## Canonical abstract syntax trees

WRLA 2006

Canonical Abstract Syntax Trees

Antoine Reilles

CNRS & LORIA Campus Scienti?que, BP 239, 54506 Vand?uvre-l` es-Nancy Cedex France

Abstract This paper presents GOM, a language for describing abstract syntax trees and generating a Java implementation for those trees. GOM includes features allowing the user to specify and modify the interface of the data structure. These features provide in particular the capability to maintain the internal representation of data in canonical form with respect to a rewrite system. This explicitly guarantees that the client program only manipulates normal forms for this rewrite system, a feature which is only implicitly used in many implementations.

1

Introduction

Rewriting and pattern-matching are of general use for describing computations and deduction. Programming with rewrite rules and strategies has been proven most useful for describing computational logics, transition systems or transformation engines, and the notions of rewriting and pattern matching are central notions in many systems, like expert systems (JRule), programming languages based on rewriting (ELAN, Maude, OBJ) or functional programming (ML, Haskell). In this context, we are developing the Tom system [10], which consists of a language extension adding syntactic and associative pattern matching and strategic rewriting capabilities to existing languages like Java, C and OCaml. This hybrid approach is particularly well-suited when describing transformations of structured entities like trees/terms and XML documents. One of the main originalities of this system is to be data structure independent. This means that a mapping has to be de?ned to connect algebraic data structures, on which pattern matching is performed, to low-level data structures, that correspond to the implementation. Thus, given an algebraic data structure de?nition, it is needed to implement an e?cient support for this de?nition in the language targeted by the Tom system, as Java or C do not provide such data structures. Tools like ApiGen [13] and Vas, which is a human readable language for ApiGen input where used previously for generating

This paper is electronically published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs

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such an implementation to use with Tom. However, experience showed that providing an e?cient term data structure implementation is not enough. When implementing computational logics or transition systems with rewriting and equational matching, it is convenient to consider terms modulo a particular theory, as identity, associativity, commutativity, idempotency, or more problem speci?c equations [9]. Then, it becomes crucial to provide the user of the data structure a way to conveniently describe such rules, and to have the insurance that only chosen equivalence class representatives will be manipulated by the program. This need shows up in many situations. For instance when dealing with abstract syntax trees in a compiler, and requiring constant folding or unboxing operators protecting particular data structures. GOM is a language for describing multi-sorted term algebras designed to solve this problem. Like ApiGen, Vas or ASDL [15], its goal is to allow the user of an imperative or object oriented language to describe concisely the algebra of terms he wants to use in an application, and to provide an (e?cient) implementation of this algebra. Moreover, it provides a mechanism to describe normalization functions for the operators, and it ensures that all terms manipulated by the user of the data structure are normal with respect to those rules. GOM includes the same basic functionality as ApiGen and Vas, and ensures that the data structure implementation it provides are maximally shared. Also, the generated data structure implementation supports the visitor combinator [14] pattern, as the strategy language of Tom relies on this pattern. Even though GOM can be used in any Java environment, its features have been designed to work in synergy with Tom. Thus, it is able to generate correct Tom mappings for the data structure (i.e. being formal anchors [6]). GOM provides a way to de?ne computationally complex constructors for a data structure. It also ensures those constructors are used, and that no raw term can be constructed. Private types [8] in the OCaml language do provide a similar functionality by hiding the type constructors in a private module, and exporting construction functions. However, using private types or normal types is made explicit to the user, while it is fully transparent in GOM. MOCA, developed by Fr? ed? eric Blanqui and Pierre Weis is a tool that implements normalization functions for theories like associativity or distributivity for OCaml types. It internally uses private types to implement those normalization functions and ensure they are used, but could also provide such an implementation for GOM. The rest of the paper is organized as follows: in Section 2, to motivate the introduction of GOM, we describe the Tom programming environment and its facilities. Section 3 presents the GOM language, its semantics and some simple use cases. After presenting how GOM can cooperate with Tom in Section 4, we expose in Section 5 the example of a prover for the calculus of structures [3] showing how the combination of GOM and Tom can help 2

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producing a reliable and extendable implementation for a complex system. We conclude with summary and discussions in Section 6.

