Invited Paper Conflict Management in Linguistic … · In the beginning of the 80’s Botvinnik,...

Invited Paper

Conflict Management in Linguistic Geometry:Applications to EngineeringBoris StilmanDepartment of Computer Science & EngineeringUniversity of Colorado at Denver, Campus Box 109Denver, CO 80217-3364, USAEmail: [email protected]: http://www.cudenver.edu/~bstilman/boris.html

AbstractTo discover the inner properties of human expert heuristics, which were successful in a certainclass of complex control systems, and apply them to different systems, we develop a formaltheory, the Linguistic Geometry*. This research relies on the formalization of search heuristicsof highly-skilled human experts. These heuristics allow to introduce (if necessary) artificialconflict and manage complex system as a two-player opposing pursuit-evasion game. Thefollowing decomposition of the complex system into the hierarchy of dynamic subsystemsallows to solve otherwise intractable problems reducing the search. The hierarchy ofsubsystems is represented as a hierarchy of formal attribute languages. This paper includes abrief survey of Linguistic Geometry, and two comprehensive examples of conflict managementin planning for autonomous aerospace robotic vehicles and in scheduling for power productionplant maintenance. These examples demonstrate that Linguistic Geometry tools generate highquality solutions and provide dramatic search reduction in comparison with conventional searchalgorithms.

1 BackgroundConflict management is required in many real-world systems including variousengineering systems. There are many problems where the conflict between thenumber of opposing sides is inherent. For example, these are problems ofplanning activities of the robotic combat or problems of managing economiccompetition. On the other hand, there is a number of manufacturing problemswhere conflict exists but must be avoided. For example, these are problems ofavoiding of immovable or mobile obstacles in robotic manufacturing like theproblem of efficient control of robotic arms with collision avoidance. In thisclass of problems, usually, there are no opposing sides but conflict exists, and,hence, should be managed. Finally, the third class of problems, which includesproblems of optimal scheduling of manufacturing processes with multipleconstraints, does not include conflict or opposing sides at all. For efficientsolution of problems from the second and third classes employing human expertheuristics we would have to introduce “artificial conflict” and manage it.

* This research is supported in part by Sandia National Laboratories,Albuquerque, NM, USA.

Transactions on Information and Communications Technologies vol 16, © 1996 WIT Press, www.witpress.com, ISSN 1743-3517

There are many real-world problems where human expert skills inreasoning, and, especially, in conflict management are incomparably higherthan the level of modern computing systems. At the same time there are evenmore areas where advances are required but human problem-solving skills cannot be directly applied. For example, there are problems of planning andautomatic control of autonomous agents such as space vehicles, stations androbots with cooperative and opposing interests functioning in a complex,hazardous environment. We encounter the same difficulties trying to solveproblems of optimal scheduling or collision avoidance in manufacturing.Reasoning about such complex systems should be done automatically, in atimely manner, and often in real time. Moreover, there are no highly-skilledhuman experts in these fields ready to substitute for robots (on a virtual model)or transfer their knowledge to them. There is no grand-master in robot controlor scheduling, although, of course, the knowledge of existing experts in thesefields should not be neglected – it is even more valuable. It is very important tostudy human expert reasoning about similar complex systems in the areaswhere the results are successful, in order to discover the keys to success, andthen apply and adopt these keys to the new, as yet, unsolved problems. One ofsuch areas where human experts are undoubtedly successful is conflictmanagement in two-player opposing games.

The question then is what language tools do we have for the adequaterepresentation of human expert skills? An application of such language to thearea of successful results achieved by the human expert should yield a formal,domain independent knowledge ready to be transferred to different areas.Neither natural nor programming languages satisfy our goal. The first areinformal and ambiguous, while the second are usually detailed, lower-leveltools. Actually, we have to learn how we can formally represent, generate, andinvestigate a mathematical model based on the abstract images extracted fromthe expert vision of the problem.

In the beginning of the 80’s Botvinnik, Stilman, and others developed oneof the most interesting and powerful heuristic hierarchical models based onsemantic networks. It was successfully applied to scheduling, planning, control,and computer chess. Application of the developed model to the chess domainwas implemented in full as program PIONEER (Botvinnik, 1984). Similarheuristic model was implemented for power equipment maintenance in anumber of computer programs being used for maintenance scheduling all overthe former USSR (Botvinnik et al., 1983; Stilman, 1985, 1993a). The semanticnetworks were introduced in (Botvinnik, 1984; Stilman, 1977) in the form ofideas, plausible discussions, and program implementations. The major thrust ofour research is to investigate the power and transfer the developed searchheuristics to different problem domains employing formal mathematical tools. In the 60’s, a formal syntactic approach to the investigation of properties ofnatural language resulted in the fast development of a theory of formallanguages by Chomsky (1963), Ginsburg (1966), and others. This developmentprovided an interesting opportunity for dissemination of this approach todifferent areas. In particular, there came an idea of analogous linguisticrepresentation of images. This idea was successfully developed into syntacticmethods of pattern recognition by Fu (1982), Narasimhan (1966), and Pavlidis(1977), and picture description languages by Shaw (1969), Feder (1971), andRosenfeld (1979).

Searching for the adequate mathematical tools formalizing human heuristicsof dynamic hierarchies, we have transformed the idea of linguisticrepresentation of complex real world and artificial images into the idea ofsimilar representation of complex hierarchical systems (Stilman, 1985).


However, the appropriate languages possess more sophisticated attributes thanlanguages usually used for pattern description. The origin of such languages canbe traced back to the research on programmed attribute grammars by Knuth(1968), Rozenkrantz (1969).

A mathematical environment (a “glue”) for the formal implementation ofthis approach was developed following the theories of formal problem solvingand planning by Nilsson (1980), Fikes and Nilsson (1971), Sacerdoti (1975),McCarthy (1980), McCarthy and Hayes (1969), and others, based on the firstorder predicate calculus.

2 Linguistic Geometry: Informal SurveyA formal theory, the Linguistic Geometry - LG, includes the syntactic tools forknowledge representation and reasoning about multiagent discrete pursuit-evasion games. It relies on the formalization of search heuristics, which allowone to decompose the game into a hierarchy of images (subsystems), and thussolve intractable problems by reducing the search.

Linguistic Geometry has been developed as a generic approach to a certainclass of complex systems that involves breaking down a system into dynamicsubsystems. This approach gives us powerful tools for reducing the search indifferent complex problems by decomposing a complex system into a hierarchyof dynamic interacting subsystems. Linguistic Geometry permits us to studythis hierarchy formally, investigating its general and particular properties. Thesetools provide a framework for the evaluation of the complexity and quality ofsolutions, for generating computer programs for specific applications. Inparticular, Linguistic Geometry allowed us to discover the inner properties ofhuman expert heuristics that are successful in a certain class of multiagentdiscrete games. This approach provides us with an opportunity to transferformal properties and constructions from one problem to another and to reusetools in a new problem domain. In a sense, it is the application of the methodsof a chess expert to robot control or maintenance scheduling and vice versa.

An introduction of the hierarchy is as follows. We substitute new multi-goal, multi-level system for the original two-goal one-level system byintroducing intermediate goals and breaking the system down into subsystemsstriving to attain these goals. The goals of the subsystems are individual butcoordinated within the main mutual goal. For example, each second-levelsubsystem includes elements of both opposing sides: the goal of one side is toattack and destroy another side's element (a target), while the opposing sidetries to protect it. In the robot control, it means the selection of a couple ofrobots of opposing sides: one - as an attacking element, and the other - as alocal target, generation of the local paths for approaching the target, as well asthe paths of other robots supporting the attack or protecting the target.

A set of dynamic subsystems is represented as a hierarchy of formallanguages where each sentence - group of words or symbols - of the lower levellanguage corresponds to a word in the higher level one. This is a routineprocedure in our natural language. For example, the phrase: "A man whoteaches students" introduces the hierarchy of languages. The symbols of thelanguage may include all the English words except "professor". The higher-level language might be the same language with addition of one extra word"professor" which is simply a short designation for "A-man-who-teaches-students".

