in Linear Logic Denis Bechet LIPN - Institut Galilee Universite Paris 13 Avenue Jean-Baptiste Clement Villetaneuse, France Abstract

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1 Second Order Connectives and roof Transformations in Linear Logic Denis Bechet LIN - Institut Galilee Universite aris 1 Avenue Jean-Baptiste Clement 90 Villetaneuse, France dbe@lipnuniv-paris1fr Abstract This article presents a method for transforming linear logic proofs that reduces the number of primitive constructions Inside formulas, some connectives are replaced by second order equivalents roofs are also transformed and cut elimination may be emulated This mechanism leads to proofs without the additives & and and the constants 1,, 0 and > and where the modality is only introduced with empty environments As CS-transformations in -calculus leads to particular reduction strategies, those transformations inhibit some reductions known as commutative conversions As a consequence, proof normalization is Church-Rosser in this system This method captures strategies of normalization used by various translation from linear logic to interaction nets It can be seen as an intermediate translation giving a theoretical justication for those coding Another interesting aspect of second order exponentials is their similarities to combinatorial exponentials 1 Introduction Linear logic [Gir8], which may be seen as a logic that takes resources into account, has a sequent calculus presentation Another presentation consists in using proof structures and proof nets composed of connectives and boxes linked together to form nets The aim of this notion is to identify certain not conceptually dierent proofs in sequent calculus However, proof nets are not perfect since we would like to identify more proofs than proof nets does This is due to boxes which delimit the inuence of certain context sensitive connectives (, >, & and ) This article presents a way of eliminating the sensitivity of these connectives by using second order equivalents: 1 0 = :X X 0 = 9X:X X 0 0 = :X > 0 U 0 V U& 0 V 0 U 0 U = 9X:X = :(X U)(X V )X = 9X:(X U)(X V ) X = :(X 0 )(X (XX))(X U)X = 9X:(X 1 0 )(X (X X))(X U) X 1

2 The principle of these new constructions is to divide each context-sensitive connective in two parts, one managing the context and the other introducing the connective This leads to formula transformations and proof transformations that preserve the elimination of main cuts However, like CS-transformations, this mechanism eliminates non-essential cuts known as commutative conversions [Gir8, GLT88] A benet of this strategy is to force cut elimination to be Church- Rosser In fact there are two motivations for those transformations The rst one is that this method inhibits commutative conversion As a consequence, this strategy can be compared to codings of linear logic into interaction nets [Laf9] where cut-elimination is done only for main reductions Those transformations can be seen as an intermediary step that gives a theoretical meaning of those codings The second motivation concerns our second order exponentials that look like the combinatorial approach of linear logic exponentials This article presents successively the transformations for the additives and &, for exponentials and nish by constants Elimination of Additives 1 Second Order Additives System F [GLT88] uses two kinds of sum type The rst version denes a new connective +, rules to introduce and eliminate it and conversions of redexes The second version emulates it as the abbreviation U + V = :(U X) (V X) X Linear logic has the same property We can use the additives and & as primitive operators or we can dene them as abbreviations Denition 1 The second order additives are dened by: U 0 V U& 0 V = :(X U)(X V )X = 9X:(X U)(X V ) X In linear logic, classical and second order additives are linearly equivalent We can prove that & and & 0 (resp and 0 ) imply each other Theorem The following sequents are provable: U V ` U 0 V U&V ` U& 0 V U 0 V ` U V U& 0 V ` U&V roof: Since proofs of the right sequents may be obtained by duality from proofs of the left ones, we only need to give proofs of U V ` U 0 V and U 0 V ` U V We have to prove the sequents ` U &V ; U 0 V and ` U & 0 V ; U V : ` X ; X ` U ; U ` X ; X ` V ; V ` U ; X U; X ` U ; (X U); (X V ); X ` U ; (X U)(X V )X ` U ; :(X U)(X V )X ` U &V ; :(X U)(X V )X ` U &V ; U 0 V ` V ; X V; X ` V ; (X U); (X V ); X ` V ; (X U)(X V )X ` V ; :(X U)(X V )X &

