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The Synthesis of Minimal-description Models for Representing Knowledge
Cop yrig ht © IFA C A rti fi cial Le n ingrad . US SR I'lHJ
ln td li g~ncc,
THE SYNTHESIS OF MINIMAL·DESCRIPTION MODELS FOR REPRESENTING KNOWLEDGE K. A. Pupkov and O. V. Evseev Mosco l\' fl w itllle of Electronic Mechanical E ngin eering ..Hosco l\ '. USS R
,
" v
.'
Abstract. We consider the problem of constructing minimaldescription models for totalities of objects, - optimal models containing only a minimum of necessary knowledge without any element of which the model as a whole becomes not adequate to the object domain. A mathematical technique is introduced to describe and analyse the structure of such models and characteristics expressing their structure are analytically exhibited. Precise and E. -approximated methods for the synthesis of optimal models having the exhibited structure are described. The method for minimal supplement of optimal models is given. Keywords. Optimal systems; models; approximation theory; graph theory; identification; tree(mathematical); linear programming. INTRODUCTION
FORMULATION OF THE PROBLEM
The so-called frameworks of minimal description have a number of useful characteristics. A model containing only a minimal necessary amount of description occupies less storage than any other model. Further, provided that the model contains redundant information about some element of enVironment, the analysis of data about the situation including this element will also be redundant. So in order to decrease the decision making time it is worth using minimal-description models that do not contain redundant information. But the available methods do not provide the means for a precise constructing of the minimal-description models and do not make it possible to estimate the difference between the heuristic solutions and optimal solutions.
Let the objects or situations {S1' ••• 'SnJ =S be adequately described by multigraphs {G 1 , ••• ,Gn l=G with coloured nodes and arcs. It is necessary to construct the minimal adequate model of knowledge (MK)n , containing information only about some subgraphs (subdescriptions) g1' ••• ~ of graphs GiEG. It is necessary to find such subdescriptions that: a) there exists MK describing S adequately only with the help of these subdescriptions, their correlationship and their connection with names of situations of S; b) the summary volume of subdescriptions {g1, ••• ,~J= G is minimal in the sense of estimation
If there is a certain initial description of objects a minimal description of each object can be obtained by removing from its description those elements without which it still remains distinguishable from other objects (Victorova, 1981). But in the general case the solution of such local problems does not provide the global minimum of the whole description of objects. The model to be found must contain the optimal system of correlated adequate minimal descriptions.
where aj,b j are the costs of nodes and arcs rj of GiEG. The costs are assigned due to a concrete problem and can represent a number of computer operations required to input information in to the system about the availability of basic
139
K.A. Pupkov a nd O. V. Evseev
140
units used to describe the situations in the given situation. One can take a j =b j =1 (V j).
MK is considered to be adequate if
there exists a procedure P for a correct identification of all situations on the basis of information stored P is a tree (Pig.1) whose nodes correspond to the checks of presence of su~s1tuations s(gi) described by gie G in the input situation; arcs correspond to positive and negative results of the checks, and leaves correspond to the situations of S. Such a procedure is a permissible procedural form of MK sought for and is given by a pair = (G,P).
inn .
n
Por fixed G a set of volume and adaquacy equivalent modelsn =(G,P) is given by hierarhical model n=(G,Q) where the tree Q is of the type shown in Pig.2. Nodes of Q correspond to subgraphs giE G and subclasses MS: G (they are rectangular in Fig.2). If successor of gi is M there is the subsituation s(gi) in all situations corresponding to graphs of M. Subclasses-successors of some class cover all this class. Permissible trees P differ one from another by permutation of checks corresponding to subgraphs-successors of a single class M.
n
In the model =(G,Q) subclasses M are subjected to the ratio of taxonomic hierarchy. Chains (from root to leaves)_point out generalized notions giEG describing the given situation and subclasses which this situation enters. Such a declarative formn =(G,Q) of MK is sought for The synthesis problem is to find ~ and Q providing minimum (1) such that modeln =(G, Q) would be adequate. is the model of minimal description as it does not permit removing a single element forming it without the loss of the model's adequacy_
n
THE STRUCTURE OF OPTIMAL MODELS Let.f(M,t)= lK 1 , ••• ,KJ, (1~2) be a covering of subset M~ G by classes K1 , ••• ,Kl where: a) for any Ki ,
-
(i=1,1-1) there exists a set
CKi~~
of subgraphs g of volume X (see (1) when G =(g)) belonging only to graphs of Ki ; b) each g does not belong to graphs of Kl • Let the so-called covering C(G) be a totality of coverings.f (M, t ) : ! (G,o) = {K1 ,··· 'IS.J is applied to G; .f (K i , Of) is applied to non one-element Ki' (1.6 i ~ 1) and so on, until all classes become oneelement (Fi~.). Any C(G) generates some MK n =C~,Q). It is sufficient to take arbitrary subgraph gi of each set CK . for obtaining the set ~ G. For formation Q it is sufficient to take a tree defining covering C(G) (Fig.) and "hang" nodes with names of chosen subgraphs to his edges.
