linux/Documentation/kbuild/kconfig-language.rst

================
Kconfig Language
================

Introduction
------------

The configuration database is a collection of configuration options
organized in a tree structure::

	+- Code maturity level options
	|  +- Prompt for development and/or incomplete code/drivers
	+- General setup
	|  +- Networking support
	|  +- System V IPC
	|  +- BSD Process Accounting
	|  +- Sysctl support
	+- Loadable module support
	|  +- Enable loadable module support
	|     +- Set version information on all module symbols
	|     +- Kernel module loader
	+- ...

Every entry has its own dependencies. These dependencies are used
to determine the visibility of an entry. Any child entry is only
visible if its parent entry is also visible.

Menu entries
------------

Most entries define a config option; all other entries help to organize
them. A single configuration option is defined like this::

  config MODVERSIONS
	bool "Set version information on all module symbols"
	depends on MODULES
	help
	  Usually, modules have to be recompiled whenever you switch to a new
	  kernel.  ...

Every line starts with a key word and can be followed by multiple
arguments.  "config" starts a new config entry. The following lines
define attributes for this config option. Attributes can be the type of
the config option, input prompt, dependencies, help text and default
values. A config option can be defined multiple times with the same
name, but every definition can have only a single input prompt and the
type must not conflict.

Menu attributes
---------------

A menu entry can have a number of attributes. Not all of them are
applicable everywhere (see syntax).

- type definition: "bool"/"tristate"/"string"/"hex"/"int"

  Every config option must have a type. There are only two basic types:
  tristate and string; the other types are based on these two. The type
  definition optionally accepts an input prompt, so these two examples
  are equivalent::

	bool "Networking support"

  and::

	bool
	prompt "Networking support"

- input prompt: "prompt" <prompt> ["if" <expr>]

  Every menu entry can have at most one prompt, which is used to display
  to the user. Optionally dependencies only for this prompt can be added
  with "if".

- default value: "default" <expr> ["if" <expr>]

  A config option can have any number of default values. If multiple
  default values are visible, only the first defined one is active.
  Default values are not limited to the menu entry where they are
  defined. This means the default can be defined somewhere else or be
  overridden by an earlier definition.
  The default value is only assigned to the config symbol if no other
  value was set by the user (via the input prompt above). If an input
  prompt is visible the default value is presented to the user and can
  be overridden by him.
  Optionally, dependencies only for this default value can be added with
  "if".

 The default value deliberately defaults to 'n' in order to avoid bloating the
 build. With few exceptions, new config options should not change this. The
 intent is for "make oldconfig" to add as little as possible to the config from
 release to release.

 Note:
	Things that merit "default y/m" include:

	a) A new Kconfig option for something that used to always be built
	   should be "default y".

	b) A new gatekeeping Kconfig option that hides/shows other Kconfig
	   options (but does not generate any code of its own), should be
	   "default y" so people will see those other options.

	c) Sub-driver behavior or similar options for a driver that is
	   "default n". This allows you to provide sane defaults.

	d) Hardware or infrastructure that everybody expects, such as CONFIG_NET
	   or CONFIG_BLOCK. These are rare exceptions.

- type definition + default value::

	"def_bool"/"def_tristate" <expr> ["if" <expr>]

  This is a shorthand notation for a type definition plus a value.
  Optionally dependencies for this default value can be added with "if".

- dependencies: "depends on" <expr>

  This defines a dependency for this menu entry. If multiple
  dependencies are defined, they are connected with '&&'. Dependencies
  are applied to all other options within this menu entry (which also
  accept an "if" expression), so these two examples are equivalent::

	bool "foo" if BAR
	default y if BAR

  and::

	depends on BAR
	bool "foo"
	default y

- reverse dependencies: "select" <symbol> ["if" <expr>]

  While normal dependencies reduce the upper limit of a symbol (see
  below), reverse dependencies can be used to force a lower limit of
  another symbol. The value of the current menu symbol is used as the
  minimal value <symbol> can be set to. If <symbol> is selected multiple
  times, the limit is set to the largest selection.
  Reverse dependencies can only be used with boolean or tristate
  symbols.

