xref

Xref is a cross reference tool that can be used for finding dependencies between functions, modules, applications and releases.

Calls between functions are either local calls like f(), or external calls like m:f(). Module data, which are extracted from BEAM files, include local functions, exported functions, local calls and external calls. By default, calls to built-in functions (BIF) are ignored, but if the option builtins, accepted by some of this module's functions, is set to true, calls to BIFs are included as well. It is the analyzing OTP version that decides what functions are BIFs. Functional objects are assumed to be called where they are created (and nowhere else). Unresolved calls are calls to apply or spawn with variable module, variable function, or variable arguments. Examples are M:F(a), apply(M, f, [a]), and spawn(m, f(), Args). Unresolved calls are represented by calls where variable modules have been replaced with the atom '$M_EXPR', variable functions have been replaced with the atom '$F_EXPR', and variable number of arguments have been replaced with the number -1. The above mentioned examples are represented by calls to '$M_EXPR':'$F_EXPR'/1, '$M_EXPR':f/1, and m:'$F_EXPR'/-1. The unresolved calls are a subset of the external calls.

Applications are collections of modules. The modules' BEAM files are located in the ebin subdirectory of the application directory. The name of the application directory determines the name and version of the application. Releases are collections of applications located in the lib subdirectory of the release directory. There is more to read about applications and releases in the Design Principles book.

Xref servers are identified by names, supplied when creating new servers. Each Xref server holds a set of releases, a set of applications, and a set of modules with module data. Xref servers are independent of each other, and all analyses are evaluated in the context of one single Xref server (exceptions are the functions m/1 and d/1 which do not use servers at all). The mode of an Xref server determines what module data are extracted from BEAM files as modules are added to the server. Starting with R7, BEAM files compiled with the option debug_info contain so called debug information, which is an abstract representation of the code. In functions mode, which is the default mode, function calls and line numbers are extracted from debug information. In modules mode, debug information is ignored if present, but dependencies between modules are extracted from other parts of the BEAM files. The modules mode is significantly less time and space consuming than the functions mode, but the analyses that can be done are limited.

An analyzed module is a module that has been added to an Xref server together with its module data. A library module is a module located in some directory mentioned in the library path. A library module is said to be used if some of its exported functions are used by some analyzed module. An unknown module is a module that is neither an analyzed module nor a library module, but whose exported functions are used by some analyzed module. An unknown function is a used function that is neither local or exported by any analyzed module nor exported by any library module. An undefined function is an externally used function that is not exported by any analyzed module or library module. With this notion, a local function can be an undefined function, namely if it is externally used from some module. All unknown functions are also undefined functions; there is a figure in the User's Guide that illustrates this relationship.

Starting with R9C, the module attribute tag deprecated can be used to inform Xref about deprecated functions and optionally when functions are planned to be removed. A few examples show the idea:

-deprecated({f,1}).

The exported function f/1 is deprecated. Nothing is said whether f/1 will be removed or not.

-deprecated({f,'_'}).

All exported functions f/0, f/1 and so on are deprecated.

-deprecated(module).

All exported functions in the module are deprecated. Equivalent to -deprecated({'_','_'})..

-deprecated([{g,1,next_version}]).

The function g/1 is deprecated and will be removed in next version.

-deprecated([{g,2,next_major_release}]).

The function g/2 is deprecated and will be removed in next major release.

-deprecated([{g,3,eventually}]).

The function g/3 is deprecated and will eventually be removed.

-deprecated({'_','_',eventually}).

All exported functions in the module are deprecated and will eventually be removed.

Before any analysis can take place, module data must be set up. For instance, the cross reference and the unknown functions are computed when all module data are known. The functions that need complete data (analyze, q, variables) take care of setting up data automatically. Module data need to be set up (again) after calls to any of the add, replace, remove, set_library_path or update functions.

