HandWiki - Recent changes [en]

Engineering:Voltage and frequency scaling

2026-08-22T14:28:18Z

← Older revision		Revision as of 09:28, 22 August 2026
Line 1:		Line 1:
	{{for\|the CPU design principle\|Frequency scaling}}		{{for\|the CPU design principle\|Frequency scaling}}
	~~{{redirect\|CPU throttling\|other uses\|Throttle (disambiguation)#Computing}}~~
	'''Dynamic frequency scaling''' and '''dynamic voltage scaling''' are techniques used primarily for [[Power management\|power management]] in [[Computer architecture\|computer architecture]]. Dynamic frequency scaling almost always appears in conjunction with [[Physics:Dynamic voltage scaling\|dynamic voltage scaling]], since higher frequencies require higher supply voltages for the digital circuit to yield correct results. The combined topic is known as '''dynamic voltage and frequency scaling''' ('''DVFS''').		'''Dynamic frequency scaling''' and '''dynamic voltage scaling''' are techniques used primarily for [[Power management\|power management]] in [[Computer architecture\|computer architecture]]. Dynamic frequency scaling almost always appears in conjunction with [[Physics:Dynamic voltage scaling\|dynamic voltage scaling]], since higher frequencies require higher supply voltages for the digital circuit to yield correct results. The combined topic is known as '''dynamic voltage and frequency scaling''' ('''DVFS''').

Engineering:Voltage and frequency scaling

2026-08-22T14:27:54Z

import

New page

{{for|the CPU design principle|Frequency scaling}}
{{redirect|CPU throttling|other uses|Throttle (disambiguation)#Computing}}
'''Dynamic frequency scaling''' and '''dynamic voltage scaling''' are techniques used primarily for [[Power management|power management]] in [[Computer architecture|computer architecture]]. Dynamic frequency scaling almost always appears in conjunction with [[Physics:Dynamic voltage scaling|dynamic voltage scaling]], since higher frequencies require higher supply voltages for the digital circuit to yield correct results. The combined topic is known as '''dynamic voltage and frequency scaling''' ('''DVFS''').

==Dynamic frequency scaling==
'''Dynamic frequency scaling''' (also known as '''CPU throttling''') is a power management technique in computer architecture whereby the [[Clock rate|frequency]] of a microprocessor can be automatically adjusted "on the fly" depending on the actual needs, to [[Engineering:Power management integrated circuit|conserve power]] and reduce the amount of heat generated by the chip. Dynamic frequency scaling helps preserve battery on mobile devices and decrease cooling cost and noise on [[Engineering:Quiet PC|quiet computing settings]], or can be useful as a security measure for overheated systems (e.g. after poor [[Engineering:Overclocking|overclocking]]).

=== Operation ===
The dynamic power (''switching power'') dissipated by a chip is ''C·V2·A·f'', where C is the [[Physics:Capacitance|capacitance]] being switched per clock cycle, V is [[Physics:Voltage|voltage]], A is the ''activity factor''<ref name="ActivityFactor">{{cite journal | title = Timing-aware power-optimal ordering of signals | author = K. Moiseev, A. Kolodny and S. Wimer | journal = ACM Transactions on Design Automation of Electronic Systems |volume=13 |issue=4 |date=September 2008| pages = 1–17 | doi = 10.1145/1391962.1391973 | s2cid = 18895687 }}</ref> indicating the average number of switching events per clock cycle by the transistors in the chip (as a unitless quantity) and f is the clock frequency.<ref>{{Cite book|first=J. M.|last= Rabaey|title= Digital Integrated Circuits|publisher= Prentice Hall|year= 1996}}</ref>

Voltage is therefore the main determinant of power usage and heating.<ref>{{cite web|url=https://software.intel.com/en-us/blogs/2014/02/19/why-has-cpu-frequency-ceased-to-grow|author= Victoria Zhislina|date=2014-02-19|title=Why has CPU frequency ceased to grow?|publisher=Intel}}</ref> The voltage required for stable operation is determined by the frequency at which the circuit is clocked, and can be reduced if the frequency is also reduced.<ref>{{Cite web |last=Le Sueur |first=Etienne |last2=Heiser |first2=Gernot |title=Dynamic Voltage and Frequency Scaling: The Laws of Diminishing Returns |url=https://www.usenix.org/legacy/events/hotpower/tech/full_papers/LeSueur.pdf |access-date=28 Oct 2025 |website=usenix.org}}</ref> Dynamic power alone does not account for the total power of the chip, however, as there is also static power, which is primarily because of various leakage currents. Due to static power consumption and asymptotic execution time it has been shown that the energy consumption of software shows convex energy behavior, i.e., there exists an optimal CPU frequency at which energy consumption is minimized.<ref>{{cite arXiv | title = The Energy/Frequency Convexity Rule: Modeling and Experimental Validation on Mobile Devices |year=2014 | eprint = 1401.4655|author1=Karel De Vogeleer |last2=Memmi |first2=Gerard |last3=Jouvelot |first3=Pierre |last4=Coelho |first4=Fabien |class=cs.OH }}</ref>
Leakage current has become more and more important as transistor sizes have become smaller and threshold voltage levels are reduced. A decade ago, dynamic power accounted for approximately two-thirds of the total chip power. The power loss due to leakage currents in contemporary CPUs and SoCs tend to dominate the total power consumption. In the attempt to control the leakage power, [[Engineering:High-κ dielectric|high-k metal-gates]] and power gating have been common methods.

Dynamic voltage scaling is another related power conservation technique that is often used in conjunction with frequency scaling, as the frequency that a chip may run at is related to the operating voltage.

The efficiency of some electrical components, such as voltage regulators, decreases with increasing temperature, so the power usage may increase with temperature. Since increasing power use may increase the temperature, increases in voltage or frequency may increase system power demands even further than the CMOS formula indicates, and vice versa.<ref>{{cite web | url = http://www.silentpcreview.com/article821-page5.html | title = Asus EN9600GT Silent Edition Graphics Card | author = Mike Chin | page = 5 | work = Silent PC Review | access-date = 21 April 2008}}</ref><ref name="SPCRNewLevels">{{cite web | url = http://www.silentpcreview.com/article814-page1.html | title = 80 Plus expands podium for Bronze, Silver & Gold | author = Mike Chin | work = Silent PC Review | date = 19 March 2008 | access-date = 21 April 2008 }}</ref>

=== Standard interface ===

[[ACPI]] 1.0 (1996) defines a way for a CPU to go to idle "C states", but defines no frequency-scaling system.

ACPI 2.0 (2000) introduces a system of ''P states'' (power-performance states) that a processor can use to communicate its possible frequency–power settings to the OS. The operating system then sets the speed as needed by switching between these states. Throttling technology such as SpeedStep, PowerNow!/Cool'n'Quiet, and PowerSaver all work through P states. There is a limit of 16 states maximum.<ref>{{cite web | url = http://www.acpi.info/DOWNLOADS/ACPIspec30.pdf | title = Advanced Configuration and Power Interface Specification, Revision 3.0, Section 2.6 Device and Processor Performance State Definitions | date = 2004-09-02 | website = ACPI.info | page = 23 | archive-date = November 28, 2015 | archive-url = https://web.archive.org/web/20151128143452/http://www.acpi.info/DOWNLOADS/ACPIspec30.pdf }}</ref>

ACPI 5.0 (2011) introduces '''collaborative processor performance control''' ('''CPPC'''), exposing hundreds of performance levels to the OS for selection in the form of a "performance level" abstracted away from the frequency. This abstraction provides leeway for the processor to adjust its workings (C-states) in ways other than just the frequency (P-states).<ref>{{cite web |title=Collaborative Processor Performance Control (CPPC) — The Linux Kernel documentation |url=https://www.kernel.org/doc/html/latest/admin-guide/acpi/cppc_sysfs.html |website=www.kernel.org }}</ref><ref>{{cite web |title=8.4. Declaring Processors |url=https://uefi.org/htmlspecs/ACPI_Spec_6_4_html/08_Processor_Configuration_and_Control/declaring-processors.html#collaborative-processor-performance-control |website=ACPI Specification 6.4 documentation}}</ref><ref>{{cite web |title=Overview about power and performance tuning for the Windows Server |url=https://learn.microsoft.com/en-us/windows-server/administration/performance-tuning/hardware/power/power-performance-tuning |website=learn.microsoft.com |date=29 August 2022 }}</ref>

=== Autonomous frequency scaling ===

A number of modern CPUs can perform frequency scaling autonomously, using a performance level range and a "efficiency/performance preference" hint from the OS.

* Intel CPUs starting with [[Skylake (microarchitecture)|Skylake]] support ''hardware-managed P-states'' aka ''Speed Shift'', It based on CPPC protocol, and it using [[Model-specific register|model-specific register]] as the control channel.<ref>{{man|8|x86_energy_perf_policy|Linux}}</ref><ref>{{cite web |title=intel_pstate CPU Performance Scaling Driver — The Linux Kernel documentation |url=https://www.kernel.org/doc/html/v4.19/admin-guide/pm/intel_pstate.html |website=www.kernel.org}}</ref>
* AMD CPUs starting with [[Engineering:Zen 2|Zen 2]] supports a similar feature. It depends on CPPC being enabled. The preferred communication channel is a MSR (different from the Intel one) introduced in Zen 3; Zen 2 units use the ACPI AML method.<ref>{{cite web |title=amd-pstate CPU Performance Scaling Driver — The Linux Kernel documentation |url=https://docs.kernel.org/admin-guide/pm/amd-pstate.html |website=docs.kernel.org}}</ref>

===Performance impact===
Dynamic frequency scaling reduces the number of instructions a processor can issue in a given amount of time, thus reducing performance. Hence, it is generally used when the workload is not CPU-bound.

Dynamic frequency scaling by itself is rarely worthwhile as a way to conserve switching power. Saving the highest possible amount of power requires dynamic voltage scaling too, because of the V2 component and the fact that modern CPUs are strongly optimized for low power idle states. In most constant-voltage cases, it is more efficient to run briefly at peak speed and stay in a deep idle state for longer time (called "[[Race to sleep|race to idle]]" or computational sprinting), than it is to run at a reduced clock rate for a long time and only stay briefly in a light idle state. However, reducing voltage along with clock rate can change those trade-offs.

A related-but-opposite technique is [[Engineering:Overclocking|overclocking]], whereby processor performance is increased by ramping the processor's (dynamic) frequency beyond the manufacturer's design specifications.

One major difference between the two is that in modern PC systems overclocking is mostly done over the [[Engineering:Front Side Bus|Front Side Bus]] (mainly because the multiplier is normally locked), but dynamic frequency scaling is done with the [[Engineering:CPU multiplier|multiplier]]. Moreover, overclocking is often static, while dynamic frequency scaling is always dynamic. Software can often incorporate overclocked frequencies into the frequency scaling algorithm, if the chip degradation risks are allowable.