2

The Tom language

Tom is a language extension which adds pattern matching primitives to existing imperative languages. Pattern-matching is directly related to the structure of objects and therefore is a very natural programming language feature, commonly found in functional languages. This is particularly well-suited when describing various transformations of structured entities like, for example, trees/terms, hierarchized objects, and XML documents. The main originality of the Tom system is its language and data-structure independence. From an implementation point of view, it is a compiler which accepts di?erent native languages like C or Java and whose compilation process consists in translating the matching constructs into the underlying native language. It has been designed taking into account experience about e?cient compilation of rule-based systems [7], and allows the de?nition of rewriting systems, rewriting rules and strategies. For an interested reader, design and implementation issues related to Tom are presented in [10]. Tom is based on the notion of formal anchor presented in [6], which de?nes a mapping between the algebraic terms used to express pattern matching and the actual objects the underlying language manipulates. Thus, it is data structure independent, and customizable for any term implementation. For example, when using Java as the host language, the sum of two integers can be described in Tom as follows:

Term plus(Term t1, Term t2) { %match(Nat t1, Nat t2) { x,zero -> { return x; } x,suc(y) -> { return suc(plus(x,y)); } } }

Here the de?nition of plus is speci?ed functionally, but the function plus can be used as a Java function to perform addition. Nat is the algebraic sort Tom manipulates, which is mapped to Java objects of type Term. The mapping between the actual object Term and the algebraic view Nat has to be provided by the user. The language provides support for matching modulo sophisticated theories. For example, we can specify a matching modulo associativity and neutral element (also known as list-matching) that is particularly useful to model the exploration of a search space and to perform list or XML based transformations. To illustrate the expressivity of list-matching we can de?ne the search of a zero in a list as follows: 3

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boolean hasZero(TermList l) { %match(NatList l) { conc(X1*,zero,X2*) -> { return true; } } return false; }

In this example, list variables, annotated by a * should be instantiated by a (possibly empty) list. Given a list, if a solution to the matching problem exists, a zero can be found in the list and the function returns true, false otherwise, since no zero can be found. Although this mechanism is simple and powerful, it requires a lot of work to implement an e?cient data structure for a given algebraic signature, as well as to provide a formal anchor for the abstract data structure. Thus we need a tool to generate such an e?cient implementation from a given signature. This is what tools like ApiGen [13] do. However, ApiGen itself only provides a tree implementation, but does not allow to add behavior and properties to the tree data structure, like de?ning ordered lists, neutral element or constant propagation in the context of a compiler manipulating abstract syntax tree. Hence the idea to de?ne a new language that would overcome those problems.

3

The GOM language

We describe here the GOM language and its syntax, and present an example data-structure description in GOM. We ?rst show the basic functionality of GOM, which is to provide an e?cient implementation in Java for a given algebraic signature. We then detail what makes GOM suitable for e?ciently implement normalized rewriting [9], and how GOM allows us to write any normalization function. 3.1 Signature de?nitions

An algebraic signature describes how a tree-like data structure should be constructed. Such a description contains Sorts and Operators. Operators de?ne the di?erent node shapes for a certain sort by their name and the names and sorts of their children. Formalisms to describe such data structure de?nitions include ApiGen [13], XML Schema, ML types, and ASDL [15]. To this basic signature de?nition, we add the notion of module as a set of sorts. This allows to de?ne new signatures by composing existing signatures, and is particularly useful when dealing with huge signatures, as can be the abstract syntax tree de?nition of a compiler. Figure 1 shows a simpli?ed syntax for GOM signature de?nition language. In this syntax, we see that a module can import existing modules to reuse its sorts and operators de?nitions. Also, each module declares the sorts it de?nes with the sorts keyword, and declares 4

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Gom M odule Imports Grammar P roduction F ield M oduleN ame SortN ame Symbol SlotN ame

::= M odule ::= module M oduleN ame [ Imports ] Grammar ::= imports ( M oduleN ame )* ::= sorts ( SortN ame )* abstract syntax ( P roduction )* ::= Symbol [ ( F ield (, F ield )* )] -> SortN ame | Symbol ( SortN ame ? ) -> SortN ame ::= SlotN ame : SortN ame ::= Identif ier ::= Identif ier ::= Identif ier ::= Identif ier