The lowest level subsystems are represented by a Language of Trajectoriesin which expressions are trajectories, the strings of parametric symbolsa(x1)a(x2)...a(xn), in which the parameters incorporate the semantics of some


problem domain. Strings of the type represent paths - trajectories - of system'selements. In this first-level subsystem for a robotic model, xi are coordinates ofthe robot's planning path. In a maintenance scheduling model, an analogousstring represents a maintenance schedule for a specific power unit, and xicorrespond to the specific days of the scheduling period.

The second-level subsystems are represented by a Language of TrajectoryNetworks in which expressions are trajectory networks, denoted as stringscomposed of parametric symbols t(p1,t1,τ1)t(p2,t2,τ2)...t(pk,tk,τk), where pi is an

element of the system, i.e., a robot, ti is an entire trajectory, τi are problemdomain-specific parameters. These networks represent a framework fordynamic tactical planning. The elements move along the network trajectoriesattempting to achieve local goals, while advancing the achievement of thestrategic goal of the entire system, such as victory in a battlefield or the optimalscheduling of a power production system. In different problems there may bemany levels of Trajectory Network Languages representing a hierarchy ofsubsystems.

The entire system operates by changing from one state to another. That is,the motion of an element from one point to another causes an adjustment of thehierarchy of languages. This adjustment is represented as a mapping or atranslation from one hierarchy to another, or to a new state of the samehierarchy. Thus, in the system operation, the search process generates a tree oftranslations of the hierarchy of languages.

In the highest-level formal language, the Language of Translations, this treeof translations is represented as a string of parametric symbols. Each arc of thesearch tree, i.e., each symbol, represents the movement of an element from onepoint to another along the trajectory network. Expressions in the Language ofTranslations are search trees for an optimal (suboptimal) operation, such as theoptimal plan for a robotic combat, or the optimal maintenance schedule.Generation in this language is controlled by interaction of trajectory networks.This generation results in a dramatically reduced search tree which yields asolution of a problem.

Various examples of problems solved employing LG tools have beenpublished (Stilman, 1985-1996). The first pilot implementation of the elementsof the generic hierarchy of formal languages for the 2D case was done at theUniversity of Colorado at Denver in 1993 by King (1993) and Mathews (1993)employing CLIPS tools (Giarratano, 1991) and C language, respectively. A fullsize prototype of the generic Hierarchy of Languages, the testbed for variousapplications, is currently being developed at Sandia National Laboratories.

The first practical application of the Linguistic Geometry was the programfor Real-Time Fire Vehicles Routing developed by R. Turek in 1995. Currentlyit is being converted into a commercial product for the city of Denver to replaceexisting software. Prototype of this program was demonstrated at NASAGoddard Conference on Space Applications of Artificial Intelligence, NASA,Greenbelt, MD, May 1995.

A simulation software prototype for Multi-Robot Cooperation with OptimalPath Planning for Intelligent Manufacturing was developed by Fletcher (1996)at the University of Colorado at Denver and demonstrated at Sandia NationalLaboratories, Albuquerque, NM, in May 1996. This prototype is animplementation of two layers of the Hierarchy of Languages.

Another prototype of the program for the real time generation of anoptimum air combat scenario is currently being developed at the U.S. Air ForcePhillips Laboratory at Kirtland AFB. It is intended to control manned and


unmanned aerial vehicles guided by the satellite based sensors to detect mobileadversarial missile mobile launchers and destroy them. It is based on the firstexample considered in this paper (Section 6).

Application of Linguistic Geometry to the Safety Critical Control Systemsfor Cruise Missiles, Planetary Exploration Vehicles, National Missile Defense(with threat and countermeasure assessments), Space Combat, and others, iscurrently being considered at Sandia National Laboratories. Application of LGto High Integrity Software Development is based on Multiagent Strategic GameApproach to parallel software processes (Yakhnis, Stilman, 1995a, 1995b). Thisapplication is currently being developed at Sandia National Laboratories and atthe University of Colorado at Denver.

Another area for application of the Linguistic Geometry tools is the set ofproblems without explicit opposing agents and mobile units. It is possible tointroduce artificial opposing agents and successfully solve problems of this kindemploying LG tools. In particular, a new approach to scheduling problemsbased on the second example considered in this paper (Section 7) is beingdeveloped at the University of Colorado at Denver.

Sections 3-5 contain a brief survey of Linguistic Geometry tools. On twoexamples we show how these tools generate solutions of the computationallyhard search problems employing very small searches (Sections 6, 7). In Section8 we evaluate solutions and discuss program implementations.

3 Class of Problems

A Complex System is the following eight-tuple:< X, P, Rp, {ON}, v, Si, St, TR>, where

X={xi} is a finite set of points; locations of elements;P={pi} is a finite set of elements; P is a union of two non-intersecting subsets

P1 and P2;Rp(x, y) is a set of binary relations of reachability in X (x and y are from X, p

from P);ON(p)=x, where ON is a partial function of placement from P into X;v is a function on P with positive integer values describing the values of

elements.The Complex System searches the state space, which should have initialand target states;

Si and St are the descriptions of the initial and target states in the language ofthe first order predicate calculus, which matches with each relation acertain Well-Formed Formula (WFF). Thus, each state from Si or St isdescribed by a certain set of WFF of the form {ON(pj) = xk};

TR is a set of operators, TRANSITION(p, x, y), of transitions of the Systemfrom one state to another one. These operators describe the transition interms of two lists of WFF (to be removed from and added to thedescription of the state), and of WFF of applicability of the transition.Here,

Remove list: ON(p) = x, ON(q) = y;Add list: ON(p) = y;Applicability list: (ON(p) = x)^Rp(x,y),

where p belongs to P1 and q belongs to P2 or vice versa. The transitionsare carried out with participation of a number of elements p from P1, P2.

According to the definition of the set P, the elements of the System are


divided into two subsets P1 and P2. They might be considered as units movingalong the reachable points. Element p can move from point x to point y if thesepoints are reachable, i.e., Rp(x, y) holds. The current location of each element isdescribed by the equation ON(p) = x. Thus, the description of each state of theSystem {ON(pj) = xk} is the set of descriptions of the locations of elements.The operator TRANSITION(p, x, y) describes the change of the state of theSystem caused by the move of the element p from point x to point y. Theelement q from point y must be withdrawn (eliminated) if p and q do not belongto the same subset (P1 or P2).

The problem of the optimal operation of the System is considered as asearch for the optimal sequence of transitions leading from the initial state of Sito a target state of St.

It is easy to show formally that a robotic system can be considered as aComplex System (see Section 7). Many different technical and human societysystems (including military battlefield systems, systems of economiccompetition, positional games) that can be represented as twin sets of mobileunits (representing two or more opposing sides) and their locations can beconsidered as Complex Systems.

A formal representation of the Complex System as a Multiagent StrategicGraph-Game is considered in (Yakhnis, Stilman, 1995a, 1995b).

To solve this class of problems, we could use formal methods like those inthe problem-solving system STRIPS (Fikes and Nilsson, 1971), nonlinearplanner NOAH (Sacerdoti, 1975), or in subsequent planning systems. However,the search would have to be made in a space of a huge dimension (for nontrivialexamples). Thus, in practice, no solution would be obtained.

We devote ourselves to finding a solution of a reformulated problem.

4 Set of Paths: Language of Trajectories

This language is a formal description of the set of lowest-level subsystems, theset of all paths between points of the Complex System. An element mightfollow a path to achieve the goal “connected with the ending point” of this path.

A trajectory for an element p of P with the beginning at x of X and the endat y of X (x ≠ y) with a length l is the following formal string of symbols a(x)with points of X as parameters:

to=a(x)a(x1)…a(xl),where xl = y, each successive point xi+1 is reachable from the previous point xi,i.e., Rp(xi, xi+1) holds for i = 0, 1,…, l–1; element p stands at the point x:ON(p)=x. We denote by tp(x, y, l) the set of all the trajectories for element p,beginning at x, end at y, and with length l.

A shortest trajectory t of tp(x, y, l) is the trajectory of minimum length forthe given beginning x, end y, and element p. Trajectories constructed of kshortest trajectories are called admissible trajectories of degree k.

