Tom2.10:Gom
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Tom2.10 : Language Reference 
Tom > Gom > Strategies > Runtime Library > Migration guide > EMF 
The basic functionality of Gom is to provide a tree implementation in Java corresponding to an algebraic specification. The syntax is inspired from the algebraic type definition of ML languages. Gom generates an efficient code which ensures maximal sharing.
To maintain the terms in a preferred canonical form, Gom provides an original hooks mechanism comparable to "private types" of Caml, or "smart constructors" of Haskell.
The tree implementations Gom generates are characterized by strong typing, immutability (there is no way to manipulate them with sideeffects), maximal subterm sharing and the ability to be used with strategies.
Contents 
Gom syntax
Gom provides a syntax to concisely define abstract syntax tree. Each Gom file consists in the definition of one or more modules. In each module we define sorts, and operators for the sorts. Additionally, we can define hooks to modify the default behavior of a constructor operator.
Example of signature
The syntax of Gom is quite simple and can be used to define manysorted abstractdatatypes. The module
section defines the name of the signature. The imports
section defines the name of the imported signatures. The abstract syntax
part defines the operators with their signature. For each argument, a sort and a name (called slotname) has to be given.
module Expressions imports String int abstract syntax Bool = True()  False()  Eq(lhs:Expr, rhs:Expr) Expr = Id(stringValue:String)  Nat(intValue:int)  Add(lhs:Expr, rhs:Expr)  Mul(lhs:Expr, rhs:Expr)
The definition of a signature in Gom has several restrictions:
 there is no overloading: two operators cannot have the same name
 for a given operator, all slotnames must be different
 if two slots have the same slotname, they must belong to the same sort. In the previous example,
Eq
,Add
, andMul
can have a slot calledlhs
of sortExpr
. But,Id
andNat
cannot have a same slot namedvalue
, since their sort are not identical (the slots are respectively of sortsString
andint
forId
andNat
).
Builtin sorts and operators
Gom supports several builtin sorts, that may be used as sort for new operators slots. To each of these builtin sorts corresponds a Java type. Native data types from Java can be used as builtin fields, as well as ATerm
and ATermList
structures. To use one of these builtin types in a Gom specification, it is required to add an import for the corresponding builtin.
Name ! Java type 

int 
String 
char 
double 
long 
float 
aterm.ATerm 
aterm.ATermList 
It is not possible to define a new operator whose codomain is one of the builtin sorts, since those sorts are defined in another module.
Note: to be able to use a builtin sort, it is necessary to declare the import of the corresponding module ( 
External Gom modules may be imported in a Gom specification, by adding the name of the module to import in the 'imports' section. Once a module imported, it is possible to use any of the sorts this module declares or imports itself as type for a new operator slot. Adding new operators to an imported sort is however not allowed.
Grammar
GomGrammar  ::=  Module 
Module  ::=  'module' ModuleName [Imports] Grammar 
Imports  ::=  'imports' (ModuleName)* 
Grammar  ::=  'abstract syntax' (TypeDefinition ∣ HookDefinition)* 
TypeDefinition  ::=  SortName '=' OperatorDefinition ('' OperatorDefinition)* 
OperatorDefinition  ::=  OperatorName'('[SlotDefinition(','SlotDefinition)*]')' 
∣  OperatorName'(' SortName'*' ')'  
SlotDefinition  ::=  SlotName':' SortName 
ModuleName  ::=  Identifier 
SortName  ::=  Identifier 
OperatorName  ::=  Identifier 
SlotName  ::=  Identifier 
HookDefinition  ::=  HookType':' HookOperation 
HookType  ::=  OperatorName 
∣  'module' ModuleName  
∣  'sort' SortName  
HookOperation  ::=  'make' '(' [Identifier (',' Identifier)*] ')' '{' TomCode '}' 
∣  'make_insert' '(' Identifier ',' Identifier ')' '{' TomCode '}'  
∣  'make_empty()' '{' TomCode '}'  
∣  'Free() {}'  
∣  'FL() {}'  
∣  ('AU' ∣'ACU') '()' '{' ['`' term] '}'  
∣  'interface()' '{' Identifier(',' Identifier)* '}'  
∣  'import()' '{' JavaImports '}'  
∣  'block()' '{' TomCode'}'  
∣  'rules()' '{' RulesCode '}'  
∣  'graphrules(' Identifier ',' ('Identity' ∣'Fail')')''{' GraphRulesCode '}'  
RulesCode  ::=  (Rule)* 
Rule  ::=  RulePattern '>' TomTerm ['if' Condition] 
RulePattern  ::=  [ AnnotedName '@' ] PlainRulePattern 
PlainRulePattern  ::=  ['!']VariableName [ '*' ] 
∣  ['!']HeadSymbolList '(' [RulePattern ( ',' RulePattern )*] ')'  
∣  '_'  
∣  '_*'  
GraphRulesCode  ::=  (GraphRule)* 
GraphRule  ::=  TermGraph '>' TermGraph ['if' Condition] 
TermGraph  ::=  [ Label ':' ] TermGraph 
∣  '&' Label  
∣  VariableName [ '*' ]  
∣  HeadSymbolList '(' [TermGraph ( ',' TermGraph )*] ')'  
Label  ::=  Identifier 
Condition  ::=  TomTerm Operator TomTerm 
∣  RulePattern '<<' TomTerm  
∣  Condition '&&' Condition  
∣  Condition '' Condition  
∣  '(' Condition ')'  
Operator  ::=  '>' ∣ '>=' ∣'<' ∣'<=' ∣'==' ∣ '!=' 
Using Gom with Tom
A first solution to combine Gom with Tom is to use Gom as a standalone tool, using the command line tool or the ant task.
