A self-dual modality for "before" in the category of coherence spaces and in the category of hypercoherences
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. A self-dual modality for “before” in the category of coherence spaces and in the category of hypercoherences Christian Retore´. N˚ 2432 De´cembre 1994. PROGRAMME 2. ISSN 0249-6399. apport de recherche.
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(4) A self-dual modality for “before” in the category of coherence spaces and in the category of hypercoherences Christian Retor´e Programme 2 — Calcul symbolique, programmation et g´enie logiciel Projet MEIJE Rapport de recherche n ˚ 2432 — D´ecembre 1994 — 15 pages. Abstract: In his paper “A new constructive logic: classical logic” Jean-Yves Girard brought up the question of a self-dual modality. This note provides a semantical solution with respect to the self-dual connective before in the category of coherence spaces, and in the category of hypercoherences. Key-words: Denotational semantics. Logic, proof theory, linear logic.. (R´esum´e : tsvp) . retore@loria.fr A more ”categorical” version is to appear in french: [Ret94b]. Unit´e de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) T´el´ephone : (33) 93 65 77 77 – T´el´ecopie : (33) 93 65 77 65.
(5) Une modalit´e autoduale pour “pr´ec`ede” dans la cat´egorie des espaces coh´erents et dans celle des hypercoh´erences R´esum´e : Dans son article “A new constructive logic: classical logic” Jean-Yves Girard pose la question d’une modalit´e autoduale. Cette note fournit une solution s´emantique relative au connecteur autodual pr´ ec` ede dans la cat´egorie des espaces coh´erents et dans celle des hypercoh´erences. Mots-cl´e : S´emantique d´enotationnelle. Logique, th´eorie de la d´emonstration, logique lin´eaire..
(6) A self-dual modality for “before”. 1. 3. Presentation. This note strongly relies on [Gir91], where the question was raised. This section is just a concise reminder. The structural rules of classical logic are responsible for the non-determinism of classical logic, and linear logic which carefully handles these rules is especially adequate for a constructive treatment of classical logic, as the afore mentioned paper shows. Linear logic handles structural rules by the modalities (a.k.a. exponentials) “?” and “!”. The modality “?” allows contraction and weakening in positive position, and the modality “!” in negative position. The formula linearly implies while the formula is linearly implied by . In semantical words this means we have the linear morphisms: . . . .
(7) . . . . . The major difficulty when dealing with classical logic are the cross-cuts, appearing in the cut elimination theorem of Gentzen as a rule called MIX [Gen34]. This rule is a generalised cut between several occurences of and several occurences of , which is in fact a cut between two formulae coming both from a contraction. This is a major cause of non-determinism: an example can be found in [Gir91], Appendix B, Example 2, p 294. In linear logic, such a cut may not happen, since contraction applies on for mulae, while their negation for applying a cut is which can not come from a contraction. Let us now quote the precise paragraph of [Gir91] p 257 which motivates this note:. “The obvious candidate for a classical semantics was of course coherence spaces which had already given birth to linear logic; the main reason for choosing them was the presence of the involutive linear negation. However the difficulty with classical logic is to accommodate structural rules (weakening and contraction ); in linear logic, this is possible by considering. RR n ˚ 2432.
(8) 4. C. Retor´e coherent spaces . But since classical logic allows contraction and weakening both on a formula and its negation, the solution seemed to require the linear negation of to be of the form , which is a nonsense (the negation of is which is by no means of isomorphic to
(9) a space ). Attempts to find a self-dual variant of (enjoying . ) sys tematically failed. The semantical study of classical logic stumbled on this problem of self-duality for years.”. Here too we focus on coherence spaces because of their tight relation to linear logic [Gir87, Tro92, Ret94a]. Once the modality is found in the category of coherence spaces, we briefly show that this modality also exists in the category of hypercoherences that Thomas Ehrhard introduced in[Ehr93]. This is encouraging since hypercoherences, which may be viewed as a refinement of coherence spaces, are a different semantics, also very close to linear logic. In our previous work on pomset logic [Ret93, Ret95], we studied a self-dual connective before , together with partially ordered multisets of formulae. This and it enlead us to the modality to be described. It is a functor, it is self-dual, joys both left and right contraction with respect to before , and is a retract of . . Fortunately it does not interprets weakening, otherwise we would have (seman tically) some strange phenomena like and for all ! Here is a picture of the canonical linear morphisms that the functorial modality enjoys:
(10) . . . . . . . ! . . " # "%$ . There is not yet any syntax extending pomset logic to this modality. We firstly need to study the basic steps of cut-elimination, in particular the contraction/contraction case and the commutative diagrams it requires, as Myriam Quatrini did in [Qua95] for the logical calculus LC of [Gir91].. INRIA.
