Following renewed interest in duploids arising from the exponential comonad of linear logic (the construction describing polarised intuitionistic translations into linear logic), I summarise here various remarks:• about a decomposition of Girard's "boring" translation as the expression of call-by-value in call-by-name, dual to how thunks are used to express call-by-name in call-by-value• about the coincidence between linear CPS translations and Girard's translations of intuitionistic logic into linear logic,• about a completeness property of historical models of linear logic in the above context• about a rational reconstruction of these translations with the Linear Call-by-Push-Value

Munch-Maccagnoni, Guillaume

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Note on models of polarised intuitionistic logic

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HAL Id: hal-01540760https://hal.inria.fr/hal-01540760Preprint submitted on 16 Jun 2017HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.Note on models of polarised intuitionistic logicGuillaume Munch-MaccagnoniTo cite this version:Guillaume Munch-Maccagnoni. Note on models of polarised intuitionistic logic. 2017. ￿hal-01540760￿Note on models of polarised intuitionistic logicGuillaume Munch-MaccagnoniTeam Gallinette, Inria Bretagne Atlantique, Univ. Nantes, LS2N, Nantes, FranceGuillaume.Munch-Maccagnoni@Inria.fr3rd June 2017Following renewed interest in duploids arising from the exponential comonad of linearlogic (the construction describing polarised intuitionistic translations into linear logic), Isummarise here various remarks:• about a decomposition of Girard’s “boring” translation as the expression of call-by-value in call-by-name, dual to how thunks is used to express call-by-name in call-by-value (object of a talk given in June 2012 at LDP, Marseille, entitled “Runnablemonads in models of the -calculus”),• about the coincidence between linear CPS translations and Girard’s translations ofintuitionistic logic into linear logic (object of a remark in my PhD thesis and of aprivate communication from Streicher in October 2013),• about a completeness property of historical models of linear logic in the above context(object of an untitled note circulated privately around July 2014),• about a rational reconstruction of these translations with the Linear Call-by-Push-Value(part of a talk given in March 2016 at IRIF, Paris, entitled “Adjunction models andpolarised calculi for modelling effects and resources”).1 Runnable monads in models of the -calculusHistorical models of linear logic satisfy an equalizing property not accounted for by traditional axiomaticaccounts of linear logic models (symmetric monoidal adjunctions, see Melliès, 2009). The followingproperty holds for the relational model (including generalisations with weights), coherence spaces (bothwith set and multiset exponentials), and hypercoherences:Proposition 1. Let us write V the linear category, ! ∶ V→ V the linear exponential comonad, and" ∶ !→ 1 the co-unit. Then " is a coequalizer of the pair "!, !" ∶ !!→ !.As a consequence, Führmann’s completeness property (1999) applies (in the dual) as described next.This property is best understood via the Curry-Howard correspondence by dualising Führmann’s con-struction of thunks.1Every model of linear logic V gives rise to a model of intuitionistic logic in the Kleisli category V!;the latter is equipped with the dual of a thunk which we call a runnable monad. Thunks express call byname in call by value (Hatcliff and Danvy, 1997). Dually, the runnable monad expresses call by valuein call by name, and therefore induces a translation of call