Fig. 1. Simpli?ed GOM syntax

operators for those sorts with productions. This syntax is strongly in?uenced by the syntax of SDF [12], but simpler, since it intends to deal with abstract syntax trees, instead of parse trees. One of its peculiarities lies in the productions using the ? symbol, de?ning variadic operators. The notation ? is the same as in [1, Section 2.1.6] for a similar construction, and can be seen as a family of operators with arities in [0, ∞[. We will now consider a simple example of GOM signature for booleans:

module Boolean sorts Bool abstract syntax True False not(b:Bool) and(lhs:Bool,rhs:Bool) or(lhs:Bool,rhs:Bool)

-> -> -> -> ->

Bool Bool Bool Bool Bool

From this description, GOM generates a Java class hierarchy where to each sort corresponds an abstract class, and to each operator a class extending this sort class. The generator also creates a factory class for each module (in this example, called BooleanFactory), providing the user a single entry point for creating objects corresponding to the algebraic terms. Like ApiGen and Vas, GOM relies on the ATerm [11] library, which provides an e?cient implementation of unsorted terms for the C and Java languages, as a basis for the generated classes. The generated data structure can then be characterized by strong typing (as provided by the Composite pattern used for generation) and maximal subterm sharing. Also, the generated class hierarchy does provide support for the visitor combinator pattern [14], allow5

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ing the user to easily de?ne arbitrary tree traversals over GOM data structures using high level constructs (providing congruence operators). 3.2 Canonical representatives

When using abstract data types in a program, it is useful to also de?ne a notion of canonical representative, or ensure some invariant of the structure. This is particularly the case when considering an equational theory associated to the terms of the signature, such as associativity, commutativity or neutral element for an operator, or distributivity of one operator over another one. Considering our previous example with boolean, we can consider the De Morgan rules as an equational theory for booleans. De Morgan’s laws state A ∨ B = A ∧ B and A ∧ B = A ∨ B . We can orient those equations to get a con?uent and terminating rewrite system, suitable to implement a normalization system, where only boolean atoms are negated. We can also add a rule for removing duplicate negation. We obtain the system: A∨B → A∧B A∧B → A∨B A →A GOM’s objective is to provide a low level system for implementing such normalizing rewrite systems in an e?cient way, while giving the user control on how the rules are applied. To achieve this goal, GOM provides a hook mechanism, allowing to de?ne arbitrary code to execute before, or replacing the original construction function of an operator. This code can be any Java or Tom code, allowing to use pattern matching to specify the normalization rules. To allow hooks de?nitions, we add to the GOM syntax the de?nitions for hooks, and add OpHook and F actoryHook to the productions: F actoryHook ::= factory { T omCode } OpHook ::= Symbol : Operation { T omCode } Operation ::= OperationT ype ( ( Identif ier )* ) OperationT ype ::= make | make before | make after | make insert | make after insert | make before insert T omCode ::= . . . A factory hook F actoryHook is attached to the module, and allows to de?ne additional Java functions. We will see in Section 5.3 an example of use for such a hook. An operator hook OpHook is attached to an operator de?nition, and allows to extend or rede?ne the construction function for this operator. Depending on the OperationT ype , the hook rede?nes the construction function (make), or insert code before (make before) or after (make after) the construction function. Those hooks take as many arguments as the operator they modify has children. We also de?ne operation types with an appended insert, used for variadic operators. Those hooks only take two arguments, when the 6