3 ` U; U ` U V; U 1 ` V; V ` (U V ); V ` (U V )U `((U V )U ) ` (U V )V `((U V )V ) ` (U V ) ; U V `((U V )U )((U V )V ) (U V ) ; U V 9X = (U V ) ` 9X:(X U )(X V ) X; U V ` U & 0 V ; U V Exchange rule has been omitted in the above proofs by considering a sequent as a multi-set of formulas roof Transformations for Additives The aim of the proof transformations consists, from a proof of a sequent using classical additives, to nd a proof of the sequent where classical additives are replaced by their second order version We write [U] the formula obtained by substituting 0 for and & 0 for & Denition If U is a formula, [U] is dened by induction by: [A] = A if A is an atom or a constant [U V ] = [U] [V ] and [U V ] = [U][V ] for multiplicatives [U V ] = [U] 0 [V ] = :(X [U])(X [V ])X with X not free in U or V [U&V ] = [U]& 0 [V ] = 9X:(X [U])(X [V ]) X with X not free in U or V [U] =[U] et [U] =[U] for exponentials [:U] = :[U] et [9X:U] = 9X:[U] for quantiers We write [U 1 ; : : : ; U n ] for [U 1 ]; : : : ; [U n ] Remark: Theorem shows that ` U 1 ; : : : ; U n is provable if and only if ` [U 1 ; : : : ; U n ] is provable Denition Let be a proof of ` U 1 ; : : : ; U n (written last inference of by: ` X ; X ` U 1 ; : : : ; Un ` ; X [U ]; X ` ; U = ` ; (X [U ]); (X [V ]); X 1 ` ; U V ` ; (X [U ])(X [V ])X ` ; :(X [U ])(X [V ])X ` ; [U V ] left rule The right rule is similar ` ; U ), is dened following the where is the premisse of the

4 1 ` ; U ` ; U &V ` ; V = & 1 ` ; U () ` C; [U ] ` C[U ] `(C[U ]) ` ; V () ` C; [V ] ` C[V ] `(C[V ]) ` ; (C[U ])(C[V ]) C 9X = C ` ; C (1) where 1 and ` ; 9X:(X [U ])(X [V ]) X ` ; [U &V ] are the two premisses of the & rule and where C = [A 1 ] [A n ] if = A 1 ; : : : ; A n (C = if n is null) 1 1 ` U 1 ; : : : ; Un k k = R are les premisses of the rule 1 1 k k ` [U 1 ]; : : : ; [Un] R for the other inference 1 ; : : : ; k Theorem If is a proof of ` U 1 ; : : : ; U n then is a proof of ` [U 1 ]; : : :; [U n ] Moreover, is a proof without using the classical additives of linear logic roof: By induction on Emulation of Cut Elimination Normalization of a proof may be emulated in its transformed proof Lemma If is a proof and reduces in one step to 0 without using commutative conversion of and & then reduces (in one or more steps) to [ 0 ]: Red - 0 Red + - [ 0 ] roof: We only need to prove that the transformed proof corresponding to the two dierent cases of cut between a and a & can reduce like the classical additives

5 For the cut between the & rule and the left rule (the other case is similar): 1 ` ; U ` ; V () () ` C; [U ] ` C; [V ] ` X ; X ` U ; ` C[U ] ` C[V ] ` X [U ]; X; (1) `(C[U ]) `(C[V ]) ` ; C `(X [U ]); (X [V ]); X; ` ; (C[U ])(C[V ]) C `(X [U ])(X [V ])X; 9X = C ` ; 9X:(X [U ])(X [V ]) X ` :(X [U ])(X [V ])X; ` ; [U &V ] ` [U V ]; Cut ` ; 1 ` ; U Red + ` ; ` U ; Cut Theorem If is a proof and reduces to 0 without using commutative conversion of and & then reduces to [ 0 ]: Red - 0 Red - [ 0 ] roof: By induction on the derivation from to 0 and using the Lemma Elimination of Environment in -intoduction For exponentials the aims of this section are to introduce second order exponentials, to nd proof transformations and to show that cut elimination may be emulated in the same scheme as for the additives 1 Second Order Exponentials In the previous section, the proof transformation of a & rule creats two proofs of `(C[V ]) and `(C[U]) As we can see, is introduced with an empty environment This transformation separates completely the context and the two alternatives The context will be given only when one of the alternatives will be chosen (during cut elimination) The classical rules for exponentials are:

