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 j ::: j 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 j ∆ ) 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 j ∆ ) A IL + LIN (A ! B) _ (B ! A) Logic of Linear Kripke Frames Every proof systemGödel hides a’32 model, Dummett of computation.