Spatial Problem Solving Using a Generalized N-Dimensional Binary Representation

Douglas Walter J. Chubb

CECOM Center for Signals Warfare Vint Hill Farms Station, Warrenton, Virginia

Abstract

A spatial transformation called a Functional Binary De- composition (FBD) is introduced and shown to produce a powerful semantic representation of the inherent cou- pling which exists within a spatial domain, a plan actor within the domain, and a plan actor action performed by the actor within the domain. The FBD may be general- ized to an N-dimensional representation called a Con- junctive FBD (CFDB). The CFBD is shown topossess an efficacious problem solving form which makes it possible to quickly solve N-dimensional FBD single-actor prob- lems. Because the FBD representation is also provably robust, stochastic functions representing time-varying events such as weather may be subsumed as part of the CFBD representation. The use and the control of these functions is described.

I. Introduction

The US Army C2SW is developing automated tech- niques to assist the DivisiodCORPS Intelligence Officer (G2) during his preparation of a tactical situation as- sessment (TSA) [6,11,121. The TSA describes the prob- able intention of the opposing forces and includes an es- timate of the location of the forces, the probable military tactic to be employed, and a temporal estimate of when the tactic is likely to occur. To develop the TSA, the G2 presumably makes use of a detailed knowledge of oppos- ing forces and tactics, the terrain and its effect upon these tactics, and other information to recognize ongoing enemy plans. Since accurate plan generation and plan recognition capabilities form the basis for effective com- mand and control, albeit military or otherwise, the de- velopment of accurate and efficient planning algorithms has become a research topic of considerable interest [3,5,6,9,10,141.

Plan generation algorithms may be classified by the rank of the planning domain, i.e., finite or infinite. Sev- eral finite domain plan generation algorithms exist [1,3,103 and finite domain planning appears to be well- understood. However, the extensibility of such tech- niques to infinite domains continues to be a research topic. Recently, Chapman 131 proved that nonlinear planning within finite domains which includes repre- sentations for conditional actions, dependency of effects upon input situations, or derived side effects is undecid- able. A paradox central to any formal theory of infinite domain plan generation and recognition is how humans generate or recognize real-world plans. Chapman and Agre [ l l have suggested that the plan domain indepen- dence criteria be relaxed and that plan logical consisten- cy be assured locally by subsumption of truth criteria specific to the local domain, i.e., a type of circumscrip- tion. Recently, Chubb [61 showed that, with the possible exception of finite domains, the existence of such criteria can not be verified a priori by the planner. Chapman and others [1,3,91 have suggested that real-world plan- ners improvise, doing something easy and observing re- sults. However, this heuristic begs the question. What discriminator is used to discern ”easy” from ”difficult” plans since adplissible real-world plans do not exist? Chubb [61 has conjectured that real-world planners must assume that,

C-1: the planning domain will be temporally and contex- tually invariant for sufficiently short periods of time to ensure that the expected value of the executed plan ac- tion is realized, and,

C-2: all of the domain features and associated values which effect plan action execution are known to him pri- or to action execution.

It can be shown that C-1 and C-2 are logically equiv- alent to Rao and FOO’S [141 Axiom of Simple Actions. We will show in this paper that this conjecture (temporal and contextual invariance and the sufficiency of a priori domain knowledge) forms the basis for a semantic repre-

605 TH0333-5/90/0000/0605$01 .OO 0 1990 IEEE

sentation called a Functional Binary Decomposition (FBD) which may be used to solve spatial and temporal problems arising during the preparation of a TSA. The mathematical foundations for the FBD are introduced in Section 11. Section I11 introduces a generalized N- dimensional form of the FBD called a conjunctive FBD (CFBD). Spatial and temporal problem solving using the CFBD is introduced and described in detail.

