32.12. dis --- Disassembler for Python bytecode

Source code: Lib/dis.py

---

The dis module supports the analysis of CPython bytecode by disassembling it. The CPython bytecode which this module takes as an input is defined in the file Include/opcode.h and used by the compiler and the interpreter.

CPython implementation detail: Bytecode is an implementation detail of the CPython interpreter. No guarantees are made that bytecode will not be added, removed, or changed between versions of Python. Use of this module should not be considered to work across Python VMs or Python releases.

Example: Given the function myfunc():

def myfunc(alist):
    return len(alist)

the following command can be used to display the disassembly of myfunc():

>>> dis.dis(myfunc)
  2           0 LOAD_GLOBAL              0 (len)
              3 LOAD_FAST                0 (alist)
              6 CALL_FUNCTION            1
              9 RETURN_VALUE

(The "2" is a line number).

32.12.1. Bytecode analysis

Baru pada versi 3.4.

The bytecode analysis API allows pieces of Python code to be wrapped in a Bytecode object that provides easy access to details of the compiled code.

class dis.Bytecode(x, *, first_line=None, current_offset=None)

Analyse the bytecode corresponding to a function, generator, method, string of source code, or a code object (as returned by compile()).

This is a convenience wrapper around many of the functions listed below, most notably get_instructions(), as iterating over a Bytecode instance yields the bytecode operations as Instruction instances.

If first_line is not None, it indicates the line number that should be reported for the first source line in the disassembled code. Otherwise, the source line information (if any) is taken directly from the disassembled code object.

If current_offset is not None, it refers to an instruction offset in the disassembled code. Setting this means dis() will display a "current instruction" marker against the specified opcode.

classmethod from_traceback(tb)

Construct a Bytecode instance from the given traceback, setting current_offset to the instruction responsible for the exception.

codeobj

The compiled code object.

first_line

The first source line of the code object (if available)

dis()

Return a formatted view of the bytecode operations (the same as printed by dis.dis(), but returned as a multi-line string).

info()

Return a formatted multi-line string with detailed information about the code object, like code_info().

Contoh:

>>> bytecode = dis.Bytecode(myfunc)
>>> for instr in bytecode:
...     print(instr.opname)
...
LOAD_GLOBAL
LOAD_FAST
CALL_FUNCTION
RETURN_VALUE

32.12.2. Analysis functions

The dis module also defines the following analysis functions that convert the input directly to the desired output. They can be useful if only a single operation is being performed, so the intermediate analysis object isn't useful:

dis.code_info(x)

Return a formatted multi-line string with detailed code object information for the supplied function, generator, method, source code string or code object.

Note that the exact contents of code info strings are highly implementation dependent and they may change arbitrarily across Python VMs or Python releases.

Baru pada versi 3.2.

dis.show_code(x, *, file=None)

Print detailed code object information for the supplied function, method, source code string or code object to file (or sys.stdout if file is not specified).

This is a convenient shorthand for print(code_info(x), file=file), intended for interactive exploration at the interpreter prompt.

Baru pada versi 3.2.

Berubah pada versi 3.4: Added file parameter.

dis.dis(x=None, *, file=None)

Disassemble the x object. x can denote either a module, a class, a method, a function, a generator, a code object, a string of source code or a byte sequence of raw bytecode. For a module, it disassembles all functions. For a class, it disassembles all methods (including class and static methods). For a code object or sequence of raw bytecode, it prints one line per bytecode instruction. Strings are first compiled to code objects with the compile() built-in function before being disassembled. If no object is provided, this function disassembles the last traceback.

The disassembly is written as text to the supplied file argument if provided and to sys.stdout otherwise.

Berubah pada versi 3.4: Added file parameter.

dis.distb(tb=None, *, file=None)

Disassemble the top-of-stack function of a traceback, using the last traceback if none was passed. The instruction causing the exception is indicated.

The disassembly is written as text to the supplied file argument if provided and to sys.stdout otherwise.

Berubah pada versi 3.4: Added file parameter.

dis.disassemble(code, lasti=-1, *, file=None)

dis.disco(code, lasti=-1, *, file=None)

Disassemble a code object, indicating the last instruction if lasti was provided. The output is divided in the following columns:

1. the line number, for the first instruction of each line
2. the current instruction, indicated as -->,
3. a labelled instruction, indicated with >>,
4. the address of the instruction,
5. the operation code name,
6. operation parameters, and
7. interpretation of the parameters in parentheses.