In that case, the module name of the Gom specification and the package
option determine where the files are generated. To make things correct, it is sufficient to import the generated Java classes, as well as the generated Tom file. In the case of a Gom module called Module
, all files are generated in a directory named module
and the Tom program should do the following:
import module.*; import module.types.*; class MyClass { ... %include { module/Module.tom } ... }
A second possibility to combine Gom with Tom is to use the %gom
construct offered by Tom. In that case, the Gom module can be directly included into the Tom file, using the %gom
instruction:
package myPackage; import myPackage.myclass.expressions.*; import myPackage.myclass.expressions.types.*; class MyClass { %gom{ module Expressions abstract syntax Bool = True()  False() ... Expr = Mul(lhs:Expr, rhs:Expr) } ... }
Note that the Java code is generated in a package that corresponds to the current package, followed by the classname and the modulename. This allows to define the same module in several classes, and avoid name clashes.
Hooks
Gom provides hooks that allow to define properties of the datastructure, in particular canonical forms for the terms in the signature in an algebraic way.
Algebraic rules
The 'rules' hook defines a set of conditional rewrite rules over the current module signature. Those rules are applied systematically using a leftmostinnermost reduction strategy. Thus, the only terms that can be produced and manipulated in the Tom program are normal with respect to the defined system.
module Expressions imports String int abstract syntax Bool = True()  False()  Eq(lhs:Expr, rhs:Expr) Expr = Id(stringValue:String)  Nat(intValue:int)  Add(lhs:Expr, rhs:Expr)  Mul(lhs:Expr, rhs:Expr) module Expressions:rules() { Eq(x,x) > True() Eq(x,y) > False() if x!=y }
Since the rules do alter the behavior of the construction functions in the term structure, it is required in a module that the rules in a 'rules' hook have as lefthand side a pattern rooted by an operator of the current module. The rules are tried in the order of their definitions, and the first matching rule is applied.
Note: it is possible to define rules on a variadic symbol. However, due to the leftmostinnermost rule application strategy, using a list variable at the left of a pattern is usually not needed, and may result in an inefficient procedure. 
Hooks to alter the creation operations
Hooks may be used to specify how operators should be created. 'make', 'make_empty' and 'make_insert' hooks are altering the creation operations for respectively algebraic, neutral element (empty variadic) and variadic operators. 'make_insert' is simply a derivative case of 'make', with two arguments, for variadic operators.
The hook operation type is followed by a list of arguments name between '()'. The creation operation takes those arguments in order to build a new instance of the operator. Thus, the arguments number has to match the slot number of the operator definition, and types are inferred from this definition.
Then the body of the hook definition is composed of Java and Tom code. The Tom code is compiled using the mapping definition for the current module, and thus allows to build and match terms from the current module. This code can also use the realMake
function, which consists in the "inner" default allocation function. This function takes the same number of arguments as the hook. In any case, if the hooks code does not perform a return
itself, this realMake
function is called at the end of the hook execution, with the corresponding hooks arguments
Using the expression
example introduced above, we can add hooks to implement the computation of Add
and Mul
when both arguments are known integers (i.e. when they are Nat(x)
)
module Expressions imports String int abstract syntax Bool = True()  False()  Eq(lhs:Expr, rhs:Expr) Expr = Id(stringValue:String)  Nat(intValue:int)  Add(lhs:Expr, rhs:Expr)  Mul(lhs:Expr, rhs:Expr) Add:make(l,r) { %match(Expr l, Expr r) { Nat(lvalue), Nat(rvalue) > { return `Nat(lvalue + rvalue); } } } Mul:make(l,r) { %match(Expr l, Expr r) { Nat(lvalue), Nat(rvalue) > { return `Nat(lvalue * rvalue); } } }
Using this definition, it is impossible to have an expression containing unevaluated expressions where a value can be calculated. Thus, a procedure doing constant propagations for Id
whose value is known could simply replace the Id
by the corresponding Nat
, and rely on this mechanism to evaluate the expression. Note that the arguments of the make
hook are themselves elements built on this signature, and thus the hooks have been applied for them. In the case of hooks encoding a rewrite system, this corresponds to using an innermost strategy.
List theory hooks
In order to ease the use of variadic operators with the same domain and codomain, Gom does provide hooks that enforce a particular canonical form for lists.

FL
, activated with<op>:FL() {}
, ensures that structures containing the operator<op>
are left to right comb, with an empty<op>()
at the right. This constitutes a particular form of associative with neutral element canonical form, with a restriction on the application of the neutral rules. This corresponds to the generation of the following normalisation rules:
make(Empty(),tail) > tail make(Cons(h,t),tail) > make(h,make(t,tail)) make(head,tail) > make(head,make(tail,Empty)) if tail!=Empty and tail!=Cons

Free
, activated with<op>:Free() {}
, ensures the variadic symbol remains free, i.e. deactivates the defaultFL
hook. 
AU
, activated with<op>:AU() {}
, ensures that structures containing the<op>
operator are left to right comb, and that neutral elements are removed. It is possible to specify an alternate neutral element to the associative with neutral theory, using<op>:AU() {`<elem>()}
, where<elem>
is a term in the signature. This will generate the following normalisation rules:
make(Empty(),tail) > tail make(head,Empty()) > head make(Cons(h,t),tail) > make(h,make(t,tail))

ACU
is similar toAU
, except that it also ensures elements in the left to right comb are sorted using the builtincompareTo
function.
make(Empty(),tail) > tail make(head,Empty()) > head make(Cons(h,t),tail) > make(h,make(t,tail)) make(head,Cons(h,t)) > make(head,Cons(h,t)) if head < h make(head,Cons(h,t)) > make(h,make(head,t)) if head >= h
If you do not define any hook of the form AU
, ACU
, FL
, Free
, or rules
, the FL
hook will be automatically declared for variadic operators whose domain and codomain are equals. In practice, this makes list matching and associative matching with neutral element easy to use.