===Support across vendors===

==== Intel ====
[[Company:Intel|Intel]]'s CPU throttling technology, [[Engineering:SpeedStep|SpeedStep]], is used in its mobile and desktop CPU lines.

==== AMD ====
[[Engineering:AMD|AMD]] employs two different CPU throttling technologies. AMD's [[Engineering:Cool'n'Quiet|Cool'n'Quiet]] technology is used on its desktop and server processor lines. The aim of Cool'n'Quiet is not to save battery life, as it is not used in AMD's mobile processor line, but instead with the purpose of producing less heat, which in turn allows the system fan to spin down to slower speeds, resulting in cooler and quieter operation, hence the name of the technology. AMD's [[Engineering:PowerNow!|PowerNow!]] CPU throttling technology is used in its mobile processor line, though some supporting CPUs like the [[Engineering:AMD K6-2|AMD K6-2]]+ can be found in desktops as well.

[[Engineering:AMD PowerTune|AMD PowerTune]] and AMD ZeroCore Power are dynamic frequency scaling technologies for GPUs.

==== VIA Technologies ====
[[Company:VIA Technologies|VIA Technologies]] and [[Company:Zhaoxin|Zhaoxin]] processors use a technology named [[Engineering:LongHaul|LongHaul]] (PowerSaver), while [[Company:Transmeta|Transmeta]]'s version was called [[Engineering:LongRun|LongRun]].

The 36-processor [[Asynchronous array of simple processors|AsAP 1]] chip is among the first multi-core processor chips to support completely unconstrained clock operation (requiring only that frequencies are below the maximum allowed) including arbitrary changes in frequency, starts, and stops. The 167-processor [[Asynchronous array of simple processors|AsAP 2]] chip is the first multi-core processor chip which enables individual processors to make fully unconstrained changes to their own clock frequencies.

According to the [[ACPI]] Specs, the C0 working state of a modern-day CPU can be divided into the so-called "P"-states (performance states) which allow clock rate reduction and "T"-states (throttling states) which will further throttle down a CPU (but not the actual clock rate) by inserting STPCLK (stop clock) signals and thus omitting duty cycles.

==== ARM ====
Different ARM-based systems on chip provide CPU and GPU throttling.

==Dynamic voltage scaling==
'''Dynamic voltage scaling''' is a [[Power management|power management]] technique in which the [[Physics:Voltage|voltage]] used in a component is increased or decreased, depending upon circumstances. Dynamic voltage scaling to increase voltage is known as '''overvolting'''; dynamic voltage scaling to decrease voltage is known as '''undervolting'''. Undervolting is done in order to [[Physics:Energy conservation|conserve power]], particularly in [[Engineering:Laptop|laptop]]s and other [[Engineering:Mobile device|mobile device]]s, where energy comes from a [[Physics:Electric battery|battery]] and thus is limited, or in rare cases, to increase reliability. Overvolting is done in order to support higher frequencies for [[Computer performance|performance]].

The term "overvolting" is also used to refer to increasing static operating voltage of [[Computer|computer]] components to allow operation at higher speed ([[Engineering:Overclocking|overclocking]]).

===Background===
[[Engineering:MOSFET|MOSFET]]-based digital circuits operate using voltages at circuit nodes to represent logical state. The voltage at these nodes switches between a high voltage and a low voltage during normal operation—when the inputs to a [[Engineering:Logic gate|logic gate]] transition, the transistors making up that gate may toggle the gate's output.

Toggling a MOSFET's state requires changing its gate voltage from below the transistor's [[Physics:Threshold voltage|threshold voltage]] to above it (or from above it to below it). However, changing the gate's voltage requires charging or discharging the [[Physics:Capacitance|capacitance]] at its node. This capacitance is the sum of capacitances from various sources: primarily transistor [[Engineering:Gate capacitance|gate capacitance]], [[Physics:Diffusion capacitance|diffusion capacitance]], and wires ([[Engineering:Coupling (electronics)|coupling capacitance]]).

Higher supply voltages result in faster [[Slew rate|slew rate]] (rate of change of voltage per unit of time) when charging and discharging, which allows for quicker transitioning through the MOSFET's threshold voltage. Additionally, the more the gate voltage exceeds the threshold voltage, the lower the resistance of the transistor's conducting channel. This results in a lower [[Engineering:RC time constant|RC time constant]] for quicker charging and discharging of the capacitance of the subsequent logic stage. Quicker transitioning afforded by higher supply voltages allows for operating at higher frequencies.

===Methods===
Many modern components allow voltage regulation to be controlled through software (for example, through the [[BIOS]]). It is usually possible to control the voltages supplied to the CPU, RAM, [[Engineering:Peripheral Component Interconnect|PCI]], and [[Engineering:PCI Express|PCI Express]] (or [[Engineering:Accelerated Graphics Port|AGP]]) port through a PC's BIOS.

However, some components do not allow software control of supply voltages, and hardware modification is required by overclockers seeking to overvolt the component for extreme overclocks. [[Engineering:Video card|Video card]]s and [[Engineering:Motherboard|motherboard]] northbridges are components which frequently require hardware modifications to change supply voltages. These modifications are known as "voltage mods" or "Vmod" in the overclocking community.

===Undervolting===
Undervolting is reducing the voltage of a component, usually the processor, reducing temperature and cooling requirements, and possibly allowing a fan to be omitted. Just like overclocking, undervolting is highly subject to the so-called silicon lottery: one CPU can undervolt slightly better than the other and vice versa.

===Power===
{{main | CPU power dissipation }}

The ''switching power'' dissipated by a chip using static [[Engineering:CMOS|CMOS]] gates is <math>\alpha \cdot C \cdot V^2 \cdot f</math>, where <math>C</math> is the [[Physics:Capacitance|capacitance]] being switched per clock cycle, <math>V</math> is the supply [[Physics:Voltage|voltage]], <math>f</math> is the switching frequency,<ref>J. M. Rabaey. Digital Integrated Circuits. Prentice Hall, 1996.</ref> and <math>\alpha </math> is the activity factor. Since <math>V</math> is squared, this part of the power consumption decreases quadratically with voltage. The formula is not exact however, as many modern chips are not implemented using 100% CMOS, but also use special memory circuits, [[Dynamic logic (digital electronics)|dynamic logic]] such as [[Domino logic|domino logic]], etc. Moreover, there is also a static leakage current, which has become more and more accentuated as feature sizes have become smaller (below 90 nanometres) and threshold levels lower.

Accordingly, dynamic voltage scaling is widely used as part of strategies to manage switching power consumption in battery powered devices such as cell phones and laptop computers. Low voltage modes are used in conjunction with lowered clock frequencies to minimize power consumption associated with components such as CPUs and DSPs; only when significant computational power is needed will the voltage and frequency be raised.

Some peripherals also support low voltage operational modes. For example, low power MMC and SD cards can run at 1.8 V as well as at 3.3 V, and driver stacks may conserve power by switching to the lower voltage after detecting a card which supports it.

When leakage current is a significant factor in terms of power consumption, chips are often designed so that portions of them can be powered completely off. This is not usually viewed as being dynamic voltage scaling, because it is not transparent to software. When sections of chips can be turned off, as for example on [[Company:Texas Instruments|TI]] [[Engineering:OMAP|OMAP3]] processors, drivers and other support software need to support that.

===Program execution speed===
The speed at which a digital circuit can switch states - that is, to go from "low" (VSS) to "high" (VDD) or vice versa - is proportional to the voltage differential in that circuit. Reducing the voltage means that circuits switch slower, reducing the maximum frequency at which that circuit can run. This, in turn, reduces the rate at which program instructions that can be issued, which may increase run time for program segments which are sufficiently CPU-bound.

This again highlights why dynamic voltage scaling is generally done in conjunction with dynamic frequency scaling, at least for CPUs. There are complex tradeoffs to consider, which depend on the particular system, the load presented to it, and power management goals. When quick responses are needed (e.g. Mobile Sensors and [[Context-aware pervasive systems|Context-Aware Computing]]), clocks and voltages might be raised together. Otherwise, they may both be kept low to maximize battery life.

===Implementations===
The 167-processor [[Asynchronous array of simple processors|AsAP 2]] chip enables individual processors to make extremely fast (on the order of 1-2ns) and locally controlled changes to their own supply voltages. Processors connect their local power grid to either a higher (VddHi) or lower (VddLow) supply voltage, or can be cut off entirely from either grid to dramatically cut leakage power.

Another approach uses per-core on-chip switching regulators for dynamic voltage and frequency scaling (DVFS).<ref>
Wonyoung Kim, Meeta S. Gupta, Gu-Yeon Wei and David Brooks.
[http://www.eecs.harvard.edu/~meeta/my_papers/hpca08_onchipvrm.pdf "System Level Analysis of Fast, Per-Core DVFS using On-Chip Switching Regulators"].
2008.
</ref>

===Operating system API===
Unix system provides a userspace governor, allowing to modify the CPU frequencies{{cn|date=March 2024}} (though limited to hardware capabilities).

===System stability===
[[Dynamic frequency scaling]] is another power conservation technique that works on the same principles as dynamic voltage scaling. Both dynamic voltage scaling and dynamic frequency scaling can be used to prevent computer system overheating, which can result in program or operating system [[Crash (computing)|crashes]], and possibly hardware damage. Reducing the voltage supplied to the CPU below the manufacturer's recommended minimum setting can result in system instability.

===Temperature===
The efficiency of some electrical components, such as voltage regulators, decreases with increasing temperature, so the power used may increase with temperature causing [[Astronomy:Thermal runaway|thermal runaway]]. Increases in voltage or frequency may increase system power demands even faster than the CMOS formula indicates, and vice versa.<ref>{{cite web | url = http://www.silentpcreview.com/article821-page5.html | title = Asus EN9600GT Silent Edition Graphics Card | author = Mike Chin | page = 5 | work = Silent PC Review | access-date = 2008-04-21}}</ref><ref name="SPCRNewLevels"/>

===Caveats===
The primary caveat of overvolting is increased heat: the power dissipated by a circuit increases with the square of the voltage applied, so even small voltage increases significantly affect power. At higher temperatures, transistor performance is adversely affected, and at some threshold, the performance reduction due to the heat exceeds the potential gains from the higher voltages. Overheating and damage to circuits can occur very quickly when using high voltages.

There are also longer-term concerns: various adverse device-level effects such as hot carrier injection and [[Chemistry:Electromigration|electromigration]] occur more rapidly at higher voltages, decreasing the lifespan of overvolted components.