The parameter interpretation recognizes local and global variable names, constant values, branch targets, and compare operators.

The disassembly is written as text to the supplied file argument if provided and to sys.stdout otherwise.

Berubah pada versi 3.4: Added file parameter.

dis.get_instructions(x, *, first_line=None)

Return an iterator over the instructions in the supplied function, method, source code string or code object.

The iterator generates a series of Instruction named tuples giving the details of each operation in the supplied code.

If first_line is not None, it indicates the line number that should be reported for the first source line in the disassembled code. Otherwise, the source line information (if any) is taken directly from the disassembled code object.

Baru pada versi 3.4.

dis.findlinestarts(code)

This generator function uses the co_firstlineno and co_lnotab attributes of the code object code to find the offsets which are starts of lines in the source code. They are generated as (offset, lineno) pairs.

dis.findlabels(code)

Detect all offsets in the code object code which are jump targets, and return a list of these offsets.

dis.stack_effect(opcode[, oparg])

Compute the stack effect of opcode with argument oparg.

Baru pada versi 3.4.

32.12.3. Python Bytecode Instructions

The get_instructions() function and Bytecode class provide details of bytecode instructions as Instruction instances:

class dis.Instruction

Details for a bytecode operation

opcode

numeric code for operation, corresponding to the opcode values listed below and the bytecode values in the Opcode collections.

opname

human readable name for operation

arg

numeric argument to operation (if any), otherwise None

argval

resolved arg value (if known), otherwise same as arg

argrepr

human readable description of operation argument

offset

start index of operation within bytecode sequence

starts_line

line started by this opcode (if any), otherwise None

is_jump_target

True if other code jumps to here, otherwise False

Baru pada versi 3.4.

The Python compiler currently generates the following bytecode instructions.

General instructions

NOP

Do nothing code. Used as a placeholder by the bytecode optimizer.

POP_TOP

Removes the top-of-stack (TOS) item.

ROT_TWO

Swaps the two top-most stack items.

ROT_THREE

Lifts second and third stack item one position up, moves top down to position three.

DUP_TOP

Duplicates the reference on top of the stack.

DUP_TOP_TWO

Duplicates the two references on top of the stack, leaving them in the same order.

Unary operations

Unary operations take the top of the stack, apply the operation, and push the result back on the stack.

UNARY_POSITIVE

Implements TOS = +TOS.

UNARY_NEGATIVE

Implements TOS = -TOS.

UNARY_NOT

Implements TOS = not TOS.

UNARY_INVERT

Implements TOS = ~TOS.

GET_ITER

Implements TOS = iter(TOS).

GET_YIELD_FROM_ITER

If TOS is a generator iterator or coroutine object it is left as is. Otherwise, implements TOS = iter(TOS).

Baru pada versi 3.5.

Binary operations

Binary operations remove the top of the stack (TOS) and the second top-most stack item (TOS1) from the stack. They perform the operation, and put the result back on the stack.

BINARY_POWER

Implements TOS = TOS1 ** TOS.

BINARY_MULTIPLY

Implements TOS = TOS1 * TOS.

BINARY_MATRIX_MULTIPLY

Implements TOS = TOS1 @ TOS.

Baru pada versi 3.5.

BINARY_FLOOR_DIVIDE

Implements TOS = TOS1 // TOS.

BINARY_TRUE_DIVIDE

Implements TOS = TOS1 / TOS.

BINARY_MODULO

Implements TOS = TOS1 % TOS.

BINARY_ADD

Implements TOS = TOS1 + TOS.

BINARY_SUBTRACT

Implements TOS = TOS1 - TOS.

BINARY_SUBSCR

Implements TOS = TOS1[TOS].

BINARY_LSHIFT

Implements TOS = TOS1 << TOS.

BINARY_RSHIFT

Implements TOS = TOS1 >> TOS.

BINARY_AND

Implements TOS = TOS1 & TOS.

BINARY_XOR

Implements TOS = TOS1 ^ TOS.

BINARY_OR

Implements TOS = TOS1 | TOS.

In-place operations

In-place operations are like binary operations, in that they remove TOS and TOS1, and push the result back on the stack, but the operation is done in-place when TOS1 supports it, and the resulting TOS may be (but does not have to be) the original TOS1.