The result of setting up module data is the Call Graph. A (directed) graph consists of a set of vertices and a set of (directed) edges. The edges represent calls (From, To) between functions, modules, applications or releases. From is said to call To, and To is said to be used by From. The vertices of the Call Graph are the functions of all module data: local and exported functions of analyzed modules; used BIFs; used exported functions of library modules; and unknown functions. The functions module_info/0,1 added by the compiler are included among the exported functions, but only when called from some module. The edges are the function calls of all module data. A consequence of the edges being a set is that there is only one edge if a function is locally or externally used several times on one and the same line of code.

The Call Graph is represented by Erlang terms (the sets are lists), which is suitable for many analyses. But for analyses that look at chains of calls, a list representation is much too slow. Instead the representation offered by the digraph module is used. The translation of the list representation of the Call Graph - or a subgraph thereof - to the digraph representation does not come for free, so the language used for expressing queries to be described below has a special operator for this task and a possibility to save the digraph representation for subsequent analyses.

In addition to the Call Graph there is a graph called the Inter Call Graph. This is a graph of calls (From, To) such that there is a chain of calls from From to To in the Call Graph, and every From and To is an exported function or an unused local function. The vertices are the same as for the Call Graph.

Calls between modules, applications and releases are also directed graphs. The types of the vertices and edges of these graphs are (ranging from the most special to the most general): Fun for functions; Mod for modules; App for applications; and Rel for releases. The following paragraphs will describe the different constructs of the language used for selecting and analyzing parts of the graphs, beginning with the constants:

• Expression ::= Constants
• Constants ::= Consts | Consts : Type | RegExpr
• Consts ::= Constant | [Constant, ...] | {Constant, ...}
• Constant ::= Call | Const
• Call ::= FunSpec -> FunSpec | {MFA, MFA} | AtomConst -> AtomConst | {AtomConst, AtomConst}
• Const ::= AtomConst | FunSpec | MFA
• AtomConst ::= Application | Module | Release
• FunSpec ::= Module : Function / Arity
• MFA ::= {Module, Function, Arity}
• RegExpr ::= RegString : Type | RegFunc | RegFunc : Type
• RegFunc ::= RegModule : RegFunction / RegArity
• RegModule ::= RegAtom
• RegFunction ::= RegAtom
• RegArity ::= RegString | Number | _ | -1
• RegAtom ::= RegString | Atom | _
• RegString ::= - a regular expression, as described in the re module, enclosed in double quotes -
• Type ::= Fun | Mod | App | Rel
• Function ::= Atom
• Application ::= Atom
• Module ::= Atom
• Release ::= Atom
• Arity ::= Number | -1
• Atom ::= - same as Erlang atoms -
• Number ::= - same as non-negative Erlang integers -

Examples of constants are: kernel, kernel->stdlib, [kernel, sasl], [pg -> mnesia, {tv, mnesia}] : Mod. It is an error if an instance of Const does not match any vertex of any graph. If there are more than one vertex matching an untyped instance of AtomConst, then the one of the most general type is chosen. A list of constants is interpreted as a set of constants, all of the same type. A tuple of constants constitute a chain of calls (which may, but does not have to, correspond to an actual chain of calls of some graph). Assigning a type to a list or tuple of Constant is equivalent to assigning the type to each Constant.

Regular expressions are used as a means to select some of the vertices of a graph. A RegExpr consisting of a RegString and a type - an example is "xref_.*" : Mod - is interpreted as those modules (or applications or releases, depending on the type) that match the expression. Similarly, a RegFunc is interpreted as those vertices of the Call Graph that match the expression. An example is "xref_.*":"add_.*"/"(2|3)", which matches all add functions of arity two or three of any of the xref modules. Another example, one that matches all functions of arity 10 or more: _:_/"[1-9].+". Here _ is an abbreviation for ".*", that is, the regular expression that matches anything.