In order to mitigate the increased heat from overvolting, it's recommended to use [[Physics:Liquid cooling|liquid cooling]] to achieve higher ceilings and thresholds than you normally would with an aftermarket cooler. Also known as 'all-in-one' (AIO) coolers, they offer a far more effective method of unit cooling by relocating heat outside a computer case via the fans on the radiator whereas air cooling only disperses heat from the affected unit, increasing overall ambient temperatures.<ref>{{cite web | url = https://www.intel.com/content/www/us/en/gaming/resources/cpu-cooler-liquid-cooling-vs-air-cooling.html | title = CPU Cooler: Liquid Cooling vs Air Cooling | author = Intel| access-date = 2024-03-31}}</ref>

== See also ==
* [[Physics:Adaptive voltage scaling|Adaptive voltage scaling]] (AVS)
* [[Engineering:AMD PowerTune|AMD PowerTune]]/[[Engineering:AMD PowerPlay|AMD PowerPlay]] (graphics)
* [[Engineering:AMD Turbo Core|AMD Turbo Core]] (CPUs)
* [[Engineering:Clock gating|Clock gating]]
* [[Engineering:Cool'n'Quiet|AMD Cool'n'Quiet]] (desktop CPUs)
* Energy–delay product (EDP)
* [[HLT (x86 instruction)]]
* [[Engineering:Intel Turbo Boost|Intel Turbo Boost]] (CPUs)
* [[Physics:Overvoltage|Overvoltage]]
* [[Engineering:Power gating|Power gating]]
* [[Engineering:Power ramp|Power ramp]]
* [[Power–delay product]] (PDP)
* [[Engineering:PowerNow!|AMD PowerNow!]] (laptop CPUs)
* [[Engineering:SpeedStep|Intel SpeedStep]] (CPUs)
* Switched-mode power supply applications (SMPS) applications
* Switching energy
* Undervoltage
* Voltage optimization

==References==
{{reflist}}

==Further reading==
* {{cite book |author-first=Vincent C. |author-last=Gaudet |chapter=Chapter 4.1. Low-Power Design Techniques for State-of-the-Art CMOS Technologies |editor-first=Bernd |editor-last=Steinbach |title=Recent Progress in the Boolean Domain |publisher=[[Company:Cambridge Scholars Publishing|Cambridge Scholars Publishing]] |publication-place=Newcastle upon Tyne, UK |location=Freiberg, Germany |edition=1 |date=2014-04-01 |orig-date=2013-09-25 |isbn=978-1-4438-5638-6 |pages=187–212 |url=https://books.google.com/books?id=_pwxBwAAQBAJ |access-date=2019-08-04}} [https://web.archive.org/web/20210228020207/https://www.tau.ac.il/~ilia1/publications/rpbd_book.pdf] (xxx+428 pages)

{{Computer processor power management technologies}}
{{CPU technologies}}

{{DEFAULTSORT:Dynamic Frequency Scaling}}
[[Category:Clock signal]]
[[Category:Energy conservation]]
[[Category:Central processing unit]]
[[Category:Computer hardware tuning]]

{{Sourceattribution|Voltage and frequency scaling}}

Software:MALDA (programming language)

2026-08-22T14:27:20Z

MainEditor moved page Software:MALDA (programming language) to MALDA (programming language) without leaving a redirect

New page

----

{{Short description|Open-source programming language for agents, web apps and workflows}}
{{Draft topics|software|computing|technology}}
{{AfC topic|other}}


{{Draft article|subject=software}}

{{Infobox programming language
| name = MALDA
| logo =
| paradigm = Multi-paradigm: [[Imperative programming|imperative]], [[Object-oriented programming|object-oriented]], [[Concurrent computing|concurrent]]
| designed_by = Andrea Maldini
| developer = Andrea Maldini
| released = {{Start date and age|2026|07|31}}
| latest release version = 1.0.1
| latest release date = {{Start date and age|2026|08|18}}
| typing = Dynamic, with optional type hints
| implementations = MALDA interpreter and C# / JavaScript transpilers
| platform = [[.NET|.NET]] 8
| operating system = [[Software:Microsoft Windows|Windows]], [[Software:Linux|Linux]], [[Software:MacOS|macOS]] (Desktop IDE is Windows-only)
| license = Dual [[Software:MIT License|MIT]] or [[Software:Apache License 2.0|Apache License 2.0]]; separate runtime exception for compiled programs
| file_ext = .malda
| website = {{URL|https://github.com/amaldini/maldalang}}
}}

'''MALDA''' ('''Multi Agent Language with Development Automation''') is an [[Open-source software|open-source]] [[Programming language|programming language]] and toolchain for writing programs that combine ordinary control flow with [[Large language model|large language model]] prompts, software agents, HTTP services and long-running workflows.<ref name="readme">{{cite web |title=MALDA README |url=https://github.com/amaldini/maldalang/blob/main/README.md |website=GitHub |access-date=18 August 2026}}</ref> Source files use the {{code|.malda}} extension. The language is dynamically typed at run time; optional type annotations are used by the editor and, in transpile/publish mode, as a compile-time check rather than as a full static type system.<ref name="readme"/><ref name="spec">{{cite web |title=Malda Language Specification 1.0 — Tier 0 Core |url=https://github.com/amaldini/maldalang/blob/main/docs/spec/malda-language-1.0.md |website=GitHub |date=12 August 2026 |access-date=18 August 2026}}</ref>

Programs may be executed by a tree-walking interpreter, [[Source-to-source compiler|transpiled]] to [[C Sharp (programming language)|C#]] and built as a .NET executable, or compiled to a [[JavaScript]] subset for the browser.<ref name="arch">{{cite web |title=MALDA architecture (OSS core) |url=https://github.com/amaldini/maldalang/blob/main/docs/architecture.md |website=GitHub |access-date=18 August 2026}}</ref> The reference implementation is written in C# on [[.NET|.NET]] 8 and is dual-licensed under the MIT License or Apache License 2.0, at the user's option.<ref name="readme"/>

== History ==
MALDA was designed and is maintained by Andrea Maldini, a business-software developer working in C#, [[Java (programming language)|Java]] and VB.NET, as a first language implementation rather than as a research compiler project.<ref name="readme"/> Public project materials describe the work as an attempt to treat prompts, tools, agents, HTTP endpoints and durable workflows as grammar instead of library glue, after first trying the same ideas as libraries.<ref name="announce">{{cite web |title=MALDA — public announcement copy |url=https://github.com/amaldini/maldalang/blob/main/docs/announcement.md |website=GitHub |access-date=18 August 2026}}</ref> Those materials also cite Geoff Huntley's "Ralph Wiggum" agent loop and Huntley's esoteric language ''cursed'' as the prompt to attempt a language at all, not as a syntactic parent.<ref name="announce"/>

A draft of the Tier 0 language specification was dated 4 June 2026.<ref name="spec"/> The open-source core was first published on GitHub on 31 July 2026 as version 0.1.0, with a runtime, C# and JavaScript transpilers, command-line interface, IDEs, language server and VS Code extension.<ref name="v010">{{cite web |title=MALDA 0.1.0 |url=https://github.com/amaldini/maldalang/blob/main/docs/releases/v0.1.0.md |website=GitHub |access-date=18 August 2026}}</ref><ref name="gitinit">{{cite web |title=Initial public MALDA core, dual licensed MIT OR Apache-2.0 |url=https://github.com/amaldini/maldalang/commit/ |website=GitHub |date=31 July 2026 |access-date=18 August 2026}}</ref> The specification was marked Final 1.0 on 12 August 2026.<ref name="spec"/> Toolchain version 1.0.0, tagged 17 August 2026, aligned the CLI and Desktop IDE version numbers with that specification and made type-hint errors refuse emit on transpile and publish.<ref name="v100">{{cite web |title=MALDA 1.0.0 |url=https://github.com/amaldini/maldalang/blob/main/docs/releases/v1.0.0.md |website=GitHub |date=17 August 2026 |access-date=18 August 2026}}</ref> Version 1.0.1 followed on 18 August 2026 as a patch release; it is not a new specification line.<ref name="v101">{{cite web |title=MALDA 1.0.1 |url=https://github.com/amaldini/maldalang/blob/main/docs/releases/v1.0.1.md |website=GitHub |date=18 August 2026 |access-date=18 August 2026}}</ref>

The public repository is the language core. Separate product applications built on MALDA are kept out of that tree.<ref name="readme"/>

== Design ==
MALDA is a general-purpose language with a C-like surface: {{code|var}} bindings, {{code|function}} declarations ({{code|fn}} and {{code|def}} are syntax errors), classes, exceptions, {{code|match}} expressions, sum types, {{code|async}}/{{code|await}}, and a namespaced standard library under {{code|io}}, {{code|math}} and {{code|str}} among others.<ref name="spec"/><ref name="readme"/> Functions are declared at top level; nested function statements are not part of the language.<ref name="spec"/> Composition across files uses parse-time {{code|include}} or runtime {{code|import}} / selective {{code|import { … } from}}.<ref name="announce"/>

The project's stated design split is that relatively stable ideas (prompt templates, actor message passing, workflow {{code|step}}/{{code|retry}}/{{code|compensate}}) are syntax, while model providers, tool protocols and model names remain in the library layer.<ref name="announce"/> An earlier {{code|chain}} keyword for LLM pipelines was removed in favour of ordinary functions and the pipe operator.<ref name="announce"/>

=== Type system ===
Run-time values have a single kind (integer, float, string, boolean, null, array, object, function, class, variant, task, actor, or actor reference).<ref name="spec"/> Integers are 32-bit signed and overflow throws; floats are IEEE doubles; strings are UTF-16.<ref name="spec"/>

Type annotations such as {{code|var n: int {{=}} 0}} and {{code|function add(a: int) -> int}} are optional. They feed the language server and desktop editor, which report mismatches as errors by default; they are not enforced by the interpreter at run time.<ref name="readme"/><ref name="spec"/> {{code|malda compile --mode transpile}} and {{code|publish}} refuse to emit when those errors are present, unless {{code|--lenient-types}} is passed.<ref name="v100"/> {{code|schema}} declarations and {{code|validate}} check JSON-shaped data; sum types and {{code|match}} cover tagged unions.<ref name="spec"/>

=== Prompts, agents and tools ===
A {{code|prompt}} is a named declaration with untyped parameters and interpolated role sections. Calling it without {{code|await}} returns the rendered template (usable without an API key); calling it with {{code|await}} sends it to a configured model. A {{code|-> SchemaName}} binding validates model JSON against a declared schema.<ref name="readme"/> Functions may be exposed as agent tools with {{code|@Tool}} or over the [[Model Context Protocol]] with {{code|@MCPTool}}.<ref name="announce"/>

LLM features use OpenRouter when an API key is configured, otherwise a small local [[GGUF]] model (Qwen2.5-0.5B) downloaded on first use as an offline fallback.<ref name="readme"/>

=== Web and user interface ===
HTTP APIs and pages are declared with decorators ({{code|@GET}}, {{code|@POST}}, {{code|@PAGE}}, {{code|@AIPAGE}} and similar) on host objects such as {{code|HttpServer}} and {{code|RestServer}}.<ref name="readme"/> Server-driven {{code|ui.*}} trees are hosted by a separate UI host runtime. This web stack is a language feature and is distinct from the browser-based MALDA IDE.<ref name="arch"/>