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operator they apply to is variadic, and allow to modify the operation of adding one element to the list of arguments of a variadic operator. Such hooks can be used to de?ne the boolean normalization system: module Boolean sorts Bool abstract syntax True -> Bool False -> Bool not(b:Bool) -> Bool and(lhs:Bool,rhs:Bool) -> Bool or(lhs:Bool,rhs:Bool) -> Bool not:make(arg) { %match(Bool arg) { not(x) -> { return ‘x; } and(l,r) -> { return ‘or(not(l),not(r)); } or(l,r) -> { return ‘and(not(l),not(r)); } } return ‘make_not(arg); } We see in this example that it is possible to use Tom in the hook de?nition, and to use the algebraic signature being de?ned in GOM in the hook code. This lets the user de?ne hooks as rewriting rules, to obtain the normalization system. The signature in the case of GOM is extended to provide access to the default construction function of an operator. This is done here with the make_not(arg) call. When using the hook mechanism of GOM, the user has to ensure that the normalization system the hooks de?ne is terminating and con?uent, as it will not be enforced by the system. Also, combining hooks for di?erent equational theories in the same signature de?nition can lead to non con?uent systems, as combining rewrite systems is not a straightforward task. However, a higher level strata providing completion to compute normalization functions from their equational de?nition, and allowing to combine theories and rules could take advantage of GOM’s design to focus on high level tasks, while getting maximal subterm sharing, strong typing of the generated code and hooks for implementing the normalization functions from the GOM strata. GOM can then be seen as a reusable component, intended to be used as a tool for implementing another language (as ApiGen was used as basis for ASF+SDF [2]) or as component in a more complex architecture.

4

The interactions between GOM and Tom

The GOM tool is best used in conjunction with the Tom compiler. GOM is used to provide an implementation for the abstract data type to be used in a 7

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ADT

GOM

mapping.tom

∪

Tom+Java

Java+Tom

Tom compiler

Tom compiler

Java

Java

javac

bytecode

Fig. 2. The interaction between Tom and GOM

Tom program. The GOM data structure de?nition will also contain the description of the invariants the data structure has to preserve, by the mean of hooks, such that it is ensured the Tom program will only manipulate terms verifying those invariants. Starting from an input datatype signature de?nition, GOM generates an implementation in Java of this data structure (possibly using Tom internally) and also generates an anchor for this data structure implementation for Tom (See Figure 2). The users can then write code using the match construct on the generated mapping and Tom compiles this to plain Java. The dashed box represents the part handled by the GOM tool, while the grey boxes highlight the source ?les the user writes. The generated code is characterized by strong typing combined with a generic interface and by maximal sub-term sharing for memory e?ciency and fast equality checking, as well as the insurance the hooks de?ned for the data structure are always applied, leading to canonical terms. Although it is possible to manually implement a data structure satisfying those constraints, it is di?cult, as all those features are strongly interdependent. Nonetheless, it is then very di?cult to let the data structure evolve when the program matures while keeping those properties, and keep the task of maintaining the resulting program manageable. In the following example, we see how the use of GOM for the data structure de?nition and Tom for expressing both the invariants in GOM and the rewriting rules and strategy in the program leads to a robust and reliable implementation for a prover in the structure calculus. 8

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? ?↓

S {?} ai↓ S [a, a]

S ([R, T ], U ) s S [(R, U ), T ]

Fig. 3. System BV

S [R, U ]; [T, V ] q↓ S [ R; T , U ; V ]

5

A full example: the structure calculus

We describe here a real world example of a program written using GOM and Tom together. We implement a prover for the calculus of structure [3] where some rules are promoted to the level of data structure invariants, allowing a simpler and more e?cient implementation of the calculus rules. Those invariants and rules have been shown correct with respect to the original calculus, leading to an e?cient prover that can be proven correct. Details about the correctness proofs and about the proof search strategy can be found in [5]. We concentrate here on the implementation using GOM. 5.1 The approach

When building a prover for a particular logic, and in particular for the system BV in the structure calculus, one needs to re?ne the strategy of applying the calculus rules. This is particularly true with the calculus of structure, because of deep inference, non con?uence of the calculus and associative-commutative structures. We describe here brie?y the system BV, to show how GOM and Tom can help to provide a robust and e?cient implementation of such a system. Atoms in BV are denoted by a, b, c, . . . Structures are denoted by R, S, T, . . . and generated by S ::= ? | a | S ; . . . ; S

