A Language Prototyping Tool based on Semantic Building Blocks

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1 A Language Prototyping Tool based on Semantic Building Blocks J. E. Labra Gayo, J. M. Cueva Lovelle, M. C. Luengo Díez, and B. M. González Rodríguez Department of Computer Science, University of Oviedo C/ Calvo Sotelo S/N, 3307, Oviedo, Spain Abstract. We present a Language Prototyping System that facilitates the modular development of interpreters from semantic specifications. The theoretical basis of our system is the integration of ideas from generic programming and modular monadic semantics. The system is implemented as a domain-specific language embedded in Haskell and contains an interactive framework for language prototyping. In the monadic approach, the semantic spscification of a programming language is captured as a function Σ M V where Σ represents the abstract syntax, M the computational monad, and V the domain value. In order to obtain more extensibility, we use folds or catamorphisms over the fixpoint of non-recursive pattern functors that capture the structure of the abstract syntax. For each pattern functor F, the semantic specifications are defined as independent F-Algebras whose carrier is M V, where M is the computational monad and V models the domain value. The copmputational monad M can itself be obtained from the composition of several monad transformers applied to a base monad, and the domain value V can be defined using extensible union types. In this paper, we also show that when the abstract syntax contains several categories, it is possible to define many-sorted algebras obtaining the same modularity. 1 Introduction E. Moggi [39] applied monads to denotational semantics in order to capture the notion of computation and the intuitive idea of separating computations from values. After his work, there was some interest in the development of modular interpreters using monads [49,43,10]. The problem was that, in general, it is not possible to compose two monads to obtain a new monad [25]. A proposed solution was the use of monad transformers [33,32] which transform a given monad into a new one adding new operations. This approach was called modular monadic semantics. In a different context, the definition of recursive datatypes as least fixpoints of pattern functors and the calculating properties that can be obtained be means

2 of folds or catamorphisms led to a complete discipline which could be named as generic programming [3,34,35]. In [9], L. Duponcheel proposed the combined use of folds or catamorphisms with modular monadic semantics allowing the independent specification of the abstract syntax, the computational monad and the domain value. Following [36], we applied monadic folds to modular monadic semantics allowing the separation between recursive evaluation and semantic specification [28,29,30]. In practice, the abstract syntax is usually formed from n mutually recursive categories, In this paper we show how we can extend our previous work to handle many-sorted algebras. The paper is organized as follows. In section 2 we give an informal presentation of modular monadic semantics defining some monad transformers. Section 3 presents the basic concepts from generic programming extending previous work to handle many-sorted algebras. In section 4 we specify the semantics of a simple imperative programming language from reusable components. Along the paper, we use Haskell syntax with some freedom in the use of mathematical operators and datatype declarations. As an example, the predefined datatype data Either a b = Left a Right b could be defined with our notation as α β L α R β We also omit the type constructors in some definitions for brevity. The notions we use from category theory are defined in the paper, so it is not a prerequisite. 2 Modular Monadic Semantics A monad M captures the intuitive notion of computation. In this way, the type M α represents a computation the returns a value of type α In functional programming, a monad can be defined as a type constructor M with 2 operations return : α M α ( =) : M α (α M β) M β which satisfy a number of laws (see [47,49,48]). Example 1. The simplest monad is the identity monad Id α α return = λx x m = f = f x