11. The FBD Transformation

A plan, P, is defined as a time-ordered sequence of plan actions, Ai, i = l , ..., n, where each action is execut- ed by a plan actor, AC, within some planning domain context expression, Ci. Each actor-action-context expres- sion, {AC, A, C}i , is called aplan tuple. That is,

where n>O. Each plan tuple is a description of a plan action, the associated plan action actor, and the associat- ed context state prior to the initiation of action execution by the actor. Actions are executed by the plan actor within domain contexts which contain features which makes the initiation of the action possible. Action ex- ecution transforms the domain context expression into a new context expression wherein a new plan action may be executed.

The Ci are represented syntactically in raster format with spatial resolution limited by the physical size of the pixel elements, czy E Ci. Associated with each cxy is a set of spatial features {fi, ..., fp} and an associated set of feature values {vfi, ..., vfpl. Each Ci may be represented by a finite set of feature tuples, {{fi,x,y, vfi,x,y}, ..., {fp,r,t, vfp,r,l}} where subscripts represent the feature type and spatial location.

Plan action execution is the actor's reasoned manipu- lation of the domain context. Plan tuple action execu- tion is described by the plan tuple execution function, E. I f C = {{fk, vfk} Iv k, k=l, ..., M}, the set of all possible features and values, then E is a mapping from a plan tu- ple in P to A E P(C), the power set of C , i.e., the non- empty set of ordered pairs ({AC, A , C]i , A) E E. Then giv- en an AC there exists a non-empty set of As which may be executed. By C-2, associated with each {AC, Ai) there exists some pi, a subset of Ci , which determines the admissibility of E{AC, A, c)i. C-2 assumes that AC is aware of the relationship between pi and action ex- ecution and has verified that pi C Ci before action execu- tion begins. Fortunately, complex military tactics re-

quire a considerable coordinated effort between men and machines, and the feature elements of pi are well-known and predictable. This must be so for the Army Com- mander to make effective use of the forces and equip- ment a t his disposal. In addition, prior to a tactical en- gagement, military equipment and forces are exhaus- tively tested and trained under a wide variety of condi- tions to ensure their tactical extensibility within any Ci. Testing and training results are readily available for most military equipment'forces [81. Then, given an ele- ment {AC, A, C)i e P, we define the Functional Binary Decomposition (FBD) Transformation on Ci as the map- ping,

for every czy E Ci

(2)

In keeping with C-1, equation (2) assumes spatial in- variance for each cxy E Ci.

The FBD Spatial Representation

Equation (2) produces a binary representation of gen- eral complexity. This representation is further pro- cessed to produce the FBD spatial representation which is used to solve both spatial and temporal problems. Oneelement values represent those czy where the actor action is not admissible. We assume that the one- element pixels are 8-connected and the zero-element pixels are 4-connected. Each path-connected one- element region is processed to develop it's boundary list, adherent list, multiply-connected (MC) boundary list(s), their associated MC adherent lists, both boundary and MC boundary cut-points, and region interior points (Fig. 1). The resulting spatial representation can be quickly developed on binary arrays of general complexity [71.

In addition, this representation is provably robust [7] which is of critical importance during tactical situations when spatial features may change quickly and abruptly. Spatial representation robustness makes i t possible to incorporate syntactic changes directly to the FBD repre- sentation. Each syntactic change is semantically inter- preted and evaluated using the parent FBD representa- tion. The completed FBD representation represents the semantic coupling between the execution of an actor ac- tion and the domain in which the action execution is em- bedded. We assume that the execution of every action in-

Interior Points \ Boundary Points

MCBoundary Points

Fig. 1. FBD representation of a 1-element, 8-connected binary region of general complexity.

volves mobility; temporal movement always occurs dur- ing action execution (note: even the "no-action'' action involves temporal movement) and every action of inter- est to the military requires some type of spatial move- ment. We therefore assume that the execution of an ac- tor's plan involves both spatial and temporal movement. The FBD representation will be used to solve spatial problems while a variant of the FBD representation is used to solve temporal problems.

