A congruence relation for restructuring classical terms

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1 A congruence relation for restructuring classical terms Dragiša Žunić1 and ierre Lescanne 2 1 Carnegie Mellon Universit in atar 2 Laboratoire de l nformatique du arallélisme, L Ecole Normale Supérieure de Lon, France Abstract. We present a congruence relation on sequent-stle classical proofs which identifies proofs up to trivial rule permutation. The stud is performed in the framework of X calculus which provides a Curr- Howard correspondence for classical logic (with eplicit structural rules) ensuring that proofs can be seen as terms and proof transformation as computation. Congruence equations provide an eplicit account for classifing proofs (terms) which are sntacticall different but essentiall the same. We are able to prove that there is alwas a representative of a congruence class which enables computation at the top level. 1 ntroduction For a long time it was considered not possible to give constructive semantics to classical logic, but onl to intuitionistic and linear logic. ecent works have shown that this is actuall possible if one gives up on the principle that the computational semantics is a confluent rewrite sstem. n this work we stud the essence of classical proofs b means of specifing its unessential content. Namel we present a list of congruence rules specifing which sntacticall different proofs/terms should be considered the same. Moreover, this congruence relation enables us to pick a representative of a congruence class so that the cut elimination can continue at the top level. This is done in the framework of X calculus - a higher order rewrite sstem designed to provide a Curr-Howard correspondence with the sequent sstem G1 [13] for classical logic (presented in Fig. 2). This sstem is characteried b the presence of structural rules weakening and contraction and it is ver close to the original formulation b Genten [5]. 2 elated work The first computational interpretation of classical logic reling on sequents was presented b Curien and Herbelin in [4] and in [8], while a direct correspondence with a standard sequent formulation of classical logic, with the intent of capturing its computational content in full richness (although with implicit structural

2 rules), was presented b Urban in [14] and b Bierman and Urban in [15]. This research further lead to the formulation of X calculus [11, 2] which was a base to introduce eplicit control over erasure and duplication b featuring eplicit structural rules weakening and contraction, ielding the X calculus [18, 6]. This kind of conservative calculus etension had been studied in the intuitionistic framework b Kesner and Lengrand [9]. ndeed there is a certain analog in going from λ [1] (featuring eplicit substitution) towards λlr [9] (eplicit substitution, erasure and duplication) in the natural deduction setting, and going from X to X in the classical sequent calculus setting. 3 The calculus X ntuitivel when we speak about X -terms we speak about classical proofs formalied in the sequent calculus with eplicit structural rules weakening and contraction. Terms are built from names. This concept differs essentiall from the one applied in λ-calculus, where variable is the basic notion. The difference lies in the fact that a variable can be substituted b an arbitrar term, while a name can be onl renamed (that is, substituted b another name). n our calculus the renaming is eplicit, which means that it is epressed within the language itself and is not defined in the meta-theor. The notation of using hats over names has been borrowed from rincipia Mathematica [17] and is used to denote the binding of a name. Definition 1. The snta of X -calculus 3 is presented in Fig. 1, where,,... range over an infinite set of in-names and,, γ... range over an infinite set of out-names., ::=. capsule (aiom rule). eporter (ritht arrow-introduction) [] ŷ importer (left arrow-introduction) cut (cut) left-eraser (left weakening) right-eraser (right weakening) < ] ŷ left-duplicator (left contraction) [ >γ right-duplicator (right contraction) Fig. 1. The snta of X Although we distinguish two categories of names (in-names and out-names) the will usuall be referred to simpl as names. There are eight constructors introduced in the snta, whose names suggest their computational role. 3 We consider onl the implicational fragment here.