3 In the rest of the paper, we will use the do-notation defined as: do { m; e } m = λ do { e } do { x m; e } m = λ x do { e } do { let exp; e } let exp in do { e } do { e } e It is possible to define monads that capture different kinds of computations, like partiality, nondeterminism, side-effects, exceptions, continuations, interactions, etc. [39,40,5]. Table 1 presents two classes of monads that will be used in the rest of the paper. Table 1. Some classes of monads Name Operations Environment Access rdenv : M Env inenv : Env M α M α State transformer update : (State State) M State fetch : M State set : State M State When describing the semantics of a programming language using monads, the main problem is the combination of different classes of monads. It is not possible to compose two monads to obtain a new monad in general [25]. Nevertheless, a monad transformer T can transform a given monad M into a new monad T M that has new operations and maintains the operations of M. The idea of monad transformer is based on the notion of monad morphism that appeared in Moggi s work [39] and was later proposed in [33]. The definition of a monad transformer is not straightforward because there can be some interactions between the intervening operations of the different monads. These interactions are considered in more detail in [31,32,33] and in [17] it is shown how to derive a backtracking monad transformer from its specification. Our system contains a library of predefined monad transformers corresponding to each class of monad and the user can also define new monad transformers. When defining a monad transformer T over a monad M, it is necessary to specify the new return and ( =), the lift : M α T M α operation that transforms any operation in M into an operation in the new monad T M, and the new operations provided by the new monad. Table 2 presents the definitions of the two monad transformers that will be used in the paper. 2.1 Extensible domains [33] defines extensible union types using multi-parameter type classes. Although we are not going to give the full details, we can assume that if α is a subtype

4 Table 2. Some monad transformers with their definitions Environment reader T Env M α Env M α return x = λρ returnx x = f = λρ (xρ) =(λa f a ρ) lift x = λρ x =return rdenv = λρ returnρ inenv ρ x = λ x ρ State transformer T State M α State M (α, State) return x = λς return(x, ς) x = f = λς (xς) =(λ(v, ς ) f v ς ) lift x = λς x =(λx return(x, ς)) update f = λς return(ς, f ς) fetch = update (λς ς) set ς = update (λ ς) of β, which will be denoted as α β, then we have the functions : α β and : β α. We also assume that α (α β) and that β (α β). As an example, if we define a domain of integers and booleans as Int Bool, then ( 3) belongs to that domain and to further extensions of it. 3 Generic Programming concepts 3.1 Functors, Algebras and Catamorphisms As in the case of monads, functors also come from category theory but can easily be defined in a functional programming setting. A functor F can be defined as a type constructor that transforms values of type α into values of type F α and a function map F : (α β) F α F β. The fixpoint of a functor F can be defined as µf In (F (µf)) In the above definition, we explicitly write the type constructor In because we will refer to it later. A recursive datatype can be defined as the fixpoint of a non-recursive functor that captures its shape. Example 2. The following inductive datatype for arithmetic expressions Term Term N Int Term + Term Term Term

5 can be defined as the fixpoint of the functor A T x N Int x + x x x where the map T is 1 : map T : (α β) (T α Tβ) map T f (N n) = n map T f (x 1 + x 2 ) = f x 1 + f x 2 map T f (x 1 x 2 ) = f x 1 f x 2 Once we have the shape functor T, we can obtain the recursive datatype as the fixpoint of T Term µt In this way, the expression can be represented as In ((In (N 2)) + (In (N 3))) : Term The sum of two functors F and G, denoted by F G can be defined as (F G) x F x G x where map F G is map F G : (α β) (F G) α (F G) β map F G f (L x) = L (map F f x) map F G f (R x) = R (map G f x) Using the sum of two functors, it is possible to extend recursive datatypes. Example 3. We can define a new pattern functor for boolean expressions B x = B Bool x == x x < x and the composed recursive datatype of arithmetic and boolean expressions can easily be defined as Expr µ(t B) Given a functor F, an F-algebra is a function ϕ F : F α α where α is called the carrier. An homomorphism between two F-algebras ϕ : F α α and ψ : F β β is a function h : α β which satisfies 1 h. ϕ = ψ. map F h In the rest of the paper we omit the definition of map functions as they can be automatically derived from the shape of the functor.

