Commutative Locative Quantifiers for Multiplicative Linear Logic

Transcription

1 Commutative Locative Quantifiers for Multiplicative Linear Logic Stefano Guerrini 1 and Patrizia Marzuoli 2 1 Dipartimento di Informatica Università degli Studi Roma La Sapienza Via Salaria, Roma, Italy guerrini@diuniroma1it 2 Dip di Scienze Matematiche ed Informatiche Roberto Magari Università degli Studi di Siena Pian dei Mantellini Siena, Italy marzuoli@unisiit Abstract The paper presents a solution to the technical problem posed by Girard after the introduction of Ludics of how to define proof nets with quantifiers that commute with multiplicatives According to the principles of Ludics, the commuting quantifiers have a locative nature, in particular, quantified formulas are not defined modulo variable renaming The solution is given by defining a new correctness criterion for first-order multiplicative proof structures that characterizes the system obtained by adding a congruence implying x(a B) = xa xb to first-order multiplicative linear logic with locative quantifiers In the conclusions we shall briefly discuss the interpretation of locative quantifiers as storage operators Keywords: ludics, linear logic, proof nets 1 Introduction Ludics is a logic based on locative principles introduced by Girard in 2001 [Gir01] In Ludics, the forall quantifier commutes over every connective but the exists The extension of that general rule of commutativity would force to accept the logical principle x(a B) xa xb or equivalently x(a B) xa xb that is not even valid in Classical Logic, and the principle x(a B) xa xb (1) that is not valid in Linear Logic, while the corresponding formula x(a B) xa xb is a valid classical principle Partially supported by the italian PRIN Project FOLLIA

2 2 S Guerrini, P Marzuoli In this paper we shall analyze what happens if we add the distributivity of the forall quantifier over the tensor in First-Order Multiplicative Linear Logic, MLL 1 for short We remark that all the other commutativity principles of forall are valid in Linear Logic [Gir87] In particular, we can derive xa xb x(a B) As a consequence, forcing the distributivity rule 1 in MLL 1 corresponds to add the equivalence x(a B) = xa xb Let MLL 1 be the logic obtained by adding to MLL 1 a -congruence on formulas that contains x(a B) xa xb A deductive system for MLL 1 can be given by means of a subtype relation st x(a B) xa xb However, the system obtained in this way does not have the subformula property and has some problems with s, unless one restricts to locative quantifiers In fact, since the commutative equivalence may introduce pairs of distinct -quantifiers corresponding to the same variable, if the witness of the -rule can be any term, we would get that distinct -quantifiers binding the same variable should be instantiated by distinct terms, leading to a contradiction The problems previously mentioned are solved by moving from sequent calculus to multiplicative proof nets [Gir87,DR89] with first-order quantifiers [Gir91] In particular, we shall define a subset of MLL 1 proof structures that corresponds to MLL 1 and for which -elimination holds The proof nets defined in this way are then one of the first synt for locative principles inspired by Ludics Anyhow, we remark that the approach followed here is quite different from those presented in [FM05,CF05], in which the authors investigate proof nets with a synt matching the operators of ludics In our approach, we stick to the usual synt of Linear Logic, incorporating into it some principles inspired by locativity We also remark that the paper solves the technical problem of a system for locative quantifiers, but does not give any semantic interpretation of them In the conclusions, section 5, we shall briefly discuss some possible interpretations of locative quantifiers as storage operators that we are investigating The reformulation of the results in the paper for second order quantifiers is straightforward In section 2 we shall recall the main definitions of the calculus and of the proof nets for MLL 1 (see [Gir91]) In section 3 we shall analyze the system MLL 1 obtained by forcing that the quantifiers commute over the multiplicatives and we shall give a deductive system based on a notion of subtype In section 4 we shall extend the proof nets for MLL 1 to MLL 1 and prove their -elimination 2 First-Order Multiplicative Linear Logic The formulas of the First-Order Multiplicative Linear Logic without unit MLL 1 [Gir91] are built from atoms p by means of the connectives and, and the quantifiers and For every atom p there is a dual atom p Duality extends to every formula by (A B) = A B, ( xa) = xa, and (A ) = A Because of the previous duality rules we can restrict to one-sided sequents Γ, where Γ is a finite multiset of formulas