3 Definition 2. We define free names and principal names of a term, written f n() and pn(), respectivel, in the following wa: f n() pn().,,. fn( )\{, } [] ŷ fn( ) fn()\{, } fn( ) fn()\{, } none fn( ) fn( ) < 1 2 ] fn( )\{ 1, 2 } [ 1 2 > fn( )\{ 1, 2 } A name which is not free is called a bound name. The notion of a principal name describes a free name that appears at the top level in the structure of term. We will use and to emphasie that and have and as principal names, respectivel. Our consideration of principal names is motivated b the nature of cut operation, which alwas refers to a pair of free names (an out-name and an in-name) - one in each corresponding term - and binds them. f these names are at the top level (accessible) then the cut elimination proceeds in one of its essential steps. However this research is concerned b static aspects onl (dealing with proof structure), but it most certainl attempts to la the foundation for revealing the computational essence as well. n the X calculus we consider onl linear terms. We sa that a term is linear if ever name has at most one free occurrence and ever binder does bind an actual occurrence of a name (and therefore onl one). We use the standard definition of linear terms, see [18] on page 43. We will specif the notions of contet, subterm and immediate subterm in order to efficientl work with the structure of terms. Definition 3. The contets are formall defined as follows: C{ } ::= { } { }. { } [] ŷ [] ŷ { } { } { } { } { } < ŷ { }] [{ } >γ C{C{ }} nformall, a contet is a term with a hole which can accept another term. Therefore C{ } denotes placing the term in the contet C{ }. Definition 4. A term is a subterm of a term, denoted as if there is a contet C{ } such that = C{} (the smbol = stands for sntactic equalit.). t is eas to show that the subterm relation is refleive, antismmetric and transitive (i.e., is an order).

4 Definition 5. A term is an immediate subterm of if = C{} and C{ } C {C { }}, C{ } { }. 4 The tpe assignment sstem Given a set T of basic tpes, a tpe is given b A, B ::= A T A B. Epressions of the form : Γ are used to present the tpe assignment, where is an X term and Γ, are contets whose domains consist of free innames and out-names of, respectivel 4. Comma in the epression Γ, Γ stands for the set union. Definition 6. We sa that a term is tpable if there eist contets Γ and such that : Γ is derivable in the sstem of rules given b Fig. 3. Γ A, Γ, B Γ, A B, (L ) ( ) Γ, Γ, A B, Γ A B, (a) A A Γ A, Γ, Γ, Γ, A (cut) Γ (weak-l) Γ, A Γ (weak-) Γ A, Γ, A, A (cont-l) Γ, A Γ A, A, (cont-) Γ A, Fig. 2. Sequent sstem G1 for classical logic eduction rules are numerous as the capture the richness and compleit of classical cut elimination, and ver fine-grained due to the presence of eplicit terms for erasure and duplication. For details see [18]. 5 The congruence relation This section presents a congruence relation on terms, denoted, which is represented b a list of equations 5. The congruence relation induced b these rules 4 Technicall contets are sets of pairs (name, formula); if we forget about labels and consider onl tpe, we are going back to Genten s classical sstem G1 (Fig. 2), where contets are multisets of formulas. 5 The list is partial due to limited space. A complete list of congruence rules can be found in [18].

5 (a). : :A :A : Γ :A, : Γ, :B (L ) [] ŷ : Γ, Γ, :A B, : Γ, :A :B, ( ). : Γ :A B, : Γ :A, : Γ, :A (cut) : Γ, Γ, : Γ (weak-l) : Γ, :A : Γ (weak-) : Γ :A, : Γ, :A, :A (cont-l) < ŷ ] : Γ, :A : Γ :A, : A, (cont-) [ >γ : Γ γ :A, Fig. 3. The tpe sstem for X is a refleive, smmetric and transitive relation closed under an contet. These rules define eplicitl and in an intuitive wa which sntacticall different terms should be considered the same. The ke propert of this congruence relation is that, given a cut term, it offers a representative of a congruence class so that the computation (cut elimination) continues at the top level (see Theorem 1). The diagrammatic representation is assigned to ever congruence rule to strengthen the reader s intuition (a formal stud of the diagrammatic calculus, which can be derived from the term calculus presented here, is in the domain of future work). A label is assigned to ever congruence rule, and thus each rule is declared as: name :. eporter-importer γ γ E E ei1 : ( γ [] ŷ ). (. ) γ [] ŷ with, fn( ) ei2 : ( γ [] ŷ ). γ [] ŷ (. ) with, fn() eporter-cut γ γ E E