  Note:
	select should be used with care. select will force
	a symbol to a value without visiting the dependencies.
	By abusing select you are able to select a symbol FOO even
	if FOO depends on BAR that is not set.
	In general use select only for non-visible symbols
	(no prompts anywhere) and for symbols with no dependencies.
	That will limit the usefulness but on the other hand avoid
	the illegal configurations all over.

- weak reverse dependencies: "imply" <symbol> ["if" <expr>]

  This is similar to "select" as it enforces a lower limit on another
  symbol except that the "implied" symbol's value may still be set to n
  from a direct dependency or with a visible prompt.

  Given the following example::

    config FOO
	tristate "foo"
	imply BAZ

    config BAZ
	tristate "baz"
	depends on BAR

  The following values are possible:

	===		===		=============	==============
	FOO		BAR		BAZ's default	choice for BAZ
	===		===		=============	==============
	n		y		n		N/m/y
	m		y		m		M/y/n
	y		y		y		Y/m/n
	n		m		n		N/m
	m		m		m		M/n
	y		m		m		M/n
	y		n		*		N
	===		===		=============	==============

  This is useful e.g. with multiple drivers that want to indicate their
  ability to hook into a secondary subsystem while allowing the user to
  configure that subsystem out without also having to unset these drivers.

  Note: If the combination of FOO=y and BAR=m causes a link error,
  you can guard the function call with IS_REACHABLE()::

	foo_init()
	{
		if (IS_REACHABLE(CONFIG_BAZ))
			baz_register(&foo);
		...
	}

  Note: If the feature provided by BAZ is highly desirable for FOO,
  FOO should imply not only BAZ, but also its dependency BAR::

    config FOO
	tristate "foo"
	imply BAR
	imply BAZ

- limiting menu display: "visible if" <expr>

  This attribute is only applicable to menu blocks, if the condition is
  false, the menu block is not displayed to the user (the symbols
  contained there can still be selected by other symbols, though). It is
  similar to a conditional "prompt" attribute for individual menu
  entries. Default value of "visible" is true.

- numerical ranges: "range" <symbol> <symbol> ["if" <expr>]

  This allows to limit the range of possible input values for int
  and hex symbols. The user can only input a value which is larger than
  or equal to the first symbol and smaller than or equal to the second
  symbol.

- help text: "help"

  This defines a help text. The end of the help text is determined by
  the indentation level, this means it ends at the first line which has
  a smaller indentation than the first line of the help text.

- module attribute: "modules"
  This declares the symbol to be used as the MODULES symbol, which
  enables the third modular state for all config symbols.
  At most one symbol may have the "modules" option set.

Menu dependencies
-----------------

Dependencies define the visibility of a menu entry and can also reduce
the input range of tristate symbols. The tristate logic used in the
expressions uses one more state than normal boolean logic to express the
module state. Dependency expressions have the following syntax::

  <expr> ::= <symbol>                           (1)
           <symbol> '=' <symbol>                (2)
           <symbol> '!=' <symbol>               (3)
           <symbol1> '<' <symbol2>              (4)
           <symbol1> '>' <symbol2>              (4)
           <symbol1> '<=' <symbol2>             (4)
           <symbol1> '>=' <symbol2>             (4)
           '(' <expr> ')'                       (5)
           '!' <expr>                           (6)
           <expr> '&&' <expr>                   (7)
           <expr> '||' <expr>                   (8)

Expressions are listed in decreasing order of precedence.

(1) Convert the symbol into an expression. Boolean and tristate symbols
    are simply converted into the respective expression values. All
    other symbol types result in 'n'.
(2) If the values of both symbols are equal, it returns 'y',
    otherwise 'n'.
(3) If the values of both symbols are equal, it returns 'n',
    otherwise 'y'.
(4) If value of <symbol1> is respectively lower, greater, lower-or-equal,
    or greater-or-equal than value of <symbol2>, it returns 'y',
    otherwise 'n'.
(5) Returns the value of the expression. Used to override precedence.
(6) Returns the result of (2-/expr/).
(7) Returns the result of min(/expr/, /expr/).
(8) Returns the result of max(/expr/, /expr/).

An expression can have a value of 'n', 'm' or 'y' (or 0, 1, 2
respectively for calculations). A menu entry becomes visible when its
expression evaluates to 'm' or 'y'.

