Complexity Reduction: theory, metrics andapplications

C. Tijus, S. Poitrenaud, E. Zibetti F. Jouen, M. Bui Ella PinskaCHArt CHArt - CSMC Eurocontrol Experimental Centre

University Paris 8 EPUE EurocontrolSaint-Denis, France Paris, France Bretigny sur Orge, France

Abstract-Generalized Galois Lattices Formalism for computing As a matter of fact, as noticed by [3], we do not have acontextual categorization allows metrics to evaluate complexity scientific definition of complexity. For some authors [4, 5],and efficiency as well as methods for simplifying or complicating the term "complexity" is generally avoided as an overused andthe external object at hand. Such methods are adapted for virtualenvironments and augmented reality devices for which it is poorly defined word, except in specific systems. Theysimple to change the distribution of features over categories. For advocate in favor of terms such as "diversity" orreal world objects, and human operators that operate on them, "complication", and are using "complexity" for real non-the online computation allows a survey of the complexity level modeled world entities.and a "simplify it first" planning of operations. Let's consider however scientific definitions such as

"multiple components", for instance in a definition given byI. WHAT COMPLEXITY IS AND WHAT COMPLEXITY Is NOT [6]: "the degree of complication of a system or system

component, determined by such factors as the number andintricacy ofinterfaces, the number and intricacy ofconditionalfor human intellectual reflection or of nervous breakdown for b

novices, complexity is lacking of a clear definition. Wordnet sructe", the MountinQe Instie defiof data

[ 1], that provides an example of the use of the word numer of pible state a usste cnstaked on, ithecomplexity" gives the ironic "he enjoyed the complexity of condition of a system, situation, or organization that ismodern computers" but is defining complexity as being "the cinterated with some degree of order but has too manyquality of being intricate and compounded". A tautological entsgand rltionships u ders in hsimpl a alydeiito sic "itiae is deie as "hvn ma,ny

elements and relationships to understand In simple analytic or

logical ways." First, if components or states are very similar,complexly arranged elements".. the number will not be an important factor of complexity.Among~~ ~~deihtr ofcmlxt,tee.raepaes Second, as understanding iS related to a cognitive system, this"Complexity in game settings is almost universally considered ' u rn . .t

desirable". And there are also coffee or wine amateurs: is for a cognitive system that there is complexity. However,"Complexity relates to the co-presence of attributes in a although complexity is complexity for a given cognitivecoffee: acidity, body, earthiness, sweetness, etc., combine to system, one would expect some measurement that takes into

mkacfeopx,ACr'Y aaccount the object to be evaluated in term of complexity.

amakesaoffe complex",w s"A Another well admitted scientific definition of complexity ofexamples of complex wines". an object is given by [7], [8]: "the length (in bits) of theOn~~~~~~~~~~~~th.ieoo-noal itain,tecmiteo shortest algorithm that completely describes the object", "thethe IEEE Computer Society [2] defines complexity as "the shretagrtmta copetl decie th obef,"htegreetIEEE icmputer society [2] d nestcomplext as"hesn complexity of a number is the length (in bits) of the shortestdegrementationwchat systemfficttoo ondensthas and designer algorithm that prints out the number". Because humans writeimplementationtat is difficult o understandan vrf." algorithms and because the computer (running the algorithm)Thus, Complexity appears to a matter of taste or of usability. can bedausa ati cognite te com itym)

But what iscomplexity? ~~~~~can be considered as an artificial cognitive system: complexityBut what iS complexity?Answer.com does not answer, using a direct tautology: "the is on the programmer's side and dependents of the cognitive

or condition of being complex." For Encarta, it is "the system, programmer or program. For instance, a language can

quaplicaty d ..'of be more or less adapted to some data. Thus, increasing thecomplicated nature: the condition of being difficult to e power of the language might render shorter some descriptions,understand, or solve; condition of havig many parts: but not others. The arguments stand for definitions that are

c based on other cognitive resources than the number ofcomplicated thing: one of the interrelated problems or alo-rithmic steps, suc as time and memory in comp.tationaldifficulties involved in a complicated matter." In summary, copext thoy[]dictionaries are of no help.

This work was supported by the Complex System Modeling andCognition Lab, European Joint Research Lab, Eurocontrol-EPHE.