Properties of the Complex System permit us to define (in general form) andstudy formal grammars for generating the shortest trajectories (Stilman, 1993a). A Language of Trajectories Lt

H(S) for the Complex System in a state S isthe set of all the trajectories of length less or equal H. Various properties of thislanguage and generating grammars were investigated in (Stilman, 1993a).

To construct and study the Language of Trajectories for the ComplexSystem we have to investigate geometrical properties of this System and learnhow to measure distances between points.


A map of the set X relative to the point x and element p for the ComplexSystem is the mapping:

MAPx,p: X —> Z+(where x is from X, p is from P), which is constructed as follows. We considera family of reachability areas from the point x, i.e., a finite set of the followingnonempty subsets {Mk

x,p} of X (Figure 1):k=1: Mk

x,p is a set of points m reachable in one step from x: Rp(x,m)=T;k>1: Mk

x,p is a set of points reachable in k steps and not reachable in k-1steps, i.e., points m reachable from points of Mk-1

x,p and not included in anyMi

x,p with i less than k.

X

x

M M M

M 1 x,p

x,p x,p x,p 2 3 4

Figure 1: Interpretation of the family of reachability areas.

Let MAPx,p(y)=k, for y from Mkx,p (the number of steps from x to y). For the

remaining points, let MAPx,p(y)=2n, if y≠x (n is the number of points in X);MAPx,p(y) = 0, if y = x.

It is easy to verify that the map of the set X for the specified element p fromP defines an asymmetric distance function on X:

1. MAPx,p(y) > 0 for x≠y; MAPx,p(x) = 0;2. MAPx,p(y)+MAPy,p(z) ≥ MAPx,p(z).

If Rp is a symmetric relation,3. MAPx,p(y) = MAPy,p(x).

In this case each of the elements p from P specifies on X its own metric.Examples of distance measurements and trajectory generation for various

problems are considered in (Stilman, 1993a, 1993c, 1994b).

5 Networks of Paths: Languages of Trajectory Networks

After defining the Language of Trajectories, we have new tools for thebreakdown of our System into subsystems. These subsystems should be varioustypes of trajectory networks, i.e., the sets of interconnected trajectories with onesingled out trajectory, called the main trajectory. An example of such networkis shown in Figure 2. The basic idea behind these networks is as follows.Element po should move along the main trajectory a(1)a(2)a(3)a(4)a(5) toreach the ending point 5 and remove the target q4 (an opposing element).Naturally, the opposing elements should try to disturb those motions bycontrolling the intermediate points of the main trajectory. They should comecloser to these points (to the point 4 in Figure 2) and remove element po afterits arrival (at point 4). For this purpose, elements q3 or q2 should move alongthe trajectories a(6)a(7)a (4) and a(8)a(9)a(4), respectively, and wait (ifnecessary) at the next to last point (7 or 9) for the arrival of element po at point


4. Similarly, element p1 of the same side as po might try to disturb the motionof q2 by controlling point 9 along the trajectory a(13)a(9). It makes sense forthe opposing side to include the trajectory a(11)a(12)a(9) of element q1 t oprevent this control.

Similar networks are used for the breakdown of complex systems indifferent areas. Let us consider a linguistic formalization of such networks. TheLanguage of Trajectories describes "one-dimensional" objects by joiningsymbols into a string employing a reachability relation Rp(x, y). To describenetworks, i.e., “multi-dimensional" objects made up of trajectories, we use therelation of trajectory connection.

A trajectory connection of the trajectories t1 and t2 is the relation C(t1,t2). Itholds if the ending link of the trajectory t1 coincides with an intermediate linkof the trajectory t2; more precisely, t1 is connected with t2 if among theparameter values P(t2)={y,y1,…,yl} of trajectory t2 there is a value yi = xk,where t1=a(xo)a(x1)…a(xk). If t1 belongs to a set of trajectories with thecommon end-point, then the entire set is said to be connected with t2.

For example, in Figure 2 the trajectories a(6)a(7)a(4) and a(8)a(9)a(4) areconnected with the main trajectory a(1)a(2)a(3)a(4)a(5) through point 4.Trajectories a(13)a(9) and a(11)a(12)a(9) are connected with a(8)a(9)a(4).

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Figure 2: Network language interpretation.

To formalize the trajectory networks, we define and use routine operationson the set of trajectories: CA

k(t1,t2), a k-th degree of connection, andCA

+(t1,t2), a transitive closure.Trajectory a(11)a(12)a(9) in Figure 2 is connected degree 2 with trajectory

a(1)a(2)a(3)a(4)a (5), i.e., C2(a(11)a(12)a (9), a(1)a(2)a(3)a(4)a(5)) holds.Trajectory a(10)a(12) in Figure 2 is in transitive closure to the trajectorya(1)a(2)a(3)a(4)a(5) because C3(a(10)a(12), a(1)a(2)a(3)a(4)a(5)) holds bymeans of the chain of trajectories a(11)a(12)a(9) and a(8)a(9)a(4).

A trajectory network W relative to trajectory to is a finite set of trajectoriesto,t1,…,tk from the language Lt

H(S) that possesses the following property: forevery trajectory ti from W (i = 1, 2,…,k) the relation CW

+(ti,to) holds, i.e., eachtrajectory of the network W is connected with the trajectory to that was singledout by a subset of interconnected trajectories of this network. If the relationCW

m(ti, to) holds, i.e., this is the m-th degree of connection, trajectory ti iscalled the m negation trajectory.


Obviously, the trajectories in Figure 2 form a trajectory network relative tothe main trajectory a(1)a(2)a(3)a(4)a(5). We are now ready to define networklanguages.

A family of trajectory network languages LC(S) in a state S of the ComplexSystem is the family of languages that contains strings of the form

t(t1, param)t(t2, param)…t(tm, param),where param in parentheses substitute for the other parameters of a particularlanguage. All the symbols of the string t1, t2,…, tm correspond to trajectoriesthat form a trajectory network W relative to t1.

Different members of this family correspond to different types of trajectorynetwork languages, which describe particular subsystems for solving searchproblems. One such language is the language that describes specific networkscalled Zones. They play the main role in the model considered here (Botvinnik,1984; Stilman, 1977, 1993b, 1993c, 1994a). A formal definition of thislanguage is essentially constructive and requires showing explicitly a methodfor generating this language, i.e., a certain formal grammar, which is presentedin (Stilman, 1993b, 1993c, 1994a). In order to make our points transparent here,we define the Language of Zones informally.

A Language of Zones is a trajectory network language with strings of theform

Z=t(po,to,τo) t(p1,t1,τ1)…t(pk,tk,τk),where to, t1,…, tk are the trajectories of elements po, p2,…, pk respectively;

τo,τ1,…,τk are nonnegative integers that “denote time allotted for the motionalong the trajectories” in agreement with the mutual goal of this Zone: toremove the target element – for one side, and to protect it – for the opposingside. Trajectory t(po,to,τo) is called the main trajectory of the Zone. Theelement q standing at the ending point of the main trajectory is called the target.The elements po and q belong to the opposing sides.

Consider the Zone corresponding to the trajectory network in Figure 2.Z=t(po,a(1)a(2)a(3)a(4)a(5), 4)t(q3,a(6)a(7)a(4), 3)t(q2, a(8)a(9)a(4), 3)

t(p1, a(13)a(9), 1)t(q1, a(11)a(12)a(9), 2) t(p2, a(10)a(12), 1)Let us assume that in this example all the elements can move simultaneously..Assume, also, that the goal of the white side is to remove target q4, while thegoal of the black side is to protect it. According to these goals, element pobegins motion to the target while elements q2 or q3 move to intercept elementpo. Actually, only those black trajectories are to be included into the Zonewhere the motion of the element makes sense, i. e., the length of the trajectoryis less than the amount of time (third parameter τ) allocated to it. For example,the motion along the trajectories a(6)a(7)a(4) and a(8)a(9)a(4) makes sense,because they are of length 2 and time allocated equals 3: each of the elementshas 3 time increments to reach point 4 to intercept element po assuming onewould go along the main trajectory without move omission. According todefinition of Zone, the trajectories of white elements (except po) could only beof the length 1, e.g., a(13)a(9) or a(10)a(12). As element p1 can intercept themotion of the element q2 at the point 9, black includes into the Zone thetrajectory a(11)a(12)a(9) of the element q1, which has enough time for motionto prevent this interception. The total amount of time allocated to the wholebundle of black trajectories connected (directly or indirectly) with the givenpoint of the main trajectory is determined by the number of that point. For


example, for the point 4, it equals 3 time increments.A language LZ

H(S) generated by the certain grammar GZ (Stilman, 1993b,1993c, 1994a) in a state S of a Complex System is called the Language ofZones.