=== Actors ===
Actors are a Tier 0 construct: {{code|actor}} declarations, {{code|spawn}}, {{code|send}}, {{code|receive}} and {{code|self}}.<ref name="spec"/> Message handlers are written with {{code|on}} over private state.<ref name="announce"/>

=== Workflows ===
Durable workflows are blocks of statements using reserved words including {{code|step}}, {{code|retry}}, {{code|compensate}}, {{code|approval}}, {{code|wait}} and {{code|timeout}}.<ref name="announce"/> Recovery is step-level memoization to a local [[Software:SQLite|SQLite]] file ({{code|./.malda/workflows.db}}): a completed step returns its persisted JSON-shaped result instead of re-running.<ref name="announce"/> Non-deterministic and side-effecting built-in names are refused outside {{code|step}} by a fixed deny-list; the implementation is a single-writer, single-machine store, not a clustered workflow engine.<ref name="announce"/><ref name="readme"/>

== Example ==
The following program prints a greeting:<ref name="hello">{{cite web |title=hello_world.malda |url=https://github.com/amaldini/maldalang/blob/main/Examples/Basics/hello_world.malda |website=GitHub |access-date=18 August 2026}}</ref>

<syntaxhighlight lang="text">
io.print("Hello, World!");
</syntaxhighlight>

A characteristic prompt example binds a template to a schema. Without {{code|await}}, the call is a rendered template; {{code|validate}} performs the same check that {{code|await}} would apply to model JSON:<ref name="readme"/>

<syntaxhighlight lang="text">
schema Review {
summary: string;
issues: string[];
}

prompt codeReview(code, language) -> Review {
system: "You are an expert reviewer of {language}.",
user: "Review this {language} code:\n\n{code}"
}

var rendered = codeReview("function add(a, b) { return a + b; }", "javascript");
io.print(rendered.user);
</syntaxhighlight>

== Implementation ==
The reference toolchain is implemented in C# with a hand-written [[Recursive descent parser|recursive descent parser]]. The pipeline is lexer, parser and abstract syntax tree, then one of three backends: the interpreter, a C# transpiler, or a JavaScript / [[Software:Progressive web app|progressive web app]] transpiler.<ref name="arch"/><ref name="announce"/> Language intelligence (completions, diagnostics, hover) is shared across the Desktop IDE, Web IDE and language server.<ref name="arch"/>

The three backends are not equivalent. The JavaScript path is a documented browser subset: it does not include agents, HTTP servers or workflows.<ref name="readme"/> C# transpile covers the built-in registry for shipping .NET executables; interpreted and transpiled programs are compared on a curated set of examples for matching standard output.<ref name="arch"/>

The standard library is large (on the order of three hundred named built-ins in the project registry), covering strings, collections, files, HTTP, JSON, SQL, graphs and process control, with .NET interop as an escape hatch. There is no public package registry comparable to [[Software:Npm|npm]] or crates.io.<ref name="announce"/>

== Tooling ==
{| class="wikitable"
|-
! Tool !! Role
|-
| {{code|malda}} CLI || Interpret, compile/transpile, templates ({{code|malda new}}), workflow commands
|-
| Desktop IDE || Full Windows reference environment (WPF)
|-
| Web IDE || Browser learning playground (Blazor / Monaco); not feature-parity with Desktop
|-
| {{code|malda-lsp}} || [[Language Server Protocol]] host
|-
| {{code|malda debug-adapter}} || Debug Adapter Protocol for interpret-mode debugging (separate from the language server)
|-
| vscode-malda || Visual Studio Code / Cursor extension
|}

Self-contained release archives are published for Windows (CLI and Desktop IDE) and Linux (CLI).<ref name="readme"/> The HTML reference manual is published in English (canonical) and Italian.<ref name="readme"/> The repository also ships a compact language pack under {{code|docs/llm/}} intended for coding agents that have not seen MALDA in training data.<ref name="readme"/>

== Licensing ==
The core (runtime, compiler, IDEs, language server, reference manual and examples) is dual-licensed MIT OR Apache-2.0.<ref name="readme"/> Because the toolchain copies runtime helpers into compiled output, a runtime exception states that this does not impose attribution obligations on user programs.<ref name="runtime">{{cite web |title=MALDA Runtime Library Exception |url=https://github.com/amaldini/maldalang/blob/main/LICENSE-RUNTIME-EXCEPTION |website=GitHub |access-date=18 August 2026}}</ref> The name "MALDA" is asserted as a trademark of Andrea Maldini and is not granted by either copyright licence.<ref name="tm">{{cite web |title=The MALDA name |url=https://github.com/amaldini/maldalang/blob/main/TRADEMARK.md |website=GitHub |access-date=18 August 2026}}</ref> There is no contributor licence agreement.<ref name="readme"/>

== See also ==
* [[Ballerina (programming language)]]
* [[Model Context Protocol]]
* [[List of programming languages]]

== References ==
{{Reflist}}

== External links ==
* {{official website|https://github.com/amaldini/maldalang}}
* [https://amaldini.github.io/maldalang/ MALDA Reference Manual] (English)
* [https://amaldini.github.io/maldalang/it/ Manuale di riferimento MALDA] (Italian)

{{Programming languages}}

[[:Category:Dynamically typed programming languages]]
[[:Category:Concurrent programming languages]]
{{Sourceattribution|MALDA (programming language)}}

Company:Axiom Software Limited

2026-08-22T14:25:47Z

import

New page

{{Infobox company
| name = Axiom Software Limited
| logo =
| type = Private limited company
| industry = Software, Human Resources
| founded = 19 January 1994
| founder = Mark Housley, Paul Brookes
| hq_location_city = Manchester
| hq_location_country= United Kingdom
| key_people = Mark Housley (Managing Director)
| products = Discus Online, myDISCprofile, Axiom Discovery
| website = {{URL|https://www.axiomsoftware.com}}
}}

'''Axiom Software Limited''' is a British software development company headquartered in Manchester, England. Established in 1994, the company specializes in web-based [[Psychometrics|psychometrics]], talent management, and behavioral profiling platforms.<ref>{{cite web |title=AXIOM SOFTWARE LIMITED overview - Find and update company information |url=https://find-and-update.company-information.service.gov.uk/company/02888933 |publisher=Companies House |access-date=2026-03-23}}</ref>

== History ==
The company was incorporated on 19 January 1994 under the initial name Brookhous Systems Limited, before adopting the name Axiom Software Limited in September 1994.<ref>{{cite web |title=AXIOM SOFTWARE LIMITED filing history |url=https://find-and-update.company-information.service.gov.uk/company/02888933/filing-history |publisher=Companies House |access-date=2026-03-23}}</ref> The organization operates from its corporate office located at Empress Buildings, 380 Chester Road in Manchester.

== Products ==
Axiom Software Limited develops and maintains several digital assessment platforms, most notably '''Discus Online''', a multi-currency software platform used for global recruitment, candidate job-matching, employee assessments, and workplace relationship mapping based on the [[Social:DISC assessment|DISC assessment]] methodology. The company also operates myDISCprofile and the Axiom Discovery skills testing system.

== References ==
{{reflist}}

== References ==

{{reflist}}
{{Sourceattribution|Axiom Software Limited}}

Software:MALDA (programming language)

2026-08-22T14:24:09Z

import

New page

Nonlocal symmetry

2026-08-22T14:23:41Z

← Older revision		Revision as of 09:23, 22 August 2026
(One intermediate revision by the same user not shown)
Line 1:		Line 1:
	~~{{Multiple issues\|~~
	~~{{coi\|date=May 2026}}~~
	}}
	In mathematical physics and the theory of differential equations, a '''nonlocal symmetry''' is a symmetry of a differential equation whose infinitesimal generator depends on integral or potential variables, rather than solely on the independent variables, the dependent variables, and a finite number of their derivatives. Unlike local symmetries (e.g., [[Lie point symmetry\|Lie point symmetries]] or contact symmetries), nonlocal symmetries cannot be detected within the standard finite-dimensional jet bundle formalism. They often arise from Bäcklund transformations, reciprocal transformations, Lax pairs, or from systematic constructions like conservation-law (CL) coverings and inverse potential systems (IPSs).<ref>{{Cite journal \|last1=Bluman \|first1=George W. \|last2=Yang \|first2=Zhengzheng \|date=2013-09-01 \|title=A symmetry-based method for constructing nonlocally related partial differential equation systems \|url=https://pubs.aip.org/jmp/article/54/9/093504/233198/A-symmetry-based-method-for-constructing \|journal=Journal of Mathematical Physics \|language=en \|volume=54 \|issue=9 \|article-number=093504 \|doi=10.1063/1.4819724 \|arxiv=1211.0100 \|bibcode=2013JMP....54i3504B \|issn=0022-2488}}</ref><ref>{{Cite journal \|last=Bazghandi \|first=Mustafa \|date=2026-04-29 \|title=A symmetry-based study of nonlocal structures in the modified Benjamin–Bona–Mahony equation \|url=https://doi.org/10.1140/epjp/s13360-026-07737-y \|journal=The European Physical Journal Plus \|language=en \|volume=141 \|issue=4 \|article-number=470 \|doi=10.1140/epjp/s13360-026-07737-y \|issn=2190-5444}}</ref>		In mathematical physics and the theory of differential equations, a '''nonlocal symmetry''' is a symmetry of a differential equation whose infinitesimal generator depends on integral or potential variables, rather than solely on the independent variables, the dependent variables, and a finite number of their derivatives. Unlike local symmetries (e.g., [[Lie point symmetry\|Lie point symmetries]] or contact symmetries), nonlocal symmetries cannot be detected within the standard finite-dimensional jet bundle formalism. They often arise from Bäcklund transformations, reciprocal transformations, Lax pairs, or from systematic constructions like conservation-law (CL) coverings and inverse potential systems (IPSs).<ref>{{Cite journal \|last1=Bluman \|first1=George W. \|last2=Yang \|first2=Zhengzheng \|date=2013-09-01 \|title=A symmetry-based method for constructing nonlocally related partial differential equation systems \|url=https://pubs.aip.org/jmp/article/54/9/093504/233198/A-symmetry-based-method-for-constructing \|journal=Journal of Mathematical Physics \|language=en \|volume=54 \|issue=9 \|article-number=093504 \|doi=10.1063/1.4819724 \|arxiv=1211.0100 \|bibcode=2013JMP....54i3504B \|issn=0022-2488}}</ref><ref>{{Cite journal \|last=Bazghandi \|first=Mustafa \|date=2026-04-29 \|title=A symmetry-based study of nonlocal structures in the modified Benjamin–Bona–Mahony equation \|url=https://doi.org/10.1140/epjp/s13360-026-07737-y \|journal=The European Physical Journal Plus \|language=en \|volume=141 \|issue=4 \|article-number=470 \|doi=10.1140/epjp/s13360-026-07737-y \|issn=2190-5444}}</ref>