by value into linear logic, in which context itcoincides with Girard’s “boring translation” (Munch-Maccagnoni, 2013). Such models of intuitionisticlogic in which the order of evaluation matters are polarised.The polarised structure is depolarised when one of the following equivalent properties are satisifed(Munch-Maccagnoni, 2014): the evaluation order does not matter (unrestricted -reduction is satisfied),the runnable monad is isomorphic to the identity functor, and  ∶ ! → !! is an isomorphism. Thiscannot happen, though, if the model of linear logic has an isomorphism !o ⊸ o ≅ o so as to interpretuntyped -calculus. Indeed, if the model is depolarised, then V! is bi-cartesian closed, and therefore theimpossibility results in Huwig and Poigné (1990) apply.2 Führmann’s completenessA runnable monad determines a notion of linearity: Führmann’s thunkability in the dual. Führmann’sresult then states that Proposition 1 is equivalent to the following:Corollary 2. The subcategory of V! whose morphisms are linear is isomorphic to V, in the category ofcategories equipped with a comonad and functors that preserve the co-monad structure.This justifies in particular the terminology “linear”: it generalizes Girard’s notion of linearity origin-ating from the historical models of linear logic to more general models of computation. (Specializationsof this theorem to a notion of isomorphisms that preserves the structure of connectives of linear logichave since been studied by Blute, Cockett, and Seely, 2015.)The phenomenon is similar to the relationship, among models of classical logic, between responsecategories and control categories (Selinger, 2001). Any response category (indirect model) gives rise toa control category (direct model), and any control category arises from a response category (the categoryof linear morphisms) in this way. This relationship between indirect and direct models is a reflection(Führmann, 1999).This result has sometimes been used to support the point of view that control categories are “the same”as response categories. However, the response category arising from a control category is not necessarilyisomorphic to the one that gave rise to the control category (Selinger, 2001). But, the additional propertygiven by Proposition 1 characterises the reflective full subcategory of indirect models equivalent to thecategory of direct models. In other words, in the case of historical models of linear logic giving rise topolarised intuitionistic logic models, one indeed has an isomorphism between the indirect model and thedirect model (which the syntaxes for linear logic do not necessarily reflect).2.1 Example in RelLet Rel be the category of sets and relations. Following Ehrhard (2012) we write multi-sets m =[a1,… , an] ∈Mfin and their union additively. The finite multi-set co-monad is written (!, ", ). -terms2are interpreted in the Kleisli category Rel!, that is, they are represented by sets f of relations (m, b) ∈ fwith m ∈Mfin(A) and b ∈ B for some sets A,B.Proposition 3. Rel! has a runnable monad (T , , ) defined with:• TA def=Mfin(A);• Tf def={([m1 +⋯ + mn], [a1,… , an])|||n ∈ ℕ and ∀i, (mi, ai) ∈ f};• Adef={([[a]], a) |||a ∈ A};• Adef= {(m,m) | m ∈Mfin(A)}.Proof. This is an instance of Führmann’s construction (1999), seen in the dual. ∎Definition 4. Amorphism f ∈ Rel!(A,B) is linear if B ◦Tf = f◦A (or equivalently f ◦!"A = f◦"!Ain Rel).Proposition 5. A morphism f ∈ Rel!(A,B) is linear if and only if (m, b) ∈ f ⟹ #m = 1. Inparticular " is a coequalizer of the pair "!, !" ∶ !!→ !.Proof. For any f ∈ Rel!(A,B), the morphisms f ∗ = B◦Tf and f◦A in Rel!