6 Natural Conflict in Robot ControlFor the robot control model the set X of the Complex System (Section 3)represents the operational district, which could be the area of combat operation,broken into smaller 2D or 3D areas, “points”, e.g., in the form of the big 2D or3D grid. It could be a space operation, where X represents the set of differentorbits, or an air force battlefield, etc., P is the set of robots or autonomousvehicles. It is broken into two subsets P1 and P2 with opposing interests;Rp(x,y) represents moving capabilities of different robots for different problemdomains: robot p can move from point x to point y if Rp(x, y) holds. Some ofthe robots can crawl, others can jump or ride, sail and fly, or even move fromone orbit to another. Some of them move fast and can reach point y (from x) in“one step”, i.e., Rp(x, y) holds, others can do that in k steps only, and many ofthem can not reach this point at all. ON(p) = x, if robot p is at the point x; v(p)is the value of robot p. This value can be determined by the technicalparameters of the robot. It might include the immediate value of this robot forthe given combat operation. Si is an arbitrary initial state of operation foranalysis, or the start state; St is the set of target states. These might be the stateswhere robots of each side reached specified points. On the other hand, St canspecify states where opposing robots of the highest value are destroyed. The setof WFF {ON(pj) = xk} corresponds to the list of robots with their coordinates ineach state. TRANSITION(p, x, y) represents the move of the robot p from thelocation x to location y; if a robot of the opposing side stands on y, a removaloccurs, i.e., robot on y is destroyed and removed.

6.1 2D Concurrent Model: Problem Statement

Robot-aircraft with various moving capabilities are shown in Figure 3. Theoperational district X is the table 8 x 8. Robot W-FIGHTER (White Fighter)standing on h8, can move to any next square (shown by arrows). The otherrobot B-BOMBER (Black Bomber) from h7 can move only straight ahead, onesquare at a time, e.g., from h7 to h6, from h6 to h5, etc. Robot B-FIGHTER(Black Fighter) standing on a6, can move to any next square similarly to therobot W-FIGHTER (shown by arrows).

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a b c d e f g hFigure 3: 2D problem with totally concurrent motions.


Robot W-BOMBER (White Bomber) standing on c6 is analogous with therobot B-BOMBER; it can move only straight ahead but in reverse direction.Thus, robot W-FIGHTER on h8 can reach any of the points y ∈ {h7, g7, g8} inone step, i.e., RW-FIGHTER(h8, y) holds, while W-BOMBER can reach only c7in one step.

Assume that robots W-FIGHTER and W-BOMBER belong to one side,while B-FIGHTER and B-BOMBER belong to the opposing side: W-FIGHTER ∈ P1, W-BOMBER ∈ P1, B-FIGHTER ∈ P2, B-BOMBER ∈ P2.Also, assume that two more robots, W-TARGET and B-TARGET, (unmovingdevices or targeted areas) stand on h2 and c8, respectively. W-TARGETbelongs to P1, while B-TARGET ∈ P2. Each of the BOMBERs can destroyunmoving TARGET ahead of the course. Each of the FIGHTERs is able todestroy an opposing BOMBER approaching its location, but it also able todestroy an opposing BOMBER if this BOMBER itself arrives at the currentFIGHTER’s location. For example, if the B-FIGHTER is at location c8 and W-BOMBER arrives there (unprotected) then during the same time increment itdestroys the TARGET and is destroyed itself by B-FIGHTER. Each BOMBERcan be protected by its friendly FIGHTER if the latter approached theBOMBER’s prospective location. In this case the joint protective power of thecombined weapons of the friendly BOMBER and FIGHTER can protect theBOMBER from interception. For example, W-FIGHTER located at d6 canprotect W-BOMBER on c6 and c7.

Each of the BOMBERs is vulnerable not only to a FIGHTER’s attack butalso to the explosion of another BOMBER. If W-FIGHTER hits B-BOMBERwhile the latter is fully armed, i.e., it is not at its final destination – square h2,and W-BOMBER is moving during the same time increment, it will bedestroyed as a result of the B-BOMBER’s explosion. If W-BOMBER is notmoving at this moment it is safe. Similar condition holds for B-BOMBER: itcan not move at the moment when W-BOMBER is being destroyed (excludingc8).

The combat considered can be broken into two local operations. The firstoperation is as follows: robot B-BOMBER should reach point h2 to destroy theW-TARGET, while W-FIGHTER will try to intercept this motion. The secondoperation is similar: robot W-BOMBER should reach point c8 to destroy the B-TARGET, while B-FIGHTER will try to intercept this motion. After destroyingthe opposing TARGET and keeping the BOMBER safe, the attacking side isconsidered as a winner of the local operation and the global combat. The onlychance for the opposing side to avenge is to hit its TARGET at the same timeincrement and this way end the battle in a draw. The conditions consideredabove give us St, the description of target states of the Complex System. Thedescription of the initial state Si follows from Figure 3 .

Assume that all the units of the opposing sides can move simultaneously.With respect to other examples of serial (Sections 7-9) and concurrent systems(Stilman, 1995a, 1996c) in this model there is no alternation of turns. It means,for example, that during the current time increment, all four vehicles, W-BOMBER, W-FIGHTER, B-BOMBER, and B-FIGHTER, three of them, two,one, or none of them can move. Hence, this is a model with incompleteinformation about the current move (before it is done). When moving, each sidedoes not know the opposing side component of the concurrent move, i.e., theimmediate opposing side motions, if they are not constrained to one or zeromotions and, thus, can be predicted. Moreover, after developing a strategy eachside can not follow it because of the uncertainty with the other side current


motions. However, if the strategy includes only variations of concurrent moveswith single “universal” component (group of motions) for one side good for allpossible components of the other side, this strategy can be actuallyimplemented. It seems that local operations (Figure 3 ) are independent, because they arelocated far from each other. Moreover, the operation of B-BOMBER from h7looks like unconditionally winning operation, and, consequently, the globalbattle can be easily won by the Black side.

Is there a strategy for the White side to make a draw?

6.2 2D Concurrent Model: Search

Consider how the hierarchy of languages works for the optimal control of thismodel. We have to generate the Language of Trajectories and the Language ofZones in each state of the search. The details of these generations areconsidered in (Stilman, 1993b, 1993c, 1993d, 1994a). We generate the string ofthe Language of Translations (Stilman, 1994a) representing it as a search tree(Figure 5) and comment on its generation. This tree is different from theconventional search trees. Every concurrent move is represented by twoconsecutive arcs. The arc outgoing the white node represents the Whitecomponent of a concurrent move, the concurrent motions of the White side,while the arc outgoing the black node represents the Black component of thesame move.

First, the Language of Zones in the start state is generated. Every unit triesto attack every opposing side unit. The targets for attack are determined withinthe limit of five steps. It means that horizon H of the language LZ(S) is equal to5, i.e., the length of main trajectories of all Zones must not exceed 5 steps. Thealgorithm for choosing the right value of the horizon is considered in (Stilman,1994c). All the Zones generated in the start state are shown in Figure 4.

Figure 4: Zones in the start state.

Zones for the FIGHTERs as attacking elements are shown in the left diagram,while Zones for BOMBERs – in the right one.

Generation begins with the concurrent move 1. c6-c7 a6-b7 in the WhiteZone with the vulnerable Black target of the highest value and the shortest maintrajectory. The order of consideration of Zones and particular trajectories isdetermined by the grammar of translations.