Line 62:		Line 59:
	[[Category:Lie groups]]		[[Category:Lie groups]]

	~~{{Improve categories\|date=August 2026}}~~
	{{Sourceattribution\|Nonlocal symmetry}}		{{Sourceattribution\|Nonlocal symmetry}}

Nonlocal symmetry

2026-08-22T14:22:48Z

import

New page

{{Multiple issues|
{{coi|date=May 2026}}
}}
In mathematical physics and the theory of differential equations, a '''nonlocal symmetry''' is a symmetry of a differential equation whose infinitesimal generator depends on integral or potential variables, rather than solely on the independent variables, the dependent variables, and a finite number of their derivatives. Unlike local symmetries (e.g., [[Lie point symmetry|Lie point symmetries]] or contact symmetries), nonlocal symmetries cannot be detected within the standard finite-dimensional jet bundle formalism. They often arise from Bäcklund transformations, reciprocal transformations, Lax pairs, or from systematic constructions like conservation-law (CL) coverings and inverse potential systems (IPSs).<ref>{{Cite journal |last1=Bluman |first1=George W. |last2=Yang |first2=Zhengzheng |date=2013-09-01 |title=A symmetry-based method for constructing nonlocally related partial differential equation systems |url=https://pubs.aip.org/jmp/article/54/9/093504/233198/A-symmetry-based-method-for-constructing |journal=Journal of Mathematical Physics |language=en |volume=54 |issue=9 |article-number=093504 |doi=10.1063/1.4819724 |arxiv=1211.0100 |bibcode=2013JMP....54i3504B |issn=0022-2488}}</ref><ref>{{Cite journal |last=Bazghandi |first=Mustafa |date=2026-04-29 |title=A symmetry-based study of nonlocal structures in the modified Benjamin–Bona–Mahony equation |url=https://doi.org/10.1140/epjp/s13360-026-07737-y |journal=The European Physical Journal Plus |language=en |volume=141 |issue=4 |article-number=470 |doi=10.1140/epjp/s13360-026-07737-y |issn=2190-5444}}</ref>

==Historical background==

During the twentieth century, researchers discovered that many important nonlinear equations possess hidden symmetries that cannot be represented solely in terms of local variables and their derivatives. Studies of integrable systems such as the Korteweg–de Vries (KdV) equation, nonlinear Schrödinger equation, and sine-Gordon equation revealed the existence of infinite-dimensional symmetry structures generated by nonlocal operators.<ref>{{Cite journal |last=Vinogradov |first=A. M. |date=1989-01-01 |title=Symmetries and conservation laws of partial differential equations: Basic notions and results |url=https://doi.org/10.1007/BF00131928 |journal=Acta Applicandae Mathematica |language=en |volume=15 |issue=1 |pages=3–21 |doi=10.1007/BF00131928 |issn=1572-9036}}</ref>

The modern geometric formulation of nonlocal symmetries emerged through the work of Vinogradov, Krasil'shchik, Bluman, Anco, Olver, and many others, who developed the theory of coverings, potential symmetries, and conservation-law-based symmetry methods .<ref>{{Cite book |last=Olver |first=Peter J. |title=Applications of Lie Groups to Differential Equations |date=1986 |publisher=Springer US |isbn=978-1-4684-0274-2 |series=Graduate Texts in Mathematics |location=New York, NY}}</ref><ref>{{Cite book |last1=Bluman |first1=George W. |title=Applications of symmetry methods to partial differential equations |last2=Cheviakov |first2=Alexei F. |last3=Anco |first3=Stephen C. |date=2010 |publisher=Springer |isbn=978-1-4614-2498-7 |series=Applied mathematical sciences |location=New York, NY Heidelberg}}</ref>

==Formal definition==
Consider a differential equation

<math>
\Delta(x,u^{(n)})=0,
</math>

where <math>x</math> denotes the independent variables, <math>u</math> the dependent variables, and <math>u^{(n)}</math> represents the collection of derivatives up to order <math>n</math>.

A classical Lie symmetry is generated by a vector field

<math>
X=\xi^i(x,u)\frac{\partial}{\partial x^i}
+\phi^\alpha(x,u)\frac{\partial}{\partial u^\alpha}.
</math>

A nonlocal symmetry extends this framework by allowing the coefficients <math>\xi^i</math> and <math>\phi^\alpha</math> to depend on additional variables:

<math>
w=(w^1,\ldots,w^m),
</math>

which satisfy auxiliary differential relations such as

<math>
w_x = F(x,u,u_x,\ldots).
</math>

The resulting symmetry generator has the form

<math>
X=\xi^i(x,u,w)\frac{\partial}{\partial x^i}
+\phi^\alpha(x,u,w)\frac{\partial}{\partial u^\alpha}.
</math>

Because the variables (w) are defined through integration or other nonlocal constructions, the corresponding symmetry is called nonlocal.

==Applications==
Nonlocal symmetries have numerous applications in mathematics and mathematical physics, including:

* construction of exact solutions;
* similarity reductions;
* classification of integrable equations;
* generation of soliton solutions.

In recent years, nonlocal symmetry methods have also been applied to fractional differential equations, nonlocal diffusion models, and reaction–diffusion systems (e.g. <ref>{{Cite journal |last1=Shi |first1=Yapeng |last2=Feng |first2=Yuqiang |last3=Yu |first3=Jicheng |date=2025-12-01 |title=Classical and nonclassical symmetry analysis of generalized time-fractional Novikov equation |url=https://www.sciencedirect.com/science/article/pii/S0577907325003958 |journal=Chinese Journal of Physics |volume=98 |pages=900–913 |doi=10.1016/j.cjph.2025.10.014 |bibcode=2025ChJPh..98..900S |issn=0577-9073}}</ref>)

== References ==
{{reflist}}

[[Category:Lie groups]]

{{Improve categories|date=August 2026}}
{{Sourceattribution|Nonlocal symmetry}}

Quinn theorem

2026-08-22T14:22:20Z

import

New page

In [[Differential topology|differential topology]] in mathematics, the '''Quinn theorem''' is a result about [[Topological manifold|topological]] [[4-manifold|4-manifolds]]. It shows the importance of the topological property of [[Compact space|compactness]] for smoothability, hence if there exists a compatible [[Smooth structure|smooth structure]]. The Quinn theorem is named after [[Biography:Frank Quinn (mathematician)|Frank Quinn]], who proved it in 1984.

== Formulation ==
Every non-compact topological 4-manifold is smoothable.<ref>{{Cite book |last=Freed |first=Daniel |title=[[Instantons and Four-Manifolds]] |last2=Uhlenbeck |first2=Karen |author-link2=Karen Uhlenbeck |publisher=[[Company:Springer Nature|Springer]] |year=1984 |isbn=978-1-4613-9705-2 |series=Mathematical Sciences Research Institute Publications |volume=1 |at=Theorem 1.4 on p. 25 |language=en |doi=10.1007/978-1-4613-9703-8}}</ref> A counterexample is the [[E8 manifold]], which is a compact and non-smoothable topological 4-manifold. A more general formulation is: A topological 4-manifold has a smooth structure in the [[Complement (set theory)|complement]] of any [[Closed set|closed set]] with at least one point in each compact component.<ref>Freedman & Quinn 1990, p. 116</ref><ref>Scorpan 05, p. 219</ref>

== See also ==

* [[Moise's theorem]], lower-dimensional cases

== Literature ==

* {{cite journal |last1=Quinn |first1=Frank |date=1984 |title=Smooth structures on 4-manifolds |journal=American Mathematical Society |series=Contemp. Math. |language=en |volume=35 |issue= |pages=473–479 |doi= |issn= |mr=}}
* {{cite book |last1=Freedman |first1=Michael |url= |title=Topology of 4-Manifolds |last2=Quinn |first2=Frank |author-link2=Frank Quinn |publisher= |year=1990 |isbn=9780691632346 |series= |volume= |location= |pages=116 |language=en |chapter=8.2 Smoothing open 4-manifolds |doi=}}
* {{cite book |last1=Scorpan |first1=Alexandru |title=The Wild World of 4-Manifolds |publisher=American Mathematical Society |year=2005 |isbn=978-1-4704-6861-3 |series=Mathematical Sciences Research Institute Publications |volume=1}}

== References ==
<references/>

== External links ==

* Quinn theorem on [[NLab|''n''Lab]]

[[Category:Differential topology]]

{{Sourceattribution|Quinn theorem}}

Strong set order

2026-08-22T14:21:50Z

import

New page

{{short description|Partial order in lattice theory}}

In [[Order theory|order theory]], the '''strong set order''' <math>\leq_s</math> is a partial order over the subsets of a [[Lattice|lattice]]. It is widely used to study monotone comparative statics of [[Parametric programming|parametrized optimization problems]] in economic theory and [[Operations research|operations research]].<ref name = "topkis1998">{{cite book |last1=Topkis |first1=Donald M. |title=Supermodularity and Complementarity |date=1998 |publisher=Princeton University Press |isbn=9780691032443 |page=32}}</ref>

The strong set order was first defined and used by Arthur F. Veinott in his unpublished lecture notes.<ref name="topkis1978">{{cite journal |last1=Topkis |first1=Donald M. |title=Minimizing a Submodular Function on a Lattice |journal=Operations Research |date=1978 |volume=26 |issue=2 |pages=305-321 |doi=10.1287/opre.26.2.305 |url=https://www.jstor.org/stable/169636?seq=1 |quote=Veinott (personal communication) introduced this relation.}}</ref><ref name = "milgrom1994">{{cite journal |last1=Milgrom |first1=Paul |last2=Chris |first2=Shannon |title=Monotone Comparative Statics |journal=Econometrica |date=1994 |volume=62 |issue=1 |pages=157-180 |doi=10.2307/2951479 |url=https://www.jstor.org/stable/2951479 |quote=(...) the strong set order <math>\leq_s</math>, introduced by Veinott (1989).}}</ref><ref>{{cite journal |last1=Veinott |first1=Arthur F. |title=Lattice Programming |journal=Unpublished notes from lectures delivered at Johns Hopkins University |date=1989}}</ref> It was later popularized by Donald M. Topkis, [[Biography:Paul Milgrom|Paul Milgrom]] and Chris Shannon via their work on monotone comparative statics, in particular through Topkis' Theorem.