>0

| [ S, . . . , S ] | ( S, . . . , S ) | S

>0 >0

where ?, the unit, is not an atom. S ; . . . ; S is called a seq structure, [S, . . . , S ] is called a par structure, and (S, . . . , S ) is called a copar structure, S is the negation of the structure S . A structure R is called a proper par structure if R = [R1 , R2 ] where R1 = ? and R2 = ?. A structure context, denoted as in S { }, is a structure with a hole. We use this notation to express the deduction rules for system BV, and will omit context braces when there is no ambiguity. The rules for system BV are simple, provided some equivalence relations on BV terms. The seq, par and copar structures are associative, par and copar being commutative too. Also, ? is a neutral element for seq, par and copar structures, and a seq, par or copar structure with only one substructure is equivalent to its content. Then the deduction rules for system BV can be expressed as in Figure 3. Because of the contexts in the rules, the corresponding rewriting rules can 9

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be applied not only at the top of a structure, but also on each subterm of a structure, for implementing deep inference. Deep inference then, combined with associativity, commutativity and ? as a neutral element for seq, par and copar structures leads to a huge amount of non-determinism in the calculus. A structure calculus prover implementation following strictly this description will have to deal with this non-determinism, and handle a huge search space, leading to ine?ciency [4]. The approach when using GOM and Tom will be to identify canonical representatives, or preferred representatives for equivalence classes, and implement the normalization for structures leading to the selection of the canonical representative by using GOM’s hooks. This process requires to de?ne the data structure ?rst, and then de?ne the normalization. This normalization will make sure all units ? in seq, par and copar structures are removed, as ? is a neutral for those structures. We will also make sure the manipulated structures are ?attened, which corresponds to selecting a canonical representative for the associativity of seq, par and copar, and also that subterms of par and copar structures are ordered, taking a total order on structures, to take commutativity into account. When implementing the deduction rule, it will be necessary to take into account the fact that the prover only manipulates canonical representatives. This leads to simpler rules, and allow some new optimizations on the rules to be performed. 5.2 The data structure

We ?rst have to give a syntactic description of the structure data-type the BV prover will use, to provide an object representation for the seq, par and copar structures ( R; T , [R, T ] and (R, T )). In our implementation, we considered these constructors as unary operators which take a list of structures as argument. Using GOM, the considered data structure can be described by the following signature: module Struct imports public sorts Struc StrucPar StrucCop StrucSeq abstract syntax o -> Struc a -> Struc b -> Struc c -> Struc d -> Struc ...other atom constants neg(a:Struc) -> Struc concPar( Struc* ) -> StrucPar 10

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concCop( Struc* ) concSeq( Struc* ) cop(copl:StrucCop) par(parl:StrucPar) seq(seql:StrucSeq)

-> -> -> -> ->

StrucCop StrucSeq Struc Struc Struc

To represent structures, we de?ne ?rst some constant atoms. Among them, the o constant will be used to represent the unit ?. The neg operator builds the negation of its argument. The grammar rule par(StrucPar) -> Struc de?nes a unary operator par of sort Struc which takes a StrucPar as unique argument. Similarly, the rule concPar(Struc*) -> StrucPar de?nes the concPar operator of sort StrucPar. The syntax Struc* indicates that concPar is a variadic-operator which takes an inde?nite number of Struc as arguments. Thus, by combining par and concPar it becomes possible to represent the structure [a, [b, c]] by par(concPar(a,b,c)). Note that this structure is ?attened, but with this description, we could also use nested par structures, as in par(concPar(a,par(concPar(b,c)))) to represent this structure. (R, T ) and R; T are represented in a similar way, using cop, seq, concCop, and concSeq. 5.3 The invariants, and how they are enforced

So far, we can manipulate objects, like par(concPar()), which do not necessarily correspond to intended structures. It is also possible to have several representations for the same structure. Hence, par(concPar(a)) and cop(concCop(a)) both denote the structure a, as R ≈ [R] ≈ (R) ≈ R. Thus, we de?ne the canonical (prefered) representative by ensuring that

?

[], and () are reduced when containing only one sub-structure: par(concP ar(x)) → x nested structures are ?attened, using the rule: par(concP ar(a1 , . . . , ai , par(concP ar(x1 , . . . , xn )), b1 , . . . , bj )) → par(concP ar(a1 , . . . , ai , x1 , . . . , xn , b1 , . . . , bj ))

?