The Black component of this move, 1. ... a6-b7, is in the same Zone alongthe first negation trajectory. The interception continues: 2. c7-c8 b7-c8/h7-h6(Figure 6, left). This is a triple move. During the second time increment W-BOMBER hit the TARGET at c8 and was destroyed by the B-FIGHTER at c8.Also, immediately, the attack Zone of the B-BOMBER from h7 to h2 wasactivated: h7-h6 is the motion during the same time increment. Here thegrammar terminates this branch with the value -1 (as a win of the Black side).


This value is given by the special branch termination procedure built into thegrammar. This procedure determined that W-FIGHTER is out of the Zone of B-BOMBER, thus, it can not intercept B-BOMBER which means that the latterwill successfully hit the TARGET on h2.

c6-c7 a6-b7

c7-c8 b7-c8-1

h8-g7

b7-c8h7-h6

-1c7-c8

c7-c8g7-f6

h6-h5

c6-c7

h7-h6a6-b7

-1c7-c8 b7-c8

c7-c8g7-f6

b7-c8h6-h5 -1

g7-f6

-1c7-c8f6-e5

h5-h4

f6-e5

h7-h6 c7-c8g7-f6

h6-h5

a6-b7 g7-h7c7-c8 b7-c8 0

g7-f6

c7-c8d6-d7

h3-h2

a6-b7 c7-c8 b7-c8-1

h8-g7c7-c8 b7-c8 -1

h8-g7

h8-g7c7-c8 -1

h8-g7h7-h6

-1c7-c8

c7-c8 h5-h4 -1f6-e5

f6-e5

-1

e5-d6 h4-h3

h7-h6

b7-c8

b7-c8 h6-h5

h6-h5 h5-h4

h5-h4

h6-h5

h6-h5

h6-h5g7-f6 -1c7-c8

c7-c8 -1f6-e5

f6-e5 e5-d6 h3-h2

h5-h4 h4-h3

h4-h3

h4-h3

0

h7-h6

-1h6-h5

-1c7-c8

h6-h5b7-c8 h5-h4

h5-h4 c7-c8d6-d7

h3-h2e5-d6 h4-h3 0

+1

h7-h6

Figure 5: Search tree for the 2D Concurrent Model.

Then, the grammar initiates the backtracking climb. Each backtrackingmove is followed by the inspection procedure, the analysis of the subtreegenerated so far. After the climb up to the move 1. c6-c7 a6-b7, the subtree tobe analyzed consists of one branch (of one move): 2. c7-c8 b7-c8/h7-h6. Theinspection procedure determined that the current minimax value (-1) can be“improved” by the improvement of the exchange on c8 (in favor of the Whiteside). This can be achieved by participation of W-FIGHTER from h8, i.e., bygeneration and inclusion of the new so-called “control” Zones with the maintrajectory from h8 to c8. These Zones were detected (within the horizon 5) inthe terminal state after the move 2. c7-c8 b7-c8/h7-h6, Figure 6 (left).Obviously they could not be detected in the start state of this problem (Figure 3)


because the main element, W-BOMBER, could not “see” the target, B-FIGHTER, within given horizon. Also, at the moment of detection it was toolate to include them into the search. These Zones have been stored and kept idlefor possible activation at the higher levels of the search tree. The set of differentZones from h8 to c8 (the bundle of Zones) is shown in Figure 6 (right). Themove-ordering procedure picks the subset of Zones with main trajectoriespassing g7. These trajectories partly coincide with the main trajectory ofanother Zone attacking the opposing W-BOMBER on its future location h6.The motion along such trajectories allows to “gain time”, i.e., to approach twogoals simultaneously.

The generation continues with the simultaneous motion of all four agents,the four-move, W-BOMBER, W-FIGHTER and B-FIGHTER, B-BOMBER, intheir respective Zones: 2. c7-c8/h8-g7 b7-c8/h7-h6. The B-FIGHTERintercepted W-BOMBER at c8 while W-FIGHTER is unable to intercept the B-BOMBER during its attack from h6 to h2. The branch termination proceduredetermined that W-FIGHTER is outside the B-BOMBER’s attack Zone,terminated this branch, evaluated it as a win for the Black (-1), and initiated thebacktracking climb. Move 2. ... was changed for the triple move 2. h8-g7 b7-c8/h7-h6 in attempt to find a better combination of White motions.

Black side, after finding b7-c8/h7-h6 to be a “good” component of theconcurrent move 2. in the previous branches, continues to include thiscomponent in the following branches. Obviously, this component is veryimportant. As it was noted above, a totally concurrent model is a model withincomplete information. Each side knows all the previous moves, the history ofoperation, and, theoretically, all possible future outcomes of the current move,the look-ahead tree. The only thing it does not know is the concurrent action ofthe opposing side as a component of the current move. Thus, for each side it isimportant to find not just a “good” own component of a concurrent move but acomponent to be “good” for all components of the opposing side.

Figure 6: States where the control Zone from h8 to c8 was detected (left) andwhere it was included into the search (right)

Figure 7: States where the control Zones from g7 to c7, c8 were detected (left)and where they were included into the search (right).


Figure 8: States where the control Zones from f6 to c7, c8 were detected (left)and where they were included into the search (right).

Such component would allow to avoid uncertainty in constructing an optimalvariation, a branch, which can be implemented. A component b7-c8/h7-h6 is acandidate to be a good one for the Black while h8-g7 is a candidate for White.

After 2. h8-g7 b7-c8/h7-h6 termination procedure did not terminate thebranch, and continued 3. c7-c8 h6-h5 in the same Black and White Zones. Thenit terminated the branch and evaluated it as a win (-1) for the Black side (Figure7, left). Indeed, W-BOMBER hit B-TARGET on c8 but it is being destroyeditself by B-FIGHTER which was waiting for it at c8. Also, W-FIGHTER againis out of the attack Zone of B-BOMBER from h5 to h2. In this state a set ofnew control Zones of W-FIGHTER from g7 to c8 were detected and stored asidle to be activated later if necessary. New climb up to the move 2. h8-g7 b7-c8/h7-h6 and execution of theinspection procedure result in the inclusion of the groups of new control Zonesfrom g7 to c7 and c8 in order to improve the exchanges at these locations. Bothgroups of Zones (to c7 and c8) have been detected earlier in the search tree. Theset of Zones with different main trajectories from g7 to c7 and from g7 to c8 isshown in Figure 7 (right). Besides that, the trajectories from g7 to h4, h3, andh2, are shown in the same Figure 7. These are “potential” first negationtrajectories of the concurrent Zone which is similar to the serial Zone (Section5). It means that beginning with the second symbol a(f6), a(g6) or a(h6) thesetrajectories become first negation trajectories in the Zone of B-BOMBER onh6. Speaking informally, from the squares f6, g6, and h6, Zone gateways, W-FIGHTER can intercept B-BOMBER. The move-ordering procedure picks thesubset of Zones with the main trajectories passing f6. These trajectories partlycoincide with the potential first negation trajectories. The motion along suchtrajectories allows to “gain time”, i.e., to approach two goals simultaneously.Thus, the new White component 3. c7-c8/g7-f6 is included with the same Blackcomponent 3. ... h6-h5, the branch was terminated with the value -1. Thefollowing climb and branching with inclusion of g7-f6 as a single motioncomponent resulted in 3. g7-f6 h6-h5, and the branch is not terminated. Itcontinues with the move 4. c7-c8 h5-h4. This state is shown in Figure 8, left.Then this branch is terminated with the value -1. As usual, this value wasassigned by the termination procedure which detected that W-FIGHTER isoutside the Zone of B-BOMBER and thus does not have enough time forinterception.

After the climb, the grammar continued branching 4. c7-c8/f6-e5 h5-h4.The component f6-e5 is selected by the move ordering procedure as the time-gaining move approaching two goals simultaneously, c7 as a goal of the controlZone of W-FIGHTER and one of the gateways (e5, f5, g5) of the Zone of B-BOMBER (Figure 8, right). But it was also terminated with the value -1. After


4. f5-e5 h4-h3, 5. e5-d6 h4-h3, and 6. c7-c8/d6-d7 h3-h2, the branch isterminated with the value of 0.