3 Commutative Locative Quantifiers 3 The rules of the sequent calculus MLL 1 are Γ, A, A A, A Γ, Γ, A, B Γ,, A B Γ, A x x FV(Γ ) Γ, xa Γ, A, B Γ, AB Γ, A[t/x] Γ, xa t Let us remark that we do not rename the variable x that we want to bind by a -quantifier The usual property of variable renaming is obtained by assuming that MLL 1 formulas are equated modulo α-congruence The variable x in a -rule is an eigenvariable Wlog, in the following, we shall assume that the eigenvariables of the universal quantifiers in a proof are distinct, that is, each -rule in the proof uses a new eigenvariable This corresponds to the assumption that the eigenvariables in the proofs combined by a -rule or by a are distinct, that is, in some cases, in order to combine two proofs with distinct eigenvariables we may have to replace the eigenvariables that occur in both the proofs with new variables 21 MLL 1 proof-nets A link between (occurrences of) formulas is a pair of (possibly empty) sequences of formulas P C: the formulas in P are the premises of the link, the formulas in C are its conclusions Every rule of the sequent calculus corresponds to a link whose premises are the main premises of the rule, while the conclusions are the main conclusions of the rule In particular, the iom link has two conclusions and no premises, the link has no conclusions and two premises, the tensor and par links have two premises and one conclusion, the forall and exists links have one premise and one conclusion A proof structure is a set of formulas and links st every formula is conclusion of exactly one link and is premise of at most one link The formulas that are not premise of any link are the conclusions of the proof structure Graphically, we can represent ioms/s by drawing an edge labeled / between the conclusions/premises of the link and the other links by drawing an edge between every premise of the link and its conclusion, as explained in the picture below A A A B A B A A[t/x] A A xa xa In a forall link, the variable x is the eigenvariable of the link In an MLL 1 proof structure, the eigenvariables of the -links in the proof structure are distinct We shall assume that the conclusions of the proof structures are closed As A B AB

4 4 S Guerrini, P Marzuoli a consequence, every variable that occurs free in some formula of the proof structure is the eigenvariable of one (and only one) -link A switching of a proof-structure consists in: (i) the choice of a position L or R for each par link; (ii) the choice of a formula B for each -link, where B is any formula of the proof structure in which the eigenvariable of the link occurs free or the premise of the link if the eigenvariable does not occur free in any formula Given a proof structure S and a switching σ, the switch σ(s) is the graph defined in the following way: (i) for each iom or link between A and A, there is an edge between A and A ; (ii) for each tensor link with conclusion A B there is an edge between the premise A and A B, and an edge between the premise B and A B; (iii) for each exists link there is an edge between its premise and its conclusion; (iv) for each par link with conclusion AB there is an edge between the premise A and AB if σ chooses L for that link, or an edge between the premise B and AB if σ chooses R; (v) for each forall with conclusion xa there is an edge between the conclusion xa and the formula B selected by σ Definition 1 (correctness criterion) An MLL 1 proof structure N is correct if the switch σ(n) is connected and acyclic, that is, it is a tree, for every switching σ A correct proof structure N is a proof net Theorem 1 (sequentialization) Given an MLL 1 proof π of a sequent Γ, there is a corresponding MLL 1 proof net pn(π) with conclusions Γ and a link for every rule in π Conversely, for every MLL 1 proof net N, there is a corresponding MLL 1 proof π st pn(π) = N Proof [Gir91] 22 Cut-elimination The proof nets of MLL 1 have a terminating -elimination defined by the reduction rules A A A A A B A B A A B B A B A B