If you use a hook rules
, there may be an interaction that can lead to nontermination. Therefore, no hook will be automatically added, and you are forced to declare a hook of the form AU
, ACU
, FL
or Free
.
Note: if you do not really understand what happens when you define a hook 
Generated API
For each Module, the Gom compiler generates an API specific of this module, possibly using the API of other modules (declared in the Imports section). This API is located in a Java package named using the ModuleName lowercased, and contains the tree implementation itself, with some additional utilities:
 an abstract class named ModuleName
AbstractType
is generated. This class is the generic type for all nodes whose type is declared in this module. It declares generic functions for all tree nodes: atoATerm()
method returning aaterm.ATerm
representation of the tree; asymbolName()
method returning aString
representation for the function symbol of the tree root; thetoString()
method, which returns a string representation.
public aterm.ATerm toATerm(); public String symbolName(); public int compareTo(Object o); public int compareToLPO(Object o); public void toStringBuilder(java.lang.StringBuilder buffer);
 in a subpackage
types
, Gom generates one class for each sort defined in the module, whose name corresponds to the sort name. Each sort class extends ModuleNameAbstractType
for the module, and declares aboolean
methodis
OperatorName()
for each operator in the module (false
by default). It declares getters (methods namedget
SlotName()
) for each slot used in any operator of the sort (throwing an exception by default). A methodSlotName fromTerm(aterm.ATerm trm)
is generated, allowing to use ATerm as an exchange format. The methodsSlotName fromString(String s)
andSlotName fromStream(InputStream stream)
allow the use of an ATerm representation in String or stream form, to store terms in file and read them back.
public boolean is<op>(); public <SlotType> get<slotName>(); public <SortName> set<slotName>(<SlotType>); public static <SortName> fromTerm(aterm.ATerm trm); public static <SortName> fromString(String s); public static <SortName> fromStream(java.io.InputStream stream) throws java.io.IOException; public int length(); public <SortName> reverse();
 given a sort (SortName), generate a class (in a package
types.
SortName) for each operator. This class extends the class generated for the corresponding sort. It provides getters for the slots of the operator, and theis
OperatorName()
method is overridden to returntrue
. Those classes implement thetom.library.sl.Visitable
interface. It is worth noting that builtin fields are not accessible from theVisitable
, and thus will not be visited by strategies. The operator class also implements a staticmake
method, building a new instance of the operator. Thismake
method is the only way to obtain a new instance.
public static <op> make(arg1,...,argn);
 for each listoperator, Operator for instance, the generated code contains two operator classes: one name
Empty
Operator which is used to represent the empty list of arity 0, and the other namedCons
Operator having two fields: one with the codomain sort, and one with the domain sort of the variadic operator, respectively namedHead
Operator andTail
Operator, leading to getter functionsgetHead
OperatorandgetTail
Operator. This allows to define lists as the composition of manyCons
and oneEmpty
objects.  for each module, one file ModuleName
.tom
providing a Tom mapping for the sorts and operators defined or imported by the module.
Example of generated API
We show elements of the generated API for a very simple example, featuring variadic operator. It defines natural numbers as Zero()
and Suc(n)
, and lists of natural numbers.
module Mod abstract syntax Nat = Zero()  Suc(pred:Nat)  Plus(lhs:Nat,rhs:Nat)  List(Nat*)
Using the command gom Mod.gom
, the list of generated files is:
mod/Mod.tom (the Tom mapping) mod/ModAbstractType.java (abstract class for the "Mod" module) mod/types/Nat.java (abstract class for the "Nat" sort) mod/types/nat/List.java \ mod/types/nat/ConsList.java \ mod/types/nat/EmptyList.java / Implementation for the operator "List" mod/types/nat/Plus.java (Implementation for "Plus") mod/types/nat/Suc.java (Implementation for "Suc") mod/types/nat/Zero.java (Implementation for "Zero")
The ModAbstractType
class declares generic methods shared by all operators in the Mod module:
public aterm.ATerm toATerm() public String symbolName() public String toString()
The mod/types/Nat.java class provides an abstract class for all operators in the Nat sort, implementing the ModAbstractType and contains the following methods. First, the methods for checking the root operator, returning false
by default:
public boolean isConsList() public boolean isEmptyList() public boolean isPlus() public boolean isSuc() public boolean isZero()
Then getter methods, throwing an UnsupportedOperationException
by default, as the slot may not be present in all operators. This is convenient since at the user level, we usually manipulate objects of sort Nat, without casting them to more specific types.
public mod.types.Nat getpred() public mod.types.Nat getlhs() public mod.types.Nat getHeadList() public mod.types.Nat getrhs() public mod.types.Nat getTailList()
The fromTerm static method allows Gom data structure to be interoperable with ATerm
public static mod.types.Nat fromTerm(aterm.ATerm trm)
The operator implementations redefine all or some getters for the operator to return its subterms. It also provides a static make method to build a new tree rooted by this operator, and implements the tom.library.sl.Visitable interface. For instance, in the case of the Plus operator, the interface is:
public static Plus make(mod.types.Nat lhs, mod.types.Nat rhs) public int getChildCount() public tom.library.sl.Visitable getChildAt(int index) public tom.library.sl.Visitable setChildAt(int index, tom.library.sl.Visitable v)
completed with the methods from the Nat class and the ModAbstractType.