It seems that the sought draw is found. The following climb with activationof the inspection procedure in every node ended at the top level. All theattempts of the Black to change the components 4. ... h5-h4, 3. ... h6-h5, 2. ...h7-h6 for a different motion failed. If B-BOMBER's motion is not included inthese concurrent moves the W-FIGHTER appears in the B-BOMBER's attackZone and these branches should be terminated with the value 0 which does notimprove the current minimax value for Black.

The Black component of 1. c6-c7 a6-b7 was changed for the doublemotions 1. c6-c7 a6-b7/h7-h6. It seems that this move almost depreciatedprevious search. The minimax value brought to the top of the subtree outgoingthis move is -1. However, the tree generation followed after the change of 1. c6-c7 a6-b7/h7-h6 for the double move 1. c6-c7/h8-g7 a6-b7/h7-h6 showed thatprevious search was very important. As a result of this search the grammarlearned key networks, Zones of W-FIGHTER with main trajectories from g8 toc8, from g7, f6 to c7 and c8. The optimal branch is shown in Figure 5 with boldlines.

A different version of this problem with concurrent motions of thecooperating units only is considered in (Stilman, 1995c).

7 Artificial Conflict in Scheduling

In this section we consider a computationally hard problem without naturalconflict. To apply Linguistic Geometry tools we map this problem into the classof Complex Systems, introduce the opposing sides, the conflict, and manage it.

Assume that energy-producing company is going to set up a maintenanceplan for power-producing equipment for a given planning period Tmax, e.g., amonth or a year. An array of m demands for maintenance work of power unitsis generated. The problem is to satisfy these demands. To do that we mustinclude the maintenance work for all the demanded units into the plan, i.e., toschedule maintenance. A maintenance work of a power unit causes turning offof this unit, and, consequently, a fall of generating power in the system. Thus, itis impossible to satisfy all the demands because of problem constraints, whichis basically the power reserve, i.e., the amount of power to be lost withoutturning off customers. This amount varies daily.

Each demand requests maintenance work for one power unit (j-th unit) andcontains three attributes: wj, the demanded power of the unit; hj, the fall (loss)in the operating power of the energy-producing system because of maintenanceof this unit (resources requirement); and xj

max, required duration ofmaintenance. For simplicity, we neglect the rest of the demand parameters. Forthe same reason we specify the only type of constraints function, f(i) of powerreserve for the energy-producing system, where i is the number of a day of theplanning period. On the i-th day of the planning period the total fall (loss) in theoperating power, because of the maintenance of some power units, can not begreater than the value f(i). The values of all the parameters are positive integernumbers. The optimum criterion of the plan is the maximum total demandedpower of the units being maintained.

Consider a simplified maintenance scheduling problem shown in Figure 9.We have to maintain one power unit. Assume that maintenance is an immediateaction without actual duration, and that maintenance work demanded for thisunit requires 3 units of one type of resources and 4 units of another type, e.g., 3Megawatt of power reserve and 4 teams of maintenance personnel. Assume also


that the power reserve in the amount of 5 Megawatt is readily available, whilethe personnel should be moved from the remote location, and this moverequires at least two days.

5

6

3

4

r2r1

Figure 9: Resources requirements as opposing side

We represent the requirements of resources as two pyramids of black disks.Both pyramids are one move away from the power unit to be maintained. Thismeans that motion of any of this disks to the location of the power unit willdestroy it: back disks “control” the maintenance. Our purpose is to remove thiscontrol. This can be done employing available resources represented aspyramids of white disks, e.g., the power reserve and the maintenance personnel.The pyramid of the power reserve is one move away from the location r1, andtwo moves away from the location of the black pyramid of the powerrequirements. Thus, in two steps white disks of power reserve can reach theblack pyramid and destroy it. But the black disks can prevent this employingthe control path leading to the location r1 of the length 1. Hence, when a whitedisk arrives on r1, a black disk of power requirements can move to the samelocation and destroy this white disk. This way, 3 white disks can be exchangedfor 3 black disks on r1, which means that we supplied required resources ofpower reserve. Analogously we can supply personnel teams exchanging 4 whitedisks with 4 black ones on r2. The only difference is that the pyramid of humanresources is 3 steps away from location r2 of the prospective exchange. This isbecause the move of the personnel teams requires two days.

7.1 Conversion into Two-Player Game

Now we are ready to introduce maintenance with the non-zero duration.Consider an example of maintenance of two power units p1 and p2 (Figure 10).This is the ”toy” maintenance scheduling problem for two units over a period of


three days where the maintenance of each of the power units requires two days:w1=5, w2=2; h1=3, h2=2; x1

max=x2max=2; Tmax=3; f(1)=4; f(2)=5; f(3)=3.

A reader should not be confused by the simplicity of the example shown inthe Figure 10. It is cited here only for clarification of our approach. For thepractical applications hundreds and even thousands of power units wereconsidered. Moreover, different kinds of resources were taken into accountincluding those which required some time to be delivered to the places ofmaintenance.

p1

Qfall

11

Pres

1P

2

res

q1

Qfall

21

Pres

31 1 1

Q12fall

22

p2

Qfall

Pres

1P2

res

q 2

Pres

32 2 2

Figure 10: Two separate networks for the two-unit scheduling problem

Unit p1 should move along the path of the length 3 and destroy the target q1,while unit p2 should destroy target q2. The values of these targets represent thedemanded power of each of the units, i.e., w1 and w2. The pyramids of blackdiscs Qij

fall represent requirements of resources, the daily fall in the operatingpower in case of the maintenance of the units p1 and p2. The black discs controlthe nodes of paths for discs p1, p2 and able to destroy each of them. The valueof white disks p1 and p2 is considered as +∞, i.e., maintenance can not takeplace without provision of resources. Elements depicted as pyramids of whitedisks Pijres represent daily stocks of resources, the power reserve for the power-producing system, f(1), f(2), f(3). It means we are forced to spend white discs ofpyramids Pij

res exchanging them with the black discs of Qijfall. These actions

can "clear away" the paths for the power units p1 and p2. The power reserve for


the third day Pi3res looks “unnecessary”. Maintenance of p1 and p2 is shown in

Figure 10 in the form of separate networks, but actually the daily stock ofresources for both power units is common. This means that white pyramidsP1jres and P2jres for j =1, 2, 3, respectively, must be the same (Figure 11).

p1

Qfall

11Q

fall

21

Q12fall

22

p2

Qfall

Pres

1P2

resP

3

res

q2

q

(1,1,r) (1,0,r) (2,1,r) (2,0,r) (3,0,r)

(0,0,p) 1

(1,0,p) 1

(2,0,p) 1

(g,0,p)1

(0,0,p) (1,0,p) (2,0,p) (g,0,p) 2 2 22

1

Figure 11: Preliminary network representation for the two-unitscheduling problem

For liquidation of the elements from Qfall we have three sets (pyramids ofdisks) P1res, P2res, and P3res at the points corresponding to the power reserve inthe system during each particular day. It is necessary to carry out a transition,i.e., to move an element from P1

res to the point of exchange, then move anelement from Q12

fall to the same point, i.e., to perform a "capture", then movethe next element from P1

res, and so forth.In the given example the pyramids are placed one step away from the points

of exchange. It means the instantaneous availability of resources in the givenproblem. For complex real-world problems the pyramids of resources have tobe placed several steps away from the points of exchange, which means thatresource delivery should start in advance, in several time intervals. Returning to our example, if at the point (1, 1, r) it is possible to exchangeall the elements from Qfall, then point (1, 0, p2) becomes traversable freely forthe element p2. In our example, however, it is not possible (as is in fact shownin Figure 12), owing to the fact that three elements of the pyramid P1

res werespent on removing the control from the point (1, 0, p1), i.e., on liquidatingQ11

fall, and the remaining single element is not sufficient for destroying the twoelements of Q12

fall. Thus, element p2 can not move along its’ path, i.e., on thefirst day of the planning period, only one of the power units (p1, for example)can be taken out for maintenance because of the insufficiency of the powerreserve.


p1

Q12fall

22

p2

Qfall

Pres

1P2

resP

3

res

q 2

q1

(1,1,r) (1,0,r) (2,1,r) (2,0,r) (3,0,r)

(0,0,p) 1

(1,0,p) 1

(2,0,p) 1

(g,0,p) 1

(0,0,p) (1,0,p) (2,0,p) (g,0,p) 2 2 2 2

Figure 12: Representation of the system state after providing resources for themaintenance of the power unit p1

The second unit p2 can be taken out on the second day. To represent this wehave to introduce a new path (Figure 13):

(0,0,p2)–>(0,1, p2)–>(1,1,p2)–>(2,1,p2)–>(g,0,p2).Thus, unit p2 is forced to move to the point (0, 1, p2) which means to stay out ofmaintenance during the first day. On the second day it can move from (0, 1, p2)to (1, 1, p2) because the amount of resources, pyramid P2res at (2, 0, r), issufficient to do away the opposite side control from both points (2, 0, p1) and(1, 1, p2 ) for units p1 and p2, respectively. Different versions of themaintenance plan are matched by different variants of movement of disksshown in Figure 13.