5 Commutative Locative Quantifiers 5 A A[t/x] A[t/x] A[t/x] xa xa Let us remark that the reduction of a / implies the substitution of the witness t of the -rule in the for the free occurrences of the eigenvariable x of the -quantifier introduced by the -rule in the Correctness is preserved by -elimination Therefore, the normal form of any proof net is a proof net too The reduction rules for -elimination are confluent Hence, the normal form of any proof net is unique 3 First-Order Commutative Quantifiers Let be the least congruence over MLL 1 formulas induced by x(a B) xa xb xa A if x FV(A) The congruence naturally extends to sequents: Γ Γ if there is a bijection between the formulas in Γ and the formulas in Γ st every formula A Γ is mapped to some formula A Γ st A A Definition 2 (MLL 1 ) A sequent Γ is derivable in MLL 1 if there is an MLL 1 proof π of some Γ Γ st 1 the α-congruence has not been used; 2 every xa in π is introduced by an t -rule with t = x; 3 π is -free The restriction on quantifiers and α-congruence are required by -elimination We remark that such a restriction makes the system much weaker, for instance, we cannot prove ( ya[y/x]), xa, if y is a variable distinct from x The study of -elimination will be pursued on proof nets in section A deductive system for MLL 1 MLL 1 can be also defined as the deductive system obtained from MLL 1 by eliminating the rule, by restricting the -rule to the case in which the witness t is the variable x bound by the rule Γ, A Γ, xa and by equating formulas modulo the -equivalence x

6 6 S Guerrini, P Marzuoli Let us recall that the part of the that is missing in MLL 1 is the distributivity of over We can then define the subtyping relation as the least partial order closed by contexts and induced by x(a B) xa xb xa A if x FV(A) Correspondingly, we have the subtyping rule Γ, A with A A Γ, A where is the anti-reflexive restriction of It is readily seen that the deductive system MLL 1 obtained by replacing the condition that formulas are equated modulo -equivalence with the previous subtyping rule is equivalent to MLL 1 In fact, let us observe first that A B implies that B, A is derivable in MLL 1 (without s and using the restricted version of the -rule only) Therefore, since for every A A there is B st A B A, the formula A is derivable in MLL 1 iff there is a -free MLL 1 proof of some B with B A that uses the restricted version of the -rule only and does not use the α-congruence 4 MLL 1 proof nets In order to define the proof structures for MLL 1, it is useful to observe that in the deductive system MLL 1 we can restrict to the case in which the -rules are applied to a single variable at a time, restricting the -rule to the case Γ, xφ(a 1,, A m, B 1,, B n ) Γ, φ( xa 1,, xa n, B 1,, B n ) φ x FV(B 1,, B n ) where φ is a -tree, that is, a tree of -connectives, whose leaves are the formulas A 1,, A m, B 1,, B n Formally, -trees are defined by: (i) every formula A is a -tree with leaf A; (ii) if φ 1 and φ 2 are -trees with leaves L 1 and L 2, then φ 1 φ 2 is a -tree with leaves L 1, L 2 It is readily seen that, under the proviso of the rule, xφ(a 1,, A m, B 1,, B n ) φ( xa 1,, xa n, B 1,, B n ) If we restrict to atomic ioms, or at least to quantifier free ioms, every -quantifier must be introduced by a corresponding -rule and we can assume that a φ -rule that distributes a x is applied immediately after the x -rule that introduces the quantifier We can then merge the x -rule and the φ -rule into a unique rule, getting the deductive system MLL 1 defined by Γ, A, B p, p Γ,, A B Γ, A, B Γ, AB Γ, A Γ, xa x Γ, φ(a 1,, A m, B 1,, B n ) Γ, φ( xa 1,, xa m, B 1,, B n ) x x FV(Γ, B 1,, B n )