(11) A self-dual modality for “before”. 2. 5. Preliminary remarks. 2.1. Before. We refer the reader to [Gir87, Tro92] for the definition of coherence spaces. Let us simply recall the multiplicative connective before studied in [Ret93, Ret95], writ! : ten . Definition 1 Given two coherence spaces defined by: web. . . ! . . . !
(12) . , the coherence space. . ! . is. . . coherence. and. whenever . . . and. . or. . From [Ret93] we know that the following easy proposition holds: Proposition 1 This connective is: self-dual, . . ! . !. associative. . admits. . . . . as a unit,. in between. and. . ! . non-commutative,. . ! . . . . . . . . . ,. . ! . ,. . ,. . : for all formulae ! . . and. , ! wehave and. ".
(13) . .. 2.2 Several trivial remarks on trees. ')(. Definition 2 We write # for $&% , #+* for the set of finite words on # , including the empty word, #), for the set of infinite words on # , with the usual lexicographical order /.10 range over #, , while 2 range over #+* . We and product topology. Letters like 2 32 4* for - 252 2 2 2 2 768676 . use the standard notation Proposition 2 (9;:=< generic trees on > ) The set 9;:< of continuous functions from # , to a set > (discrete topology) is in a one-to-one correspondence with the set of finite binary trees on > such that any two sister leaves have distinct labels. Thus, an element of 9&: < may be described:. RR n ˚ 2432.
(14) 6. C. Retor´e. 2 676867 32 (
(15) #&* > satisfying:. 2 - (a) - #), - # , 2 #;*
(16) 2 25% and 2 2 ' (b) In this formalism 3- is computed as follows: applying (a), there exists a unique. such that there exists - with 2 - , let - 32 3% =* .. $ (1) as a finite set . (2) as the normal form of a term of the grammar:. < > < "< ! < means &% 0 0 where the reduction is $#" > # # !('*,) + -# where 0 a term of < having an occurrence of the subterm < . In this formalism - is computed as follows: /. (. ! %&- (. 3- ! '- - - Proof: Straightforward, see appendix if not familiar with 0 , . 1 As it is very simple, an example will avoid us to be too formal:. (. . $ Example 1 Let > . Here are the three description of the same element of 9&: (0) as a function . 3%;%;%&. . 3%;% ' . . 3% ' %&. . 2 32 2 $ % ' %;% . . 3% &' ' ' %&%;' ' % ';' %; ';'&' . :. . $ (1) as a finite set of pairs . <. . . . # * and 4 > ( : ' '& ';' % ';';'& ( %. INRIA.
(17) A self-dual modality for “before”. <. (2) as a normal term of. :. 7. !. . . such a term is the normal form of, e.g.. ! . . . !. . ! . ! . .
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(29) . (3) the best way to think of it is the most unpleasant to type:. . One more easy remark: Proposition 3 Let. $. . Proof:. 3 3.1. . -. 9&: <. . $. . . . If. $. . and. , then there exists -. - -. - . . $. #&,. such that. - . 1. As for proposition 2., see appendix if necessary.. The modality and its properties The modality . The first attempt to find such a modality, inspired by the product of Q copy of lead me to consider sequences of finitely many tokens in , indexed by the rational num' bers of % . It works but has too many ( # !) contraction isomorphisms. Achim Jung told me binary trees should work and avoid this drawback, and thanks to his suggestion, I arrived to the following:. . Definition 3 Let A be a coherence space. We define . ! of continuous functions from #;, coherence Two functions and of 9;: " web The set 9&: whenever. . $. - # , - $ 3- . RR n ˚ 2432. and. . to. . . as follows:. (discrete topology). . are said to be strictly coherent. - -. . - . $ . - .
(30) 8. C. Retor´e. 3.2. Properties. The following is clear from proposition 2: Proposition 4 (denumerable web) If. is denumerable, so is . . . And next come the key property: Proposition 5 (self-duality) The modality is self-dual, i.e. Proof:. . . . . . . .. These two coherence spaces obviously have the same web.. Hence it is equivalent to show that, given two distinct tokens in the following exclusive disjunction holds:. .