INPLACE_POWER

Implements in-place TOS = TOS1 ** TOS.

INPLACE_MULTIPLY

Implements in-place TOS = TOS1 * TOS.

INPLACE_MATRIX_MULTIPLY

Implements in-place TOS = TOS1 @ TOS.

Baru pada versi 3.5.

INPLACE_FLOOR_DIVIDE

Implements in-place TOS = TOS1 // TOS.

INPLACE_TRUE_DIVIDE

Implements in-place TOS = TOS1 / TOS.

INPLACE_MODULO

Implements in-place TOS = TOS1 % TOS.

INPLACE_ADD

Implements in-place TOS = TOS1 + TOS.

INPLACE_SUBTRACT

Implements in-place TOS = TOS1 - TOS.

INPLACE_LSHIFT

Implements in-place TOS = TOS1 << TOS.

INPLACE_RSHIFT

Implements in-place TOS = TOS1 >> TOS.

INPLACE_AND

Implements in-place TOS = TOS1 & TOS.

INPLACE_XOR

Implements in-place TOS = TOS1 ^ TOS.

INPLACE_OR

Implements in-place TOS = TOS1 | TOS.

STORE_SUBSCR

Implements TOS1[TOS] = TOS2.

DELETE_SUBSCR

Implements del TOS1[TOS].

Coroutine opcodes

GET_AWAITABLE

Implements TOS = get_awaitable(TOS), where get_awaitable(o) returns o if o is a coroutine object or a generator object with the CO_ITERABLE_COROUTINE flag, or resolves o.__await__.

GET_AITER

Implements TOS = get_awaitable(TOS.__aiter__()). See GET_AWAITABLE for details about get_awaitable

GET_ANEXT

Implements PUSH(get_awaitable(TOS.__anext__())). See GET_AWAITABLE for details about get_awaitable

BEFORE_ASYNC_WITH

Resolves __aenter__ and __aexit__ from the object on top of the stack. Pushes __aexit__ and result of __aenter__() to the stack.

SETUP_ASYNC_WITH

Creates a new frame object.

Miscellaneous opcodes

PRINT_EXPR

Implements the expression statement for the interactive mode. TOS is removed from the stack and printed. In non-interactive mode, an expression statement is terminated with POP_TOP.

BREAK_LOOP

Terminates a loop due to a break statement.

CONTINUE_LOOP(target)

Continues a loop due to a continue statement. target is the address to jump to (which should be a FOR_ITER instruction).

SET_ADD(i)

Calls set.add(TOS1[-i], TOS). Used to implement set comprehensions.

LIST_APPEND(i)

Calls list.append(TOS[-i], TOS). Used to implement list comprehensions.

MAP_ADD(i)

Calls dict.setitem(TOS1[-i], TOS, TOS1). Used to implement dict comprehensions.

For all of the SET_ADD, LIST_APPEND and MAP_ADD instructions, while the added value or key/value pair is popped off, the container object remains on the stack so that it is available for further iterations of the loop.

RETURN_VALUE

Returns with TOS to the caller of the function.

YIELD_VALUE

Pops TOS and yields it from a generator.

YIELD_FROM

Pops TOS and delegates to it as a subiterator from a generator.

Baru pada versi 3.3.

IMPORT_STAR

Loads all symbols not starting with '_' directly from the module TOS to the local namespace. The module is popped after loading all names. This opcode implements from module import *.

POP_BLOCK

Removes one block from the block stack. Per frame, there is a stack of blocks, denoting nested loops, try statements, and such.

POP_EXCEPT

Removes one block from the block stack. The popped block must be an exception handler block, as implicitly created when entering an except handler. In addition to popping extraneous values from the frame stack, the last three popped values are used to restore the exception state.

END_FINALLY

Terminates a finally clause. The interpreter recalls whether the exception has to be re-raised, or whether the function returns, and continues with the outer-next block.

LOAD_BUILD_CLASS

Pushes builtins.__build_class__() onto the stack. It is later called by CALL_FUNCTION to construct a class.

