This note is about using computational effects for scalability. With this
method, the specification gets more and more complex while its semantics gets
more and more correct. We show, from two fundamental examples, that it is
possible to design a deduction system for a specification involving an effect
without expliciting this effect

Duval, Dominique

English

arXiv

This note is about using computational effects for scalability. With this method, the specification gets more and more complex while its semantics gets more and more correct. We show, from two fundamental examples, that it is possible to design a deduction system for a specification involving an effect without expliciting this effect

Hal - Université Grenoble Alpes

Scalability using effectsDominique DuvalTo cite this version:Dominique Duval. Scalability using effects. 2013. <hal-00840140>HAL Id: hal-00840140https://hal.archives-ouvertes.fr/hal-00840140Submitted on 1 Jul 2013HAL 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, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Scalability using effectsDominique Duval ∗10 June 2013 — SLS 2013 — Extended abstractAbstractThis note is about using computational effects for scalability. With thismethod, the specification gets more and more complex while its semanticsgets more and more correct. We show, from two fundamental examples,that it is possible to design a deduction system for a specification involvingan effect without expliciting this effect.1 IntroductionA well-known pedagogical trick for teaching complex features is to “lure” thestudents by first providing a simplified version of this feature, before adding therequired corrections to this approximate version. Typically: “The plural formof most nouns is created by adding the letter ’s’ to the end of the word, but thereare some exceptions. . . ”. In computer science, such an approach is used in themechanism of exceptions, as well as in many other computational effects. Theaim of this note is to present computational effects as a tool for solving somescalability issues in the design of specifications.There are several approaches for managing a large specification (or program).For instance, components and modules can be used for breaking down the spec-ification into smaller pieces; the intended semantics of the whole specificationis obtained by “merging” the semantics of its components. Then each moduleis “right”, in the sense that its semantics describes some “part” of the intendedsemantics. Another method is the stepwise refinement, where one builds pro-gressively the required specification from more abstract specifications; duringthe refinement process, the specification gets more and more complex while itssemantics gets more and more precise, until the intended semantics is obtained.Then each step is “right”, in the sense that the intended semantics is one of thepossible semantics of each intermediate specification.In this note we focus on using computational effects for scalability. Withthis method, the specification gets more and more complex while its semanticsgets more and more correct. Thus, each step is “wrong”, in the sense thatthe intended semantics is not a semantics of any intermediate specification.∗Universite´ de Grenoble, Laboratoire Jean Kuntzmann, Dominique.Duval@imag.fr.1It follows that the main issue for using computational effects for scalabilityis whether it is possible to use the intermediate specifications for testing orverifying or proving non-trivial properties of the final specification. In the nextsections we argue in favour of a positive answer to this question, from twoexamples which both rely on a common general algebraic framework. The keypoint is that each intermediate specification may become “right” simply byclassifying its features according to the way they behave with respect to thecorresponding effect, without describing explicitly this behaviour.This note is based on work with J.-C. Reynaud, J.-G. Dumas, L. Fousse andC. Domı´nguez.2 ExceptionsThe mechanism of exceptions has two parts: raising exceptions (with keywordsthrow or raise) and handling them (with keywords try/catch or handle).The lure in the raising of exceptions lies in the fact that a function f whichtakes an argument of type A and returns a value of type B is not interpreted asa map from [[A]] to [[B]] (where [[T ]] denotes the interpretation of a type T ) butas a map [[f ]] : [[A]]→ [[B]] + Exc, where Exc is the set of exceptions and “+”denotes the disjoint union. This can be expressed in a categorical framework [9]:a function f : A → B is interpreted as a map [[f ]] : [[A]] → [[B]] in the Keislicategory of the monad X 7→ X + Exc on the category of sets. The lure in thehandling of exceptions is that a function f : A→ B inside a catch clause mayrecover from an exception (it may also raise an exception), so that is interpretedas a map [[f ]] : [[A]] + Exc → [[B]] + Exc.Thus, in order to deal with exceptions, our proposal is to add decorations tothe syntax: a catcher f (2) : A→ B is interpreted as [[f ]] : [[A]] +Exc → [[B]] +Exc while a propagator f (1) : A→ B is interpreted as [[f ]] : [[A]]→ [[B]] + Excand a pure function f (0) : A → B is interpreted simply as [[f ]] : [[A]] → [[B]].In order to prove properties of programs involving exceptions, we must also adddecorations to the equations: when f, g : A → B are catchers (which is thegeneral case), a strong equation f ≡ g means that [[f ]] = [[g]] : [[A]] + Exc →[[B]]+Exc and a weak equation f ∼ g means that [[f ]]◦ in1 = [[g]]◦ in1 : [[A]]→[[B]] + Exc, where in1 : [[A]]→ [[A]] + Exc is the left coprojection.A major point is the existence of a deduction system associated to thesedecorations, which can be used for proving properties of specifications usingexceptions. For instance, there are rules for the obvious hierarchies: each purefunction can be seen as a propagator, each