?

subterms are sorted (according to a given total lexical order <): concP ar(. . . , xi , . . . , xj , . . .) → concP ar(. . . , xj , . . . , xi , . . .) if xj < xi .

This notion of canonical form allows us to e?ciently check if two terms represent the same structure with respect to commutativity of those connectors, neutral elements and reduction rules. The ?rst invariant we want to maintain is the reduction of singleton for seq, par and copar structures. If we try to build a cop, par or seq with an empty list of structures, then the creation function shall return the unit o. Else if the list contains only one element, it has to return this element. Otherwise, it will just build the requested structure. As all manipulated terms are canonical forms, we do not have for this invariant to handle the case of a structure list 11

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containing the unit, as it will be enforced by the list invariants. This behavior can be implemented as a hook for the seq, par and cop operators. par(parl:StrucPar) -> Struc par:make (l) { %match(StrucPar l) { concPar() -> { return ‘o(); } concPar(x)-> { return ‘x; } } return ‘make_par(l); } This simple hook implements the invariant for singletons for par, and use a call to the Tom constructor make_par(l) to call the intern constructor (without the normalization process), to avoid an in?nite loop. Similar hooks are added to the GOM description for cop and seq operators. We see here how the pattern matching facilities of Tom embedded in GOM can be used to easily implement normalization strategies. The hooks for normalizing structure lists are more complex. They ?rst require a total order on structures. This can be easily provided as a function, de?ned in a factory hook. The comparison function we provide here uses the builtin translation of GOM generated data structures to text to implement a lexical total order. A more speci?c (and e?cient) comparison function could be written, but for the price of readability. factory { public int compareStruc(Object t1, Object t2) { String s1 = t1.toString(); String s2 = t2.toString(); int res = s1.compareTo(s2); return res; } } Once this function is provided, we can de?ne the hooks for the variadic operators concSeq, concPar and concCop. The hook for concSeq is the simplest, since the structures are only associative, with ? as neutral element. Then the corresponding hook has to remove the units, and ?atten nested seq. concSeq( Struc* ) -> StrucSeq concSeq:make_insert(e,l) { %match(Struc e) { o() -> { return l; } seq(concSeq(L*)) -> { return ‘concSeq(L*,l*); } } return ‘make_concSeq(e,l); } 12

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This hook only checks the form of the element to add to the arguments of the variadic operator, but does not use the shape of the previous arguments. The hooks for concCop and concPar are similar, but they do examine also the previous arguments, to perform sorted insertion of the new argument. This leads to a sorted list of arguments for the operator, providing a canonical representative for commutative structures. concPar( Struc* ) -> StrucPar concPar:make_insert(e,l) { %match(Struc e) { o() -> { return l; } par(concPar(L*)) -> { return ‘concPar(L*,l*); } } %match(StrucPar l) { concPar(head,tail*) -> { if(!(compareStruc(e, head) < 0)) { return ‘make_concPar(head,concPar(e,tail*)); } } } return ‘make_concPar(e,l); } The associative matching facility of Tom is used to examine the arguments of the variadic operator, and decide whether to call the builtin construction function, or perform a recursive call to get a sorted insertion. As the structure calculus verify the De Morgan rules for the negation, we could write a hook for the neg construction function applying the De Morgan rules as in Section 3.2 to ensure only atoms are negated. This will make implementing the deduction rules even simpler, since there is then no need to propagate negations in the rules. 5.4 The rules

Once the data structure is de?ned, we can implement proof search in system BV in a Tom program using the GOM de?ned data structure by applying rewriting rules corresponding to the calculus rules to the input structure repeatedly, until reaching the goal of the prover (usually, the unit ?). Those rules are expressed using Tom’s pattern matching over the GOM data structure. They are kept simple because the equivalence relation over structures is integrated in the data structure with invariants. In this example, [] and () structures are associative and commutative, while the canonical representatives we use are sorted and ?attened variadic operators. For instance, the rule s of Figure 3 can be expressed as the two rules [(R, T ), U ] → ([R, U ], T ) and [(R, T ), U ] → ([T, U ], R), using only associative matching instead of associative commutative matching. Then, those rules are 13