6 ec1 : ( γ ŷ). (. ) γ ŷ with, fn( ) ec2 : ( γ ŷ). γ ŷ(. ) with, fn() importer-importer t t (3) t ii1 : ( [] ŷ ) [] t ( [] t ) [] ŷ with, fn( ) ii2 : ( [] ŷ ) [] t [] ŷ ( [] t ) with, fn() ii3 : ( [] t ) [] t ( [] ŷ ) with, t fn() cut-cut (3) cc1 : ( ) ŷ ( ŷ) with, fn( ) cc2 : ( ) ŷ ( ŷ) with, fn() cc3 : ( ŷ) ŷ( ) with, fn() cut-importer ci1 : ( [] ŷ ) ẑ ( ẑ) [] ŷ with, fn( ) ci2 : ( [] ŷ ) ẑ [] ŷ ( ẑ) with, fn() (3) (4) ci3 : ( [] ẑ ) ( ) [] ẑ with, fn() ci4 : ( [] ẑ ) [] ẑ ( ) with, fn()

7 cut-contraction cc t 1 : < 1 2 ŷ] (< 1 2 ]) ŷ with 1, 2 fn( ) cc t 2 : < 1 2 ŷ] ŷ(< 1 2 ]) with 1, 2 fn(), (3) (4) cc t 3 : [ ŷ 1 2 > ([ 1 2 >) ŷ cc t 4 : [ ŷ 1 2 > ŷ([ 1 2 >) with 1, 2 fn( ), with 1, 2 fn() weak-general C C wg1 : (C{ }) C{ } / fn(c{ }) wg2 : (C{ }) C{ } / fn(c{ }) The relation induces congruence classes on terms. We use cl( ) to denote the congruence class of a term with respect to the relation. Notice that each congruence class has finitel man terms, and thus finitel man possibilities to select a representative of a class. Congruence rules satisf some standard properties such as preservation of free names (interface preservation, as in Lafont s interaction nets [10]) and preservation of tpes. t has been argued in [12] that two sequent proofs induce the same proof net if and onl if one can be obtained from the other b a sequence of transpositions of independent rules, without giving details about these operations. We proceed further as we have presented eplicitl how this transposition is done at the static level. This detailed stud, presented here in the framework of X calculus, will enable the search for the essence of computational aspect as well, b considering reduction as reducing modulo congruence relation.

8 n what follows we show that the congruence rules allow us to perform restructuring of X -terms (i.e. sequent proofs), so that the cut-names (names bound b a cut operation) are brought to the top level. Take for eample an arbitrar X -term of the form: The name might not be directl accessible (that is, is mabe not principal name for ). Furthermore, this might hold for both names involved in the cut, and, which are possibl deep inside the structure of their corresponding terms. We prove that it is possible to transform the above term using congruence rules defined in the previous section 6 - to: C{ } where C is a contet, and and are principal names of and, respectivel. n other words we can alwas pick at least one representative of a congruence class cl( ), which allows us to continue the computation at the top level. n what follows we first formulate two lemmas which focus on a single cutname (either or ) at a time, followed b the main theorem which addresses both cut-names. n proving the results we will use the transitivit of relation: f and then. Lemma 1 (Left-restructuring). For ever term of the form, there eists a contet C and a term, where is a principal name for and, such that C{ } roof. B induction on the structure of and case analsis. The base case is when is a principal name for. Then we have = and C = { }. Assume that the propert holds for the immediate subterms of. The possible cases for are 1. = ŷ M. γ, γ 2. = M [] ẑ N 3. = M ŷn 4.&5. = < M] and = [M 2 >, 6.&7. = M and = M, We anale the first 3 cases 7 : 6 For the cases we prove here the set of congruence rules presented is sufficient. resenting a complete proof would require to use all congruence rules. 7 Due to a limited space.