(31) . . . . 0. If because of the previous proposition 3, there is an infinite word of , such that . The following exclusive disjunction holds: and . . . . . . . . . . . . . 1. and is part-wise equivalent to the previous exclusive disjunction.. Proposition 6 (contraction isomorphism) There is a canonical linear isomorphism ! . between and Proof:. Consider the following subset of !" $# :. . . +-, % '& ()( *( /.. *0 ,. (0
(32) . .. . -3 ( and (*12 ( . We show that it is the trace of a linear isomorphism between and 4# . Firstly,. 4#. %. clearly defines a bijection, between the webs and. 56 .. Secondly, let us see that, given (7(. (. ( . 9;:. . *( + and 878 . + in % we have . ( <( ( . *( $# . INRIA.
(33) A self-dual modality for “before”. 9. . We assume that ( ( , i.e. . 0
(34) or Two cases may occur: either 8 $# have ( *(. :. . . . . . . .. .. . .. ( and ( .. 12 , and we show that in both cases we. 0 ,. . . . . 50 we have ( and ( , and thus ( <( 8 4# . . ( Indeed we have ( (05 ( and ( 0
(35) and we know that ( . ( Indeed for all we have 0 0
(36) and therefore ( (0 0 . . all , we have 1 ( Indeed for 05 and therefore ( (*1 *1 .. and thus ( *( 8 $# . (1) If 12 , then we have ( Indeed we have ( (*12 ( and ( <12 and we know that ( . ( Indeed for all we have 1 12 and therefore ( (*1 <1 . ( *( 8 $# and therefore we either have. 9 We assume that ( and (
(37) or ( . We show that in both cases we have ( . then there exists such that ( and ( (0) If ( and. . Let 0
(38) . ( for all ( Indeed we have ( (0
(39) ( and 0
(40) and we know that ( ( . Let . and If 0 then therefore ( (0 ( 0 . . If 1 then ( (<1 ( *1 . then there exists such that ( and ( (1) If ( . Let 12 . for all ( and ( Indeed we have ( (*12 . . <12 and we know that ( . . ( If then 1 with . Therefore ( (<1 ( *1 . (0) If . . . . . . .. .. . .. .. .. .. .. . . . . . . . . . . .. . .. . . . . . . . . . . . .. .. .. .. .. .. . . . . . . .. .. . . . RR n ˚ 2432. . . . . . . 1.
(41) 10. C. Retor´e . Proposition 7 ( retract of . . . ) Any coherence space . . . is a linear retract of . . Consider the following subset of , where constant function mapping any element of , to .. 0. & -,. :. . . Proof:. . ." . .. stands for the. 3. . It is a + linear trace both from to and from to . The compound is & , ." 3 which is a strict subset of
(42) , while the compound is exactly . . $ $ 2 - #,. Proposition 8 ( is a functor) If wing trace:. . . $ - & ( 3- . . defines . . . 1. . by the follo-. . This makes an endo-functor. Proof: (1) Firstly,let us show that be in . . defines a linear map from . to . . Let . . and. + . . Thus there exists such that and + . Since we know for all that and . are in which is linear, we have and since both + and + Now let . We have . are in which is linear we have Applying proposition 3, there exists an such that and for all . We necessarily have and therefore . . Hence . . For all , both + and + are in which Assume is linear. Therefore, for all , proposition . Applying 3, either. or there exists an such that and for all . In the second case we have since for all . . In both case we have . . Assume that . . . . INRIA.
(43) A self-dual modality for “before”. (2) It is easily seen that . . . . 11. (3) Let us now show that commutes with linear composition. Let , be two linear functions. . . . . . . and . . . Assuming that *( is in it is easily seen that *( is in . . Indeed there exists a in is in. such that and + 8<( is in . Thus, for all the+pair is in and the pair <( is in , so that *( is in for all , i.e. <( is in . We now assume that *( is in and show that it is in too.. If *( is in then <( - is in for all , i.e. for + all there is in and +8 *( is in . The exists some such that point is to show that among the functions from , to that this existential quantifier defines, one is an element of . Consider the function <( from , to with the discrete topology on . It is continuous, because the product topology of a finite product of discrete spaces is the discrete topology on the (finite) product of the involved sets. Therefore it may be described as a finite ++3 binary tree, with leaves in with the properties (a) . We write it as a finite set & 7 and (b), like in 2. For all , there exists such that (1) of proposition and ( take e.g. 0 * . Therefore for all there exists in . such that is in and is in , and for each we choose one (there are finitely many ). Now, we can define with the help of (a). For each . there exists a unique such that , and we define to be , and thus is clearly a continuous function +3 from , to . Notice that the generic tree of is not necessarily & but its normal form according to (2) of proposition 2: the property (b) may fail since there possibly exist , and 0 , 1 while such that . . Now it is easily seen that is in and 8<( in . Indeed for all there. exists ,+ a unique such that + and we then have , ( and thus is in and 8 <( . is in .. . . . 0. . . . . . RR n ˚ 2432. . .. . . . . . . . . . 0. . 0. . . . . 1.
