL.
Rules of HNGL
Rules for NJ plus
k[Γ ], ∆ Γ
A A r :k Spt r : Com Γ ,∆ A A,B B Γ
r : Rep A A
A prederivation is a well-formed derivation tree based on the rules of HNGL.
Rules of HNGL
Rules for NJ plus
k[Γ ], ∆ Γ
A A r :k Spt r : Com Γ ,∆ A A,B B
Γ ∆
A A r : Ctr A A prederivation is a well-formed derivation tree based on the rules of HNGL.
Rules of HNGL
Rules for NJ plus
k[Γ ], ∆ Γ
A A r :k Spt r : Com Γ ,∆ A A,B B
Γ ∆ Γ
A A A r : Ctr r : Rep A A Rules of HNGL
Rules for NJ plus
k[Γ ], ∆ Γ
A A r :k Spt r : Com Γ ,∆ A A,B B
Γ ∆ Γ
A A A r : Ctr r : Rep A A
A prederivation is a well-formed derivation tree based on the rules of HNGL. Hyper rule h-r for NJ rule r
Γ ∆ · · R · R · 1 · 2 · A → B A h-→-e Γ ∆ · · · · R1 R2 · · A → B A →-e B
Hyper rules
Applies to k prehyper deductions and produces another prehyper deduction:
R1 ··· Rk h-r R 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
Γ ∆ · · R · R · 1 · 2 · A → B A h-→-e Γ ∆ · · · · R1 R2 · · A → B A →-e B Hyper communication rule
Γ ∆ · · R · R · 1 · 2 · A B h-Com Γ ∆ · · · · R1 R2 · · A B x: ComA,B x¯: ComB,A B A Hyper splitting rule
Γ , ∆ · R · · A h-Spt k[Γ ], ∆ Γ , l[∆] · · · · R · · k A l A x: SptΓ ,∆ x¯: Spt∆,Γ A 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 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.
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),
⇒ ⇒ com A AB B 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 x: Com A,B B A,B A 1→-i 2→-i A → B B → A ∨-i ∨-i C C y: Ctr C HNGL deduction
AB h-Com A B x: ComA,B x¯: ComB,A B A h-→-i 1[A] x: ComA,B B B x¯: ComB,A 1 →:i A A → B h-→-i 1[A] 2[B] x: ComA,B x¯: ComB,A B A 1 →:i 2 →:i A → B B → A h-∨-i 1[A] 2 x: ComA,B [B] B x¯: ComB,A 1 →:i A A → B 2 →:i ∨:i B → A C h-∨-i 1[A] 2[B] x: ComA,B x¯: ComB,A B A 1 →:i 2 →:i A → B B → A ∨:i ∨:i C C h-Ctr
1[A] 2[B] x: ComA,B x¯: ComB,A B A 1 →:i 2 →:i A → B B → A ∨:i ∨:i C C y: Ctr C 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. ñ but: procedural definition (like BCF system): ñ difficult to check whether a given figure forms a proof ñ difficult to reason on normalisation (needs reshuffling of proof trees)
We need criteria to check whether a set of trees forms a proof!
Discussion
ñ Hyper rules – derivations are completely in ND style
ñ Hyper rules mimic HLK/BCF system
ñ natural style of deduction We need criteria to check whether a set of trees forms a proof!
Discussion
ñ Hyper rules – derivations are completely in ND style
ñ Hyper rules mimic HLK/BCF system
ñ natural style of deduction
ñ but: procedural definition (like BCF system): ñ difficult to check whether a given figure forms a proof ñ difficult to reason on normalisation (needs reshuffling of proof trees) Discussion
ñ Hyper rules – derivations are completely in ND style
ñ Hyper rules mimic HLK/BCF system
ñ natural style of deduction
ñ but: procedural definition (like BCF system): ñ difficult to check whether a given figure forms a proof ñ 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: ComB,A x¯: Com A A,B B
E F E ∧ F 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.
Equivalence classes
c1 c¯2 cn c2
c3 c¯1 ∧ ... Equivalence classes
c1 c¯2 cn c2
c3 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. Criterion 2: There is a total order on communication and split labels that is compatible with the order on the branches.
Another criteria
What about this:
B F x: ComB,A x¯: ComF,E A E
x: Com E x¯: Com A E,F F A,B B Another criteria
What about this:
B F x: ComB,A x¯: ComF,E A E
x: Com E x¯: Com A E,F F A,B B
Criterion 2: There is a total order on communication and split labels that is compatible with the order on the branches. 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.
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 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. And the following intended HND proof: [C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C)
Motivation of these concepts
Consider the following hyper-sequent derivation: B ⇒ B C ⇒ C C,B ⇒ BA ⇒ A C,B ⇒ CA ⇒ A com1 com2 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 ⇒ BA ⇒ A C,B ⇒ CA ⇒ A com1 com2 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] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) x1 x2 x¯2 x¯1
y z w
Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).
Motivation of these concepts II
[C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).
Motivation of these concepts II
[C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) and the associated graph
x1 x2 x¯2 x¯1
y z w u v Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).
Motivation of these concepts II
[C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) connectivity condition does not hold for u
x1 x2 x¯2 x¯1
y z w Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later).