7 Commutative Locative Quantifiers 7 41 Proof structures The links of the proof nets for MLL 1 are the same given for MLL 1, with the restriction that in the -link t = x However, in order to define MLL 1 proof nets, we need to consider proof nets that correspond to MLL 1 proofs too In particular, we have to replace the -link with the link corresponding to the - rule The -link becomes then a particular case of the -link, the case in which the -tree φ has only one leaf, that is, φ(a) = A Let us recall that in the definition of MLL 1 proof structures we have assumed that the eigenvariables in a proof structure are distinct Such a condition is too strong in the case of MLL 1, where the α-congruence does not hold in the case with that we shall analyze in subsection 45, such a condition is unsatisfiable as soon as we have a formula that contains quantifiers Therefore, we have to weaken the assumption on the eigenvariables in a proof, reformulating such a condition in a way that allows to define the switching positions of foralll links (ie, the formulas that in a switch may be connected to the conclusion of the forall) without forcing the renaming of distinct forall links that bind variables with the same name 42 Scope and eigenvariables Let us say that two formulas are connected if there is an edge between them (eg, if they are the two conclusions/premises of an iom/ link or one is the premise and the other one is the conclusion of a link) An x-path between two formulas A 0 and A n is a path from A 0 to A n in which the variable x occurs free in all the inner formulas of the path; more precisely, an x-path is a sequence of formulas A 0 A 1 A n 1 A n st, for every 0 i < n, the formulas A i, A i+1 are connected and, for every 0 i n, x FV(A i ) or A i = xa or A i is the conclusion of a x -link Let A be the conclusion of a x -link A formula B is x-bound to A, or x- bound to the -link with conclusion A, if there is an x-path between B and A The scope of A, or of the x -link with conclusion A, is the set of the formulas B bound to A st x FV(B) Let us take a x -link with conclusion A and a y -link with conclusion B (We remark that, since x and y range over the set of the variables, x and y might also denote the same variable) We shall say that the the two forall links have the same (distinct) eigenvariable(s) when they have the same scope (their scopes differ) and we shall write x y (x y) It is readily seen that x y iff x y or, when x = y, the scopes of A and B are disjoint 43 Folded and unfolded proof structures Let us say that an MLL 1 proof structure is unfolded when every -link in it is a -link An unfolded MLL 1 proof structure is also an MLL1 proof structure An MLL 1 proof structure is folded when the eigenvariables of all the -links in the proof structure are distinct

8 8 S Guerrini, P Marzuoli 44 Cut-free proof nets Any -free folded proof structure S is isomorphic to an MLL 1 proof structure S obtained by replacing the conclusion A = φ( xa 0,, xa m, B 1,, B n ) of every -link with A = xφ(a 0,, A m, B 1,, B n ) and consistently replacing A for A in every formula that contains A as a subformula In fact, the restriction on the eigenvariables of S implies that we can find a renaming of the variables in S st the variables bound by the -links in S are distinct By construction, if Γ and Γ are the conclusions of S and S, respectively, then Γ Γ This immediately implies that, if S is correct, then Γ is derivable in MLL 1 Let us say that the -free folded MLL 1 proof structure S is correct when S is correct The definition of correct -free folded MLL 1 proof structure can be also given directly, it suffices to adapt the definition of switch, forcing that a - quantifier can jump to a formula in its scope only, or to the premise of its -link Definition 3 (MLL 1 switching) An MLL1 switching of a folded MLL1 proof structure is a map that assigns a position L or R for each par link and a formula B to every -link with conclusion A, where B is a formula in the scope of A or the premise of the -link Switches and correctness are defined as for MLL 1 proof structures Proposition 1 Given an MLL 1 derivation π of a sequent Γ, there is a corresponding MLL 1 correct -free MLL1 proof structure pn(π) with conclusions Γ and a link for every rule in π Conversely, for every correct -free folded MLL 1 proof structure S, there is a corresponding MLL 1 derivation π st pn(π) = S Correctness of unfolded MLL 1 proof structures is defined in terms of correctness for folded ones, in particular, an unfolded structure will be correct only if there is an equivalent correct folded structure Let us define the following transformation on MLL 1 proof structures: A A B x A B A B x FV(B) EV(B) x A B A B A A B x B x A B A B x FV(A) FV(B) x or A B x x A B