The operators implementing the variadic operator both extend the List
class, which provides list related methods, such as length
, fromArray
, getCollection
and reverse
. The getCollection
method produces a collection of objects of the codomain type corresponding to the list elements, while the reverse
method returns the list with all elements in reverse order. The static fromArray
method produces a list from an array of objects of the codomain type. The List
class for our example then contains:
public int length() public mod.types.Nat reverse() public java.util.Collection<mod.types.Nat> getCollection() public static mod.types.Nat fromArray(mod.types.Nat[] array)
The List
class implements also the java.util.Collection
interface:
public int size() public boolean containsAll(java.util.Collection c) public boolean contains(Object o) public boolean isEmpty() public java.util.Iterator<mod.types.Nat> iterator() public Object[] toArray() public <T> T[] toArray(T[] array)
Note that all the methods of the java.util.Collection
that make the list mutable (for example, the removeAll
method) are not implemented and thus throw an UnsupportedOperationException
.
For the ConsList class, we obtain:
/* the constructor */ public static ConsList make(mod.types.Nat _HeadList, mod.types.Nat _TailList) { ... } public String symbolName() { ... } /* From the "Nat" class */ public boolean isConsList() { ... } public mod.types.Nat getHeadList() { ... } public mod.types.Nat getTailList() { ... } /* From the "ModAbstractType" class */ public aterm.ATerm toATerm() { ... } public static mod.types.Nat fromTerm(aterm.ATerm trm) { ... } /* The tom.library.sl.Visitable interface */ public int getChildCount() { ... } public tom.library.sl.Visitable getChildAt(int index) { ... } public tom.library.sl.Visitable setChildAt(int index, tom.library.sl.Visitable v) { ... } /* The MuVisitable interface */ public tom.library.sl.Visitable setChilds(tom.library.sl.Visitable[] childs) { ... }
Hooks to alter the generated API
There exist four other hooks 'import', 'interface', 'block' and 'mapping' that offer possibilities to enrich the generated API. Contrary to 'make' and 'make_insert', these hooks have no parameters. Moreover, they can be associated not only to an operator but also to a module or a sort.
HookDefinition  ::=  HookType ':' HookOperation 
HookType  ::=  OperatorName 
∣  'module' ModuleName  
∣  'sort' SortName  
HookOperation  ::=  'interface()' '{' Identifier (',' Identifier )* '}' 
∣  'import()' '{' JavaImports '}'  
∣  'block()' '{' TomCode '}'  
∣  'mapping()' '{' TomMapping '}' 
There are few constraints on the form of the code in these hooks:
 for 'import', the code is a wellformed block of Java imports.
 for 'interface', the code is a wellformed list of Java interfaces.
 for 'block', the code is a wellformed Java block which can contain Tom code.
 for 'mapping', the code is a wellformed Tom block composed only of mappings.
The code given in the hook is just added at the correct position in the corresponding Java class:
 for a module ModuleName, in the abstract class named ModuleName
AbstractType
(for now, you can only use this hook with the current module),  for a sort SortName, in the abstract class named SortName in the package types,
 for an operator OperatorName of sort SortName, in the class named SortName in the package
types/
SortName.
In the case of 'mapping' hooks, the corresponding code is added to the mapping generated for the signature.
module Expressions imports String int abstract syntax Bool = True()  False()  Eq(lhs:Expr, rhs:Expr) Expr = Id(stringValue:String)  Nat(intValue:int)  Add(lhs:Expr, rhs:Expr)  Mul(lhs:Expr, rhs:Expr) True:import() { import tom.library.sl.*; import java.util.HashMap; } sort Bool:interface() { Cloneable, Comparable } module Expressions:block() { %include{ util/HashMap.tom } %include{ sl.tom } %strategy CollectIds(table:HashMap) extends Identity() { visit Expr { Id(value) > { table.put(`value,getEnvironment().getPosition()); } } } public static HashMap collect(Expr t) { HashMap table = new HashMap(); `TopDown(CollectIds(table)).apply(t); return table; } }
Gom Antlr Adaptor
Combining Tom, Gom and Antlr is easy. The tool GomAntlrAdaptor takes a Gom signature as input and generates an adaptor to convert an AST built by Antlr into a Gom tree.