7.2 Formal Representation

In terms of Complex Systems (Section 3), this problem can be representedas a twin-set of elements and points, as shown in Figure 13. Here points form anetwork which is used by elements as a "railroad" to reach certain nodes. Thereare two classes of elements. The first one includes power units, depicted aswhite discs p1, p2, striving to reach nodes (g, 0, p1) and (g, 0, p2) and therebygain opposite elements q1, q2 ( i.e., the ones to be maintained).

The general formal representation of the Complex System for themaintenance problem is as follows:

X = (Y ∪ {g}) × Y × (Pdem∪ Qdem∪ {r}), where Y={0,1,...,Tmax},

Pdem is the set of power units included in the demands, |Pdem| is the number ofdemands.


(0,0,p) 1

(1,0,p) 1

(2,0,p) 1

(g,0,p) 1

p1

Qfall11

Qfall

21

(1,1,r) (1,0,r) (2,1,r) (2,0,r) (3,1,r) (3,0,r)

(1,0,q) (2,0,q) 1 1

2 2 2(1,0,q) (2,0,q) (3,0,q)

(0,0,p) (1,0,p) (2,0,p) (g,0,p) 2 2 2 2

(0,1,p) (1,1,p) (2,1,p) 2 2 2

Q12fall

22

p2

Qfall

Qfall

23

Pres1

P2

resP

3

res

q 2

q1

Figure 13: Network for the two-unit maintenance scheduling problem

It is introduced a duplicate set Qdem of the elements qj, and one-to-onecorrespondence qj <—> pj is established between the elements of Qdem andPdem,

P = P1 ∪ P2, P1 and P2 are not intersected and

P1=Pdem ∪ Preserve, P2 = Qdem∪ Qfall,

Tmax Tmax Qdem

Preserve= ∪ Pires, Qfall = ∪ ∩ Qij

fall

i=1 i=1 j=1

To show the number of elements |Preserve| and |Qreserve| we have to definevo. It is the quantum of power fall (loss), the common factor of all values f(i) ofpower reserve and all values hj of power fall (for all demanded units); forexample, vo=1 Megawatt. We can now determine |Preserve| and |Qfall|, havinggiven |Pi

res| and |Qijfall|. Thus, |Pi

res|=f(i)/vo and |Qijfall|=hj/ vo.

The relation of reachability Rp(x,y) can be given explicitly by setting thevalues for all the triples of p, x, y:


((x=(0, y2, p)) ∧ (y - (0, y2+1, p))) ∨ ((x=(y1, y2, p)) ∧

((y=(y1+1, y2, p))) ∧ ((x=(ximax, y2, p)) ∧ (p=pi) ∧

(y=(g, 0, p))), if p ∈ Pdem;

((x=((y1, 0, qj)) ∧ (y1>0)) ∧ (((y=(y1-y2, y2, pj)) ∧

Rp(x,y) = ((y1-y2) > 0)) ∨ (y=(y1, 1, r))), if p ∈ Qy1j

fall ⊂ Qfall;

(((x=(y1, 0, r)) ∧ (y=(y1, 1, r))) ∨ ((x=(y1, 1, r)) ∧

(y=(y1, 0, qj)) ∧ (y1>0) ∧ (qj ∈ Qdem ), if p ∈ Preserve;

F (false), if p ∈ Qdem.Note that here the reachability relation Rp is asymmetric, i.e., there exist p,

x, and y such that Rp(x, y)≠Rp(y, x). To specify the partial function ON(p), it issufficient to write out its values in the initial state So:

(0, 0, p), if p ∈ Pdem;

(g, 0, pj), if p=qj ∈ Qdem;

ON(p) = (y1, 0, r) if p ∈ Py1res ⊂ Preserve;

(y1, 0, qj) if p ∈ Qy1j

fall ⊂ QfallFunction v(p) for target elements p=qi is equal to the demanded power of

separate power unit pi; for the elements pi, striving to reach targets, it is equal tothe total power of all the demands, and for elements p of power reserve and fallv(p) equals to vo, the quantum of power fall (see above).

wi, if p=qi ∈ Qdem;

|Qdem|

v(p) = ∑ wi, if p ∈ Pdem;

i=1

vo, if p ∈ Preserve ∪ Pfall.The system operation can easily be described using formulas for the

TRANSITION operator.Si, the initial state, corresponds to the state of the energy-producing system

in the "zero day" of planning period, whileSt, the target states, correspond to the state of the system with the maximum

total demanded power of units being maintained. Thus, states from St can bedescribed as states of the energy-producing system by the end of the planningperiod, in which the WFF ON(pi)=(g, 0, pi) are true for numbers i such that


∑v(qi) is the maximum (qi from Qdem ).T, the set of transitions, consists of the “moves” of the elements along the

network. Following are the meanings of some transitions (see Figure 13): — TRANSITION(pi, x, (g, 0, pi)) with removal of the WFF ON(qi)=(g, 0,pi) means completion of the maintenance of the unit pi.

— TRANSITION(pi, x, (1, y2, pi)) with addition of the WFF ON(pi)=(1, y2,pi) means the unit pi being taken out for the maintenance work on the day y2.

— TRANSITION(pi, (0, y2, pi), (0, y2+1, pi)) with addition of the WFFON(pi) = (0, y2+1, pi) and removal of ON(pi)=(0, y2, pi ) means that on the dayy2+1 unit pi has not yet been taken out for maintenance in the given planvariant.

7.3 Trajectories in Scheduling

Following Sections 4, 5 we can construct the Hierarchy of Languages forthis problem. Here we consider several examples from the Language ofTrajectories.

Due to the asymmetry of the relation Rp for this model function MAPx,p(y)is specified following the definition from the Section 4 as asymmetric functionof distance. In particular, from Figure 13

MAP(0, 0, p2), p2(1, 0, p2)=1, MAP(0, 0, p1),p1(g, 0, p1)=3, etc.As an example of shortest trajectory here we havetp2((0, 0, p2), (g, 0, p2), 3) = a(0, 0, p2)a(1, 0, p2)a(2, 0, p2)a(g, 0, p2),

i.e., the unit p2 being taken out for maintenance on the first day and thismaintenance was completed on the second day. Another example:

tpres((1, 0, r), (1, 0, q2), 2)=a(1, 0, r)a(1, 1, r)a(1, 0, q2),The trajectory of the elements of power reserve, the pyramid P1

res, is targeted toliquidate the elements of the pyramid Q11

fall. Of course, the movement alongthis trajectory will not be included into the optimal variant of the System: theopposite side (elements from Q11

fall) would not be waiting for being capturedon the point (1, 0, q2); after the

TRANSITION(pres, (1, 0, r), (1, 1, r))one of the opposite elements qfall should move to the point (1, 1, r) —

TRANSITION(qfall, (1, 0, q2), (1, 1, r)), and remove the element pres, starting the exchange.