6 We consider a new category with F-algebras as objects and homomorphisms between F-algebras as morphisms. In this category, In : F(µF) µf is an initial object, i.e. for any F-algebra ϕ : F α α there is a unique homomorphism ([ϕ]) : µf α satisfying the above equation. ([ϕ]) is called fold or catamorphism and satisfies a number of calculational properties [3,6,35,42]. It can be defined as: ([ ]) : (Fα α) (µf α) ([ϕ]) (In x) = ϕ ( map F ([ϕ]) x) Example 4. We can obtain a simple evaluator for arithmetic expressions defining an T-algebra whose carrier is the type m v, where m is, in this case, any kind of monad, and Int is a subtype of v. ϕ T : (Monad m, Int v) T(m v) m v ϕ T (Num n) = return ( n) ϕ T (e 1 + e 2 ) = do v 1 e 1 v 2 e 2 return( ( v 1 + v 2 )) ϕ T (e 1 e 2 ) = do v 1 e 1 v 2 e 2 return( ( v 1 v 2 )) Applying a catamorphism over ϕ T terms: we obtain the evaluation function for eval Term : (Monad m, Int v) Term m v eval Term = ([ϕ T ]) The operator allows to obtain a (F G)-algebra from an F-algebra ϕ and a G-algebra ψ : (F α α) (G α α) (F G)α α (ϕ ψ)(l x) = ϕ x (ϕ ψ)(r x) = ψ x Example 5. The above definition allows to extend the evaluator of example 4 to arithmetic and boolean expressions. We can specify the semantics of boolean expressions with the following B- algebra ϕ B : (Monad m, Bool v) B(m v) m v ϕ B (B b) = return ( b)

7 ϕ B (e 1 == e 2 ) = do v 1 e 1 v 2 e 2 return( ( v 1 == v 2 )) ϕ B (e 1 < e 2 ) = do v 1 e 1 v 2 e 2 return( ( v 1 < v 2 )) Now, the new evaluator of boolean and arithmetic expressions is automatically obtained as a catamorphism over the (T B)-algebra. eval Expr : (Monad m, Int v, Bool v) Expr m v eval Expr = ([ϕ T ϕ B ]) The theory of catamorphisms can be extended to monadic catamorphisms as described in [12,19,28,30]. 3.2 Many-sorted algebras and catamorphisms The abstract syntax of a programming language is usually divided in several mutually recursive categories. It is possible to extend the previous definitions to handle many-sorted algebras. In this section, we present the theory for n = 2, but it can be defined for any number of sorts [11,37,21,41]. A bifunctor F is a type constructor that assigns a type F α β to a pair of types α and β and an operation bimap F : (α γ) (β δ) (F α β F γ δ) The fixpoint of two bifunctors F and G is a pair of values (µ 1 FG,µ 2 FG) that can be defined as: µ 1 FG In 1 (F (µ 1 FG) (µ 2 FG)) µ 2 FG In 2 (G (µ 1 FG) (µ 2 FG)) Given two bifunctors F and G, a two-sorted F, G-algebra is a pair of functions (ϕ, ψ) such that: ϕ : F α β α ψ : G α β β where α, β are called the carriers of the two-sorted algebra. It is possible to define F, G-homomorphisms and a new category where (In 1, In 2 ) form the initial object. This allows the definition of bicatamorphisms as: ([, ]) 1 : (F α β α) (G α β β) (µ 1 FG α) ([ϕ, ψ]) 1 (In 1 x) = ϕ (bimap F ([ϕ, ψ]) 1 ([ϕ, ψ]) 2 x)