Let us consider a simple grammar:
grammar Gram; @header { package parser; } @lexer::header { package parser; } ruleset : rule (rule)* EOF ; rule : 'a'  'b' ; WS : (' ''\t''\n')+ { $channel=HIDDEN; } ; SLCOMMENT : '//' (~('\n''\r'))* ('\n''\r'('\n')?)? { $channel=HIDDEN; } ;
Our idea is to use the "rewrite rule" mechanism provided by Antlr to build an AST. Therefore, we consider the following signature (i.e. the node of the AST):
module parser.Rule abstract syntax Term = A()  B()  Conc(a:Term,b:Term)
Three things have to be done:
 set
output=AST
, andASTLabelType=Tree
 declare the list of nodes (tokens) that will be used in the rewrite rules
 define the rewrite rules
grammar Gram; options { output=AST; ASTLabelType=Tree; tokenVocab=RuleTokens; } @header { package parser; } @lexer::header { package parser; } ruleset : (rule > rule) (a=rule > ^(Conc $ruleset $a))* EOF ; rule : 'a' > ^(A)  'b' > ^(B) ; WS : (' ''\t''\n')+ { $channel=HIDDEN; } ; SLCOMMENT : '//' (~('\n''\r'))* ('\n''\r'('\n')?)? { $channel=HIDDEN; } ;
In the Main
file, note the use of GramRuleAdaptor.getTerm(b)
to convert the tree built by Antlr into a Gom tree:
package parser; import org.antlr.runtime.*; import org.antlr.runtime.tree.*; import parser.rule.RuleAdaptor; public class Main { public static void main(String[] args) { try { GramLexer lexer = new GramLexer(new ANTLRInputStream(System.in)); CommonTokenStream tokens = new CommonTokenStream(lexer); GramParser parser = new GramParser(tokens); // Parse the input expression Tree b = (Tree) parser.ruleset().getTree(); System.out.println("Result = " + RuleAdaptor.getTerm(b)); // name of the Gom module + Adaptor } catch (Exception e) { System.err.println("exception: " + e); return; } } }
To compile this example:
 create a directory named
parser
,  create all three files described above (
Rule.gom
,Gram.g
andMain.t
),  execute the following commands:
gom d gen parser/Rule.gom gomantlradaptor d gen p parser parser/Rule.gom java org.antlr.Tool o gen lib gen/parser/rule parser/Gram.g tom d gen parser/Main.t cd gen javac parser/Main.java
Note: On WIndows systems, replace the ANTLR command by : 
Strategies support (*)
The data structures generated by Gom do provide support for the strategy language of Tom. We assume in this section the reader is familiar with the support of Tom as described in the Strategies chapter, and illustrated in the tutorial.
Basic strategy support
The data structure generated by the Gom compiler provides support for the sl library implementing strategies for Tom. It is thus possible without any further manipulation to use the strategy language with Gom data structures.
This strategy support is extended by the Gom generator by providing congruence and construction elementary strategies for all operators of the data structure. Those strategies are made available through a Tom mapping _<module>.tom
generated during Gom compilation.
Congruence strategies
The congruence strategies are generated for each operator in the Gom module. For a module containing a sort
Term = a()  f(lt:Term,rt:Term)  l(Term*)
congruence strategies are generated for a
, f
and l
operators, respectively called _a
, _f
and _l
. The semantics of those elementary strategies is as follows:
(_a)[t]  ⇒  t if t equals a , failure otherwise

(_f(s1,s2))[t]  ⇒  f(t1’,t2’) if t = f(t1,t2), with (s1)[t1] ⇒t1’ and (s2)[t2] ⇒t2’, 
failure otherwise  
(_l(s))[t]  ⇒  l(t1’,...,tn’) if t = l(t1,...,tn), with (s)[ti] ⇒ti’, 
failure otherwise 
Thus, congruence strategies allows to discriminate terms based on how they are built, and to develop strategies adopting different behaviors depending on the shape of the term they are applied to.
Congruence strategies are commonly used to implement a specific behavior depending on the context (thus, it behaves like a complement to pattern matching). For instance, to print all first children of an operator f
, it is possible to use a generic Print()
strategy under a congruence operator.
Strategy specPrint1 = `TopDown(_f(Print(),Identity()));
When applied to variadic operators, a congruence strategy acts over all theirs elements. So, the strategy Print()
is applied under a congruence operator printing all elements of a variadic operator l
.
Strategy specPrint2 = `TopDown(_l(Print()));
Also, congruence strategies are used to implement map
like strategies on tree structures.
Consider a signature with List = Cons(e:Element,t:List)  Empty()
, then we can define a map
strategy as:
Strategy map(Strategy arg) { return `mu(MuVar("x"), Choice(_Empty(),_Cons(arg,MuVar("x"))) ); }
The congruence strategy generated for variadic operators is similar to the map
strategy. It applies the strategy passed as argument to all subterms of the considered variadic operator.
Construction strategies
Gom generated strategies’ purpose is to allow to build new terms for the signature at the strategy level. Those strategies do not use the terms they are applied to, and simply create a new term. Their semantics is as follows:
(Make_a)[t]  ⇒  a 
(Make_f(s1,s2))[t]  ⇒  f(t1,t2) if (s1)[null] ⇒t1 and (s2)[null] ⇒t2, 
failure otherwise 
We can note that as the substrategies for Make_f
are applied to the null
term, it is required that those strategies are themselves construction strategies and do not examine their argument.
These construction strategies, combined with congruence strategies can be used to implement rewrite rules as strategies. For instance, a rule f(a,b) → g(a,b) can be implemented by the strategy:
Strategy rule = `Sequence( _f(_a(),_b()), _Make_g(_Make_a(),_Make_b()) );
Fresh Gom (**)
Fresh Gom is an extension of Gom which adds capabilities very similar to that of Cαml developed by François Pottier for the Cαml programming language. In short, it extends the syntax of Gom in order to allow the specification of binding information (like : "this constructor field is in fact a variable bound in these other fields") and generates all the boring machinery to deal with fresh variables, alphaconversion, etc. Moreover, it integrates smoothly with Tom by generating mappings which ensure that every time a constructor is matched, its concerned variables are refreshed. As an example, assume we have defined the grammar of lambda expressions using Fresh Gom. Then, when the following match instruction is run,
%match( `Lambda(abs(x,u)) ) { Lambda(abs(y,v)) > { ... } }
a fresh y is generated (fresh here means an unique identifier that has never been generated before), and v is u where all instances of x have been replaced by y.