9 1. (ŷ M. γ) ec1 ŷ (M ). γ i.h. ŷ (C {M }). γ C{M }, with C{ } = ŷ (C { }). γ 2. (a) Case M. (M [] ẑ N) ci1 (M ) [] ẑ N i.h. (C {M }) [] ẑ N C{M }, with C{ } = (C { }) [] ẑ N (b) Case N. (M [] ẑ N) ci2 (M ) [] ẑ N i.h. (C {M }) [] ẑ N C{M }, with C{ } = M [] ẑ (C { }) 3. (a) Case M. (M ŷn) cc1 (M ) ŷn i.h. (C {M }) ŷn C{M }, with C{ } = (C { }) ŷn (b) Case N. (M ŷn) cc2 M ŷ(n ) i.h. M ŷ(c {N }) C{N }, with C{ } = M ŷ(c { }) The proof goes similarl for other cases. Lemma 2 (ight-restructuring). For ever term of the form, there eists a contet C and a term, whose principal name is and, such that C{ } roof. Similarl as previous lemma, b induction on the structure of and case analsis. The following theorem confirms that the top level cut evaluation is complete, in a sense that there alwas eists a representative of a congruence class which enables cut evaluation at the top level relative to its cut-names. t also shows that in the presence of congruence rules we do not need propagation rules. ropagation rules in the X calculus can be seen as means to perform restructuring of terms. However their eistence blurs the core part of classical computation, and at the same time don t enable restructuring of normal forms (terms that do not contain cuts). Theorem 1. For ever term of the form there eists a contet C and terms and whose principal names are, respectivel, and,

10 , such that C{ } roof. B using the two previous lemmas, we construct the proof as follows: Lem. 1 And thus we are done. C 1 { } Lem. 2 C 1 {C 2 { }} C { }, with C { } = C 1 {C 2 { }} 6 Basic properties of The congruence relation describes the wa to perform restructuring of terms and thus an important propert is preservation of free names. Second important propert is tpe preservation, which ensures that term-restructuring defined b can be seen as proof-transformation. Theorem 2 (Free names preservation). f, are X -terms, then the following holds: f then fn( ) = fn() roof. The propert can be checked b treating carefull each rule. Theorem 3 (Tpe preservation). Let S be an X -term and Γ, contets. Then the following holds: f S : Γ and S S, then S : Γ roof. The proof goes b checking that the propert holds for equations inducing. We first write the tping derivation for the term on the left-hand side, and then for the term on the right-hand side of the equation. We prove several non-trivial cases. Observe the first rule of eporter-importer group of rules: ( γ [] ŷ ). ei1 (. ) γ [] ŷ, with, fn( ),. On the one hand we have: : Γ, : A : B, γ : C, : Γ, : D ( L) γ [] ŷ : Γ, Γ, : A, : C D : B,, ( ) ( γ [] ŷ ). : Γ, Γ, : C D : A B,,

11 On the other hand, : Γ, : A : B, γ : C, ( ). : Γ : A B, γ : C, : Γ, : D ( L) (. ) γ [] ŷ : Γ, Γ, : C D : A B,, The proof goes similarl for the other cases. 7 Conclusion and future work We presented a detailed account of a possible step forward, towards the unveiling of the essential content of sequent classical proofs. This is done b declaring which sntacticall different terms should be considered the same, i.e., b means of congruence rules which induce a congruence relation on terms. The framework for this is classical logic formalied in a standard sequent sstem with eplicit structural rules, namel weakening and contraction. ndeed it was necessar to have all the details, structural as well as logical, to be able to state what are the unessential aspects and thus etract the essence in a sstematic wa. Clearl this is a base for studing the operational semantics towards defining a sstem with onl the essential reduction rules, where reducing is reducing modulo congruence relation. The congruence relation is proved to satisf the minimal requirement of free names preservation, as well as that of tpe preservation. t is also possible to define a translation, call it D, from terms to diagrams in a natural wa, inductivel on the structure of terms (basic stud is given in [18]). A similar approach was taken in [9], when interpreting intuitionistic λlr-terms (featuring eplicit weakening and contraction) into proof-nets [7]. t is reasonable to epect the translation D to satisf 1 2 then D( 1 ) = D( 2 ) 1 D 2 then D( 1 ) D D( 2 ) where 1 and 2 are X -terms, and where D is a suitable reduction relation on diagrams. However this should be carefull studied and belongs in the domain of future work. Moreover, in terms of future work, this computational model ma be relevant in relation to process session tpes and concurrenc, as recent studies have shown that both linear intuitionistic [3] and linear classical logic [16], have a natural interpretation as protocols for concurrent process communication.

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