(TA,B) are definedwith:f ∗ ={([m], a) |||(m, a) ∈ f}f◦B ={([[a1],… , [an]], b)|||([a1,… , an], b) ∈ f}Thus f ∗ = f◦B if and only if (m, b) ∈ f ⇒ #m = 1. This shows that for any f ∈ Rel(!A,B) such thatf◦"!A = f◦!"A, there exists g ∈ Rel(A,B) such that f = g◦"A, determined uniquely by(a, b) ∈ g ⟺ ([a], b) ∈ f . ∎3 The polarised caseThe generalisation of Führmann’s result with polarities is provided with duploids arising from adjunc-tions (Munch-Maccagnoni, 2014). One motivation of duploids is to relax the self-duality of dialoguecategories (Melliès, 2009) so as to account for settings in which negation is not involutive (e.g. Munch-Maccagnoni, 2013, Section II.2.6). If one is interested in seeing models of intuitionistic logic arise fromones of linear logic:1. one can start from the decomposition of response categories into dialogue categories with resourceadjunctions (Melliès, 2009; Melliès and Tabareau, 2010), but now composing with the resourceadjunction on one side only:M//⊥oo V//⊥ Vopoo(Data: V a distributive dialogue category, M a cartesian category, V //⊥ Vopoo the adjunction ofnegation with itself, M //⊥ Voo a symmetric monoidal adjunction.)32. alternatively, if V above is closed, as is the case for models of linear logic, one can also considerthe duploid arising from the resource adjunction M //⊥ Voo alone. (As is well-known in this casethe monad on M is commutative.)The duploid construction subsumes call-by-value and call-by-name translations. The variety associatedto the adjunctions above describe:1. for the adjunction M //⊥ Vopoo : CPS translations of “non-linear languages with linear control”summarised in Hasegawa (2004) (encompassing notably linearly-used continuations by Berdine,O’Hearn, Reddy, and Thielecke, 2000),2. for the adjunction M //⊥ Voo : polarised translations of intuitionistic logic described by Girard(1993, 2007); Liang and Miller (2007, 2009),as we will see in more detail in the next section.Remark 6. In fact, when V is ∗-autonomous, as in the case of models of linear logic, the duploidsarising from the intuitionistic polarised translation and from the linear CPS are in an obvious equivalence,because the adjunctions we started with differ up to composition by an adjunction V //⊥ Vopoo whichhappens to be an equivalence.I believe that this remark captures the essence of a note by Streicher (2013) suggesting the coincidencebetween Girard’s translations and linear CPS (in call by name and by value), communicated privatelyafter Munch-Maccagnoni, 2013, Chapter III based on this coincidence an analysis of delimited CPStranslations as polarised translations into linear logic.Equivalence between duploids is in the sense of the 2-category of duploids, duploid functors, and nat-ural transformations whose components are linear and thunkable, which provides the appropriate notion.But in the case where negation is strictly involutive, as is usually the case in syntaxes of linear logic,then the equivalence is an isomorphism, and one already has an isomorphism between the adjunctionsM//⊥ Voo and M //⊥ Vopoo .3.1 CompletenessThe case of the runnable monad mentioned previously corresponds to taking M = V!. Another naturalchoice, considered in Ehrhard (2016) is M = V! the category of coalgebras. Either way, it is a generalfact that the monad arising on V! and V! satisfies the equalizing property dual to Proposition 1 (seee.g. Barr and Wells, 1985, Proposition 3.3.7 in the dual). This property and its dual form the criterion togeneralise Führmann’s completeness to duploids. Thus:Corollary 7. In the case of historical models of linear logic, the indirect polarised model M //⊥ Voo isisomorphic, as an adjunction, to the one between the categories of thunkable