9 Commutative Locative Quantifiers 9 where x and x denote two occurrences of the same variable, and where EV(B) denotes the set of the eigenvariables of the quantifier links associated to the quantified subformulas in B (therefore, x FV(B) EV(B) means that x cannot occur at all in B, neither free nor bound) Let us remark that the above transformations are well-defined: they transform a proof structure into a proof structure with the same conclusions Definition 4 A -free MLL 1 proof structure N is correct, it is a -free MLL 1 proof net, if there is a correct -free folded MLL 1 proof structure S st N S By reversing the direction of the transformation defined above, we see that, given a folded -free MLL 1 proof structure S, there is a (unique) -free unfolded MLL 1 proof structure N st N S Therefore, we can state Propo- sition 1 for the folded case too Proposition 2 Given an MLL 1 derivation π of a sequent Γ, there is a corresponding -free MLL 1 proof net with conclusions Γ Conversely, for every -free MLL 1 proof net N, there is a corresponding MLL 1 derivation π st pn(π) = N 45 MLL 1 proof nets with Unfortunately, the extension to the case with is not straightforward In fact, let P be the least set of MLL 1 proof structures (with ) that contains the -free MLL 1 proof nets and is closed by -composition The reduction rules defined in subsection 22 (with t = x in the case of the / -) defines a terminating and convergent -elimination procedure for P But, P is not closed by -elimination Eg, by reducing the proof structure obtained by composing the -free MLL 1 proof net N 1 that proves ( x(a B)), xa xb and the proof net N 2 that proves ( xa xb), xa B (such a proof net is an MLL 1 proof net indeed), we get the incorrect proof structure A A B x x(a B ) B A x xa B xa B that proves ( x(a B)), xa B, which is not derivable in MLL 1 The main issue is that in N 1 the -links corresponding to the two subformulas xa and xb of the conclusion xa xb have the same eigenvariable Therefore, if we want to that formula with the conclusion of another proof

10 10 S Guerrini, P Marzuoli net N, we have to ask that in N the variables bound by the -links corresponding to the subformulas xa and xb of the -formula ( xa xb) are bound to the same -link, or at least, that the structure of N is consistent with the assumption that the variables bound by xa and xb correspond to the same eigenvariable, in particular, we cannot combine a formula in which the variable x occurs free with a formula in which it occurs bound The example shows that, in the presence of s, the definition of x-path (sequence of connected formulas in which the variable x occurs free) does not suffice to characterize the dependencies between the occurrences of a variable x in the proof structure In particular, because of the restriction on quantifiers, the renaming of an eigenvariable may induce the renaming of variables that are not in its scope, creating in this way a connection between formulas in which the variable x occurs and that are not connected by an x-path Let us say that two formulas A and B of an MLL 1 proof structure S in which the variable x occurs are x-correlated if the renaming of some occurrence of x in A induces a corresponding renaming of some occurrences of x in B Let us say that an occurrence α of a variable x in the formula A is correlated to an occurrence β of x in the formula B, denoted by α = β, if the renaming of α induces the renaming of β It is readily seen that = is an equivalence relation Two quantifier links are correlated if the occurrence of the variable in the binders in their conclusions are correlated In particular, when two -links are correlated we shall say that their eigenvariables are correlated Two correlated -links are siblings if they are leaves of a tree of -links Definition 5 (correct folded MLL 1 proof structures) A folded MLL1 proof structure S is correct when every switch of S is connected and acyclic and the following sibling condition holds: if two -links of S are siblings, then they are bound to the same -link (eg, to the same eigenvariable) Since in a -free MLL 1 proof structure two -links may be correlated iff they are bound to the same -link, the sibling condition trivially holds for free MLL 1 proof structures Therefore, in the -free case, Definition 5 reduces to the definition of correct folded -free proof structure given in subsection 44 As in the -free case, correctness of unfolded proof structures is defined by means of the transformation rules on page 8 with the addition that the bottom rule can be applied also when x and x are correlated eigenvariables and not only when they are the same eigenvariable In any case, let us remark that = contains and that, in the -free case x = x iff x x Definition 6 (MLL 1 proof nets) An MLL 1 proof structure N is correct, that is to say it is an MLL 1 proof net, if there is a correct folded MLL 1 proof structure S st N S By the equivalence between = and in the -free case and by the fact that the sibling condition trivially holds in -free MLL 1 proof structures, in the -free case, the previous definition reduces to Definition 4