6 `; U `; U ` ` ; U W ` ; U; U ` ; U C ` ; U ` ; U D Thus, as the two dierent rules give the two sub-formulas X U et X V rules give three sub-formulas for 0 of & 0, the three The weakening rule corresponds to X 1 which erazes an X The contraction rule corresponds to X (X X) which duplicates X The dereliction rule corresponds to X U which transforms X to U Denition 8 The second order exponentials 0 and 0 are dened by: 0 U = :(X )(X (XX))(X U)X 0 U = 9X:(X 1)(X (X X))(X U) X Contrary to the additives, those second order exponentials are not equivalent to the classical ones However, they are equivalents to the properties of weakening and contraction Thus, they are a weak form of the classical exponentials Theorem 9 Let D 8 and U be two formulas >< ` D; 1 If ` D; D D are provable then ` D; 0 U is provable >: ` D; U roof: If we have three proofs 1, and of the sequents ` D; 1, ` D; D D and ` D; U, we can build a proof of ` D; 0 U: 1 ` D; 1 ` D1 `(D1) ` D; D D ` D(D D ) `(D(D D )) ` D; U ` DU `(DU ) ` D; D ` D; (D1)(D(D D ))(DU ) D 9X = D ` D; 9X:(X 1)(X (X X))(X U ) X ` D; 0 U Theorem 10 If U is a formula then the following sequents are provable: roof: roof W (U) of ` 0 U; 1: 8 >< >: ` 0 U; 1 ` 0 U; 0 U 0 U ` 0 U; U

7 roof C (U) of ` 0 U; 0 U 0 U : ` X ; X ` ; 1 ` X ; X; 1 `(X ); (X (XX)); (X U ); X; 1 `(X )(X (XX))(X U )X; 1 ` :(X )(X (XX))(X U )X; 1 ` 0 U; 1 ; `(X ); (X (XX)); (X U ); X; X; 0 U 0 U `(X ); (X (XX)); (X U ); XX; 0 U 0 U X ; X `(X ); (X (XX)); (X U ); X (XX); X; 0 U 0 U `(X ); (X (XX)); (X U ); X; 0 U 0 U `(X )(X (XX))(X U )X; 0 U 0 U ` :(X )(X (XX))(X U )X; 0 U 0 U ` 0 U; 0 U 0 U with the proof of `(X ); (X (XX)); (X U); X; 0 U : `(X ); (X1) `(X (XX)); (X(X X )) `(X U ); (XU ) ` X; X `(X ); (X (XX)); (X U ); X; (X1)(X(X X ))(XU ) X 9X = X `(X ); (X (XX)); (X U ); X; 9X:(X1)(X(X X ))(XU ) X `(X ); (X (XX)); (X U ); X; 0 U roof D (U) of ` 0 U; U : ` X ; X ` U; U ` X U; X; U `(X ); (X (XX)); (X U ); X; U `(X )(X (XX))(X U )X; U ` :(X )(X (XX))(X U )X; U We can extend those proofs to a sequence of formulas: ` 0 U; U Theorem 11 If = A 1 ; : : :; A n is a sequence of formulas then the following sequents are provable: ( ` 0 ; 1 ` 0 ; C C where C = 0 A 1 0 A n ( if n = 0) We write W () the proof of ` 0 ; 1 and C () the proof of ` 0 ; C C

8 roof: By induction on n in the same spirit as the proofs of the theorem 10 Those proofs, except the substitution of A 1 ; : : :; A k by other formulas depend only on the number of formulas of They contain no cut Transformations of Exponentials These second order exponentials allow us to dene a transformation on formulas, sequents and nally on proofs Denition 1 If U is a formula, [U] is dened by induction by: [A] = A if A is an atom or a constant [U V ] = [U] [V ] and [U V ] = [U][V ] for the multiplicatives [U&V ] = [U]&[V ] and [U V ] = [U] [V ] for the additives [U] = 0 [U] = 9X:(X 1)(X (X X))(X [U]) X where X is not free in U [U] = 0 [U] = :(X )(X (XX))(X [U])X where X is not free in U [:U] = :[U] et [9X:U] = 9X:[U] for the quantiers We write [U 1 ; : : : ; U n ] for [U 1 ]; : : :; [U n ] Now, we can introduce proof transformations such that if is a proof of ` U 1 ; : : : ; U n then is a proof of ` [U 1 ; : : : ; U n ] Denition 1 If is a proof of a ` U 1 ; : : : ; U n the main rule of : (written ` U 1 ; : : : ; Un ), is dened following ` ` X ; X ` ; ` = ` ; X ; X W ` ; (X ); (X (XX)); (X [U ]); X ` ; U where is the premise ` ; (X )(X (XX))(X [U ])X ` ; :(X )(X (XX))(X [U ])X ` ; [U ] of the weakening rule ` ; U; U = ` C ([U ]) ; U; U where is the premise of C ` ; [U ][U] ` [U ] [U ]; [U ] ` ; U Cut ` ; [U ] the contraction rule 8