Just like Cαml, Fresh Gom allows for complex binding specifications involving several variable sorts (called atoms in the nominal logic jargon) and for nested patterns (think for instance of the metarepresentation of a rewrite rule, where the variables of the lefthand side are bound in the righthand side: you do not statically know how many  neither at which depth  variables will appear in a lefthand side). The semantics of these constructs, although intuitive enough to be grasped as is, is formally defined in this paper by François Pottier.
Grammar
As mentioned, Fresh Gom extends the Gom grammar exposed above.
Updated Rules  
Grammar  ::=  'abstract syntax' (... ∣ AtomDecl)* 
TypeDefinition  ::=  ... ∣ SortName ['binds' AtomName+] '=' PatternDefinition 
OperatorDefinition  ::=  ... ∣ OperatorName'(' '<' SortName '>' '*' ')'SlotDefinition)*]')' 
SlotDefinition  ::=  ... ∣ SlotName':' '<' SortName '>' 
Additional Rules  
AtomDecl  ::=  'atom' AtomName 
AtomName  ::=  Identifier 
PatternDefinition  ::=  OperatorName'('[PatternSlotDefinition(','PatternSlotDefinition)*]')' 
∣  OperatorName'(' SortName'*' ')'  
PatternSlotDefinition  ::=  ['inner' ∣ 'outer' ∣ 'neutral'] SlotName':' SortName 
Description
Atoms
For each atom declared, Fresh Gom generates a class and a Tom sort of the same name. For example, a Gom module containing the line
atom LambdaVar
will generate a class LambdaVar
along with the declaration of
a tom sort
%typeterm LambdaVar { implements LambdaVar }
at the usual places. Everything is as it were an usual Gom sort for which no constructors had been declared.
The implementation of these atoms remains opaque to the user. The unique way to create an atom of name AtomName
is to use the following generated static method which creates a fresh identifier.
public abstract class AtomName { public static AtomName freshAtomName() { ... } }
Raw Sorts and Constructors
Each sort concerned by Fresh Gom (i.e. connected to an atom in the sort dependency graph) is replaced by two other ones:
 a normal one, which is identical, except that every field whose sort is an atom is replaced by a String;
 a raw one, which has exactly the same definition, but whose name and constructor names have been prefixed by "Raw".
As an example, everything is as the following declarations
atom EVar Expr = Plus(e1:Expr,e2:Expr)  Lit(n:int)  Var(v:EVar)
were replaced by the code below.
EVar = /* non legit gom */ Expr = Plus(e1:Expr,e2:Expr)  Lit(n:int)  Var(v:EVar) RawExpr = RawPlus(e1:RawExpr,e2:RawExpr)  RawLit(n:int)  RawVar(v:String)
The "raw" sorts are meant to be used at parse and prettyprinting time and do not generate any additional method than regular Gom sorts except that for conversion to "normal" sorts. The "normal" sorts are the ones to work with for all tasks jeopardized by variable capture problems: evaluation, typechecking, etc. They generate several useful methods used by the mechanisms described in the next section.
Conversion methods for converting one format into the other are generated in the classes representing the sorts.
public abstract class SortName { public RawSortName export() { ... } } public abstract class RawSortName { public SortName convert() { ... } }
While export()
always succeeds, a call to convert()
may miserably fail (RuntimeException
for the current release) if the subject contains free variables. If one wishes to provide a dictionary from the free variables to objects of "normal" sort in order to convert a nonclosed term, specialized _convert(...)
versions of convert()
are also generated. Their signature depends on the involved atoms.
For example, the following method is generated for the signature above.
public abstract class RawExpr { public abstract Expr _convert(tom.library.freshgom.ConvertMap<EVar> EVarMap); }
The tom.library.freshgom.ConvertMap
is fully documented in the Tom library API.
Informal Semantics
As in Cαml, the sorts defined in a Fresh Gom module are of two distinct kinds:
 the sorts the definition of which is preceded by
'binds' id1 id2 ..
are called pattern sorts;  the other ones are called expression sorts.
When pattern sorts are mentioned in an expression sort constructor, they must be placed inside brackets '<' '>'. This indicates a refresh point, i.e. a field that will be refreshed whenever the enclosing constructor is deconstructed (using match). For example, consider this excerpt of a module defining a simply typed lambdacalculus:
atom LVar Type = Atomic(p:String)  Arrow(ty1:Type,ty2:Type) LTerm = App(t1:LTerm,t2:LTerm)  Lam(a:<Abs>)  Var(x:LVar) Abs binds LVar = abs(x:LVar, neutral ty:Type, inner u:LTerm)
Since the pattern sort Lam
is declared to bind the atoms of sort LVar
, every time the constructor Lam
is
deconstructed, the atoms of sort LVar
in the field a
are refreshed. The canonical way to deconstruct such a constructor is thus to use a nested pattern as follows.
%match(t) { App(u,v) > { /* nothing special happens */ } Lambda(abs(x,ty,u)) > { /* x is fresh */ } Var(x) > { /* nothing special happens */ } }
Who binds what and where is specified using the 'neutral', 'inner' and 'outer' keywords. In the example above, in the abs
constructor, x
is in pattern position, which means that the atoms of sort LVar
it contains (in that case, x
itself) may be bound in the other fields of abs
. If it is the case and how is specified the following way:
 'inner' means that it is bound;
 'outer' means that it is not bound (useful for representing let statements for instance : in
let x = u in t
,x
is not bound in u);  'neutral' means it is irrelevant (here since there are no variables in types).
Compilation
The Fresh Gom mode is enable using the fresh
option of Gom, as described in the tool usage documentation. It is not compatible with the termgraph
option.