Spatial Problem Solving Using the FBD Representation

A spatial problem is a hypothesis about the spatial mobility requirements of an executed plan tuple action. Action execution begins spatially and temporally a t the actor's spatial location prior to action execution, i.e., some sxy E Ci. At the conclusion of the action execution the actor is hypothesized to be spatially located a t one or more goal positions, {grs, ..., gkh}. A spatial problem is satisfied if there exists a 4-connected, zero-element path, Pfs, {gdy}A between the zero-element points (sxy, gdy) such that every element of P is an element of an FBD re- presentation. Goals may be specified as either disjunc- tive or conjunctive, e.g., P(sxy, {grs A gkh}) requires that both P(sv, {grs}) and P(sxy, {gkh}) be satisfied.

The Straight Line Path Algorithm. Given an FBD re- presentation, FBDi, and any two zero-element points, (sxy, gdy), belonging to FBDi, then a set of spatial paths between these points is developed as follows. 1. Set a=s, , b=gdy.

Call SLF!A(a, b)

ENTER SLPA(a, b) 1: Attempt to construct an unobstructed

StraighbLine-Path between 0-element points (a, b). Function returns unobstructed 0-element path between points (a, 0).

2: If 0 =b, Save (a, b) Goto END,

Else, 0 =boundary element for some 1-element region in FBDi. Set a to path-connected adherent list element belonging to FBDi. Save (a, a) Goto 3.

Else 0 = MC-boundary element for some 1-element region in FBDi. Goto TERMINATE.

3: Set a to the FBDi adherent list element which is path-connected to a and is reached by rotating clockwise about FBDi from Q. Call SLPA(a, b).

4: Set a to the FBDi adherent list element which is path-connected to a and is reached by rotating counterclockwise about FBDi from U. Call SLPA(a, b).

END. return set of path vectors between points a, b. TERMINATE: no spatial path exists between points a,b.

What the SLPA returns is a list of branch points be- tween sxy, and gdy or between one or more regions within FBDi and points sxy or gdy. These path vectors are reas- sembled in spatial order from sxy to gdy using each re- gion's adherent list data. The Straight Line Path Algo- rithm (SLPA) solution has some interesting properties [41,

1. No two SLPA solution paths are topologically

2. A SLPA solution set is complete. That is, no other equivalent,

topologically equivalent SLPA solution exists for the same set ofpoints and the same FBDi.

3. The minimal (shortest) path between points sxy and gdy i i a member ofthe SLPA solution set.

Fig. 2. An SLPA solution set. Two 1-element regions occlude a single goal point. The use of the region's adherent list is evident

The SLPA is capable of quickly generating single ac- tor spatial solutions within the FBDi. However, the FBDi has some basic representational weaknesses which limit its usefulness.

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1. The spatial resolution is (usually) insufficient to accurately represent extended line objects such as roads and railroads.

2. Since the representation is binary, action execution preference is not represented.

3. Multiple actor action execution can not be represented.

Each of the these weaknesses can be eliminated, howev- er, with a spatial representation of multiple FBDi 's as follows,

1 . Generate an FBDi for each roadhailroad. 2. Generate an FBDi for each action preference value. 3 . Generate an FBDi for each actor action execution.

Since the two-dimensional FBD representation may be generalized, as suggested above, an N-dimensional FBD representation appears to be an efficacious representation for real-world spatial problem solving. The N- dimensional FBD representation is realized with a co- planar stack of N > l FBDi, where each FBDi represents a (different) contextual mapping from Ci to {O,l}. For ex- ample, the first k FBDi could represent important road and railroads; several FBDi could be used to represent weather patterns which might affect action execution; spatial mobility differences and a variety of terrain pref- erences might also be represented with several FBDi.

111. The Conjunctive FBD Spatial Representation

The SLPA, using a coplanar stack of N FBDi, is simi- lar to the two-dimensional process but with N degrees of freedom. In addition, associated with each cxy element is a coplanar FBD,. SLPA step 2 is now changed to,

2. If 0 = b, Save (a, b, FBD,) Goto END,

If 0 = 0-element for some FBDk, Save (a, 0, FBD,) Call SLPA(0, b, FBDk).