SETUP_WITH(delta)

This opcode performs several operations before a with block starts. First, it loads __exit__() from the context manager and pushes it onto the stack for later use by WITH_CLEANUP. Then, __enter__() is called, and a finally block pointing to delta is pushed. Finally, the result of calling the enter method is pushed onto the stack. The next opcode will either ignore it (POP_TOP), or store it in (a) variable(s) (STORE_FAST, STORE_NAME, or UNPACK_SEQUENCE).

WITH_CLEANUP_START

Cleans up the stack when a with statement block exits. TOS is the context manager's __exit__() bound method. Below TOS are 1--3 values indicating how/why the finally clause was entered:

• SECOND = None
• (SECOND, THIRD) = (WHY_{RETURN,CONTINUE}), retval
• SECOND = WHY_*; no retval below it
• (SECOND, THIRD, FOURTH) = exc_info()

In the last case, TOS(SECOND, THIRD, FOURTH) is called, otherwise TOS(None, None, None). Pushes SECOND and result of the call to the stack.

WITH_CLEANUP_FINISH

Pops exception type and result of 'exit' function call from the stack.

If the stack represents an exception, and the function call returns a 'true' value, this information is "zapped" and replaced with a single WHY_SILENCED to prevent END_FINALLY from re-raising the exception. (But non-local gotos will still be resumed.)

All of the following opcodes expect arguments. An argument is two bytes, with the more significant byte last.

STORE_NAME(namei)

Implements name = TOS. namei is the index of name in the attribute co_names of the code object. The compiler tries to use STORE_FAST or STORE_GLOBAL if possible.

DELETE_NAME(namei)

Implements del name, where namei is the index into co_names attribute of the code object.

UNPACK_SEQUENCE(count)

Unpacks TOS into count individual values, which are put onto the stack right-to-left.

UNPACK_EX(counts)

Implements assignment with a starred target: Unpacks an iterable in TOS into individual values, where the total number of values can be smaller than the number of items in the iterable: one of the new values will be a list of all leftover items.

The low byte of counts is the number of values before the list value, the high byte of counts the number of values after it. The resulting values are put onto the stack right-to-left.

STORE_ATTR(namei)

Implements TOS.name = TOS1, where namei is the index of name in co_names.

DELETE_ATTR(namei)

Implements del TOS.name, using namei as index into co_names.

STORE_GLOBAL(namei)

Works as STORE_NAME, but stores the name as a global.

DELETE_GLOBAL(namei)

Works as DELETE_NAME, but deletes a global name.

LOAD_CONST(consti)

Pushes co_consts[consti] onto the stack.

LOAD_NAME(namei)

Pushes the value associated with co_names[namei] onto the stack.

BUILD_TUPLE(count)

Creates a tuple consuming count items from the stack, and pushes the resulting tuple onto the stack.

BUILD_LIST(count)

Works as BUILD_TUPLE, but creates a list.

BUILD_SET(count)

Works as BUILD_TUPLE, but creates a set.

BUILD_MAP(count)

Pushes a new dictionary object onto the stack. Pops 2 * count items so that the dictionary holds count entries: {..., TOS3: TOS2, TOS1: TOS}.

Berubah pada versi 3.5: The dictionary is created from stack items instead of creating an empty dictionary pre-sized to hold count items.

BUILD_TUPLE_UNPACK(count)

Pops count iterables from the stack, joins them in a single tuple, and pushes the result. Implements iterable unpacking in tuple displays (*x, *y, *z).

Baru pada versi 3.5.

BUILD_LIST_UNPACK(count)

This is similar to BUILD_TUPLE_UNPACK, but pushes a list instead of tuple. Implements iterable unpacking in list displays [*x, *y, *z].

Baru pada versi 3.5.

BUILD_SET_UNPACK(count)

This is similar to BUILD_TUPLE_UNPACK, but pushes a set instead of tuple. Implements iterable unpacking in set displays {*x, *y, *z}.

Baru pada versi 3.5.

BUILD_MAP_UNPACK(count)

Pops count mappings from the stack, merges them into a single dictionary, and pushes the result. Implements dictionary unpacking in dictionary displays {**x, **y, **z}.

Baru pada versi 3.5.

BUILD_MAP_UNPACK_WITH_CALL(oparg)

This is similar to BUILD_MAP_UNPACK, but is used for f(**x, **y, **z) call syntax. The lowest byte of oparg is the count of mappings, the relative position of the corresponding callable f is encoded in the second byte of oparg.

Baru pada versi 3.5.