1-4244-0695-1/07/$25.00 ©2007 IEEE. 65

Now, let's consider a more crucial definition: "complexity is internal relevant descriptors that could be applied to thethe property of a real world system that is manifest in the object, from both the knowledge of the natural or artificialinability of any one formalism being adequate to capture all cognitive system and from the task at hand. For instance,its properties. It requires that we find distinctly diferent ways consider the following external object: "RIVF". Either the taskof interacting with systems. distinctly different in the sense of counting the number elements, or an identification task ofthat when we make successful models, the formal systems each element based on the single descriptor "letter" makeneeded to describe each distinct aspect are not derivable from "RIVF" equivalent to "RRRR". When distinguishing amongeach other" [10]. A definition that has been generalized to vowels and consonants, the equivalence does not stand. Samedomains of emergent properties and non-linear relationships requirement is necessary for components of the objects: it issince living systems are not the sole domain of emergent necessary to know how much and which parts areproperties: physical systems, in general, exhibit also such differentiated by the cognitive systems.properties [11]. Here again, however, complexity appears to Second, knowing the cognitive system, its sensory inputsbe a characteristic of the cognition system that analyses the and the corresponding internal relevant descriptors, itsoutside object, a system, situation, or organization, being said perception of components, the allocated resources, and theto be complex. task at hand, one would expect a complexity measure that does

not depend mainly on the number elements, but much more on11. COMPLEXITY As A CoGNITIvE MATTER the number of different elements, thus on the number of

All of the above definition accord to the fact that different operations: for a task being "what that's?"complexity is the result of interaction between a cognitive "RRRRRRRR" would be as simpler than "RRRR" (a group ofsystem and an object. This is no doubt, the starting step of Rs) and simpler that "RROO".defining complexity. How much complex is a chess problem Third, complexity being a matter of degree, we are lookingwill depend of which cognitive system is analyzing it. Much for a measurement of complexity that has the followingmore, complexity is not in the real world but is relevant to the characteristics:resources a cognitive system has to allocate to the processing - to be independent of the object to be evaluated, in such aof a given object. way that different objects can be compared for a given

In this line, a simple task in usual conditions can turn to be cognitive system, for instance the content of one image vs. thecomplex if required resources are allocated to another task. content of a speech,For the human cognitive system, there is a fundamental - to be independent of the language to be used, given thatdistinction between automatic processes and controlled the language can be selected according to the internalprocesses in the study of the situations of training and description of the object, the goal of processing and theexecution of tasks. According to [12], a process is automatic operations to be done: for instance, a simple confident butwhen it occurs without intention, when it does not cause longer computation could be preferred to a more complex butconscious attention, when it does not interfere with another approximate computation,mental activity. What is of interest is that human processing - to provide a unit of complexity as well as a metrics ofcommute from automatic to controlled processing with complexity and derived metrics: for instance a metrics thatadditional tasks [13] showing that a processing that was relates complexity with the number of components andsimple turn out to be a complex one. relations.

Another crucial fact is the fact that a same outside to be Fourth, a model of complexity and its correspondingprocessed object can present itself to be processed either with metrics should provide insight about how to increase ora simple or complex procedure. Imagine a friend apartment to decrease complexity since science provides tools for changingbe reached as being either "at the seven floor without real world objects.numbered floors" or as being "at the last floor" [14]. The The model of complexity we propose is based on contextualprocedure will be much more simpler in the second case (to categorization [16]. Categorization is the basic mechanism ofclimb until you cannot climb any more vs. to climb while cognition allowing to associate a same response to a group ofcounting the floors until the reached floor is the seventh). examples, but mainly to differentiate among examples.These considerations are at the foundation of situated Categorization is usually associated to long-term semanticcognition [15]. memory [17]. In opposite, contextual categorization is the

However, for a natural or artificial cognitive system in a making of categories for the current situation in which eachgiven state, with the resources that can be allocated to, and a object, or component, is categorized, which means groupedgiven task: how to measure the complexity a given object and and differentiated, according to descriptors in long-termhow to measure the difference of complexity of two different semantic memory but mainly according to the others objects,objects? We are Olooking for a measure that bypasses the or components that form its context. As a simple example,processing criteria (an object is complex if its processing is "A" in the set "A3" will be categorized as a letter while in thecomplex) and take into account the external object. set "AF", "A" will be categorized as a vowel. The basic

First, our proposal is that it is necessary to know the set of principle is that an object is categorized at the level of