There are two types of symbols: constant and non-constant symbols.
Non-constant symbols are the most common ones and are defined with the
'config' statement. Non-constant symbols consist entirely of alphanumeric
characters or underscores.
Constant symbols are only part of expressions. Constant symbols are
always surrounded by single or double quotes. Within the quote, any
other character is allowed and the quotes can be escaped using '\'.

Menu structure
--------------

The position of a menu entry in the tree is determined in two ways. First
it can be specified explicitly::

  menu "Network device support"
	depends on NET

  config NETDEVICES
	...

  endmenu

All entries within the "menu" ... "endmenu" block become a submenu of
"Network device support". All subentries inherit the dependencies from
the menu entry, e.g. this means the dependency "NET" is added to the
dependency list of the config option NETDEVICES.

The other way to generate the menu structure is done by analyzing the
dependencies. If a menu entry somehow depends on the previous entry, it
can be made a submenu of it. First, the previous (parent) symbol must
be part of the dependency list and then one of these two conditions
must be true:

- the child entry must become invisible, if the parent is set to 'n'
- the child entry must only be visible, if the parent is visible::

    config MODULES
	bool "Enable loadable module support"

    config MODVERSIONS
	bool "Set version information on all module symbols"
	depends on MODULES

    comment "module support disabled"
	depends on !MODULES

MODVERSIONS directly depends on MODULES, this means it's only visible if
MODULES is different from 'n'. The comment on the other hand is only
visible when MODULES is set to 'n'.


Kconfig syntax
--------------

The configuration file describes a series of menu entries, where every
line starts with a keyword (except help texts). The following keywords
end a menu entry:

- config
- menuconfig
- choice/endchoice
- comment
- menu/endmenu
- if/endif
- source

The first five also start the definition of a menu entry.

config::

	"config" <symbol>
	<config options>

This defines a config symbol <symbol> and accepts any of above
attributes as options.

menuconfig::

	"menuconfig" <symbol>
	<config options>

This is similar to the simple config entry above, but it also gives a
hint to front ends, that all suboptions should be displayed as a
separate list of options. To make sure all the suboptions will really
show up under the menuconfig entry and not outside of it, every item
from the <config options> list must depend on the menuconfig symbol.
In practice, this is achieved by using one of the next two constructs::

  (1):
  menuconfig M
  if M
      config C1
      config C2
  endif

  (2):
  menuconfig M
  config C1
      depends on M
  config C2
      depends on M

In the following examples (3) and (4), C1 and C2 still have the M
dependency, but will not appear under menuconfig M anymore, because
of C0, which doesn't depend on M::

  (3):
  menuconfig M
      config C0
  if M
      config C1
      config C2
  endif

  (4):
  menuconfig M
  config C0
  config C1
      depends on M
  config C2
      depends on M

choices::

	"choice" [symbol]
	<choice options>
	<choice block>
	"endchoice"

This defines a choice group and accepts any of the above attributes as
options. A choice can only be of type bool or tristate.  If no type is
specified for a choice, its type will be determined by the type of
the first choice element in the group or remain unknown if none of the
choice elements have a type specified, as well.

While a boolean choice only allows a single config entry to be
selected, a tristate choice also allows any number of config entries
to be set to 'm'. This can be used if multiple drivers for a single
hardware exists and only a single driver can be compiled/loaded into
the kernel, but all drivers can be compiled as modules.

A choice accepts another option "optional", which allows to set the
choice to 'n' and no entry needs to be selected.
If no [symbol] is associated with a choice, then you can not have multiple
definitions of that choice. If a [symbol] is associated to the choice,
then you may define the same choice (i.e. with the same entries) in another
place.

comment::

	"comment" <prompt>
	<comment options>

This defines a comment which is displayed to the user during the
configuration process and is also echoed to the output files. The only
possible options are dependencies.

menu::

	"menu" <prompt>
	<menu options>
	<menu block>
	"endmenu"

This defines a menu block, see "Menu structure" above for more
information. The only possible options are dependencies and "visible"
attributes.

if::

	"if" <expr>
	<if block>
	"endif"

This defines an if block. The dependency expression <expr> is appended
to all enclosed menu entries.

source::

	"source" <prompt>

This reads the specified configuration file. This file is always parsed.

mainmenu::

	"mainmenu" <prompt>

This sets the config program's title bar if the config program chooses
to use it. It should be placed at the top of the configuration, before any
other statement.