-
Theorem. Por any adequate G and (G,Q) there is C(Q) generating adequate MK n'=(G' ,Q') not of larger volume than
n=
n•
Conclusion: optimal adequate MK is generated by covering of the form C(G); C(G) is a "carrier" of structure of optimal adequate 1~ in the first order approximation; ~ and Q can be searched only among G,Q generated by coverings of the form C(G). Let us study ) types of extremum of objective function (1): 1) conditional, 2) inconditionally local, ) global. Minimum (1) corresponds to conditionally optimal MK no=(~o,Qo) under the following restriction: the volume of subdescriptions gi does not vanish along chains of the tree Q from root to the leaves. We introduce a particular type E~(G) of covering C(G) a totality of partially ordered partions .f°(M, 0)= {K 1 , ••• ,Klj , (Kif1Kj=~' i~j). The partition,P°is
a particular type of covering.f : subgraphs gE CK1 , (1=1,1-1) are not a subgraphs of graphs of class Kq , (q~i, q=1,1-1). Partitions ~oare ordered in E~(G): the partition J~G, 00) (where min{o;j'p°(G,(n])
00=
~
is applied to G, the partition ,O(K,~) is applied to non one-element classes K of .p o(G, ~) and so on
The Synthesis of Minimal-Description Models
Figure 1.
Figure 2.
Figure 3.
141
K.A. Pupkov and O.V. Evseev
142
,
until we have classes K1, ••• ,K1
in-
divisible by no JO(K~,Oo) at given to; then
t1 >00) 0,=
min {a: '0
31° «,0),J
are chosen for non one-element Kj and J> o (Ki' , 'If,) is applied, and so on until all classes become one-element (Fig.3). Theorem. There is a partition E~(G) for any adequate G, and any partition E~(G) generates adequate conditionally optimal MK. Conclusion: the structure of conditionally optimal MK is o exact1y given by the properties of E*(G). The problem of synthesis congerges to constructing arbitrary E*(G) and taking arbi trary subgraphs gi E CKi where CKi correspond to pOeM, 0) in E~(G). The so-called inconditiona11y 10ca1o -0 0 1y optimal MK =(G ,Q ) is certain to be obtained from no=(Go,Qo) by permutation subgraphs gie Go in the tree Qo• The ordering of subdescriptions gi in Qo over decreasing their volume becomes not valid and the set Go serves as local domain of searChing the extremum.
n
The particular form of C(G) is a partition C~(G) which consists of partitions .f;(M,O) of subsets M~ G into two classes K1 ,K2, ~ is a minimal value for which there is the partition .P °01' set M into classes K1 ,K2 or into classes K2,K1 exactly. Theorem. There is a partition C~(G) generating adequate globally optimal model n*=(G*,Q*) for any adequate G. Conclusion: the structure of globally optimal MK is given the properties of the partition C~(G); the search domain converges to the set of omode1s generated by the partitions C*(G). OPTIMAL-MODEL SYNTHESIS ALGORITHMS To obtain the conditionally optimal
no=(Go,Qo) it is necessary to be able to construct partitionsfo(M,~)
3pD(M, o)J. But partition poeM, 0') if
where 0,,= min{o: '0
?f
there exists and only if in some graph of M there is a subgraph of volume '0 which is not a subgraph of all graphs of M. Therefore constructing ofr (M, if) converges to solving the problem: in graphs Gi E M to find subgraph go of minimal volume ~o =V(go) which is not a subgraph of all graphs of M. = min V(g)
(2)
gc G. -
l.