Else, 0 =boundary element for N-K FBDi. Set a to pathconnected adherent list element boundar list element about each of the N-K FyBDi.

Else, search remaining N-1 FBDi for 0-element

Save each (a, a, FBD;). For each (a, a, FBDi), goto 3.

Else, 0 =MC-boundary element for K FBDi. IfK=N-1, goto TERMINATE.

An equivalent spatial problem solving representation called a Conjunctive FBD (CFBD) is developed by noting that SLPA TERMINATION is possible if and only if K=N-1. Ifcxy,i represents the cZy e FBDi, i=l, ..., N,

then define the value of every oxy element belonging to CFBDNas follows,

for every x, y in CFBDN. By definition, if P is an SLPA path within CFBDN, then for every oxy e P there exists an FBDh such that value cxy,h = 0 where 1 sh sN. Us- ing (3), every SLPA path element (oxy) can be described in terms of some local measure of context, i.e., an FBDh and the decision process used to produce the binary map- ping from Ci to FBDh,

Associated with each FBDi coplanar element of the N-dimensional CFBDN is a rule-based system which monitors the incoming data processed by the G2. Each on-line FBDi change to a cxy,i (may) produce an associ- ated change to a oxy e CFBDN. Since temporal and spatial invariance is assumed at the pixel level (by C-2), action processing within the CFBDN is commutative when using the temporal ordering of the incoming data to rank order actor actions and changes within the FBDi.

In summary, the CFBD provides a context-rich envi- ronment which is simple enough to provide a framework for efficacious spatial problem solving (SLPA) while likewise providing a wealth of contextual information a t the pixel level (oxy) through the associated mappings be- tween the oxy and the cxy,i as described in (3). In addi- tion, since changes to the FBDi are data driven, a natu- ral ordering is imposed upon the CFBD processing of (3) which ensures that action execution is commutative, whenever appropriate.

Temporal Problem Solving Using the CFBD

A spatial problem solution is a measure of the feasi- bility of a spatially constrained action execution. Given that action execution is possible, temporal problem solv- ing measures the feasibility of a temporally constrained spatial problem. In the present CBSW TSA system, temporal constraints are developed by D. Nobel's tem- poral reasoner [131 . Nobel's original work for the Navy is being modified for Army usage. Basically, Nobel's temporal reasoner is a tactical planner which hypoth- esizes a temporally ordered sequence of plan tuples along with a temporal estimate for each tuple hypoth- esized. The estimate consists of three time intervals, i . e . , [ T l , T21, [T2, T31, a n d [T3, T41, where T l ~ T 2 5 T 3 sT4. Temporal interval [T2 T31 represents the current best estimate when the tuple action execu-

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tion will begin. Since time intervals are rarely single- valued, a temporal solution set similar to the SLPA solu- tion set is developed. The temporal solution set algo- rithm (TSSA) is a two pass algorithm. Unlike the SLPA where solution set elements are spatial paths, the ele- ments of the temporal solution set (TSS) are 0-element points. Each point selected is guaranteed to be an ele- ment of at least one path-connected path from start to goal positions which satisfies the temporal constraint. The TSS can represent either a topological manifold or an algebraic set. Set operations such as intersection and union can be meaningfully performed between TSSi. Finally, the TSSA can be used recursively to yield the minimal (shortest) temporal path. Note that the mini- mal temporal path is not necessarily the minimal spatial path.