8 ([, ]) 2 : (F α β α) (G α β β) (µ 2 FG β) ([ϕ, ψ]) 2 (In 2 x) = ψ (bimap G ([ϕ, ψ]) 1 ([ϕ, ψ]) 2 x) The sum of two bifunctors F and G is a new bifunctor F G and can be defined as: (F G) α β F α β G α β where the bimap operator is bimap F G : (α γ) (β δ) ((F G) α β ((F G) γ δ) bimap F G f g (L x) = L (bimap F G f g x) bimap F G f g (R x) = R (bimap F G f g x) In order to extend two-sorted algebras, we define the operators 1 and 2 as: ( 1 ) : (F α β α) (G α β α) (F G) α β α (φ 1 1 φ 2 ) (L x) = φ 1 x (φ 2 1 φ 2 ) (R x) = φ 2 x ( 2 ) : (F α β β) (G α β β) (F G) α β β (ψ 1 2 ψ 2 ) (L x) = ψ 1 x (ψ 2 2 ψ 2 ) (R x) = ψ 2 x 3.3 From functors to bifunctors When specifying several programming languages, it is very important to be able to share common blocks and to reuse the corresponding specifications. For example, arithmetic expressions should be specified in one place and their specification should be reused between different languages. In order to reuse specifications made using single-sorted algebras in a twosorted framework, it is necessary to extend functors to bifunctors. Given a functor F, we define the bifunctors F 2 1 and F 2 2 as: F 2 1 α β F α F 2 2 α β F β where the bimap operations are defined as bimap F 2 1 f g x = f x bimap F 2 2 f g x = g x Given an F-algebra, the operators ɛ 2 1 and ɛ 2 2 obtain the corresponding twosorted algebras ɛ 2 1 : (F α α) F 2 1 α β α ɛ 2 1 ϕ x = ϕ x ɛ 2 2 : (F β β) F 2 2 α β β ɛ 2 2 ϕ x = ϕ x

9 4 Specification of a simple imperative language 4.1 Abstract syntax A typical imperative programming language can be divided in two different worlds: expressions and commands. In our example, the expressions will be arithmetic, boolean and variables. The abstract syntax of arithmetic and boolean expressions are captured by the functors T and B defined in examples 2 and 3. Variables are defined using the functor V V x V Name We will define commands in two steps. Firstly, sequence and assignments are defined using the bifunctor S S e c c ; c String := e Secondly, control structures (conditional and loops) are defined using the bifunctor R R e c If e c c While e c In order to define the imperative languge, we need a bifunctor that represents the shape of expressions and another one representing commands. The bifunctor of expressions can be defined as an extension of the functor obtained as the sum of T, B and V E (T B V) 2 1 The bifunctor of commands is defined as the sum of the bifunctors S and R C S R Finally, the imperative language is the fixpoint of E and C Imp µ 2 E R 4.2 Computational structure In this simple language, the computational structure needs to access the environment and to transform a global state. We will use the monad Comp which is obtained by transforming the identity monad using the monad transformers T State and T Env defined in table 2. Comp (T State. T Env ) Id The domain value of expressions consist of integer and boolean values

10 Value Int Bool and the domain value of commands is the null type () 2 indicating that commands do not return any value. The state and environment are defined as: Env Name Loc State Loc Value where Loc represent memory locations. We will also use the notation ς {x/v} to represent the updated state ς which assigns v to x. 4.3 Semantic functions The semantic specification of arithmetic and boolean expressions were defined in the examples 4 and 5. We will reuse those specifications in the imperative language. With regard to variables, the V-algebra is ϕ V : V(Comp Value) Comp Value ϕ V (Var x) = do ρ rdenv ς fetch return(ς (ρ x)) The specification of sequence and assignment is ψ S : S (Comp Value) (Comp ()) Comp () ψ S (c 1 ; c 2 ) = do c 1 c 2 ψ S (x := e) = do v e ρ rdenv ς fetch set (ς {ρ x/v}) return () In the same way, the specification of conditional and repetitive commands is: ψ R : R (Comp Value) (Comp ()) Comp () ψ R (If e c 1 c 2 ) = do v e if v then c 1 else c 2 2 () is a predefined Haskell datatype that only contains the value ()