gftr
Here r is a maximal isomorpha1 subgraph for graphs of M. To obtain..? 0 one can take J'" (M, 00) ={K1 Kd where
f
K1 = graphs of M, ~ontaining gol, K2=M' K1 • To form Go one can take an arbitrary subgraph from CK 1 , for instance, the subgraph go. The problem (2) can be po1ynomia11y converged to a problem of linear 0,1programming. But we consider a more efficient method using a modificated model proposed by V.G.Vizing when estimating the chromatic a1sss of graph (P1esnevitch,Saparov,1981; Vizing, 1965). Let us describe the polynomial desmant1ing procedure for graphs Gi , r without using index i of graph Gi • Let i,j be the nodes of
r ;
K,l - nodes of G; r(i,j),r(K,l) - colours of arcs connecting i,j in rand K,l in G (if i,j are not connected then r(i,j)=O). Let us define a set of pairs P1=(i,K), ••• ,Pm=(j,1) where i,K form the pair if their colours are identical. Let us assosiate every pair with node of graph B. If i=j or K=l then ps=(i,K),Pq=(j,l) are connected in B by an arc called a rigid one. If i~j, K~l, r(i,j)~r(K,l) ~O then Ps,Pq are connected with an arc of colour r(K,l) called a soft one. If i~j, K~l, r(i,j)=r(K,l)~ then Ps,Pq are connected with an arc of
Th e Synthe sis o f Minimal-Description Models
colour r(K,I) called a coloured one. Let us ~rk nodes of G presented in pairs P1, ••• ,Pm. Let us remove coloured arcs r(K,I) and isolated marked nodes from G. Let G' be the graph obtained. Let us remove arcs and non-marked nodes of G' from G. Let G" be the graph obtained. Let us remove from graph B soft arcs of colour r connecting nodes (i,K),(j,l) where nodes K,l of G' are connected by an arc of colour r. Let B(r,G) be the graph obtained. Lemma. The subgraph ~o being the sorutron of problem (2) is among those and only those graphs that are of~be following form: 1) a non~rked node of G'; 2) a subgraph of "node-arc-node" form of G' where both nodes are marked; 3) t+1 nodes of the same colour of G" where there are only t nodes in 4) a subgraph f~ G", rro with single jointed arc r of G" connecting nodes of f that is absent in Here r"., is one of subgraphs r of G.
r;
rm.
The subgraphs of 1,2,3 type are the simple solutions of the problem (2) that can be found quickly in polynomial number of steps. Here it is not necessary to construct B,B(r,G) it is sufficient to construct G',G". The graph B(r,G) is used for searching solutions of type 4. Theorem. An arbitrary coloured graphs rand G contain a subgraph f 5; r, G if and only if there is a set of nodes I={(i1,K1),(i2'~)' ••• ' (it,Kt )} in graph B(r,G) which are not connected with each other by rigid arcs - the so-called rigid independent set (RIS). Here f contains nodes K1,~, ••• ,Kt and nodes Kq,K s are connected by arc of colour r in f only if (i q ,K q ),(i s ,Ks )E! are connected by coloured arc of colour r in B( r ,G). Let us consider that RIS I generates f S;r,G. Let nodes (i1'K 1 ), (i2'~)~ I be connected by soft arc of colour r=r(K1'~). If we add r connecting K1 ,K2 to f, the subgraph g of type 4 will be constructed. If g $. r , it can be a solution of problem (2).
14 3
Lemma. If I consists of 2 elements, g=(K 1rK2),"r. If RIS I={(i1'K 1 ), ••• , (it,Kt )], (t ~ J) generates a subgraph fS; G with set R of arcs,then g!/: r if and only if an adjoint RIS I= { (i 1,K 1 ), ••• , (it ,Kt) generating subgraph f', G with arcs R,r does not exist.
J
An algorithm for searching 4-type solutions of problem (2) for each soft arc rE B( r , G) constructs sets I (RIS with nodes connected by r) increasing number of nodes in I. A set R of arcs of f c r,G is constructed according to the last theorem. If the conditions of the last lemma are satisfied the subgraph f is constructed, r is added to it and the 4-type g is obtained. The simplest algorithm for synthesis of the conditionally optimal MK =(~,Qo) searches the subgraph go solving problem (2) for graphs of M~ G(M=G on the first step), includes go in Go' Ki={grafhS of M, containing gol, K2 =M' K1 , constructs the part of the tree Qo (M is a root; go'~ - successors of M; K1 successor of go). Further these operations are applied to non one-element classes obtained if any. The upper bound for a number of problems (2) to be solved here is
no
A
N
=n
([log2n] + 1).