and linear morphisms inits associated duploid.We refer the reader to Munch-Maccagnoni (2014) for the details of notions and constructions. Thismotivates, I believe, the study of linear logic via their associated Call-by-Push-Value model.44 Models of Call-by-Push-Value arising from linear logicThe notion of Call-by-Push-Valuemodel (Levy, 2005) coincides with that of (indirect) polarised intuition-istic logic LJp model (Curien, Fiore, and Munch-Maccagnoni, 2016), so that one can interpret polarisedintuitionistic logic into any CBPV model, not just ones arising from linear logic. We now recall how theprevious polarised intuitionistic situations give rise to CBPV models following general considerations.We use the notions and results from Curien, Fiore, and Munch-Maccagnoni, 2016 where more detail canbe found.• Any dialogue category (Melliès, 2009) with distributive structure Vgives rise to a Linear CBPVmodel _V //⊥ _Vopoo .• If in addition V is closed, in other words is a distributive SMCC, then it gives rise to a LinearCBPV model _V //⊥ _Voo .• Any symmetric monoidal adjunction M L //⊥ VMoo , such as for M = V! for Seely categories andM = V! for linear categories—in which cases M is cartesian—(Bierman, 1995; Melliès, 2009),enriches into an adjunction _M //⊥ L⋆_Voo (where L⋆ denotes precomposition of presheaves withL).• Any ILLp model (M //⊥ Voo ,_V //⊥ _Soo ) (a symmetric monoidal adjunction over a Linear CBPVadjunction where M is cartesian) gives rise to a CBPV model _M //⊥ L⋆_Soo by composing theimage of the LCBPV adjunction by L⋆ with the previous one.Therefore we have recipes for CBPV models:1. for any distributive dialogue category with a cartesian resource modality, as the adjunction_M //⊥ L⋆_Vopoogiven by the bijectionsM(Γ × P ,M(Q0 ⊸ ⊥)) ≅ V(LΓ⊗Q0, LP ⊸ ⊥)natural in Γ, P ∈M and in Q0 ∈ V,2. for any SMCC with a cartesian resource modality, as the adjunction_M //⊥ L⋆_Voogiven by the bijectionsM(Γ × P ,MN) ≅ V(LΓ⊗LP ,N)natural in Γ, P ∈M and inN ∈ V,5which therefore enrich the adjunctions in Section 3 using the fact that the left adjointL is strongmonoidal.As previously, they differ by the composition of an adjunction L⋆_V //⊥ L⋆_Vopoo which, when the closedstructure of V is ∗-autonomous, is an equivalence.More specifically, one has:1. CBPV models_V!↑//⊥ F⋆!_Vop↓oowhere:↓N = Nop ⊸ ⊥ ↑P = (!P ⊸ ⊥)opand where powers are given with:P ⇾N = (!P ⊗Nop)opCall-by-value and call-by-name sequents and arrows are therefore found as follows:P(Γ+ ⊗ P ,Q) ≅_Vop!Γ(↑P , ↑Q) = V(!Γ⊗ (!Q ⊸ ⊥), (!P ⊸ ⊥))P →CBV Q = ↓(P ⇾ ↑Q) = (!P ⊗ (!Q ⊸ ⊥)) ⊸ ⊥N(Γ⊝ &N,M) ≅_Vop!↓Γ(↑↓N,M) = V(!(Γop ⊸ ⊥)⊗Mop, !(Nop ⊸ ⊥) ⊸ ⊥)N →CBN M = ↓N ⇾M = (!(Nop ⊸ ⊥)⊗Mop)opwhere we have written ⋅op the formal dual for legibility.2. CBPV models_V!↑//⊥ U⋆_V↓oowhere↓N = !N ↑P = Pand where powers are given with:P ⇾N = P ⊸ NCall-by-value and call-by-name sequents and arrows are therefore found as follows:P(Γ+ ⊗ P ,Q) ≅ _VΓ(↑P , ↑Q) = V(Γ⊗ P ,Q)P →CBV Q = ↓(P ⇾ ↑Q) = !(P ⊸ Q)N(Γ⊝ &N,M) ≅ _V↓Γ(↑↓N,M) = V(!Γ⊗ !N,M)N →CBN M = ↓N ⇾M = !N ⊸M63. CBPV models_V!↑//⊥ F⋆! _V↓oowhere↓N = N ↑P = !Pand where powers are given with:P ⇾N = !P ⊸ NCall-by-value and call-by-name sequents and arrows are therefore found as follows:P(Γ+ ⊗ P ,Q) ≅ _V!Γ(↑P , ↑Q) = V(!Γ⊗ !P , !Q)P →CBV Q = ↓(P ⇾ ↑Q) = !P ⊸ !QN(Γ⊝ &N,M) ≅ _V!↓Γ(↑↓N,M) = V(!Γ⊗ !N,M)N →CBN M = ↓N ⇾M = !N ⊸MAs one can see, one finds back CPS of non-linear languages with linear control (Hasegawa, 2004), andtwo formulations of the translations of call-by-value and call-by-name intuitionistic logic into linear logic(Girard, 1987). Generalising the consideration of the Kleisli categories P and N to that of the duploidarising from the adjunction accounts for the more general polarised translation in Girard (2007, 12.B.1).ReferencesMichael Barr and Charles Wells. 