The syntax of variables is simple:

• Expression ::= Variable
• Variable ::= - same as Erlang variables -

There are two kinds of variables: predefined variables and user variables. Predefined variables hold set up module data, and cannot be assigned to but only used in queries. User variables on the other hand can be assigned to, and are typically used for temporary results while evaluating a query, and for keeping results of queries for use in subsequent queries. The predefined variables are (variables marked with (*) are available in functions mode only):

E

Call Graph Edges (*).

V

Call Graph Vertices (*).

M

Modules. All modules: analyzed modules, used library modules, and unknown modules.

A

Applications.

R

Releases.

ME

Module Edges. All module calls.

AE

Application Edges. All application calls.

RE

Release Edges. All release calls.

L

Local Functions (*). All local functions of analyzed modules.

X

Exported Functions. All exported functions of analyzed modules and all used exported functions of library modules.

F

Functions (*).

B

Used BIFs. B is empty if builtins is false for all analyzed modules.

U

Unknown Functions.

UU

Unused Functions (*). All local and exported functions of analyzed modules that have not been used.

XU

Externally Used Functions. Functions of all modules - including local functions - that have been used in some external call.

LU

Locally Used Functions (*). Functions of all modules that have been used in some local call.

LC

Local Calls (*).

XC

External Calls (*).

AM

Analyzed Modules.

UM

Unknown Modules.

LM

Used Library Modules.

UC

Unresolved Calls. Empty in modules mode.

EE

Inter Call Graph Edges (*).

DF

Deprecated Functions. All deprecated exported functions and all used deprecated BIFs.

DF_1

Deprecated Functions. All deprecated functions to be removed in next version.

DF_2

Deprecated Functions. All deprecated functions to be removed in next version or next major release.

DF_3

Deprecated Functions. All deprecated functions to be removed in next version, next major release, or later.

These are a few facts about the predefined variables (the set operators + (union) and - (difference) as well as the cast operator (Type) are described below):

• F is equal to L + X.
• V is equal to X + L + B + U, where X, L, B and U are pairwise disjoint (that is, have no elements in common).
• UU is equal to V - (XU + LU), where LU and XU may have elements in common. Put in another way:
• V is equal to UU + XU + LU.
• E is equal to LC + XC. Note that LC and XC may have elements in common, namely if some function is locally and externally used from one and the same function.
• U is a subset of XU.
• B is a subset of XU.
• LU is equal to range LC.
• XU is equal to range XC.
• LU is a subset of F.
• UU is a subset of F.
• range UC is a subset of U.
• M is equal to AM + LM + UM, where AM, LM and UM are pairwise disjoint.
• ME is equal to (Mod) E.
• AE is equal to (App) E.
• RE is equal to (Rel) E.
• (Mod) V is a subset of M. Equality holds if all analyzed modules have some local, exported, or unknown function.
• (App) M is a subset of A. Equality holds if all applications have some module.
• (Rel) A is a subset of R. Equality holds if all releases have some application.
• DF_1 is a subset of DF_2.
• DF_2 is a subset of DF_3.
• DF_3 is a subset of DF.
• DF is a subset of X + B.

An important notion is that of conversion of expressions. The syntax of a cast expression is:

• Expression ::= ( Type ) Expression

The interpretation of the cast operator depends on the named type Type, the type of Expression, and the structure of the elements of the interpretation of Expression. If the named type is equal to the expression type, no conversion is done. Otherwise, the conversion is done one step at a time; (Fun) (App) RE, for instance, is equivalent to (Fun) (Mod) (App) RE. Now assume that the interpretation of Expression is a set of constants (functions, modules, applications or releases). If the named type is more general than the expression type, say Mod and Fun respectively, then the interpretation of the cast expression is the set of modules that have at least one of their functions mentioned in the interpretation of the expression. If the named type is more special than the expression type, say Fun and Mod, then the interpretation is the set of all the functions of the modules (in modules mode, the conversion is partial since the local functions are not known). The conversions to and from applications and releases work analogously. For instance, (App) "xref_.*" : Mod returns all applications containing at least one module such that xref_ is a prefix of the module name.