Motivation of these concepts II
[C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) cut at the contraction, conn. comp. fall apart
x1 x2 x¯2 x¯1
y z w Motivation of these concepts II
[C] [B] x1: ComC,A x2: ComB,A A A y: Ctr A z B → A w C → (B → A) [A] [A] x¯1: ComA,C x¯2: ComA,B C B u:∧-i B ∧ C v A → (B ∧ C) cut at the contraction, conn. comp. fall apart
x1 x2 x¯2 x¯1
y z w
Expresses an implicit ordering between the conjunction (introduced first) and the contraction (introduced later). ñ 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. ñ Local dependence of contraction premises r : The connected components in Cut(Drop(G(R), r )) of premises of r are equal.
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 ñ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . . ñ Local dependence of contraction premises r : The connected components in Cut(Drop(G(R), r )) of premises of r are equal.
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 ñ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . . ñ 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. 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 ñ some obvious consistency conditions are satisfied; like occurrence of dual labels, compatibility with fixed label order, . . . ñ 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. ñ Local dependence of contraction premises r : The connected components in Cut(Drop(G(R), r )) of premises of r are equal. Core lemma
Chain lemma – in a GLHD the following figure cannot appear.
x1 x2 x`
y1 y2 y` ∗ ∗ ∗ t ∗ t ∗ t ∗ t t t u1 u2 ... u` t ∗ t ∗ t ∗ + + + u1 u2 u` Normalisation Γ
converts to A
B
[A] Γ B →-i → →-e A B A B
Effect of normalisation: hourglass form of derivation, eliminations followed by introductions.
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule: Γ
converts to A
B
Effect of normalisation: hourglass form of derivation, eliminations followed by introductions.
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule:
[A] Γ B →-i → →-e A B A B Effect of normalisation: hourglass form of derivation, eliminations followed by introductions.
Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule:
[A] Γ Γ B converts to A →-i → →-e A B A B B Normalisation
Idea: Reorder deductions where an introduction rule is followed by an elimination rule:
[A] Γ Γ B converts to A →-i → →-e A B A B B
Effect of normalisation: hourglass form of derivation, eliminations followed by introductions. converts to (similar to cut-elimination in HLK)
Γ 1[Π] 2[Γ ] Π σ0 σ2 σ0 σ2 ∆ A → B A A → B A σ1 →-e →-e 1 y B 2 y¯ B C S S x¯: ComC,B Γ ,Π B Π,Γ B B x: ComB,C contr C B
Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ ∆ σ0 σ1 Π A → B C σ2 x: ComA→B,C x¯: Com C C,A→B → →-e A B A B Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ ∆ σ0 σ1 Π A → B C σ2 x: ComA→B,C x¯: Com C C,A→B → →-e A B A B
first try: use dual labels as channels to communicate sub-derivations Γ Π ∆ σ0 σ2 σ1 A → B A C →-e x¯: ComC,B B B x: ComB,C C Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ ∆ σ0 σ1 Π A → B C σ2 x: ComA→B,C x¯: Com C C,A→B → →-e A B A B
first try: use dual labels as channels to communicate sub-derivations Γ Π ∆ σ0 σ2 σ1 A → B A C →-e x¯: ComC,B B B x: ComB,C C Permutation Conversions for hyper natural deduction
Example conversion for normalisation in hyper natural deduction: Γ ∆ σ0 σ1 Π A → B C σ2 x: ComA→B,C x¯: Com C C,A→B → →-e A B A B converts to (similar to cut-elimination in HLK)
Γ 1[Π] 2[Γ ] Π σ0 σ2 σ0 σ2 ∆ A → B A A → B A σ1 →-e →-e 1 y B 2 y¯ B C S S x¯: ComC,B Γ ,Π B Π,Γ B B x: ComB,C contr C B Theorem Contraction, communication and splitting permutation conversions convert hyper natural deductions into hyper natural deductions.
Conversions
ñ proof follows Troelstra/Schwichtenberg proof ñ detour conversions, simplification conversion and permutation conversions as there, with cases for cut and split added ñ branches and tracks ñ double induction on cut-rank and ordinal sum of critical label sequences Conversions
ñ proof follows Troelstra/Schwichtenberg proof ñ detour conversions, simplification conversion and permutation conversions as there, with cases for cut and split added ñ branches and tracks ñ 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. 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 .
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. 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 . " modularity of deductions: reorder/restructure deductions " analyticity (sub-formula property)
normalisation " procedural normalisation via conversion steps
Returning to our wishlist
(semi) local
" construction of deductions: apply ND inspired rules to extend a HND deductions " analyticity (sub-formula property)
normalisation " procedural normalisation via conversion steps
Returning to our wishlist
(semi) local
" construction of deductions: apply ND inspired rules to extend a HND deductions " modularity of deductions: reorder/restructure deductions normalisation " procedural normalisation via conversion steps
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) 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 Thanks for your attention!
Ref: Beckmann, A. and P., N. Hyper Natural Deductions, to appear in Journal of Logic and Computation.
Further steps
ñ Extend hyper natural deduction to first order ñ Reconsidering BCF system in the light of our procedural definition ñ Develop term systems (“parallel λ”) and establish Curry-Howard correspondences ñ Investigate confluence of normalisation ñ Connections to process algebra or other systems ñ Extension to other hyper sequent systems Further steps
ñ Extend hyper natural deduction to first order ñ Reconsidering BCF system in the light of our procedural definition ñ Develop term systems (“parallel λ”) and establish Curry-Howard correspondences ñ Investigate confluence of normalisation ñ Connections to process algebra or other systems ñ 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.