(44) 12. C. Retor´e. 4. The modality in the category of hypercoherences. Since the construction is very similar, we shall be brief, and closely refer to [Ehr93]. Definition 4 Let and be hypercoherences. The hypercoherence is the hypercoherence whose web is and whose strict atomic coherence is defined by:. -. * . % 3 - & * . . iff . . and. #. '. - &. or . * . '. It is easily seen that this connective is associative, self-dual, non-commutative and in between the tensor product and the par — just like in the category of coherence spaces. Now, the self-dual modality enjoying the wanted properties is defined in the category of hypercoherences by: Definition 5 Let be an hypercoherence. The hypercoherence herence whose web is:. . . $. . is the hyperco-. # , 2 # , &
(45) (. and whose strict atomic coherence is defined by:. $ 676867 ( . * . 2 676768 2 ( * . iff . 2 # , and . 2 2 $ 2 676867 2 (.
(46) . $. . '. The proofs that it enjoys the same properties as in the category of coherence spaces are so similar that we skip them.. INRIA.
(47) A self-dual modality for “before”. 13. Appendix Here we write down the proofs of proposition 2. and 3., which are completely straightforward for anybody familiar with , .. 0. . General remarks ,. 3 Since we consider the product topology on , , the subsets & where range over * , are an open basis of , . Given two open sets of the basis and . we either have or
(48) or 3 . These sets are clopen sets: the complement of is . . where & . Notice that Any clopen set may be written as the finite union of disjoint : as it is open it is the union of , as it is closed in a compact it is compact, hence only a finite number of the are needed, and because the are either disjoint or included one in the other, we can assume that they are disjoint. #" "$ , thus %'&)(*! A clopen set always possesses a maximum element: ! %+&,( " "$ <1 * .. . . 0. 0. 0. . .. . 0. . . . Proof of proposition 2 We prove that any continuous function from , to a set - with the discrete topology may be described by a unique binary tree whose leaves are labelled by elements of - , such that no sister leaves have the same label:. 0. .
(49) 0. + 3 '& /. 01. 2 * 3 $ (a) . , 54 6 (b) 8 7 96 0 and. *0.
(50). . . $ . with. 1 :. 0.
(51). 0 $ .. It is obvious that such binary trees correspond to normal terms of the grammar given in (2) of proposition 2.. . . F OR ALL SUCH THERE EXISTS SUCH A BINARY TREE DESCRIBING 3 : < We have , ; <0 . But each ; <0 is open ( & 0 is an open of the discrete topology on - ) and , is compact, hence there exists a finite number of elements of - say 0 >=?=?= 0 such that , " " 3 ; @0 — by the way takes finitely many values. As ; <0 is a clopen set ( & 0 is closed too) it is the union of finitely many disjoint clopen sets of the basis, say >==?=?= ) BA , and we can assume that for all 7 C6 with $ 1EDF 7 C G 6 H D 6 , for all we do not have both 0 and 1 — because I . J ; <0 ; @0 becomes , J =?==. ON Hence , . . 0. .. . . 0. 0. Let . RR n ˚ 2432. . & . 0. . . . K 0 +3 ##"P" K@" "M$ .. . 0. .
(52). . " " "LK@"M$ . .
(53) 14. C. Retor´e. These N are easily seen indeed we have seen that is dis to be pairwise disjoint: joint from A , and 0 while A 0 . Hence for all there exists a unique K. that . N , i.e. for all there exists a unique such that there exists with N such K. , i.e. (a) is fulfilled. Moreover, as we have seen, for all 7 96 with 1 D 7 C6 D 6 , $ for all we do not have both 0 and 1 , i.e. (b) is fulfilled too.. . IF. . . .