== Definition ==

Given a [[Lattice (order)|lattice]] <math>(X, \leq)</math>, consider the its [[Power set|power set]] <math>\mathcal P(X)</math>. The '''strong set order''' <math>\leq_s</math> is a partial order on <math>\mathcal P(X)</math> given by:

:<math>
A \leq_s B \iff \left\{
\begin{array}{ll}
a \wedge b \in A, \\[0.1em]
a \vee b \in B
\end{array}
\ \ \ \ \ \ \ \forall a \in A, b \in B,
\right.
</math>

where <math>a \vee b</math> and <math>a \wedge b</math> are respectively the [[Join and meet|join and meet]] of <math>a, b</math>.

== Examples and non-examples ==

=== Real numbers ===

Consider the real numbers <math>\mathbb R</math> with the usual order <math>\geq</math>. Let <math>A = [\underline a, \overline a]</math> and <math>B = [\underline b, \overline b]</math>. Then

:<math>A \leq_s B \iff \underline a \leq \underline b \ \text{ and } \ \overline a \leq \overline b</math>.

More generally, for any <math>A, B \subseteq \mathbb R</math>, we have

:<math>A \leq_s B \iff \inf A \leq \inf B \ \text{ and } \ \sup A \leq \sup B</math>.

=== Euclidean space ===

Consider the [[Euclidean space]] <math>\mathbb R^n</math> with the usual pointwise order: <math>\mathbf x \leq \mathbf y \iff x_i \leq y_i \ \ \forall i \leq n</math>. For <math>n = 2</math>, let

:<math>A = [0,2] \times [0, 3],</math>

:<math>B = [1,3] \times [1, 4].</math>

Then :<math> A \leq_s B</math>. But now consider

:<math>B' = [-1, 3] \times [1,4]</math>;

Then <math>A \nleq_s B'</math>, since <math>(0, 0) \in A </math>, <math> (-1, 1) \in B'</math> but

:<math>(0, 0) \wedge (-1, 1) = (-1, 0) \notin B' </math>.

=== Power set ===

Consider the [[Power set|power set]] of the natural numbers <math>\mathcal P(\mathbb N)</math> under the [[Inclusion order|set-inclusion order]]: <math>A \leq B \iff A \subseteq B</math>. Let

:<math>A = \{\{1\}\}</math>,

:<math>B = \{\{1\}, \{2\}, \{1,2,3\}\}</math>.

It it ''not true'' that <math>A \leq_s B</math>. Indeed, we have <math>\{1\} \in A</math>, <math>\{2\} \in B</math>, but

:<math>\{1\} \wedge \{2\} = \{1\} \cap \{2\} = \varnothing \notin A,</math>

whence <math>A \nleq_s B</math>.

== Applications ==

=== Topkis' Theorem ===

The strong set order is widely used in monotone comparative statics, in particular via Topkis' Theorem. Given a lattice <math>X</math>, a poset <math>\Theta</math>, a [[Constrained optimization|constraint]] [[Set-valued function|correspondence]] <math>D : \Theta \rightrightarrows X</math> and a function <math>f : X \times \Theta \to \mathbb R</math>, the theorem gives sufficient conditions for the correspondence

<math>x^*(\theta) = \arg\max_{x \in D(\theta)} f(x, \theta)</math>

the be increasing in the strong set order, that is, for the statement

:<math>\theta \geq \theta' \implies x^*(\theta) \geq_s x^*(\theta')</math>

to hold.

=== Monotone selections ===

Increasigness in the strong set order can be used to obtain monotone [[Choice function|selection]]s from [[Set-valued function|correspondence]]s. Indeed, if <math>X </math> is a poset and <math> Y</math> is a lattice, the correspondence <math>\Gamma : X \rightrightarrows Y</math> is nonempty-valued and increasing in the strong set order, then a monotone selection <math>g</math> from <math>\Gamma</math> exists if any of the following hold:

* <math>\Gamma(x) \subseteq Y</math> has a minimal element for every <math>x \in X</math> (in particular, if <math>\Gamma</math> is [[Complete lattice|complete-lattice]]-valued). One can thus take :<math>g : X \to Y</math> by putting <math>g(x) = \min \Gamma(x)</math>. The same works if a maximal element is available.

* <math>Y</math> is a [[Lattice (order)#Sublattices|sublattice]] of a finite [[Cartesian product|product]] of [[Total order#Chains|chain]]s.<ref name="kukushkin2013">{{cite journal |last1=Kukushikin |first1=Nikolai S. |title=Increasing Selections from Increasing Multifunctions |journal=Order |date=2013 |volume=30 |pages=541–555 |doi=10.1007/s11083-012-9260-6 |url=https://link.springer.com/article/10.1007/s11083-012-9260-6}}</ref> This covers, for example, <math>Y \subseteq \mathbb R^n</math>.

* <math>X</math> is countable.<ref name="kukushkin2013"></ref>

=== Zhou's Fixed-Point Theorem for correspondences ===

The strong set order can also be used for a generalization of [[Knaster–Tarski theorem|Tarski's fixed-point theorem]] to [[Set-valued function|correspondences]] known as Zhou's fixed-point theorem:<ref name="zhou1994">{{cite journal |last1=Lin |first1=Zhou |title=The Set of Nash Equilibria of a Supermodular Game Is a Complete Lattice |journal=Games and Economic Behavior |date=1994 |volume=7 |issue=2 |pages=295-300 |doi=10.1006/game.1994.1051 |url=https://www.sciencedirect.com/science/article/pii/S0899825684710517}}</ref><ref>{{cite journal |last1=Yu |first1=Lu |title=Fixed point theorems for increasing correspondences on lattices |journal=Economic Theory |volume=68 |pages=1-19 |doi=10.1007/s00199-026-01702-7 |url=https://link.springer.com/article/10.1007/s00199-026-01702-7#citeas}}</ref>

'''Theorem:''' let <math>X</math> be a nonempty [[Complete lattice|complete lattice]] and <math>\Gamma : X \rightrightarrows X</math> a nonempty-valued correspondence. If <math>\Gamma</math> is increasing int the strong set order and <math>\Gamma(x)</math> is a [[Complete lattice#Complete sublattices|subcomplete sublattice]] for all <math>x \in X</math>, then <math>\Gamma</math> has a fixed point. Moreover, the set <math>\text{Fix}(\Gamma)</math> of such fixed points is a nonempty complete lattice.

== Weak set order ==

The strong set order if often too restrictive for some applications, in particular because it assumes that the underlying space <math>X</math> is a lattice.<ref>{{cite arXiv |last1= Che|first1=Yeon-Koo |last2= Kim|first2=Jinwoo |author-link2= | last3 = Kojima |first3 = Fuhito |author-link3 = Fuhito Kojima |date=2021 |year= |title=Weak Monotone Comparative Statics |pages=2-3|arxiv=1911.06442v4 |class= |quote= |mode= |ref= |postscript=}}</ref> A useful weaker ordering which can be used on the subsets of any poset <math>X</math> is the weak set order <math>\leq_w</math>, defined by:<ref name = "topkis1998"></ref>

:<math>
A \leq_w B \iff \left\{
\begin{array}{ll}
\forall a \in A \ \ \exists b \in B \ \text{ such that } a \leq b, \text{ and } \\[0.1em]
\forall b \in B \ \ \exists a \in A \ \text{ such that } b \geq a.
\end{array}
\right.
</math>

This order can moreover be broken down into two weaker orders: the upper and lower weak set orders <math>\leq_{uw}, \leq_{lw}</math>, respectively defined by:

:<math>
A \leq_{uw} B \iff \forall a \in A \ \ \exists b \in B \ \text{ such that } a \leq b,
</math>

:<math>
A \leq_{lw} B \iff \forall b \in B \ \ \exists a \in A \ \text{ such that } b \leq a.
</math>

Clearly <math>A \leq_w B \iff A \leq_{uw} B \text{ and } A \leq_{lw} B</math>.

== See also ==

*Topkis's Theorem

*Monotone comparative statics

==References==
{{reflist}}

{{Order theory}}

[[Category:Lattice theory]]

{{Sourceattribution|Strong set order}}

Feature Importance

2026-08-22T14:21:33Z

← Older revision		Revision as of 09:21, 22 August 2026
(One intermediate revision by the same user not shown)
Line 1:		Line 1:
	{{Short description\|Set of techniques in machine learning}}		{{Short description\|Set of techniques in machine learning}}
	~~{{MI\|~~
	~~{{AI-generated\|reason=some weird prose in previous versions and a lot of bold listing\|date=August 2026}}~~
	~~{{Copy edit\|date=August 2026 \|reason=lots of unnecessary caps}}~~
	}}
	'''Feature Importance''' (or '''Variable Importance''' or '''Feature Attribution''') refers to a set of techniques and mathematical frameworks used in [[Machine learning\|machine learning]], statistics, and their applications to quantify the contribution of input variables (features) to a model's output or to the underlying data-generating process. It is a component of [[Explainable Artificial Intelligence]] (XAI) and Interpretable Machine Learning (IML) and is used for model improvement, scientific inference, and [[Feature selection\|feature selection]].<ref name="Holzinger2022">{{cite conference		'''Feature Importance''' (or '''Variable Importance''' or '''Feature Attribution''') refers to a set of techniques and mathematical frameworks used in [[Machine learning\|machine learning]], statistics, and their applications to quantify the contribution of input variables (features) to a model's output or to the underlying data-generating process. It is a component of [[Explainable Artificial Intelligence]] (XAI) and Interpretable Machine Learning (IML) and is used for model improvement, scientific inference, and [[Feature selection\|feature selection]].<ref name="Holzinger2022">{{cite conference
	\|last1=Holzinger \|first1=Andreas		\|last1=Holzinger \|first1=Andreas
Line 50:		Line 47:
	\|pages=135–145		\|pages=135–145
	\|doi=10.1098/rspl.1888.0082		\|doi=10.1098/rspl.1888.0082
	}}</ref>~~{{primary inline\|date=August 2026}}~~ [[Biography:Sewall Wright\|Sewall Wright]] further developed variable importance, by introducing path analysis in 1921, to understand causal influences in complex systems.<ref>{{cite journal		}}</ref> [[Biography:Sewall Wright\|Sewall Wright]] further developed variable importance, by introducing path analysis in 1921, to understand causal influences in complex systems.<ref>{{cite journal
	\|last=Wright		\|last=Wright
	\|first=S.		\|first=S.
Line 58:		Line 55:
	\|year=1921		\|year=1921
	\|pages=557–585		\|pages=557–585
	}}</ref>~~{{primary inline\|date=August 2026}}~~ Path analysis determines correlative influences along direct paths by decomposing [[Correlation\|correlation]] and [[Partial correlation\|partial correlation]] coefficients into their path-based components.~~{{cn\|date=August 2026}}~~		}}</ref> Path analysis determines correlative influences along direct paths by decomposing [[Correlation\|correlation]] and [[Partial correlation\|partial correlation]] coefficients into their path-based components.