Now assume that the interpretation of Expression is a set of calls. If the named type is more general than the expression type, say Mod and Fun respectively, then the interpretation of the cast expression is the set of calls (M1, M2) such that the interpretation of the expression contains a call from some function of M1 to some function of M2. If the named type is more special than the expression type, say Fun and Mod, then the interpretation is the set of all function calls (F1, F2) such that the interpretation of the expression contains a call (M1, M2) and F1 is a function of M1 and F2 is a function of M2 (in modules mode, there are no functions calls, so a cast to Fun always yields an empty set). Again, the conversions to and from applications and releases work analogously.

The interpretation of constants and variables are sets, and those sets can be used as the basis for forming new sets by the application of set operators. The syntax:

• Expression ::= Expression BinarySetOp Expression
• BinarySetOp ::= + | * | -

+, * and - are interpreted as union, intersection and difference respectively: the union of two sets contains the elements of both sets; the intersection of two sets contains the elements common to both sets; and the difference of two sets contains the elements of the first set that are not members of the second set. The elements of the two sets must be of the same structure; for instance, a function call cannot be combined with a function. But if a cast operator can make the elements compatible, then the more general elements are converted to the less general element type. For instance, M + F is equivalent to (Fun) M + F, and E - AE is equivalent to E - (Fun) AE. One more example: X * xref : Mod is interpreted as the set of functions exported by the module xref; xref : Mod is converted to the more special type of X (Fun, that is) yielding all functions of xref, and the intersection with X (all functions exported by analyzed modules and library modules) is interpreted as those functions that are exported by some module and functions of xref.

There are also unary set operators:

• Expression ::= UnarySetOp Expression
• UnarySetOp ::= domain | range | strict

Recall that a call is a pair (From, To). domain applied to a set of calls is interpreted as the set of all vertices From, and range as the set of all vertices To. The interpretation of the strict operator is the operand with all calls on the form (A, A) removed.

The interpretation of the restriction operators is a subset of the first operand, a set of calls. The second operand, a set of vertices, is converted to the type of the first operand. The syntax of the restriction operators:

• Expression ::= Expression RestrOp Expression
• RestrOp ::= |
• RestrOp ::= ||
• RestrOp ::= |||

The interpretation in some detail for the three operators:

|

The subset of calls from any of the vertices.

||

The subset of calls to any of the vertices.

|||

The subset of calls to and from any of the vertices. For all sets of calls CS and all sets of vertices VS, CS ||| VS  is equivalent to CS | VS * CS || VS.

Two functions (modules, applications, releases) belong to the same strongly connected component if they call each other (in)directly. The interpretation of the components operator is the set of strongly connected components of a set of calls. The condensation of a set of calls is a new set of calls between the strongly connected components such that there is an edge between two components if there is some constant of the first component that calls some constant of the second component.

The interpretation of the of operator is a chain of calls of the second operand (a set of calls) that passes throw all of the vertices of the first operand (a tuple of constants), in the given order. The second operand is converted to the type of the first operand. For instance, the of operator can be used for finding out whether a function calls another function indirectly, and the chain of calls demonstrates how. The syntax of the graph analyzing operators:

• Expression ::= Expression GraphOp Expression
• GraphOp ::= components | condensation | of

As was mentioned before, the graph analyses operate on the digraph representation of graphs. By default, the digraph representation is created when needed (and deleted when no longer used), but it can also be created explicitly by use of the closure operator:

• Expression ::= ClosureOp Expression
• ClosureOp ::= closure

The interpretation of the closure operator is the transitive closure of the operand.