Examples
We present two commented examples: an interpreter for System F and a signature for a subset of ML. While the first example illustrates the use of several atom sorts, the second illustrates nested pattern sorts.
System F
Let us represent Girard's System F, also known as the secondorder lambda calculus. This calculus is used as the intermediate typed representation of many functional programming languages. It has the particularity of involving two kinds of variables : terms (x,y,z...) and types (X,Y,Z...) variables, which will illustrate this particular aspect of Fresh Gom. The language is generated by the following grammar:
t,u ::= x  λx:A.t  ΛX.t  (t u)  (t A) A,B ::= X  ∀X.A  A → B
Formal presentations of the system usually go on with
 in
λx:A.t
, the term variablex
is bound in the termt
;  in
ΛX.t
, the type variableX
is bound in the termt
;  in
∀X.A
, the type variableX
is bound in the typeA
.
We will represent these three kind of abstraction using three pattern sorts: TermTermAbs
, TypeTermAbs
and TypeTypeAbs
. The previous three points will be expressed using the 'inner' keyword. Note that in λx:A.t
, the question of wether the term variable x
is bound in the type A
is irrelevant. We will express this fact using the 'neutral' keyword. The remaining of the grammar is standard Gom code.
module SystemF imports String int abstract syntax atom TermVar atom TypeVar LTerm =  LVar(x:TermVar)  LLam(a1:<TermTermAbs>)  LApp(u:LTerm,v:LTerm)  TLam(a2:<TypeTermAbs>)  TApp(t:LTerm,A:Type) TermTermAbs binds TermVar = abs1(x:TermVar, neutral A:Type, inner t:LTerm) TypeTermAbs binds TypeVar = abs2(X:TypeVar, inner t:LTerm) Type =  TVar(X:TypeVar)  Forall(a3:<TypeTypeAbs>)  Arrow(A:Type,B:Type) TypeTypeAbs binds TypeVar = abs3(X:TypeVar, inner A:Type)
Let us now write a prettyprinter for the lambdaterms. Since prettyprint needs to manipulate raw variable names (strings), we will work in raw mode.
import systemf.types.*; public class Pretty { %include { systemf/SystemF.tom } public static String pretty(RawLTerm s) { %match(s) { RawLVar(x) > { return `x; } RawLLam(Rawabs1(x,T,t)) > { return %[(fn @`x@:@`pretty(T)@ > @`pretty(t)@)]%; } RawLApp(t,u) > { return %[(@`pretty(t)@ @`pretty(u)@)]%; } RawTLam(Rawabs2(T,t)) > { return %[(FN @`T@ > @`pretty(t)@)]%; } RawTApp(t,T) > { return %[(@`pretty(t)@ @`pretty(T)@)]%; } } throw new RuntimeException("nonexhaustive patterns"); } public static String pretty(RawType s) { %match(s) { RawTVar(X) > { return `X; } RawForall(Rawabs3(X,T)) > { return %[(forall @`X@, @`pretty(T)@)]%; } RawArrow(A,B) > { return %[(@`pretty(A)@ > @`pretty(B)@)]%; } } throw new RuntimeException("nonexhaustive patterns"); } public static void main(String args[]) { RawLTerm z = `RawLVar("z"); RawLTerm s = `RawLVar("s"); RawLTerm n = `RawLVar("n"); RawType A = `RawTVar("A"); RawType B = `RawTVar("B"); RawLTerm zero = `RawTLam(Rawabs2("A", RawLLam(Rawabs1("z",A, RawLLam(Rawabs1("s",RawArrow(A,A),z)))))); RawLTerm succ = `RawLLam(Rawabs1("n",RawForall(Rawabs3("A",RawArrow(RawArrow(A,A),RawArrow(A,A)))), RawTLam(Rawabs2("B", RawLLam(Rawabs1("z",B, RawLLam(Rawabs1("s",RawArrow(B,B), RawLApp(RawLApp(RawTApp(n,B),RawLApp(s,z)),s))))))))); System.out.println(pretty(zero)); System.out.println(pretty(succ)); } }
Let us now write an interpreter. The reduction rules of System F are the termlevel and typelevel βreductions.
(λx:A.t) u → t{x := u} (ΛX.t) A → t{X := A}
The key point is the reduction system HeadBeta
. Thanks to the mapping generated by Fresh Gom, every time an abstraction is matched, the variable are refreshed so that we avoid their potential capture. This time, we have to convert the raw term into a "normal" one to benefit of this feature.