LOAD_ATTR(namei)

Replaces TOS with getattr(TOS, co_names[namei]).

COMPARE_OP(opname)

Performs a Boolean operation. The operation name can be found in cmp_op[opname].

IMPORT_NAME(namei)

Imports the module co_names[namei]. TOS and TOS1 are popped and provide the fromlist and level arguments of __import__(). The module object is pushed onto the stack. The current namespace is not affected: for a proper import statement, a subsequent STORE_FAST instruction modifies the namespace.

IMPORT_FROM(namei)

Loads the attribute co_names[namei] from the module found in TOS. The resulting object is pushed onto the stack, to be subsequently stored by a STORE_FAST instruction.

JUMP_FORWARD(delta)

Increments bytecode counter by delta.

POP_JUMP_IF_TRUE(target)

If TOS is true, sets the bytecode counter to target. TOS is popped.

POP_JUMP_IF_FALSE(target)

If TOS is false, sets the bytecode counter to target. TOS is popped.

JUMP_IF_TRUE_OR_POP(target)

If TOS is true, sets the bytecode counter to target and leaves TOS on the stack. Otherwise (TOS is false), TOS is popped.

JUMP_IF_FALSE_OR_POP(target)

If TOS is false, sets the bytecode counter to target and leaves TOS on the stack. Otherwise (TOS is true), TOS is popped.

JUMP_ABSOLUTE(target)

Set bytecode counter to target.

FOR_ITER(delta)

TOS is an iterator. Call its __next__() method. If this yields a new value, push it on the stack (leaving the iterator below it). If the iterator indicates it is exhausted TOS is popped, and the byte code counter is incremented by delta.

LOAD_GLOBAL(namei)

Loads the global named co_names[namei] onto the stack.

SETUP_LOOP(delta)

Pushes a block for a loop onto the block stack. The block spans from the current instruction with a size of delta bytes.

SETUP_EXCEPT(delta)

Pushes a try block from a try-except clause onto the block stack. delta points to the first except block.

SETUP_FINALLY(delta)

Pushes a try block from a try-except clause onto the block stack. delta points to the finally block.

LOAD_FAST(var_num)

Pushes a reference to the local co_varnames[var_num] onto the stack.

STORE_FAST(var_num)

Stores TOS into the local co_varnames[var_num].

DELETE_FAST(var_num)

Deletes local co_varnames[var_num].

LOAD_CLOSURE(i)

Pushes a reference to the cell contained in slot i of the cell and free variable storage. The name of the variable is co_cellvars[i] if i is less than the length of co_cellvars. Otherwise it is co_freevars[i - len(co_cellvars)].

LOAD_DEREF(i)

Loads the cell contained in slot i of the cell and free variable storage. Pushes a reference to the object the cell contains on the stack.

LOAD_CLASSDEREF(i)

Much like LOAD_DEREF but first checks the locals dictionary before consulting the cell. This is used for loading free variables in class bodies.

STORE_DEREF(i)

Stores TOS into the cell contained in slot i of the cell and free variable storage.

DELETE_DEREF(i)

Empties the cell contained in slot i of the cell and free variable storage. Used by the del statement.

RAISE_VARARGS(argc)

Raises an exception. argc indicates the number of arguments to the raise statement, ranging from 0 to 3. The handler will find the traceback as TOS2, the parameter as TOS1, and the exception as TOS.

CALL_FUNCTION(argc)

Calls a callable object. The low byte of argc indicates the number of positional arguments, the high byte the number of keyword arguments. The stack contains keyword arguments on top (if any), then the positional arguments below that (if any), then the callable object to call below that. Each keyword argument is represented with two values on the stack: the argument's name, and its value, with the argument's value above the name on the stack. The positional arguments are pushed in the order that they are passed in to the callable object, with the right-most positional argument on top. CALL_FUNCTION pops all arguments and the callable object off the stack, calls the callable object with those arguments, and pushes the return value returned by the callable object.