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hierarchical segmental that permits distinctiveness. For "AF", R I V Fyou cannot use "letter" to designate "A" since "A" and "F" are ter 1 1 1 1both letters. 71 1,1

Let's consider the set of letters "RIVF" with the task of v] 1Iidentifying components having "alphanumeric", "digit", c 1 1 1"letter", "vowels", "consonants", "uppercase" and"lowercase" as descriptors. Result is a set of "uppercaseletters" that has "consonant" with 3 instances and "vowel" R i V fwith 1 instance as subclasses. Note that we get a hierarchical tte 1 1 1 1segmentation of the set that alights "letters" and "uppercase" cOILaIt 1 1 1and isolate the vowel "I". vwe1 1

The set of letters "Rivf' would provide another hierarchical ppercase 1segmentation. Result is a set of "letters" with "consonant" and l 1 1 1 E i"lower case" as subclasses and "R" being an "uppercase" assubclass of "consonant", "i" being a "vowel" as subclass of r I v 1"lowercase" and "v" and "f' being instances of a subclass of et ter 1 1 1 1both "consonant" and "lowercase". IsWoIL t 1 1 1Now, consider "Irvf'. Hierarchical segmentation is a set of r 1 1 __

"letter" having two subclasses: a set of 3 "lowercaseconsonants" and "I" as an uppercase vowel". Note, in this pr 1case, that vowel "i" has more distinctiveness in "Irvf' than in 1"RIVF" and "pops out" according to this differentiation. Fig. 1. The boolean matrix and corresponding Generalized Galois Lattices

For the same set of descriptors, it becomes clear that "Rivf' for each of the three sets of letters: RIVF, Rivfand rlvf.is more complex than "Irvf'. However, we would like to know The On x Dm Boolean matrix, which indicates whetherwhich of "RIVF" and "Irvf' is the more complex although each of n objects has or not each ofm descriptors, enables thethey do not have the same set of descriptors. creation of one hierarchy of categories with transitivity,We propose a metrics for complexity measurement that is asymmetry and reflexivity. The maximum number of

based on Galois lattices. Galois lattices [18] are used for categories is either 2n- 1, or m if m < 2n- 1, given that eachclustering, data analysis, and formal concept analysis [19, 20]. meaningful category should have at least one descriptor, whatAlthough Galois lattices computation is known for renders the "combinatorial explosion" relative to the number"combinatorial explosion" since the number of categories is of descriptors. The Galois Lattice is said to be "generalized"2n-1, where n is the number of objects. Although not new, because descriptors are of various types: features, properties,Galois lattice has never been used as the basis of a complexity dimension as well as basic actions, procedures or tasks. Themetrics. categories form a lattice whose complexity depends on the

way descriptors are distributed over objects according to theIII. COMPLEXITY MEASUREMENT AND REDUCTIONfortpsfasciin(ige1)THROUGH GENRALIZED GALIsLATTICESfour types of association (figure 1).

The metrics of complexity is derived from the lattice. ItContextual categorization, according to the descriptors provides a measure of the complexity, of the power, and of the

(features, properties, rules, procedures,) applied to objects, or efficiency of a given real world object: that is a system, acomponents, is based on of four types of associations: situation, an organization, but also a task or a procedure byindependence (presence of one descriptor does not allow us to using operations (goal and action) as descriptor ofthe object.infer presence of the other: conjunctive case as for consonant Complexityand lowercase in Rivf), equivalence (presence of one brings More the description lattice is composed of categories thatabout the presence ofthe other and vice-versa, as for letter and behave differently and/or more are numerous the descriptorsuppercase in IRVF), exclusion (presence of one precludes the that are necessary, more the object is complex. Unit ofpresence of the other, as uppercase and consonant in Irvf), complexity is the one couple category-descriptor. Complexityimplication (presence of one brings about the presence of the is defined by the product of the number of categories by theother, though the reverse is not necessarily true, as for number of descriptors: COMP.obj = Nc x Nduppercase and consonant in Rivf). Descriptive Power

These associations are computed in the Generalized Galois Some descriptors are more useful than other because theyLattices formalism, STONE, developed by [21, 20]. They are apply to many subcategories. Thus, we can define usefulnessachieved by creating subcategories and, in doing so, Of a descriptor as its power. What is the power of a descriptor?hierarchies of categories. A descriptor is all the more powerful that it applies to a great

Hierarchies of categories have a mathematical formalism number of categories (its applicability), given the total numberlabeled Galois Lattices [1, 11]. of potential categories to which it could be applied.