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encoded by the following match construct, which is placed into a strategy implementing rewriting in arbitrary context (congruence) to get deep inference, the c collection being used to gather multiple results: %match(Struc t) { par(concPar(X1*,cop(concCop(R*,T*)),X2*,U,X3*)) -> { if(‘T*.isEmpty() || ‘R*.isEmpty() ) { } else { StrucPar context = ‘concPar(X1*,X2*,X3*); if(canReact(‘R*,‘U)) { StrucPar parR = cop2par(‘R*); // transform a StrucCop into a StrucPar Struc elt1 = ‘par(concPar( cop(concCop(par(concPar(parR*,U)),T*)),context*)); c.add(elt1); } if(canReact(‘T*,‘U)) { StrucPar parT = cop2par(‘T*); Struc elt2 = ‘par(concPar( cop(concCop(par(concPar(parT*,U)),R*)),context*)); c.add(elt2); } } } } We ensure that we do not execute the right-hand side of the rule if either R or T are empty lists. The other tests implement restrictions on the application of the rules reducing the non-determinism. This is done by using an auxiliary predicate function canReact(a,b) which can be expressed using all the expressive power of both Tom and Java in a factory hook. The interested reader is referred to [5] for a detailed description of those restrictions. Also, the search strategy can be carefully crafted using both Tom and Java constructions, to achieve a very ?ne grained and evolutive strategy, where usual algebraic languages only allow breadth-?rst or depth-?rst strategies, but do not let the programmer easily de?ne a particular hybrid search strategy. While the Tom approach of search strategies may lead to more complex implementations for simple examples (as the search space has to be handled explicitly), it allows us to de?ne ?ne and e?cient strategies for complex cases. The implementation of a prover for system BV with GOM and Tom leads not only to an e?cient implementation, allowing to cleanly separate concerns about strategy, rules and canonical representatives of terms, but also to an implementation that can be proven correct, because most parts are expressed with the high level constructs of GOM and Tom instead of pure Java. As the data structure invariants in GOM and the deduction rules in Tom are de?ned algebraically, it is possible to prove that the implemented system is correct and complete with respect to the original system [5], while bene?ting from the expressive power and ?exibility of Java to express non algebraic concerns (like 14

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building a web applet for the resulting program, or sending the results in a network).

6

Conclusion

We have presented the GOM language, a language for describing algebraic signatures and normalization systems for the terms in those signatures. This language is kept low level by using Java and Tom to express the normalization rules, and by using hooks for describing how to use the normalizers. This allows an e?cient implementation of the resulting data structure, preserving properties important to the implementation level, such as maximal subterm sharing and a strongly typed implementation. We have shown how this new tool interacts with the Tom language. As Tom provides pattern matching, rewrite rules and strategies in imperative languages like C or Java, GOM provides algebraic data structures and canonical representatives to Java. Even though GOM can be used simply within Java, most bene?ts are gained when using it with Tom, allowing to integrate formal algebraic developments into mainstream languages. This integration can allow to formally prove the implemented algorithms with high level proofs using rewriting techniques, while getting a Java implementation as result. We have applied this approach to the example of system BV in the structure calculus, and shown how the method can lead to an e?cient implementation for a complex problem (the implemented prover can tackle more problems than previous rule based implementation [5]). As the compilation process of Tom’s pattern matching is formally veri?ed and shown correct [6], proving the correctness of the generated data structure and normalizers with respect to the GOM description would allow to expand the trust path from the high level algorithm expressed with rewrite rules and strategies to the Java code generated by the compilation of GOM and Tom. This allows to not only prove the correctness of the implementation, but also to show that the formal parts of the implementation preserve the properties of the high level rewrite system, such as con?uence or termination. ? Acknowledgments: I would like to thank Claude Kirchner, Pierre Etienne Moreau and all the Tom developers for their help and comments. Special thanks are due to Pierre Weis and Frederic Blanqui for fruitful discussions and their help in understanding the design issues.

References

[1] Comon, H. and J.-P. Jouannaud, Les termes en logique et en programmation (2003), master lectures at Univ. Paris Sud. URL http://www.lix.polytechnique.fr/Labo/Jean-Pierre.Jouannaud/ articles/cours-tlpo.pdf

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