	With [[Biography:Leo Breiman\|Leo Breiman]]’s introduction of Random Forests, feature importance expanded to more nonlinear models and explanations. He introduced "permutation importance" sometimes measured by gini index.<ref name="Boulesteix2012">{{cite journal		With [[Biography:Leo Breiman\|Leo Breiman]]’s introduction of Random Forests, feature importance expanded to more nonlinear models and explanations. He introduced "permutation importance" sometimes measured by gini index.<ref name="Boulesteix2012">{{cite journal

Feature Importance

2026-08-22T14:20:05Z

import

New page

{{Short description|Set of techniques in machine learning}}
{{MI|
{{AI-generated|reason=some weird prose in previous versions and a lot of bold listing|date=August 2026}}
{{Copy edit|date=August 2026 |reason=lots of unnecessary caps}}
}}
'''Feature Importance''' (or '''Variable Importance''' or '''Feature Attribution''') refers to a set of techniques and mathematical frameworks used in [[Machine learning|machine learning]], statistics, and their applications to quantify the contribution of input variables (features) to a model's output or to the underlying data-generating process. It is a component of [[Explainable Artificial Intelligence]] (XAI) and Interpretable Machine Learning (IML) and is used for model improvement, scientific inference, and [[Feature selection|feature selection]].<ref name="Holzinger2022">{{cite conference
|last1=Holzinger |first1=Andreas
|last2=Goebel |first2=Randy
|last3=Fong |first3=Ruth
|last4=Moon |first4=Taesup
|last5=Müller |first5=Klaus-Robert
|last6=Samek |first6=Wojciech
|title=xxAI - Beyond Explainable Artificial Intelligence
|conference=International Workshop on Extending Explainable AI Beyond Deep Models and Classifiers
|pages=3–10
|publisher=Springer International Publishing
|location=Cham
|year=2022
|doi=10.1007/978-3-031-04083-2_1
|isbn=978-3-031-04082-5
}}</ref>

Feature importance is a context-dependent score that varies based on the scope of the explanation, the model (or data) of interest, and the ultimate purpose of the analysis.<ref name="Holzinger2022">{{cite conference
|last1=Holzinger |first1=Andreas
|last2=Goebel |first2=Randy
|last3=Fong |first3=Ruth
|last4=Moon |first4=Taesup
|last5=Müller |first5=Klaus-Robert
|last6=Samek |first6=Wojciech
|title=xxAI - Beyond Explainable Artificial Intelligence
|conference=International Workshop on Extending Explainable AI Beyond Deep Models and Classifiers
|pages=3–10
|publisher=Springer International Publishing
|location=Cham
|year=2022
|doi=10.1007/978-3-031-04083-2_1
|isbn=978-3-031-04082-5
}}</ref>

== History ==

The earliest forms of feature importance assessed the strength of the relationships between pairs of variables within animal biology or human psychology using methods such as [[Biography:Francis Galton|Francis Galton]]'s [[Correlation|correlation]] coefficient.<ref>{{cite journal
|last=Galton
|first=Francis
|title=Co-relations and their measurement, chiefly from anthropometric data
|journal=Proceedings of the Royal Society of London
|volume=45
|year=1889
|issue=273–279
|pages=135–145
|doi=10.1098/rspl.1888.0082
}}</ref>{{primary inline|date=August 2026}} [[Biography:Sewall Wright|Sewall Wright]] further developed variable importance, by introducing path analysis in 1921, to understand causal influences in complex systems.<ref>{{cite journal
|last=Wright
|first=S.
|title=Correlation and Causation
|journal=Journal of Agricultural Research
|volume=20
|year=1921
|pages=557–585
}}</ref>{{primary inline|date=August 2026}} Path analysis determines correlative influences along direct paths by decomposing [[Correlation|correlation]] and [[Partial correlation|partial correlation]] coefficients into their path-based components.{{cn|date=August 2026}}

With [[Biography:Leo Breiman|Leo Breiman]]’s introduction of Random Forests, feature importance expanded to more nonlinear models and explanations. He introduced "permutation importance" sometimes measured by gini index.<ref name="Boulesteix2012">{{cite journal
|last1=Boulesteix |first1=Anne-Laure
|last2=Janitza |first2=Silke
|last3=Kruppa |first3=Jochen
|last4=König |first4=Inke R.
|date=November 2012
|title=Overview of random forest methodology and practical guidance with emphasis on computational biology and bioinformatics
|journal=Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery
|volume=2 |issue=6 |pages=493–507
|doi=10.1002/widm.1072
}}</ref> Simultaneously in 2001, Lipovetsky and Conklin applied the Shapley value from cooperative game theory to regression, providing a consistent method for variance attribution in the presence of multicollinearity.<ref name="Babic2026">{{cite journal
|last1=Babic |first1=Boris
|last2=Cohen |first2=I. Glenn
|last3=Savulescu |first3=Julian
|year=2026
|title=Justifiability and AI: putting explainability in its place
|journal=AI & SOCIETY
|pages=1–13
|doi=10.1007/s00146-026-03030-9
}}</ref>

== Axioms for Feature Importance ==

Formal feature importance frameworks use mathematical axioms to define consistent scoring criteria and validate the theoretical properties of different estimation methods.

=== The Shapley Axioms (S1–S4) ===

Shapley values are the unique solution that satisfies four core game-theoretic axioms, which describe how to fairly distribute the "total gain" of a model's prediction among the participating features.<ref name="Verdinelli2023">{{cite journal |last1=Verdinelli |first1=Isabella |last2=Wasserman |first2=Larry |title=Feature Importance: A Closer Look at Shapley Values and LOCO |journal=Statistical Science |date=1 November 2024 |volume=39 |issue=4 |doi=10.1214/24-STS937 |url=https://projecteuclid.org/journals/statistical-science/volume-39/issue-4/Feature-Importance--A-Closer-Look-at-Shapley-Values-and/10.1214/24-STS937.short}}</ref> <ref name="SHAP">{{cite conference
|last1=Lundberg
|first1=S. M.
|last2=Lee
|first2=S. I.
|title=A unified approach to interpreting model predictions
|conference=Advances in Neural Information Processing Systems
|volume=30
|year=2017
}}</ref>

# '''Efficiency (Local Accuracy)''': The sum of the importance scores <math>\phi_j</math> for all features must equal the difference between the model's prediction for an instance <math>f(x)</math> and the expected prediction <math>E[f(X)]</math>.
# '''Symmetry''': If two features <math>i</math> and <math>j</math> contribute exactly the same value to every possible subset of other features, they must receive the same importance score: <math>\phi_i = \phi_j</math>.
# '''Dummy (Null Player)''': If a feature contributes nothing to the value function for any subset of features, its importance score must be zero. This is crucial for identifying irrelevant features.
# '''Additivity (Linearity)''': If the value function is the sum of two functions <math>v = v_1 + v_2</math>, then the importance scores must be the sum of the scores calculated for each function: <math>\phi_j(v) = \phi_j(v_1) + \phi_j(v_2)</math>.

=== Axioms for Explaining the Data ===

While the Dummy (Null Player) axiom is desirable for obtaining feature importance scores that can be used to optimize a model, this is not the case for explaining the data, since variables may be highly correlated (and hence offer little value to the existing model), yet remain causally linked to the response variable. Alternative axioms have been proposed for feature importance methods that intend to explain the data.<ref name="Catav2021">{{cite conference
|last1=Catav
|first1=Amnon
|last2=Fu
|first2=Boyang
|last3=Zoabi
|first3=Yazeed
|last4=Meilik
|first4=Ahuva Libi Weiss
|last5=Shomron
|first5=Noam
|last6=Ernst
|first6=Jason
|last7=Sankararaman
|first7=Sriram
|last8=Gilad-Bachrach
|first8=Ran
|title=Marginal contribution feature importance: an axiomatic approach for explaining data
|conference=International Conference on Machine Learning
|pages=1324–1335
|publisher=Proceedings of Machine Learning Research
|year=2021
}}</ref>

* '''Blood Relation (Causal Graph Association)''': A feature should have non-zero importance if and only if the feature is blood related (associated) with the response in the ground-truth causal graph.
* '''Marginal Contribution''': The importance of a feature must be at least as high as the gain it provides when added to the set of all other features.
* '''Elimination''': Removing other features from the feature set can only decrease (or leave unchanged) the importance of a remaining feature. It cannot increase it.
* '''Invariance under Redundant Information and Symmetry under Duplication''': Adding a redundant feature should not change the importance of preexisting features, and identical features should receive equal importance.

== Classification of Methods ==

=== By Purpose ===

The choice of method is categorized based on the user's objective.

* '''Model Explanation''': The goal is to understand the "logic" of a black-box model to ensure safety, fairness, and reliability. Methods like SHAP, LIME, and [[Accumulated local effects]] are standard here.
* '''Data Explanation (Scientific Inference)''': The goal is to learn about the real world. Researchers prioritize methods that handle redundancy and correlation in a way that reflects the true underlying relationships (e.g., MCI, UMFI).<ref name="UMFI">{{cite conference
|last1=Janssen
|first1=J.
|last2=Guan
|first2=V.
|last3=Robeva
|first3=E.
|title=Ultra-marginal feature importance: Learning from data with causal guarantees
|conference=International Conference on Artificial Intelligence and Statistics
|pages=10782–10814
|publisher=Proceedings of Machine Learning Research
|year=2023
}}</ref>
* '''Model Optimization (Feature Selection)''': The goal is to improve the model's performance by removing irrelevant or redundant features. Techniques like Recursive Feature Elimination (RFE) use importance scores as a selection criterion.<ref name="RFE_Lit">"A SYSTEMATIC LITERATURE REVIEW: RECURSIVE FEATURE ELIMINATION ALGORITHMS". [https://ejournal.nusamandiri.ac.id/index.php/jitk/article/view/5015]</ref>

=== By Scope ===

Feature importance methods have been classified into four distinct settings based on two axes: Global vs. Local<ref name="Holzinger2022"/> and Data vs. Model.<ref name="Inconsistencies">{{cite conference
|last1=Harel
|first1=N.
|last2=Obolski
|first2=U.
|last3=Gilad-Bachrach
|first3=R.
|title=Inherent Inconsistencies of Feature Importance
|conference=XAI in Action: Past, Present, and Future Applications
}}</ref> <ref>{{cite arXiv
|last1=Chen
|first1=H.
|last2=Janizek
|first2=J. D.
|last3=Lundberg
|first3=S.
|last4=Lee
|first4=S. I.
|title=True to the model or true to the data?
|eprint=2006.16234
|year=2020
|class=cs.LG
}}</ref> <ref name="GlobalCovert">{{cite conference
|last1=Covert
|first1=I.
|last2=Lundberg
|first2=S. M.
|last3=Lee
|first3=S. I.
|title=Understanding global feature contributions with additive importance measures
|conference=Advances in Neural Information Processing Systems
|volume=33
|year=2020
|pages=17212–17223
}}</ref>