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The applicability of a descriptor to categories is defined by Appledji nnmaci ntoshits scope or extension, which means by the number ofcategories to which it applies: EXT.descrip = Nc.descripThe power of a descriptor reflects the relationship between d ppletel journaliste,num6roterr6 pertoi re

its extension and the total number of categories: POW.descripEXT.descrip / Nc Appledji nn,macintoshFinally, the power of a description corresponds to the sum Mac

of each description power: POW.obj = , POW.descripEfficiency A ppletel

A description is all the more "efficient" that its complexity j ojournaliste,num6roter r6pertoireis compensated by its power. We express this efficiency by the BUT21

pplecom FsxTermi nsl FaxViewer word5relationship between power and complexity. The efficiency of ada description of an object, EFF.obj = POW.obj / COMP.objThe value of efficiency of an object reflects its cognitiver

complexity: higher is the efficiency value, lesser is the pplecom,xTermi nal ,FaxViewer,word5cognitive cost for processing. Examples of figure 1 (RIVF, ad num6roter,repertoireRivf and Irvf) have the corresponding values:

1fr escom posi nefface ri c neT616p honeMMi ni tel n u m6 ro3s r6 po nde u r,[dfrt journa1iste

COMP.RIVF= 12 BUT2 1EXT. letter=~3, POW.letter= 1 ,T lppledji nn,Appletel ,maci ntosh,num6roter, r6 pe rtoi reEXT. uppercase =3, POW.uppercase 1IEXT. vowel =1, POW.vowel = .33EXT. consonant = 1, POW.uppercase = .3 3 cejournaliste STEP 4POW.RIVF = 2.66 BUT21EFF.RIVF =.22 .emodifier,noms,suppri mer

P [ 8t~~~~~~~~a/at I rt

COMP.Rivf= 30EXT. letter =3, POW.letter =1i fdlatEXT. uppercase =1, POW.uppercase .16 Fig. 2. Four steps in a dynamic communication interface and theEXT. lowercase =3, POW.lowercase =.5 corresponding Generalized Galois Lattice with increasing complexityEXT. vowel =1, POW.vowel = .16EXT. consonant =3, POW.uppercase = .5 deleted, and new components could appear.POW.Rivf= 2.32 These changes, that are not under the control of theEFF.Rivf= .07 cognitive system, could increase or decrease the complexity of

the dynamic system. On line construction of GeneralizedCOMP.Irvf= 15 Galois Lattices (GGL) allows the on line evaluation ofEXT. letter =3, POW.letter= 1 complexity and the monitoring of thresholds not to beEXT. uppercase =1, POW.uppercase .33 exceeded. Figure 2 shows four steps in the evolving GGL ofEXT. lowercase =1, POW.lowercase = .33 an interface of communication.EXT. vowel =1, POW.vowel = .33 The hierarchical segmentation for computing complexity inEXT. consonant =1, POW.uppercase = .33 GGL provides also solution for simplicity. Facing complexity,POW.Irvf 2.32 usual solution is to reduce the space problem by deletingEFF.Irvf= .15 components or descriptors. Given that components or features

of real world objects cannot be deleted, this does not changeT more complex than "RIVF", and "Irvf' is the real world object but its processing: processing a part of

more complex than"VF"Inaddtion,themericthe complex object, a part ofthe objects dimensions, or both.better efficiency for "RIVF". .better efficiencyfor"RIVF". ~Hierarchical segmentation theory advocates for carefullyAs seen is the application of the metrics to "RIVF, Rivf and cHangin al w or obeby generli the object

Irvf', little changes can have large effects on complexity. And features tha respond obdectors.the computing of contextual categorization in Generalized for tance,pRiv ca beschgi FonvfGalois Lattices corresponds to the complexity of the gnraingther upprcase or lowRcase, w i crefuly.. A generalizing either uppercase or lowercase, while carefullydistribution of descriptors over categories.There is also dynamicity as a property of some external asesin the efet.fsuhcagsThis iS the way that usability and learnability are improved

world objects. For instance, in an external dynamic system, nHmnCmue nefcs[1 2,adtewypolvalues of dimensions could change, components can be usal'ov polm 2]

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al A2 a3 A a5 afa 11

IV. FOR A MoRE SIMPLE FUTURE: THE CASE OF AIR 111TRAFFICCONTROL 1 1 1

b II 11According to the Hierarchical Segmentation Theory of

contextual categorization, the future world could be simplifiedfor ergonomics or complicated for fun.