The Temporal Solution Set Algorithm

We assume there exists some CFBDN and the associ- ated set of FBDi, i = l , ..., N . By (3) we noted that associ- ated with each FBDi was a rule-based system which con- trolled the values (0,l) of each element of FBDi based upon feature 2hanges within Ci. Thus the CFBDN was indirectly affected by a series of local feature changes. Each FBDi however, represents the admissibility of an action execution, not the efficiency of the action execu- tion. For temporal problem solving we need a pixel- level measure of how easily the action execution is ex- pressed. Since action execution involves mobility, the metric used is a real-valued coefficient, wry, which de- scribes the domain compliance to action mobility. A wry is associated with each element in CFBDN, where O < W , ~ S ~ . The time, tXy, required to execute an action within a pixel cry is computed t o be, t,, = Rry+( V,,*w,,) where Rzy is the pixel resolution (meters) and Vm, is the maximal actor action velocity (meterdunit time) assuming the domain does not impede action ex- ecution.

The TSS is developed as follows. TSSA Passl exam- ines every 4-connected, zero-element point, starting with the START point, and develops and saves two val- ues for each point:

1) the local minimum action execution time from START to that point, TS,i,, and, 2) an unobstructed, best-case (wxy= 1.0) estimate of the time needed to get to GOAL, TGest.

If TS,in + TGest I Time Constraint, then continue TSSA Passl processing, else remove that point from fur-

ther processing. Computed TSmin values less than the local TSmin value are posted on the applicable point, otherwise the point is removed from further consider- ation; Likewise for points whose location=GOAL. Giv- en that at least one point remains such that TSmjn + TGeSt s Time Constraint, a TSSA Pass2 begins.

The TSSA Pass2 works backwards from GOAL to START, this time depositing a computed value, TGmin, for TG,,t. Every point in CFBDN which meets the crite- ria TS,i, +TGmin 5 Time Constraint becomes an ele- ment of the TSS.

Fig 3. Example of a TSS generated using a CFBD. Represented is a network of roads and off-road vegetation. Each TSS point is part of

a path-connected temporal solution.

Modeling Stochastic Events

The CFBDN has been further generalized to include actor actions which effect the values of the wrr. Stochas- tic events, such as time-evolving weather patterns or the presence of other actors and their spatial and temporal movement within Ci, are easily expressed using an FBDi which alters both the CFBDN and its associated wxy. Pattern position and motion is expressed using an FBDi while the effects of the pattern upon other actor ac- tions is expressed by changing the values of the wzy. Since the CFBDN is a hypothesis tester, not the hypothe- sizer, CFBDs are generated for each actor being consid- ered within the TSA. For example, an Army logistics problem involving tanks and trucks would minimally require a tank C'FBD, a truck CFBD, and a weather pat- tern CFBD. Interactions between CFBDs would be con- trolled either directly, by the TSA hypothesizer (i.e., No- bel's planner) or indirectly by changing the values of the wxy. For example, repeated tank movement over terrain affects the future movement of trucks over the same ter- rain. Thus the wxy values for the truck CFBD are low- ered by a tank action. In addition, if it rains this action may further lower both the tank and truck wzy values.

In summary, the CFBD is an efficacious representation for spatial and temporal problem solving. However, ac- tion interactions between actors (how, when, and where) must be information which is supplied to the problem solver by the TSA planner. Because of the richness of the CFBD representation, spatial or temporal problem solving results may be interpreted using a variety of contextual-based explanations. Properly exploited, this capability should suggest a variety of additional plan tuple hypotheses to the planner. The legitimacy of each hypothesis, with respect to the logical corisistency of the total plan, is however, the responsibility of the TSA planner.

IV. Summary and Conclusions

The focus of this research has been to provide robust and accurate automated assistance to the G2 as he pre- pares a TSA. To this end, we have introduced a general- ized N-dimensional binary representation called a CFBD. We have shown that simple spatial and tempo- ral problems can be easily expressed using this repre- sentation and solved, using either the SLPA or the TSSA. All of the spatial and temporal problem solving algorithms described in this paper have been written in Common Lisp for a TI Explorer AI Workstation. Re- maining research issues include an examination of the applicability of the CFBD t o multi-actor conjunctive plans involving nondeterministic actor action execu- tion.

Acknowledgement

I wish to thank Dr. David McNeil for his helpful sug- gestions and criticism, particularly with the form of the TSSA.

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