{| class="wikitable"
|-
! !! Reference: Model !! Reference: Data
|-
! Global
| '''Global-Model Importance''': Explains how a trained model behaves across the entire dataset. It identifies which features the model generally relies on for its predictions.
| '''Global-Data Importance''': Explains the true relationships in the underlying phenomenon. It seeks to identify the intrinsic predictive power of features within the population, regardless of a specific model's choices.
|-
! Local
| '''Local-Model Importance''': Explains why a specific prediction was made for a single instance. It quantifies the influence of each feature on that a specific model's specific outcome.
| '''Local-Data Importance''': Explains the role of a feature for a specific individual in the real world (e.g., why a specific patient developed a disease), focusing on the causal or statistical dependencies for that point.
|}

Methods like SHAP and LIME are local-model. Global-model importance metrics include permutation importance and SAGE.<ref name="GlobalCovert" /> Global-data methods include MCI and UMFI. <ref name="Inconsistencies"/>

=== By Treatment of Correlated Features ===

Feature importance methods often differ in how they treat the correlation between features. In the presence of correlated features, predictive credit can be distributed in a variety of ways. Methods can therefore be classified based on how they assign credit to correlated features, where one end of the spectrum is given by marginal feature importance, and the other is given by conditional feature importance.<ref name="CPI">{{cite journal
|last1=Strobl
|first1=C.
|last2=Boulesteix
|first2=A. L.
|last3=Kneib
|first3=T.
|last4=Augustin
|first4=T.
|last5=Zeileis
|first5=A.
|title=Conditional variable importance for random forests
|journal=BMC Bioinformatics
|volume=9
|issue=1
|year=2008
|pages=307
|doi=10.1186/1471-2105-9-307
|doi-access=free
}}</ref> <ref name="GrompingReview">{{cite journal
|last=Grömping
|first=U.
|title=Variable importance in regression models
|journal=Wiley Interdisciplinary Reviews: Computational Statistics
|volume=7
|issue=2
|year=2015
|pages=137–152
|doi=10.1002/wics.1346
}}</ref> <ref>{{cite journal
|last1=Molnar
|first1=C.
|last2=König
|first2=G.
|last3=Bischl
|first3=B.
|last4=Casalicchio
|first4=G.
|title=Model-agnostic feature importance and effects with dependent features: a conditional subgroup approach
|journal=Data Mining and Knowledge Discovery
|volume=38
|issue=5
|year=2024
|pages=2903–2941
|doi=10.1007/s10618-022-00901-9
}}</ref>

* '''Conditional Feature Importance''': Evaluates a feature by conditioning on the values of all other features. This approach respects the dependence structure of the data and measures the unique information a feature provides that is not already captured by other variables. Methods in this category include conditional permutation importance<ref name="CPI" />, Leave-One-Covariate-Out, and partial correlation.

* '''Marginal Feature Importance''': Relies on associations between the response and predictors, regardless of multicollinearity. Purely marginal methods assign the same high importance to all features. Examples include correlation, marginal contribution feature importance, and ultra-marginal feature importance.<ref name="Catav2021" />

Methods such as SHAP and permutation importance are somewhere in between the two extremes as importance is shared among correlated features.

=== Gradient vs. Non-Gradient ===

The technical implementation of the importance measure often dictates its applicability to different model architectures.<ref name="MASE">"MASE: Interpretable NLP Models via Model-Agnostic Saliency Estimation", arXiv. [https://arxiv.org/html/2512.04386v1]</ref>

{| class="wikitable"
|-
! Mechanism !! Description !! Examples
|-
| '''Gradient-Based'''
| Utilizes the derivatives (gradients) of the model's output with respect to its input features. These are typically model-specific and used for differentiable models like neural networks.
| Saliency Maps, Integrated Gradients, Grad-CAM, DeepLIFT.
|-
| '''Non-Gradient Based'''
| Treats the model as a "black box" and relies on perturbations, shuffling, or submodel training. These are typically model-agnostic and applicable to any algorithm.
| Permutation Importance, KernelSHAP, LOCO, MCI.
|}
=== Other Classifications ===

Achen (1982) introduced a classification of linear regression-based feature importance methods: "dispersion importance" (explained variance), "level importance" (impact on the mean), and "theoretical importance" (the change in response for a given change in regressor).<ref>{{cite book
|last=Achen
|first=C. H.
|title=Interpreting and Using Regression
|publisher=Sage
|location=Newbury Park, CA
|year=1982
}}</ref>
== Applications ==
*[[Medicine:Medicine|Medicine]]<ref>{{cite conference
|last1=García Andreu
|first1=A.
|last2=Costa
|first2=M.
|last3=Pastor
|first3=O.
|year=2025
|title=Feature Selection in Medical Imaging: A Comprehensive Review
|book-title=International Conference on Advanced Information Systems Engineering
|pages=142–154
|publisher=Springer Nature Switzerland
|location=Cham
}}</ref><ref>{{cite journal
|last1=Vimbi
|first1=V.
|last2=Shaffi
|first2=N.
|last3=Mahmud
|first3=M.
|year=2024
|title=Interpreting artificial intelligence models: a systematic review on the application of LIME and SHAP in Alzheimer’s disease detection
|journal=Brain Informatics
|volume=11
|issue=1
|page=10
}}</ref>
*Earth Science<ref>{{cite journal
|last=Ibebuchi
|first=C. C.
|year=2025
|title=Uncertainty in machine learning feature importance for climate science: a comparative analysis of SHAP, PDP, and gain-based methods
|journal=Theoretical and Applied Climatology
|volume=156
|issue=9
|page=476
}}</ref>
*Computer Science<ref>{{cite journal
|last1=Samek
|first1=W.
|last2=Montavon
|first2=G.
|last3=Lapuschkin
|first3=S.
|last4=Anders
|first4=C. J.
|last5=Müller
|first5=K.-R.
|year=2021
|title=Explaining deep neural networks and beyond: A review of methods and applications
|journal=Proceedings of the IEEE
|volume=109
|issue=3
|pages=247–278
}}</ref>

==See also==
*[[Feature selection]]
*[[Explainable Artificial Intelligence]]
*[[Shapley value]]

== References ==
{{Reflist}}

[[Category:Machine learning]]

{{Sourceattribution|Feature Importance}}

Morava E-theory

2026-08-22T14:19:32Z

import

New page

In the theory of [[Spectrum (topology)|spectra]] in mathematics, a '''Morava E-theory''' (also called '''Lubin–Tate theory''' or '''Johnson–Wilson theory''') is a particular spectrum or equivalently generalized (co)homology theory. For a [[Prime number|prime number]] <math>p</math> (obmitted in notation), these form a sequence <math>E(n)</math> of spectra, hence one for each [[Natural number|natural number]] <math>n\in\mathbb{N}</math>. Morava E-theories form the foundation of [[Chromatic homotopy theory|chromatic homotopy theory]], in which a spectrum is separated into so-called (mono)chromatic layers by using its various Bousfield localizations with respect to the different Morava E-theories, so-called chromatic localizations <math>L_{E(n)}</math>. Morava E-theory is named after [[Biography:Jack Morava|Jack Morava]], who introduced it in unpublished preprints in the 1970s.

== Definition ==
For a [[Perfect field|perfect field]] <math>\mathbb{K}</math>, hence in which every [[Irreducible polynomial|irreducible polynomial]] is [[Separable polynomial|separable]], of characteristic <math>p</math>, let <math>W(\mathbb{K})</math> be its ring of [[Witt vector|Witt vectors]].<ref>Hopkins 1999, Definition 16.4</ref> Let <math>R=W(\mathbb{K})[v_1,\ldots,v_{n-1}]</math> be the [[Polynomial ring|polynomial ring]] in <math>n-1</math> variables, called Lubin–Tate ring. According to [[Landweber exact functor theorem|Landweber's exact functor theorem]], a [[Formal group law|formal group law]] over the Lubin–Tate ring defines a generalized cohomology theory, which according to [[Brown's representability theorem]] is represented by a spectrum, which is Morava E-theory <math>E(n)</math>. It fulfills:

: <math>\pi_0E(n)
=R[\beta^{\pm 1}]</math>

for its ring of coefficients with <math>\beta\in R</math> in second degree.

== Properties ==

* The chromatic localization <math>L_{E(n)}</math> preserves [[Smash product|smash products]].<ref>Lurie 2010, Lecture 22</ref>
* <math>E(n)</math> is Bousfield equivalent to <math>E(n-1)\times K(n)</math>.<ref>Lurie 2010, Lecture 23</ref> (<math>K(n)</math> is [[Morava K-theory]]. Two spectra are Bousfield equivalent, when their generalized homology theories vanish on the exact same [[Topological space|topological spaces]]. Hence for every topological space <math>X</math>, one has <math>E(n)_*(X)=0</math> if and only if <math>E(n-1)_*(X)=0</math> and <math>K(n)_*(X)=0</math>.)
* <math>E(n)</math> is <math>K(n)</math>-local.<ref>Lurie 2010, Lecture 35</ref> (<math>K(n)</math> is [[Morava K-theory]]. <math>K(n)</math>-local means that every spectrum morphism <math>X\rightarrow E(n)</math> from a <math>K(n)</math>-acyclic spectrum <math>X</math> is nullhomotopic. <math>K(n)</math>-acyclic means that <math>K(n)\wedge X\simeq 0</math>.)
* A connected [[Closed manifold|closed]] spin 10-manifold is generalized oriented in second Morava E-theory <math>E(2)</math> (for <math>p=2</math>) if and only if its seventh integral [[Stiefel–Whitney class]] <math>W_7</math> vanishes.

== E(n)-local spheres ==
For the [[Sphere spectrum|sphere spectrum]] <math>\mathbb{S}</math> (with <math>\mathbb{S}_n=S^n</math>), its chromatic localizations <math>L_{E(n)}\mathbb{S}</math> are also called <math>E(n)</math>-local spheres.

== See also ==

* [[Morava K-theory]]

== Literature ==

*{{cite web |last=Morava |first=Jack |title=Boarding School Lecture Notes |url=https://math.mit.edu/~hrm/manuscripts/bording-school.pdf |access-date=2026-08-21 |website=math.mit.edu |location=[[Organization:Princeton University|Princeton University]] |date=1973}}
*{{cite web|last=Hopkins|first= Mike
*{{cite web|last=Lurie|first= Jacob
*{{cite web|last=Lurie|first= Jacob
*{{cite web|last=Lurie|first= Jacob

== References ==
<references/>

== External links ==

* Morava E-theory on the [[NLab|''n''Lab]]

[[Category:Algebraic topology]]
[[Category:Cohomology theories]]

{{Sourceattribution|Morava E-theory}}

File:Nessia monodactyla.jpg

2026-08-22T11:52:19Z

Maintenance script uploaded File:Nessia monodactyla.jpg == Summary == Submitted to commons.wikimedia.org == Licensing == {{CC BY-SA 3.0}}

New page