For instance, air traffic controllers are managing aircrafts a1 2 i A. M5 Afoperated by individual pilots, moving in different directions at I1different altitudes and with different destinations, without I 1losing track of any, keeping them on course and on time, and I 1 1 1 1not letting any of them collide, or even violate the several very ~strict rules on altitude and distance separation. Air TrafficControl (ARC) is a complex task that several studies haveinvestigated [24, 25].Among sources of complexity, [26] reported factors such as Fig. 1. An academic case of simplifying a test interface: changing the value

overall monitoring, Vertical evolution of aircraft, Incoming of one descriptor of one of its component object greatly simplified theflow, Conflicts, etc. For the evaluation of air traffic external object. Top: the test-interface with aircrafts and the correspondingflow,Conflicts, Boolean matrix and Galois Lattice. Bottom: a similar display with a singlecomplexity, derived variables such as stress and workload [27] change affecting aircraft a4 that reduces complexity.were used as well as controllers vocal measures [28], or Taskrealization: splitting sectors, doing nothing (don't put morecomplexity). For measuring through modeling air traffic V. CONCLUSIONcontrol, dynamic density [29], [30], Non-linear approaches As noticed by [5], there is an understandable wish to[31], Times series [32] and Neural nets [26] were used. measure the complexity of real systems instead than just

Complexity can be reduced. For instance, [24] used color- models of systems. Contextual categorization which is thecoding as a design intervention meant to mitigate complexity categorization of elements according to the others contextualand aid controllers in safe handling of aircraft. As color elements is the computing of the relational links amongintroduced differentiation, more colors should provide more elements of real systems according to the distribution ofbenefit. This was found for three to six coding colors, but not descriptors over the whole set of elements. We adoptedfor six to nine coding colors, that contribute to errors of STONE as the Generalized Galois Lattices Formalism foromission. The authors did not provide how they distributed computing contextual categorization. This allows metrics forcolor to categories of planes. It could be that complexity was computing objective complexity and subjective efficiency andimproved. Complexity metrics helps making prediction about methods for simplifying or complicating the external object atsuch complexity increasing, and help designing more safely hand. Such methods are adapted for virtual environments andinterfaces of risky management of external objects. augmented reality devices for which it is simple to change theOf interest is the possibility of alerting human operators way features are distributed over categories. For real world

about which of the component is a significant contributor of objects, and human operators operating on them, the onlinecomplexity. Such an online survey would have two effects: computation allows a survey of the complexity level and afirst is to make the human operator paying attention to the less planning of operations that can be oriented by a "simplify itdiscriminative component: the one s/he should not correctly first" strategy.process and thus avoiding errors. This could be done by the Generalized Galois Lattices has been applied in perception,underlining of visual properties: for instance having this problem solving and device conception [21], [22], [16]. Aircomponent in 3D. Traffic Control might the kind of dynamic system that can be

Second effect, is about planning. If the operator has to act used for online measure of complexity. Aircrafts, from theon components, then acting on the more problematic case, controller task point of view, have descriptors such sector'swould simplified the rest of the task at hand. The "simplify it" control, next to sector's control, altitude, groundspeed,based-planning is one human strategy that could aid human heading, climbing, or descending level, direction and changingoperators dealing with complex problem. For instance, in direction, clearances, impacted by weather, and so on. Suchfigure 3-bottom, the aircrafts that are sources of complexity systems, with many descriptors, are somewhat much moreare aircrafts a5 and a6 (according to this set of descriptors). complex than the academic cases we used in this paper forAs "Complexity in game settings is almost universally understanding the principles of the contextual categorization

considered desirable", contextual categorization in GGL can approach. Their Generalized Galois Lattices need somebe used to adjust complexity to the game-player desirable simplification rules that are going to be next implemented.level.

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For instance, a rule to be implemented is as follows: in case [16] C. Tijus, "Contextual Categorization and Cognitive Phenomena," inof multiple inheritances, increase the value of links to Modeling and Using Context, V. Akman, P. Bouquet, R. Thomason, and

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