1985. Toposes, triples and theories. Vol. 278. Springer-Verlag NewYork. 4Josh Berdine, Peter W O’Hearn, Uday S Reddy, and Hayo Thielecke. 2000. Linearly used continuations.In Proceedings of the Third ACM SIGPLAN Workshop on Continuations (CW’01). Citeseer, 47–54. 4Gavin Bierman. 1995. What is a categorical model of Intuitionistic Linear Logic?. In Proc. TLCA (Lec-ture Notes in Computer Science), Vol. 902. Springer-Verlag, 78–93. 5R Blute, JRB Cockett, and RAG Seely. 2015. Cartesian differential storage categories. Theory andApplications of Categories 30, 18 (2015), 620–686. 2Pierre-Louis Curien, Marcelo Fiore, and Guillaume Munch-Maccagnoni. 2016. A Theory of Effects andResources: Adjunction Models and Polarised Calculi. In Proc. POPL. https://doi.org/10.1145/2837614.2837652 5Thomas Ehrhard. 2012. The Scott model of linear logic is the extensional collapse of its relational model.Theor. Comput. Sci. 424 (2012), 20–45. 27Thomas Ehrhard. 2016. Call-by-push-value from a linear logic point of view. In European Symposiumon Programming Languages and Systems. Springer, 202–228. 4Carsten Führmann. 1999. Direct Models for the Computational Lambda Calculus. Electr. Notes Theor.Comput. Sci. 20 (1999), 245–292. 1, 2, 3Jean-Yves Girard. 1987. Linear Logic. Theoretical Computer Science 50 (1987), 1–102. 7Jean-Yves Girard. 1993. On the Unity of Logic. Ann. Pure Appl. Logic 59, 3 (1993), 201–217. 4Jean-Yves Girard. 2007. Le Point Aveugle, Cours de logique, Tome II: Vers l’imperfection. Hermann.published subsequently in English Girard (2011). 4, 7Jean-Yves Girard. 2011. The Blind Spot: Lectures on Logic. European Mathematical Society. 8Masahito Hasegawa. 2004. Semantics of linear continuation-passing in call-by-name. In InternationalSymposium on Functional and Logic Programming. Springer, 229–243. 4, 7John Hatcliff and Olivier Danvy. 1997. Thunks and the lambda-Calculus. J. Funct. Program. 7, 3 (1997),303–319. 2Hagen Huwig and Axel Poigné. 1990. A Note on Inconsistencies Caused by Fixpoints in a CartesianClosed Category. Theor. Comput. Sci. 73, 1 (1990), 101–112. https://doi.org/10.1016/0304-3975(90)90165-E 2Paul Blain Levy. 2005. Adjunction models for call-by-push-value with stacks. In Proc. Cat. Th. andComp. Sci., ENTCS, Vol. 69. 5Chuck Liang and Dale Miller. 2007. Focusing and Polarization in Intuitionistic Logic. In CSL. 451–465.4Chuck Liang and Dale Miller. 2009. Focusing and polarization in linear, intuitionistic, and classicallogics. Theor. Comput. Sci. 410, 46 (2009), 4747–4768. 4Paul-André Melliès. 2009. Categorical semantics of linear logic. Panoramas et Synthèses, Vol. 27. So-ciété Mathématique de France, Chapter 1, 15–215. 1, 3, 5Paul-André Melliès and Nicolas Tabareau. 2010. Resource modalities in tensor logic. Ann. Pure Appl.Logic 161, 5 (2010), 632–653. 3GuillaumeMunch-Maccagnoni. 2013. Syntax andModels of a non-Associative Composition of Programsand Proofs. Ph.D. Dissertation. Univ. Paris Diderot. 2, 3, 4Guillaume Munch-Maccagnoni. 2014. Models of a Non-Associative Composition. In Proc. FoSSaCS(LNCS), A. Muscholl (Ed.), Vol. 8412. Springer, 397–412. 2, 3, 4Peter Selinger. 2001. Control Categories and Duality: On the Categorical Semantics of the Lambda-MuCalculus. Math. Struct in Comp. Sci. 11, 2 (2001), 207–260. 28Thomas Streicher. 2013. Direct Semantics is Continuation Semantics, Linearly. (2013). private commu-nication. 49

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Note on models of polarised intuitionistic logic

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