'#' Kconfig source file comment:

An unquoted '#' character anywhere in a source file line indicates
the beginning of a source file comment.  The remainder of that line
is a comment.


Kconfig hints
-------------
This is a collection of Kconfig tips, most of which aren't obvious at
first glance and most of which have become idioms in several Kconfig
files.

Adding common features and make the usage configurable
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
It is a common idiom to implement a feature/functionality that are
relevant for some architectures but not all.
The recommended way to do so is to use a config variable named HAVE_*
that is defined in a common Kconfig file and selected by the relevant
architectures.
An example is the generic IOMAP functionality.

We would in lib/Kconfig see::

  # Generic IOMAP is used to ...
  config HAVE_GENERIC_IOMAP

  config GENERIC_IOMAP
	depends on HAVE_GENERIC_IOMAP && FOO

And in lib/Makefile we would see::

	obj-$(CONFIG_GENERIC_IOMAP) += iomap.o

For each architecture using the generic IOMAP functionality we would see::

  config X86
	select ...
	select HAVE_GENERIC_IOMAP
	select ...

Note: we use the existing config option and avoid creating a new
config variable to select HAVE_GENERIC_IOMAP.

Note: the use of the internal config variable HAVE_GENERIC_IOMAP, it is
introduced to overcome the limitation of select which will force a
config option to 'y' no matter the dependencies.
The dependencies are moved to the symbol GENERIC_IOMAP and we avoid the
situation where select forces a symbol equals to 'y'.

Adding features that need compiler support
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

There are several features that need compiler support. The recommended way
to describe the dependency on the compiler feature is to use "depends on"
followed by a test macro::

  config STACKPROTECTOR
	bool "Stack Protector buffer overflow detection"
	depends on $(cc-option,-fstack-protector)
	...

If you need to expose a compiler capability to makefiles and/or C source files,
`CC_HAS_` is the recommended prefix for the config option::

  config CC_HAS_FOO
	def_bool $(success,$(srctree)/scripts/cc-check-foo.sh $(CC))

Build as module only
~~~~~~~~~~~~~~~~~~~~
To restrict a component build to module-only, qualify its config symbol
with "depends on m".  E.g.::

  config FOO
	depends on BAR && m

limits FOO to module (=m) or disabled (=n).

Compile-testing
~~~~~~~~~~~~~~~
If a config symbol has a dependency, but the code controlled by the config
symbol can still be compiled if the dependency is not met, it is encouraged to
increase build coverage by adding an "|| COMPILE_TEST" clause to the
dependency. This is especially useful for drivers for more exotic hardware, as
it allows continuous-integration systems to compile-test the code on a more
common system, and detect bugs that way.
Note that compile-tested code should avoid crashing when run on a system where
the dependency is not met.

Architecture and platform dependencies
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Due to the presence of stubs, most drivers can now be compiled on most
architectures. However, this does not mean it makes sense to have all drivers
available everywhere, as the actual hardware may only exist on specific
architectures and platforms. This is especially true for on-SoC IP cores,
which may be limited to a specific vendor or SoC family.

To prevent asking the user about drivers that cannot be used on the system(s)
the user is compiling a kernel for, and if it makes sense, config symbols
controlling the compilation of a driver should contain proper dependencies,
limiting the visibility of the symbol to (a superset of) the platform(s) the
driver can be used on. The dependency can be an architecture (e.g. ARM) or
platform (e.g. ARCH_OMAP4) dependency. This makes life simpler not only for
distro config owners, but also for every single developer or user who
configures a kernel.