11 ψ R (While e c) = loop where loop = do v e if v then do { c ; loop } else return() Finally, the interpreter is automatically obtained as a bicatamorphism Inter Imp : Imp Comp () Inter Imp = ([ɛ 2 1(ϕ T ϕ B ϕ V ), ϕ S 2 ϕ R ]) 2 Although in the above definition we have explicitily written the particular algebras, it is not necessary to do so in the implementation because the overloading mechanism of Haskell allows to detect which is the corresponding algebra. 5 Conclusions and future work We have presented an integration of modular monadic semantics and generic programming concepts that allows the definition of programming languages from reusable semantic especifications. This approach has been implemented in a Language Prototyping System which allows to share semantic building blocks and provides an interactive framework for language testing. The system can be considered as another example of a domain-specific language embedded in Haskell [46,26,20]. This approach has some advantages: The development is easier as we can rely on the fairly good type system of Haskell, it is possible to obtain direct access to Haskell libraries and tools, and we do not need to define a new language with its syntax, semantics, type system, etc. At the same time, the main disadvantages are the mixture of error messages from the domain-specific language and the host language, Haskell type system limitations and the Haskell dependency which impedes the development of interpreters implemented in different languages. It would be interesting to define an independent domain specific meta-language for semantic specifications following [5,7,38]. On the theoretical side, [17] shows how to derive a backtracking monad transformer from its specification. That approach should be applied to other types of monad transformers and it would be interesting to define a general framework for the combination many-sorted algebras and monadic catamorphisms. It would also be fruitful to study the combination of algebras, coalgebras, monads and comonads in order to provide the semantics of interactive and object-oriented features [4,23,22,27,45]. Another line of research is the automatic derivation of compilers from the interpreters built. This line has already been started in [14,15].

12 With regard to the implementation, we have also made a simple version of the system using first-class polymorphism [24] and extensible records [13]. This allows the definition of monads as first class values and monad transformers as functions between monads without the need of type classes. However, this feature is still not fully implemented in current Haskell systems. Recent advances in generic programming would also improve the implementation [18,16]. At this moment, we have specified simple imperative, functional, objectoriented and logic programming languages. The specifications have been made in a modular way reusing common components of the different languages. The original goal of our research was to develop prototypes for the abstract machines underlying the integral object-oriented operating System Oviedo3 [2] whith the aim to test new features as security, concurrency, reflectiveness and distribution [8,44]. More information on the Language Prototyping System can be obtained at [1]. References 1. Language Prototyping System D. Álvarez, L. Tajes, F. Álvarez, M. A. Díaz, R. Izquierdo, and J. M. Cueva. An object-oriented abstract machine as the substrate for an object-oriented operating system. In J. Bosch and S. Mitchell, editors, Object Oriented Technology ECOOP 97, Jÿvaskÿla, Finland, June Springer Verlag, LNCS Roland Backhouse, Patrik Jansson, Johan Jeuring, and Lambert Meertens. Generic programming - an introduction. In S. D. Swierstra, P. R. Henriques, and J. N. Oliveira, editors, Advanced Functional Programming, volume 1608 of Lecture Notes in Computer Science. Springer, L. S. Barbosa. Components as processes: An exercise in coalgebraic modeling. In S. F. Smith and C. L. Talcott, editors, FMOODS Formal Methods for Open Object-Oriented Distributed Systems, pages , Stanford, USA, September Kluwer Academic Publishers. 5. N. Benton, J. Hughes, and E. Moggi. Monads and effects. In International Summer School On Applied Semantics APPSEM 2000, Caminha, Portugal, R. Bird and Oege de Moor. Algebra of Programming. Prentice Hall, Pietro Ceciarelli and Engenio Moggi. A syntactic approach to modularity in denotational semantics. In 5th Biennial Meeting on Category Theory and Computer Science, volume CTCS-5. CWI Technical Report, M. A. Díaz, D. Álvarez, A. García-Mendoza, F. Álvarez, L. Tajes, and J. M. Cueva. Merging capabilities with the object model of an object-oriented abstract machine. In S. Demeyer and J. Bosch, editors, Ecoop 98 Workshop on Distributed Object Security, volume 1543, pages LNCS, Luc Duponcheel. Writing modular interpreters using catamorphisms, subtypes and monad transformers. Utrecht University, David Espinosa. Semantic Lego. PhD thesis, Columbia University, Maarten M. Fokkinga. Law and Order in Algorithmics. PhD thesis, University of Twente, February Maarten M. Fokkinga. Monadic maps and folds for arbitrary datatypes. Memoranda Informatica 94-28, Dept. of Computer Science, Univ. of Twente, June 1994.

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