import systemf.types.*; import tom.library.sl.*; public class Eval { %include { systemf/SystemF.tom } %include { sl.tom } /* returns t{x := u} */ public static LTerm subst(LTerm t, TermVar x, LTerm u) { %match(t) { LVar(y) > { return `y == x ? u : `LVar(y); } LLam(abs1(y,T,v)) > { return `LLam(abs1(y,T,subst(v,x,u))); } LApp(v,w) > { return `LApp(subst(v,x,u),subst(w,x,u)); } TLam(abs2(T,v)) > { return `TLam(abs2(T,subst(v,x,u))); } TApp(v,T) > { return `TApp(subst(v,x,u),T); } } throw new RuntimeException("nonexhaustive patterns"); } /* returns t{X := A} */ public static LTerm subst(LTerm t, TypeVar X, Type A) { %match(t) { LVar(x) > { return `LVar(x); } LLam(abs1(x,T,u)) > { return `LLam(abs1(x,subst(T,X,A),u)); } LApp(u,v) > { return `LApp(subst(u,X,A),subst(v,X,A)); } TLam(abs2(T,u)) > { return `TLam(abs2(T,subst(u,X,A))); } TApp(u,T) > { return `TApp(subst(u,X,A),subst(T,X,A)); } } throw new RuntimeException("nonexhaustive patterns"); } /* returns A{X := B} */ public static Type subst(Type A, TypeVar X, Type B) { %match(A) { TVar(Y) > { return `Y == X ? B : `TVar(Y); } Forall(abs3(X,T)) > { return `Forall(abs3(X,subst(T,X,B))); } Arrow(T,U) > { return `Arrow(subst(T,X,B),subst(U,X,B)); } } throw new RuntimeException("nonexhaustive patterns"); } /* beta reductions */ %strategy HeadBeta() extends Fail() { visit LTerm { LApp(LLam(abs1(x,T,t)),u) > subst(t,x,u) TApp(TLam(abs2(T,t)),A) > subst(t,T,A) } } /* call by name evaluation */ public static LTerm eval(LTerm t) { try { return `Outermost(HeadBeta()).visit(t); } catch(VisitFailure e) { throw new RuntimeException("never happens"); } } public static void main(String args[]) { RawLTerm zero = ... // as before RawLTerm succ = ... // as before RawLTerm rawThree = `RawLApp(succ,RawLApp(succ,RawLApp(succ,zero))); // convert to internal representation LTerm three = rawThree.convert(); // eval LTerm res = eval(three); // export to raw representation RawLTerm rawRes = res.export(); System.out.println(Pretty.pretty(rawRes)); } }
It is worth noticing that by changing Outermost
to Innermost
we obtain a callbyvalue interpreter.
Although this approach is highly inefficient for writing a real interpreter, Fresh Gom suits the needs of prototypes and is perfectly adapted to the description of sourcetosource transfomations (e.g. CPS, see the examples of the Tom distribution).
Mini ML
The full example can be found in the Tom distribution. We focus here on how to encode (case .. of ..)
expressions with the help of Fresh Gom.
module lambda imports int String abstract syntax atom LVar LType = Atom(n:String)  Arrow(t1:LType,t2:LType)  TypeVar(i:int) LTerm = App(t1:LTerm,t2:LTerm)  Abs(a:<Lam>)  Let(b:<Letin>)  Fix(c:<Fixpoint>)  Var(x:LVar)  Constr(f:String, children:LTermList)  Case(subject:LTerm,rules:Rules) Rules = RList(<Clause>*) /* all the variables of p are bound in t */ Clause binds LVar = Rule(p:Pattern, inner t:LTerm) Pattern binds LVar = PFun(neutral f:String, children:PatternList)  PVar(x:LVar, neutral ty:LType) LTermList = LTList(LTerm*) PatternList binds LVar = PList(Pattern*) Lam binds LVar = lam(x:LVar, neutral ty:LType, inner t:LTerm) Letin binds LVar = letin(x:LVar, outer u:LTerm, inner t:LTerm) Fixpoint binds LVar = fixpoint(x:LVar, neutral ty:LType, inner t:LTerm)
The key definition is that of Clause
, which declares that all the atoms of sort LVar
present in the pattern p
are bound in the term t
. Therefore, when deconstructing a Clause
using the match construct of Tom, all the variables in the lefthand side of the fetched clauses are refreshed in their corresponding righthand side.
TermGraph Rewriting (**)
A termgraph is a term where subterms can be shared and where there may be cycles. Gom offers support to define termgraphs and termgraph rule systems. There exist several ways to define termgraphs but in our case, we propose to represent termgraphs by terms with pointers. These pointers are defined by a relative path inside the term. All the formal definitions can be found in this paper.
In order to use termgraph rewriting, it is necessary to compile the Gom signatures with the termgraph
option, as described in the tool usage documentation.
TermGraph DataStructures
When defining a Gom algebraic signature, it is possible to construct termgraphs on these signature using the option termgraph
. In this case, the signature is automatically extended to manage labels. For every sort, Term
for instance, two new constructors are added:
LabTerm(label:String,term:Term) RefTerm(label:String)
With these two new constructors, users can define termgraphs as labelled terms:
Term cyclicTerm = `LabTerm("l",f(RefTerm("l"))); Term termWithSharing = `g(RefTerm("a"),LabTerm("a",a()));
From this labelled term, users can obtain the termgraph representation with paths using the expand
method. This method must be called before applying a termgraph strategy.
TermGraph Rules
Using the hook graphrules
, it is possible to define a set of termgraph rules.
The first parameter (MyGraphStrat) is the name of the strategy associated to the set of rules. The second parameter is the default strategy (Identity in the following example).
The lefthand and righthand sides of these rules are termgraphs. A set of rules can only be associated to a given sort.
sort Term: graphrules(MyGraphStrat,Identity) { g(l:a(),&l) > f(b()) f(g(g(a(),&l),l:x)) > g(ll:b(),&ll) if b()<<x }
In the rules, sharings and cycles are not represented by the constructor LabTerm
and RefTerm
but using a light syntax. l:t
is equivalent to LabTerm(l,t)
and &l
corresponds to RefTerm(l)
.
Contrary to classical termgraph rewriting, it is possible to reuse a label from the lefthand side in the righthand side in order to obtain side effects. This feature is inspired from Rachid Echahed’s formalism.
sort Term: graphrules(SideEffect,Identity) { f(l:a()) > g(&l,l:b()) }
This set of rules is translated into a Tom %strategy
that can be used in a Tom program:
Term t = (Term) `g(RefTerm("a"),LabTerm("a",a())).expand(); `TopDown(Term.MyGraphStrat()).visit(t)
Language Reference 
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