Let us construct the unconditionally locally optimal MK n°. Let there be I ~ 2 identical nodes g of volume (f in Qo • Moving in Qo from nodes g along the chains to the root, let's define class K where these chains converge. Let Q(K) be a subtree with the root K, 'lf1 < 1/ be the volume of successors of K. Let us describe a polynomial transformation 0. of MK J no... (Go,Qo)· a) Construct K1={ graphs of K, containing g} , K2 =K' K1 and a tree T: K is a root; g1K2 are successors of K; K1 is successor of g. b) Construct trees Q(Ki ) giving conditionally optimal models for Ki ,(i=1,2). For
K.A. Pupkov and O.V. Evse ev
144
this purpose it is necessary to remove graphs being absent in Ki from the nodes-classes of Q(K). c) Substitute for subtree obtained by attaching roots Ki of Q(Ki ) to the leaves of T the subtree Q(K) in Qo • If Q(K) gives a conditionally optimal model for K, K is referred to as a dominant class. Then, the application of to the tree Qo results in the tree Q' = Qo and the volume of MK is decreased. An unconditionally local minimum of (1) corresponds to a sequence of transformations tSj" (5 gi ' • • • , 6" gin giving maximal vo-
Cl,
og
2
lume compression of
-V(~;11
C·
(c5ji2
MK
A ~V( Q
(63'1 Q o ) ... )).
o
)-
n
Here V(Q) is the volume of MK = (G,Q). The problem is solved by E- approximated method of "branches and bounds" ( E. ~ 0). Branching OCCUl!'S by all possible sequences, bounds are calculated in polynomial number of steps. Let us denote Qi ••• jq= 639"C6jj
(...
. = ••• Jq ... V(Q )-V(Q . . ). Let in the tree o l:"J q Qi ••• jq there is some sets of Is
(03i Qo ) ... )) ,
identical nodes gs'
A V.
~
(s~i, •••
,j,q).
Let U be a set of all different gs which have a dominant class in Qi ••• j q' be the volume of suc-
rS
cessor of K , (f'$ be a volume of g • Let us marksnodes-classes whose suScessors are our gs in the tree with the root Ks. Let us find a node-subgraph of minimal volume ~$ with marked su~cessor. Lemma. If a sought for sequence begrns-with transformations ()g, , ••• 03;, Ot9" the upper and lower bounds of the A value A V* are: f i ••• jq=.AVi ••• Jq . +
These bounds also make it possible to evaluate.tl V* and to neglect obtaining too little compression of the model's volume. Let us consider the E -approximated method based on the "branches and bounds" technique for the synthesis of globally optimal MK n~=(~*,Q*). The method consists in constructing tree B~(G) which c~tains as a root subtree the tree Q* corresponding globally optimal MK exactly to given t~ 0; then Q! is selected in Be< G) by means of a linear algorithm. n~
is generated by the partition
C~(G). Therefore, the subgraphs
gic G* are among those subgraphs which can belong to the sets CKi of parti tions .1'". 0 (M, ~) in C~ (G) (it is sufficient to single any subgraph in every CKi ). Let us denote Z(G) the set of subgraphs which can form G*. Z(G) can be found when constructing conditionally optimal MK if all the solutions of each problem (2) are found. Let us construct a tree B(G) with a root G. The successor of non one-element node-class M f: G (M=G on the first step) are nodes-subgraphs gjcZ(G) for which M~M Nj~~ where
J;
Nj = [graphs of G, containing gj gj has 2 successors M(1N j , M'(MlrN j ). The leaves of B(G) are one-element classes. Let QCB(G) be a subtree with the root G where any class has only one successor. There is a one-to-one correspondence between subrees of the form Q and partitions of the form C~(G). Hence the sought for Q* is among the subtrees QCB(G). Let us assign oosts to nodes of B(G): a) cost of any leaf is 0; b) cost of node gj is the volume of gj plus costs of its successors-classes; c) the cost of a class which is not a leaf is the least of costs of its successors gj. For any olass in B(G) let us leave only one successor with the least cost. Then in B(G) a sought for tree Q* will be the single tree connected with the root G.