Such a dependency can be relaxed by combining it with the compile-testing rule
above, leading to:

  config FOO
	bool "Support for foo hardware"
	depends on ARCH_FOO_VENDOR || COMPILE_TEST

Optional dependencies
~~~~~~~~~~~~~~~~~~~~~

Some drivers are able to optionally use a feature from another module
or build cleanly with that module disabled, but cause a link failure
when trying to use that loadable module from a built-in driver.

The most common way to express this optional dependency in Kconfig logic
uses the slightly counterintuitive::

  config FOO
	tristate "Support for foo hardware"
	depends on BAR || !BAR

This means that there is either a dependency on BAR that disallows
the combination of FOO=y with BAR=m, or BAR is completely disabled.
For a more formalized approach if there are multiple drivers that have
the same dependency, a helper symbol can be used, like::

  config FOO
	tristate "Support for foo hardware"
	depends on BAR_OPTIONAL

  config BAR_OPTIONAL
	def_tristate BAR || !BAR

Kconfig recursive dependency limitations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If you've hit the Kconfig error: "recursive dependency detected" you've run
into a recursive dependency issue with Kconfig, a recursive dependency can be
summarized as a circular dependency. The kconfig tools need to ensure that
Kconfig files comply with specified configuration requirements. In order to do
that kconfig must determine the values that are possible for all Kconfig
symbols, this is currently not possible if there is a circular relation
between two or more Kconfig symbols. For more details refer to the "Simple
Kconfig recursive issue" subsection below. Kconfig does not do recursive
dependency resolution; this has a few implications for Kconfig file writers.
We'll first explain why this issues exists and then provide an example
technical limitation which this brings upon Kconfig developers. Eager
developers wishing to try to address this limitation should read the next
subsections.

Simple Kconfig recursive issue
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Read: Documentation/kbuild/Kconfig.recursion-issue-01

Test with::

  make KBUILD_KCONFIG=Documentation/kbuild/Kconfig.recursion-issue-01 allnoconfig

Cumulative Kconfig recursive issue
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Read: Documentation/kbuild/Kconfig.recursion-issue-02

Test with::

  make KBUILD_KCONFIG=Documentation/kbuild/Kconfig.recursion-issue-02 allnoconfig

Practical solutions to kconfig recursive issue
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Developers who run into the recursive Kconfig issue have two options
at their disposal. We document them below and also provide a list of
historical issues resolved through these different solutions.

  a) Remove any superfluous "select FOO" or "depends on FOO"
  b) Match dependency semantics:

	b1) Swap all "select FOO" to "depends on FOO" or,

	b2) Swap all "depends on FOO" to "select FOO"

The resolution to a) can be tested with the sample Kconfig file
Documentation/kbuild/Kconfig.recursion-issue-01 through the removal
of the "select CORE" from CORE_BELL_A_ADVANCED as that is implicit already
since CORE_BELL_A depends on CORE. At times it may not be possible to remove
some dependency criteria, for such cases you can work with solution b).

The two different resolutions for b) can be tested in the sample Kconfig file
Documentation/kbuild/Kconfig.recursion-issue-02.

Below is a list of examples of prior fixes for these types of recursive issues;
all errors appear to involve one or more "select" statements and one or more
"depends on".

============    ===================================
commit          fix
============    ===================================
06b718c01208    select A -> depends on A
c22eacfe82f9    depends on A -> depends on B
6a91e854442c    select A -> depends on A
118c565a8f2e    select A -> select B
f004e5594705    select A -> depends on A
c7861f37b4c6    depends on A -> (null)
80c69915e5fb    select A -> (null)              (1)
c2218e26c0d0    select A -> depends on A        (1)
d6ae99d04e1c    select A -> depends on A
95ca19cf8cbf    select A -> depends on A
8f057d7bca54    depends on A -> (null)
8f057d7bca54    depends on A -> select A
a0701f04846e    select A -> depends on A
0c8b92f7f259    depends on A -> (null)
e4e9e0540928    select A -> depends on A        (2)
7453ea886e87    depends on A > (null)           (1)
7b1fff7e4fdf    select A -> depends on A
86c747d2a4f0    select A -> depends on A
d9f9ab51e55e    select A -> depends on A
0c51a4d8abd6    depends on A -> select A        (3)
e98062ed6dc4    select A -> depends on A        (3)
91e5d284a7f1    select A -> (null)
============    ===================================

(1) Partial (or no) quote of error.
(2) That seems to be the gist of that fix.
(3) Same error.

Future kconfig work
~~~~~~~~~~~~~~~~~~~

Work on kconfig is welcomed on both areas of clarifying semantics and on
evaluating the use of a full SAT solver for it. A full SAT solver can be
desirable to enable more complex dependency mappings and / or queries,
for instance one possible use case for a SAT solver could be that of handling
the current known recursive dependency issues. It is not known if this would
address such issues but such evaluation is desirable. If support for a full SAT
solver proves too complex or that it cannot address recursive dependency issues
Kconfig should have at least clear and well defined semantics which also
addresses and documents limitations or requirements such as the ones dealing
with recursive dependencies.