propagator as a catcher, and eachstrong equation as a weak one. There are also rules expressing the facts thatcomposition preserves the decorations, that the strong equations generate acongruence, and that the weak equations generate a kind of “weak” congruence:if f1 ∼ f2 then f1 ◦ g ∼ f2 ◦ g (but in general h ◦ f1 6∼ h ◦ f2). More details canbe found in [5].23 StatesImperative programming relies on the notion of side-effect for states: a state ofthe memory can be observed thanks to lookup functions and it can be modifiedthanks to update functions.The lure in this situation is that a function f which takes an argumentof type A and returns a value of type B may be interpreted either as [[f ]] :[[A]] × St → [[B]] if f is an observer or as [[f ]] : [[A]] × St → [[B]] × St if fis a modifier (where St is the set of states and “×” the cartesian product). Inorder to deal with imperative programs, we add decorations to the syntax: amodifier f (2) : A → B is interpreted as [[f ]] : [[A]] × St → [[B]] × St whilean observer f (1) : A → B is interpreted as [[f ]] : [[A]] × St → [[B]]. Aswith exceptions, a pure function f (0) : A → B is interpreted simply as [[f ]] :[[A]] → [[B]]. We also add decorations to the equations: when f, g : A → Bare modifiers (this is the general case), a strong equation f ≡ g means that[[f ]] = [[g]] : [[A]] × St → [[B]] × St and a weak equation f ∼ g means thatpr1 ◦ [[f ]] = pr1 ◦ [[g]] : [[A]] × St → [[B]] where pr1 : [[B]] × St → [[B]] isthe left projection. This can also be expressed in a categorical framework: anobserver f (1) : A → B is interpreted as a map [[f ]] : [[A]] → [[B]] in the co-Keisli category of the comonad X 7→ X × St on the category of sets, whilea modifier f (2) : A → B may be interpreted as a map [[f ]] : [[A]] → [[B]]in the Keisli category of the monad X 7→ (X × St)St [9]. For instance, whendealing with a toy class for bank accounts in an object oriented language, theexpressions “deposit(7); balance();” and “7 + balance();” have differenteffects but they return the same integer. They can be seen as decorated termsf (2) = balance(1) ◦ deposit(2) ◦ 7(0) and g(1) = +(0) ◦ (7(0), balance(1)), andindeed it can be proved that f (2) ∼ g(1) without introducing any type of states.In fact, our approach reveals a duality between exceptions and states [3].Thus, we get rules for proving properties of imperative programs which are dualto the rules mentioned above for exceptions. More details are given in [4]. Inthis duality, the exception monad X + E corresponds to the comonad X × St ,but there is no comonad on sets corresponding to the states monad (X × St)St ;from our point of view this is not an issue, in contrast with the point of view ofmonads and Lawvere theories where there is a need for specific tools for handlers[10, 11].4 ConclusionIn the previous sections we have outlined the construction of deduction systemsfor dealing with exceptions and states in an implicitly way, i.e., without usingany “type of exceptions” or “type of states”; we have simply added some deco-rations to the syntax of the language. This can be done for other computationaleffects: several examples are given in [6], where we propose a formalization of thefact that the order of evaluation of the arguments of a binary function becomescrucial in presence of effects.3This work relies on categorical tools [2]: mainly on adjunction, more preciselyon categories of fractions [8], and on limit sketches [7, 1].We have described one step in our approach to scalability. Thanks to thecategorical framework it should be easy to express the composition of effects,thus getting a stepwise scalability method. The combination of this methodwith other scalability methods still has to be studied.References[1] Michael Barr, Charles Wells. Category Theory for Computing Science, 3rdedition. Publications du Centre de recherches mathe´matiques, Universite´de Montre´al (1999).[2] Ce´sar Domı´nguez, Dominique Duval. Diagrammatic logic applied to a pa-rameterization process. Mathematical Structures in Computer Science 20,p. 639-654, 2010.[3] Jean-Guillaume Dumas, Dominique Duval, Laurent Fousse, Jean-ClaudeReynaud. A duality between exceptions and states. Mathematical Struc-tures in Computer Science 22, p. 719-722 (2012).[4] Jean-Guillaume Dumas, Dominique Duval, Laurent Fousse, Jean-ClaudeReynaud. Decorated proofs for computational effects: States. ACCAT2012. Electronic Proceedings in Theoretical Computer Science 93, p. 45-59(2012).[5] Jean-Guillaume Dumas, Dominique Duval, Laurent Fousse, Jean-ClaudeReynaud. Adjunctions for exceptions. arXiv:1207.1255 (2012).[6] Jean-Guillaume Dumas, Dominique Duval, Jean-Claude Reynaud. Carte-sian effect categories are Freyd-categories. Journal of Symbolic Computa-tion 46, p. 272-293 (2011).[7] Charles Ehresmann. Esquisses et types de structures alge´briques. Bull. In-stit. Polit. Ias¸i XIV (1968).[8] Peter Gabriel, Michel Zisman. Calculus of Fractions and Homotopy Theory.Springer (1967).[9] Eugenio Moggi. Notions of Computation and Monads. Information andComputation 93(1), p. 55-92 (1991).[10] Gordon D. Plotkin, John Power. Notions of Computation Determine Mon-ads. FoSSaCS 2002, Lecture Notes in Computer Science 2303, p. 342-356,2002.[11] Gordon D. Plotkin, Matija Pretnar. Handlers of Algebraic Effects. ESOP2009. Springer-Verlag Lecture Notes in Computer Science 5502, p. 80-94(2009).4

Scalability using effects

This note is about using computational effects for scalability. With this method, the specification gets more and more complex while its semantics gets more and more correct. We show, from two fundamental examples, that it is possible to design a deduction system for a specification involving an effect without expliciting this effect. 

Dominique Duval

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Scalability using effects

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