The restriction operators are defined for closures as well; closure E | xref : Mod is interpreted as the direct or indirect function calls from the xref module, while the interpretation of E | xref : Mod is the set of direct calls from xref. If some graph is to be used in several graph analyses, it saves time to assign the digraph representation of the graph to a user variable, and then make sure that every graph analysis operates on that variable instead of the list representation of the graph.

The lines where functions are defined (more precisely: where the first clause begins) and the lines where functions are used are available in functions mode. The line numbers refer to the files where the functions are defined. This holds also for files included with the -include and -include_lib directives, which may result in functions defined apparently in the same line. The line operators are used for assigning line numbers to functions and for assigning sets of line numbers to function calls. The syntax is similar to the one of the cast operator:

• Expression ::= ( LineOp) Expression
• Expression ::= ( XLineOp) Expression
• LineOp ::= Lin | ELin | LLin | XLin
• XLineOp ::= XXL

The interpretation of the Lin operator applied to a set of functions assigns to each function the line number where the function is defined. Unknown functions and functions of library modules are assigned the number 0.

The interpretation of some LineOp operator applied to a set of function calls assigns to each call the set of line numbers where the first function calls the second function. Not all calls are assigned line numbers by all operators:

• the Lin operator is defined for Call Graph Edges;
• the LLin operator is defined for Local Calls.
• the XLin operator is defined for External Calls.
• the ELin operator is defined for Inter Call Graph Edges.

The Lin (LLin, XLin) operator assigns the lines where calls (local calls, external calls) are made. The ELin operator assigns to each call (From, To), for which it is defined, every line L such that there is a chain of calls from From to To beginning with a call on line L.

The XXL operator is defined for the interpretation of any of the LineOp operators applied to a set of function calls. The result is that of replacing the function call with a line numbered function call, that is, each of the two functions of the call is replaced by a pair of the function and the line where the function is defined. The effect of the XXL operator can be undone by the LineOp operators. For instance, (Lin) (XXL) (Lin) E is equivalent to (Lin) E.

The +, -, * and # operators are defined for line number expressions, provided the operands are compatible. The LineOp operators are also defined for modules, applications, and releases; the operand is implicitly converted to functions. Similarly, the cast operator is defined for the interpretation of the LineOp operators.

The interpretation of the counting operator is the number of elements of a set. The operator is undefined for closures. The +, - and * operators are interpreted as the obvious arithmetical operators when applied to numbers. The syntax of the counting operator:

• Expression ::= CountOp Expression
• CountOp ::= #

All binary operators are left associative; for instance, A | B  || C is equivalent to (A | B) || C. The following is a list of all operators, in increasing order of precedence:

• +, -
• *
• #
• |, ||, |||
• of
• (Type)
• closure, components, condensation, domain, range, strict

Parentheses are used for grouping, either to make an expression more readable or to override the default precedence of operators:

• Expression ::= ( Expression )

A query is a non-empty sequence of statements. A statement is either an assignment of a user variable or an expression. The value of an assignment is the value of the right hand side expression. It makes no sense to put a plain expression anywhere else but last in queries. The syntax of queries is summarized by these productions:

• Query ::= Statement, ...
• Statement ::= Assignment | Expression
• Assignment ::= Variable := Expression | Variable = Expression

A variable cannot be assigned a new value unless first removed. Variables assigned to by the = operator are removed at the end of the query, while variables assigned to by the := operator can only be removed by calls to forget. There are no user variables when module data need to be set up again; if any of the functions that make it necessary to set up module data again is called, all user variables are forgotten.

Types

application() = atom()
arity() = int() | -1
bool() = true | false
call() = {atom(), atom()} | funcall()
constant() = mfa() | module() | application() | release()
directory() = string()
file() = string()
funcall() = {mfa(), mfa()}
function() = atom()
int() = integer() >= 0
library() = atom()
library_path() = path() | code_path
mfa() = {module(), function(), arity()}
mode() = functions | modules
module() = atom()
release() = atom()
string_position() = int() | at_end
variable() = atom()
xref() = atom() | pid()