9 ` ; U ` ; U = D of the dereliction rule `; U `; U = ` X ; X ` ; U ` ; X [U ]; X ` ; (X ); (X (XX)); (X [U ]); X ` ; (X )(X (XX))(X [U ])X ` ; :(X )(X (XX))(X [U ])X ` ; [U ] W ([)] ` ; 1 () ` C; 1 ` C1 `(C1) C ([)] ` ; C C () ` C; C C ` C(C C ) `(C(C C )) ` ; [U ] ` C; [U ] ` C[U ] `(C[U ]) where is the premise () ` ; C ` ; (C1)(C(C C ))(C[U ]) C 9X = C (1) ` ; 9X:(X 1)(X (X X))(X [U ]) X ` ; [U ] where is the premise of the rule and where C = [A 1 ] [A n ] if = A 1 ; : : :; A n (C = if n=0) 1 1 ` U 1 ; : : : ; Un of the rule k k = R 1 1 k k ` [U 1 ]; : : : ; [Un] R for the other rules 1 ; : : : ; k are the premises Theorem 1 If is a proof of ` U 1 ; : : : ; U n then is a proof of ` [U 1 ]; : : : ; [U n ] Moreover, rule only appears with empty environments in roof: By induction on Emulation of Cut Elimination Like additives, normalization of a proof may be emulated in its transformed proof Lemma 1 If is a proof and reduces in one step to 0 without using commutative conversion for exponentials then reduce (in one or more step) to [ 0 ] Red - 0 Red + - [ 0 ] 9

10 roof: As with additives, we only need to prove that the three dierent kinds of cut between 0 U and 0 U reduce as the classical exponentials do For instance, with a cut between a rule and a dereliction: W ([)] ` ; 1 ` C ; 1 ` C 1 `(C 1) () C ([)] ` ; C C ` C ; C C ` C (C C) `(C (C C)) () h ` ; [U] i ` C ; [U] ` C [U] `(C [U]) ` ; (C 1)(C (C C))(C [U]) C () ` ; 9X:(X 1)(X (X X))(X [U]) X ` ; [U] ` ; C 9X = C (1) ` ; ` X ; X h ` U ; ` X [U ]; X; `(X ); (X (XX)); (X [U ]); X; `(X )(X (XX))(X [U ])X; ` :(X )(X (XX))(X [U ])X; ` [U ]; Cut i h 1 ` ; U i Red + h ` ; ` U ; i Cut Theorem 1 If reduces to without using commutative conversion of exponentials then reduces to [ 0 ] Red - 0 Red - [ 0 ] roof: By induction on the derivation from to 0 and by using Lemma 1 Constant Elimination The second order quantiers allow us to eliminate the multiplicative and additive constants of linear logic in the same spirit as the previous transformations Denition 1 The second order constants are dened by: 1 0 = :X X 0 = 9X:X X 0 0 = :X > 0 = 9X:X The new constants are linearly equivalent to the classical ones They behave like classic ones as respects to cut elimination (without commutative conversion of and >) 10

11 Conclusion We can apply the dierent transformations all together on formulas, sequents and proofs The result is linear logic proofs without any constant and additive and where rules have empty environments Normalization (without commutative conversion) of a proof can be emulated in the transformed one This mechanism may be compared to dierent codings of linear logic proofs (and -calculus) into reduction net systems [Laf9] All those translations emulate cut elimination without commutative conversion Our linear logic proof transformations act similarly for cut elimination but take place inside linear logic The strategies used by those systems are globally the same as ours Thus, the proof transformations give a theoretical justication for those codings: they can be obtained rst by transforming a proof then coding it in the object language This decomposition divides those emulations in two mechanisms The rst one (our proof transformations) inhibits commutative conversions and the second one uses a simple translation (because proof constructions are less numerous in transformed proofs) References [Gir8] Jean-Yves Girard Linear logic Theoretical Computer Science, 0:1{10, 198 [GLT88] Jean-Yves Girard, Yves Lafont, and Taylor roofs and Types Cambridge Tracts in Theoretical Computer Science Cambridge University ress, 1988 [Laf9] Yves Lafont From proof nets to interaction nets In J-Y Girard, Y Lafont, and L Regnier, editors, Advances in Linear Logic, pages { Cambridge University ress, 199 roceedings of the Workshop on Linear Logic, Ithaca, New York, June