The trajectorytp2((0, 0, p2), (g, 0, p2), 4)=a(0, 0, p2)a(0, 1, p2)a(1, 1, p2)a(2, 1, p2)a(g, 0, p2)

can serve as an example of an admissible trajectory of degree 2 (Section 4).The movement along this trajectory corresponds to the variant of the plan withunit p2 being taken out for maintenance on the second day.

7.4 Networks in Scheduling

From Figure 13 it is seen that, for setting up the maintenance plan, theelements pi have to go from the points (0, 0, pi) to the points (g, 0, pi). Inparticular, for element p2 to get through to the point (g,0,p2) along any of thepaths

(0,0,p2) –> (1,0,p2) –> (2,0,p2)–>(g,0,p2) or (0,0,p2)–>(0,1, p2)–>(1,1,p2)–>(2,1,p2)–>(g,0,p2),

it is necessary to do away with the elements of the set (pyramid) Q12fall at the


point (1, 0, q2), as well as the elements of the pyramids Q22fall, Q23

fall at thepoints (2, 0, q2), (3, 0, q2). The elements of these pyramids control the points ofthe path of the element p2 to the target. Pyramids of elements from Qfall

correspond to the fall in power of the energy-producing system during the timeof the power units’ maintenance.

Due to the instantaneous availability of resources mentioned above all theZones generated for this example are very “simple.” They have 1-st negationtrajectories of the length 1 only. Nevertheless this example is interestingbecause the Language of Zones here corresponds to the network of Zonessubordinate to each other. The highest level of this hierarchy consists of threeZones Z1, Z2, and Z3 of the power units p1 and p2. The first one Z1 includes themain trajectory

a(0, 0, p1)a(1, 0, p1)a(2, 0, p1)a(g, 0, p1)and 1-st negation trajectories

a(1, 0, q1)a(1, 0, p1) for the elements from Q11fall anda(2, 0, q1)a(2, 0, p1) from Q21fall.

Zone Z2 and Z3 include main trajectoriesa(0, 0, p2)a(1, 0, p2)a(2, 0, p2)a(g, 0, p2) anda(0, 0, p2)a(0, 1, p2)a(1, 1, p2)a(2, 1, p2)a(g, 0, p2),

respectively, and 1-st negation trajectories:a(1, 0, q2)a(1, 0, p2) for the elements from Q12fall ,a(2, 0, q2)a(2, 0, p2) anda(2, 0, q2)a(1, 1, p2) for the elements from Q22fall,a(3, 0, q2)a(2, 1, p2) for the elements from Q23fall.

The next level of this hierarchy includes many Zones Zijres which provideresources for maintenance work. For example, Zones Zi1res include the maintrajectories a(1,0,r)a(1,1,p2)a(1,0,q1) for different elements from P1res whileZi2res include a(1,0,r)a(1,1,p2)a(1,0,q2). Zones Zi1res and Zi2res are intended forliquidation of the elements from Q11fall and Q12fall, respectively, which meansthat providing of resources is required. First negation trajectories for Zi1res area(1,0,q1)a(1,1,r) while for Zi2res - a(1,0,q2)a(1,1,r). The trajectories supportingthe attack include a(1,0,r)a(1,1,r). These Zones are subordinate to Zones Z1,Z2,and Z3. Similar Zones are generated for P2res and P3res.

For complex real-world problems subordinate Zones usually have longermain trajectories, which means that resources delivery in this case requiressome time.

8 Evaluation and Implementation

The first example considered in the paper (Section 6) is the problem of optimalbehavior of four robotic vehicles in a 2D space participating in a two-playergame. The problem is represented as four aircraft discrete event pursuit-evasiongame in the on-surface projection. In our earlier examples, the mobile units(i.e., aircraft, spacecraft, etc.) moved in a serial mode, one unit at a time.Moreover, the motions of the opposing sides alternated. In the given examplethe aircraft, both cooperating and opposing, can move concurrently. Thisexample involves 2 opposing teams, 2 aircraft in each team. The introduction ofconcurrency results in a significant growth of the unreduced branching factor(up to 324). Although the depth of the search is merely 6, it still gives us a giantunreduced search tree of about 3246 moves (i.e., of the order of 1015). Incontrast, the search tree generated by the Linguistic Geometry tools for thistotally concurrent model contains just 34 moves. Since the maximum depth


reached is 6, the branching factor of this tree is 1.53. This is a tremendousreduction in comparison with a 1015 move tree that would have to be generatedby the conventional search procedures, or even with the theoretical minimum ofthe minimax search with alpha-beta cutoffs which is (1015)1/2 ≅ 3*107. Thisresult also suggests that the growth of the computational complexity from theserial case to the concurrent one for the Linguistic Geometry models may beabout linear.

Evaluating the accuracy of the solution we should refer to solution of thesimilar serial problem with alternating motions. In (Stilman, 1996d) it wasproved that the Linguistic Geometry solution for this problem is optimal. Itappears that LG tools are able to distinguish and significantly expand theislands of potential stability, the sets of states (positions) with known value, inthe ocean of all states of unknown value, the state space. Moreover, we canconsider the LG search like an optimal navigation of the ship from the startstate through the ocean of unknown states to the expanded islands employingthe shortest path. It is likely that similar ideas work in all the LG examples, inparticular, in the example with concurrent motions considered in this paper(Section 6). If this is the case then the solution to this example is also optimal.We can speculate that LG tools allow for a very efficient break of the statespace that drives the search directly to the optimum.

A dramatic search reduction achieved in the serial and concurrent games(Stilman, 1994-1996) allowed us to initiate the development of the prototype ofthe system for optimal planning and control of the real world aerospace combatwith participation of air fighters, satellites, and unmanned aerial vehicles -UAVs. This work is currently under way at Phillips Lab, Kirtland AFB, NM,USA.

The techniques of conversion of scheduling problem into the two-playeropposing game (Section 7) were developed, in particular, on the basis of themaintenance scheduling programs. The program for monthly schedulinggenerated different search trees depending on the number of demands in eachmonth and a list of other constraints (Stilman, 1985), (Reznitskiy and Stilman,1983). The number of demands varied from 118 to 405 in different months. Thetotal number of nodes never exceeded 165. With 31 as the maximum length ofthe solution, a reduced branching factor in these problems never exceeded 1.06.(To understand these results we should take into account that the programaggregated some of the demands. The unreduced branching factor varied from50 to 100.)

The experiments with the program for annual maintenance schedulingshowed that even this higher dimensional problem can be solved employing theproposed approach (Botvinnik et al., 1983). The power equipment maintenanceplan for the USSR United Power System was computed for 1121 demands.Each demand contained 12 parameters, including resources requirements anddifferent types of constraints. Two types of resources were considered: thepower reserve and the maintenance personnel. The last one was broken intodifferent specialties. Obviously, for the annual plan the length of the solutionwas 365. The reduced branching factor never exceeded 1.005.

Evaluation of the accuracy of a solution for maintenance scheduling is aproblem. The optimal plan is usually unknown but the results achieved can beevaluated according to the optimum criterion: maximum total demanded powerof the units being actually maintained. For monthly scheduling the totaldemanded power of the solutions varied from 91% to 99% of the theoreticaloptimum value. For the annual scheduling the total demanded power of thesolutions was equal to 83% of the total demand while a theoretical optimum


was unknown. The comparison with analogous scheduling programs based onbranch-and-bound (or dynamic programming) search strategies showed theadvantage of the Linguistic Geometry approach for monthly planning; thequality of the plan was about the same, but the computation time in our casewas essentially shorter. In all experiments the branching factor of the treesgenerated by conventional programs was substantially higher. For yearlyplanning problems the competition could not even happen, becauseconventional programs could not overcome in a reasonable time the“combinatorial explosion” for such a higher-dimensional problem.

The results demonstrated on two examples in solving complex planning andscheduling problems indicate that implementations of the dynamic hierarchyresulted in the extremely goal-driven algorithms generating search trees withthe branching factor close to 1.

Various directions of research in Linguistic Geometry include formalinvestigation of the complexity of the hierarchy of languages which representseach state in the search process, investigation of classes of problems where LGtools generate optimal or suboptimal solutions, further development of thetotally concurrent version of LG tools, etc.

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