11 Commutative Locative Quantifiers Cut-elimination Cut-elimination is defined on MLL 1 proof nets, that is, on unfolded proof structures Unfortunately, correctness is not preserved by one step of -elimination In fact, let us take the following proof net N p p q x (p q ) p q p q p q x p x q x p x q x p x q x p x q q x p x q x p x q The two framed -formulas in N have the same eigenvariable In one step, N reduces to a proof structure M in which the two framed quantifiers are with the matching -formulas M is in normal form for but contains two -quantifiers with the same eigenvariable Therefore, M is not a proof net Even if correctness is not preserved by a single -elimination step, we can define a big-step procedure that reduces an MLL 1 proof net to another proof net proof struc- Given an MLL 1 proof net N, let S be the correct folded MLL 1 ture st N S Every conclusion of a -link in S is the image of a corresponding formula in N that we shall call a folding root of N Every folding root of N is a -tree φ( xa 1,, xa m, B 1, B n ) with x 1 i n FV(B i) EV(B i ) Let us say that φ is the folding tree of the folding root A Let A be a formula of an MLL 1 proof net N We shall say that N reduces by a big-step that eliminates A, written N b M, when: 1 A is not a folding root and N M by the application of the -elimination rule that reduces the of A; 2 A is a folding root and N M by a sequence of -elimination rules that eliminate the -links corresponding to the the folding tree of A, moving the from the folding root A to the leaves of its folding tree Proposition 3 (big-step -elimination) Let N be an MLL 1 proof net If N b M, then M is an MLL 1 proof net Big-step -elimination is a particular reduction strategy for Therefore, by the uniqueness of the normal-form of, Proposition 3 proves that elimination is sound and that every sequent derivable by an MLL 1 proof net (with ) is derivable in MLL 1 Theorem 2 (-elimination) For every MLL 1 proof net N, there is a unique MLL 1 -free MLL 1 proof net M st N M

12 12 S Guerrini, P Marzuoli 5 Conclusions and further work The paper solves the problem of how to characterize the proof nets for a logical system with locative quantifiers that commutes over the multiplicative connectives However, this is just a first step, because in the paper we have not addressed at all any semantics for locative quantifiers Our ongoing research suggests that locative quantifiers can be interpreted as storage operators In particular, x states a constraint on or assigns a value to a location x of the store, while x operates on all the values that may be stored in x We are also trying to rel the restrictions on the quantifier rules, reintroducing generic terms as witnesses in the -rule A first possibility that we are considering is to associate a store to every proof, assuming that an x -rule with witness t can be applied only if the term t is compatible with the value, if any, in the location x Another approach is to assume that the language of terms contains an intersection operator st A(t 1 ) and A(t 2 ) imply A(t 1 t 2 ) In this way, we might get rid of the problem of how to instantiate with two distinct terms two -quantifiers with the same eigenvariable, as this would correspond to replacing both with the intersection of the two terms Acknowledgments We wish to thank Claudia Faggian and Laurent Regnier who have been the reviewer of the PhD thesis [Mar06] that contains a preliminary version of the work presented in the paper Their detailed comments and suggestions have been very useful in writing the paper References [CF05] Pierre-Louis Curien and Claudia Faggian L-nets, strategies and proof-nets In C-H Luke Ong, editor, Computer Science Logic, 19th International Workshop, CSL 2005, 14th Annual Conference of the EACSL, Oxford, UK, August 22-25, 2005, Proceedings, pages Springer, 2005 [DR89] V Danos and L Regnier The structure of multiplicatives Archive for Mathematical Logic, 28: , 1989 [FM05] Claudia Faggian and François Maurel Ludics nets, a game model of concurrent interaction In 20th IEEE Symposium on Logic in Computer Science (LICS 2005), June 2005, Chicago, IL, USA, Proceedings, pages IEEE Computer Society, 2005 [Gir87] Jean-Yves Girard Linear logic Theoretical Computer Science, 50(1):1 102, 1987 [Gir91] J-Y Girard Quantifiers in linear logic II Prépublications de l Équipe de Logique 19, Université Paris VII, Paris, 1991 [Gir01] Jean-Yves Girard Locus solum: From the rules of logic to the logic of rules Mathematical Structures in Computer Science, 11(3): , 2001 [Mar06] Patrizia Marzuoli Ludics and Proof-Nets a New Correctness Criterion PhD Thesis, Dottorato di Ricerca in Logica Matematica e Informtaica Teorica, Università di Siena, February 2006