MAKE_FUNCTION(argc)

Pushes a new function object on the stack. From bottom to top, the consumed stack must consist of

• argc & 0xFF default argument objects in positional order, for positional parameters
• (argc >> 8) & 0xFF pairs of name and default argument, with the name just below the object on the stack, for keyword-only parameters
• (argc >> 16) & 0x7FFF parameter annotation objects
• a tuple listing the parameter names for the annotations (only if there are any annotation objects)
• the code associated with the function (at TOS1)
• the qualified name of the function (at TOS)

MAKE_CLOSURE(argc)

Creates a new function object, sets its __closure__ slot, and pushes it on the stack. TOS is the qualified name of the function, TOS1 is the code associated with the function, and TOS2 is the tuple containing cells for the closure's free variables. argc is interpreted as in MAKE_FUNCTION; the annotations and defaults are also in the same order below TOS2.

BUILD_SLICE(argc)

Pushes a slice object on the stack. argc must be 2 or 3. If it is 2, slice(TOS1, TOS) is pushed; if it is 3, slice(TOS2, TOS1, TOS) is pushed. See the slice() built-in function for more information.

EXTENDED_ARG(ext)

Prefixes any opcode which has an argument too big to fit into the default two bytes. ext holds two additional bytes which, taken together with the subsequent opcode's argument, comprise a four-byte argument, ext being the two most-significant bytes.

CALL_FUNCTION_VAR(argc)

Calls a callable object, similarly to CALL_FUNCTION. argc represents the number of keyword and positional arguments, identically to CALL_FUNCTION. The top of the stack contains keyword arguments (if any), stored identically to CALL_FUNCTION. Below that is an iterable object containing additional positional arguments. Below that are positional arguments (if any) and a callable object, identically to CALL_FUNCTION. Before the callable object is called, the iterable object is "unpacked" and its contents are appended to the positional arguments passed in. The iterable object is ignored when computing the value of argc.

Berubah pada versi 3.5: In versions 3.0 to 3.4, the iterable object was above the keyword arguments; in 3.5 the iterable object was moved below the keyword arguments.

CALL_FUNCTION_KW(argc)

Calls a callable object, similarly to CALL_FUNCTION. argc represents the number of keyword and positional arguments, identically to CALL_FUNCTION. The top of the stack contains a mapping object containing additional keyword arguments. Below this are keyword arguments (if any), positional arguments (if any), and a callable object, identically to CALL_FUNCTION. Before the callable is called, the mapping object at the top of the stack is "unpacked" and its contents are appended to the keyword arguments passed in. The mapping object at the top of the stack is ignored when computing the value of argc.

CALL_FUNCTION_VAR_KW(argc)

Calls a callable object, similarly to CALL_FUNCTION_VAR and CALL_FUNCTION_KW. argc represents the number of keyword and positional arguments, identically to CALL_FUNCTION. The top of the stack contains a mapping object, as per CALL_FUNCTION_KW. Below that are keyword arguments (if any), stored identically to CALL_FUNCTION. Below that is an iterable object containing additional positional arguments. Below that are positional arguments (if any) and a callable object, identically to CALL_FUNCTION. Before the callable is called, the mapping object and iterable object are each "unpacked" and their contents passed in as keyword and positional arguments respectively, identically to CALL_FUNCTION_VAR and CALL_FUNCTION_KW. The mapping object and iterable object are both ignored when computing the value of argc.

Berubah pada versi 3.5: In versions 3.0 to 3.4, the iterable object was above the keyword arguments; in 3.5 the iterable object was moved below the keyword arguments.

HAVE_ARGUMENT

This is not really an opcode. It identifies the dividing line between opcodes which don't take arguments < HAVE_ARGUMENT and those which do >= HAVE_ARGUMENT.

32.12.4. Opcode collections

These collections are provided for automatic introspection of bytecode instructions:

dis.opname

Sequence of operation names, indexable using the bytecode.

dis.opmap

Dictionary mapping operation names to bytecodes.

dis.cmp_op

Sequence of all compare operation names.

dis.hasconst

Sequence of bytecodes that access a constant.

dis.hasfree

Sequence of bytecodes that access a free variable (note that 'free' in this context refers to names in the current scope that are referenced by inner scopes or names in outer scopes that are referenced from this scope. It does not include references to global or builtin scopes).

dis.hasname

Sequence of bytecodes that access an attribute by name.

dis.hasjrel

Sequence of bytecodes that have a relative jump target.

dis.hasjabs

Sequence of bytecodes that have an absolute jump target.

dis.haslocal

Sequence of bytecodes that access a local variable.

dis.hascompare

Sequence of bytecodes of Boolean operations.