(54). . . . TWO SUCH BINARY TREES DIFFER. EITHER DO THE CONTINUOUS FUNCTIONS THEY DEFINE. . +3. Assume now we have two such binary trees, i.e. two finite sets & .20 . and +3. ' & 0 both satisfying (a) and (b). Write for the projection of to * . This defines two continuous functions from , to - , by setting 0 where is. . Indeed (a) makes the unique index for which there exists and such that sure that for a given . in there exists exactly one 0 such that .)0 is in , and the : :
(55) , which is a clopen inverse image of a subset of - is the finite union
(56) set of , , hence an open set.. . 6 such that $ 0 $ is in but not in . Assume that , i.e. that there exists It is easily noticed that the
(57) (resp. ) are the biggest basis open on which (resp. ) & 0 3 , we obtain and is constant. Indeed, assuming the contrary, say and therefore there are two sister leaves of above , say 0 and that 1 , but as their labels are the images by of any infinite word extending them, their labels are the same, namely 0 , and this conflicts with (b). Because 3 of the previous paragraph, if then . Otherwise, if i.e. if 3 are equal, up to some renumbering, let us say that . . .,Since the sets & /. and & . $ 0 * , and , there exists 6 such that 0 $ 0 $ . Thus $ 0 * 0 $ 0 $ . Proof of proposition 3 Now, let us see that whenever two continuous functions from , to a set - (discrete 6 . topology) differ, there exists such that and for all The product of the discrete topological space - by itself, is the discrete topological from to defined by space - too. Hence the function < 0 1 iff 0 + - defined by - is continuous. The function from , to is continuous — product topology on - 3- . Therefore set, ; 0 is a clopen which has a greatest element (of the shape <1 * ). Thus and whenever .. . . 0. . . . 0. . . 0. . . . . . . . . . . . . 0. 0. . 0. . Acknowledgements: Thanks to Thomas Ehrhard, Jean-Yves Girard, Catherine Gourion, Achim Jung, Myriam Quatrini and Thomas Streicher for their comments. The commutative diagrams are drawn with Paul Taylor’s LATEX package diagrams. INRIA.
(58) A self-dual modality for “before”. 15. References [Ehr93] Thomas Ehrhard. Hypercoherences: a strongly stable model of linear logic. Mathematical Structures in Computer Science, 3(4):365–385, December 1993. [Gen34] Gehrard Gentzen. Untersuchungen u¨ ber das logische Shließen II. Mathematische Zeitschrift, 39:405–431, 1934. Traduction fran¸caise de J. Ladri`ere et R. Feys: Recherches sur la d´eduction logique, Presses Universitaires de France, Paris, 1955. [Gir87]. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1–102, 1987.. [Gir91]. Jean-Yves Girard. A new constructive logic: classical logic. Mathematical Structures in Computer Science, 1(3):255–296, November 1991.. [Qua95] Myriam Quatrini. S´emantique coh´erente des exponentielles: de la logique lin´eaire a` la logique classique. Th`ese de Doctorat, sp´ecialit´e Math´ematiques, Universit´e Aix-Marseille 2, January 1995. [Ret93]. Christian Retor´e. R´eseaux et S´equents Ordonn´es. Th`ese de Doctorat, sp´ecialit´e Math´ematiques, Universit´e Paris 7, February 1993.. [Ret94a] Christian Retor´e. On the relation between coherence semantics and multiplicative proof nets. Rapport de Recherche 2430, I.N.R.I.A. Sophia-Antipolis, December 1994. [Ret94b] Christian Retor´e. Une modalit´e autoduale pour le connecteur “pr´ec`ede”. In Pierre Ageron, editor, Cat´egories, Alg`ebres, Esquisses et N´eo-Esquisses, Publications du Groupe de Recherche en Algorithmique et Logique. D´epartement de Math´ematiques, Universit´e de Caen, September 1994. [Ret95]. Christian Retor´e. Pomset logic. Rapport de Recherche, I.N.R.I.A. Lorraine, January 1995.. [Tro92]. Anne Sjerp Troelstra. Lectures on Linear Logic, volume 29 of Center for the Study of Language and Information (CSLI) Lecture Notes. University of Chicago Press, 1992.. RR n ˚ 2432.
(59) Unite´ de recherche INRIA Lorraine, Technopoˆle de Nancy-Brabois, Campus scientifique, 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LE`S NANCY Unite´ de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unite´ de recherche INRIA Rhoˆne-Alpes, 46 avenue Fe´lix Viallet, 38031 GRENOBLE Cedex 1 Unite´ de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unite´ de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. E´diteur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) ISSN 0249-6399.
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