Further work on both of these areas is welcomed on Kconfig. We elaborate
on both of these in the next two subsections.

Semantics of Kconfig
~~~~~~~~~~~~~~~~~~~~

The use of Kconfig is broad, Linux is now only one of Kconfig's users:
one study has completed a broad analysis of Kconfig use in 12 projects [0]_.
Despite its widespread use, and although this document does a reasonable job
in documenting basic Kconfig syntax a more precise definition of Kconfig
semantics is welcomed. One project deduced Kconfig semantics through
the use of the xconfig configurator [1]_. Work should be done to confirm if
the deduced semantics matches our intended Kconfig design goals.
Another project formalized a denotational semantics of a core subset of
the Kconfig language [10]_.

Having well defined semantics can be useful for tools for practical
evaluation of dependencies, for instance one such case was work to
express in boolean abstraction of the inferred semantics of Kconfig to
translate Kconfig logic into boolean formulas and run a SAT solver on this to
find dead code / features (always inactive), 114 dead features were found in
Linux using this methodology [1]_ (Section 8: Threats to validity).
The kismet tool, based on the semantics in [10]_, finds abuses of reverse
dependencies and has led to dozens of committed fixes to Linux Kconfig files [11]_.

Confirming this could prove useful as Kconfig stands as one of the leading
industrial variability modeling languages [1]_ [2]_. Its study would help
evaluate practical uses of such languages, their use was only theoretical
and real world requirements were not well understood. As it stands though
only reverse engineering techniques have been used to deduce semantics from
variability modeling languages such as Kconfig [3]_.

.. [0] https://www.eng.uwaterloo.ca/~shshe/kconfig_semantics.pdf
.. [1] https://gsd.uwaterloo.ca/sites/default/files/vm-2013-berger.pdf
.. [2] https://gsd.uwaterloo.ca/sites/default/files/ase241-berger_0.pdf
.. [3] https://gsd.uwaterloo.ca/sites/default/files/icse2011.pdf

Full SAT solver for Kconfig
~~~~~~~~~~~~~~~~~~~~~~~~~~~

Although SAT solvers [4]_ haven't yet been used by Kconfig directly, as noted
in the previous subsection, work has been done however to express in boolean
abstraction the inferred semantics of Kconfig to translate Kconfig logic into
boolean formulas and run a SAT solver on it [5]_. Another known related project
is CADOS [6]_ (former VAMOS [7]_) and the tools, mainly undertaker [8]_, which
has been introduced first with [9]_.  The basic concept of undertaker is to
extract variability models from Kconfig and put them together with a
propositional formula extracted from CPP #ifdefs and build-rules into a SAT
solver in order to find dead code, dead files, and dead symbols. If using a SAT
solver is desirable on Kconfig one approach would be to evaluate repurposing
such efforts somehow on Kconfig. There is enough interest from mentors of
existing projects to not only help advise how to integrate this work upstream
but also help maintain it long term. Interested developers should visit:

https://kernelnewbies.org/KernelProjects/kconfig-sat

.. [4] https://www.cs.cornell.edu/~sabhar/chapters/SATSolvers-KR-Handbook.pdf
.. [5] https://gsd.uwaterloo.ca/sites/default/files/vm-2013-berger.pdf
.. [6] https://cados.cs.fau.de
.. [7] https://vamos.cs.fau.de
.. [8] https://undertaker.cs.fau.de
.. [9] https://www4.cs.fau.de/Publications/2011/tartler_11_eurosys.pdf
.. [10] https://paulgazzillo.com/papers/esecfse21.pdf
.. [11] https://github.com/paulgazz/kmax