Th e Synthesis of Minima l-Des c ription Mode ls
The prqcess of directed constructing tree Bi(G)C B(G) provides the availability of the sought for tree Q* in it. We continue to construct successors only for those leaves K of B(G) which are the leaves of subtree Q'cB(G) pos~sing the property: the lower bound V of value v*=V(n*) is minimal for Q' under condition that the root part of the tree Q* is Q', ~/' and E =V-V is minimal for Q' where ",. V is the upper bound of V* under the same condition. Bounds V,, are obtained by the volume of nodessubgraphs in Q' and values of bounds '" A V(K)~ V(K) of volume V*(K) of globa!lY optimal models for leavescl~~es K of Q'. Bounds '" V(K), -A V(K) are calculated in polynomial number of steps according to the results of synthesis of conditionally optimal MK. Let no(K)= =(G~,Q~) be conditionally optimal MK for graphs of K. Let there are subgraphs of volume ~ ~ in G~. Let us find in Q~ for each
to, , ...,
1E W(K)=b'f'~t··· ~m J the class K such that there is the greatest number m ( 6) of subgraphs of volume just below it. Let ~o (K) be a volume of successors of K in Q', and I K I be a number of elements 8f K. K'~
t
~.
Values ~(K)=
Lm(!)·(o(K»).
L m(~)'1+(IKI-1-
~EW(K)
~
, V(K)=
oe:W(K)
= V(no(K» are upper and lower bounds of value V*(K) of volume of globally optimal-model for graphs of K: V(K)~ V*(K)~ V(K). These bounds also make it possible to evaluate the difference V( no)-V( n1fo ) in polynomial number of steps and to abandon obtaining too little reducing of the MK volume in constructing n~. ,,~ Besides, if V(G)=v(G), no =(Go,Qo) is
-
globally optimal MK. If Go consists of subgraphs of identical volume, is globally optimal MK.
nD
145
SUPPLEMENT OF OPTTIlIAL KNOWLEDGE MODEL of the formn(G)=(G,P) converges to inputing of data about the new situation Sn+1 f= S= { S1' ••• 'Sn} described by graph Gn+1~ G= {G 1 , ••• ,Gn ) into the model. If we input Sn+1 to the entry of procedure P, Sn+1 is certain to be erroneously assosiated with situation Sib S. Let us use the result of this erroneous identification for the correction of n (G). It is necessary to distinguish Sn+1 from Si for P to provide a correct identification of all S1, ••• ,Sn+1. For this purpose let us add in set a subgraph go of one of the graphs Gi , Gn +1 such that another of these graphs does not contain go' and let us substitute for node go with successors Si,Sn+1 the leaf Si in the tree P. The resultant new tree P' provides correct identification of all S1, ••• ,Sn+1. For obtaining a minimal supplement of the model's volume it is necessary to take a minimal in volume go' i.e. to solve problem (2). As a result we have a new , adequate model n =(GUgo'P') for situations S1' ••• Sn+1. There is a minimal difference in the volume between this procedural model and the ini tial procedural model (G). Ap¥lications. The main application of me hods eleborated is the synthesis of mjnimal-description models for robots with microprocessor control. The methods were used to construct a subsystem for planning a path connection of a mobile robot (Evseev,Solonin, 1982). Two minimal-description models were used. The first model contained information about applicability conditions of local plan operators - operators of constructing paths going round obstacles. The second model contained knowledge about the form of typical obstacles and estimations of complexity of overcoming them by robot. It helped us to
~
-
n
K. A. Pupk ov a nd O.V. Evs eev
146
construct a plan of the robot's movement with minimal executing difficulty in concrete situations. The models were synthesised in declara ti ve form (the volume cutting down obtained in comparison with the initial description was about 6 times). Each model was transformed to equivalent procedural form ~= (G*,P*) where tree P* represents the procedure for the identification of situations with a minimal summary cost of elementary operations in checks. These procedures formed the basis of the software of a robot's decision-making system. As a result, the very strict requirements of storage and speed of decision-making system were met. Experience of using the presented methods shows that in a number of practical problems optimal solutions may be found manually without a computer. Apparently, this is connected with the fact that objects usually have such descriptions that problem (2) has the simplest solutions for them i.e. 1,2,3 (given in the lemma) that are easily found. REFERENCES Vizing, V.G. (1965). Chromatic class of a multigraph. Cybernetics.3. Victorova, N.P. (1981). Generalization and classification methods of level structures of special kind. proceed~s of Academf of Sciences. Tec~cal Cyberne ics,
2·
Evseev, a.v., and A.V.Solonin (1982). Branches and boundy method in the problem of planning of optimal ronte of a robot motion. In D.E.Okhotsimsky (Ed.), Advances in Information and Control Systems of robots. Aplied mathematics Institute of Academy of Sciences, Moscow. Evseev, a.v., and K.A.Pupkov (1981). Minimum resentation of external worls model in decision problems of stimUlus-reaction kind. Proceedings of Acadeil of Sciences, Technical Cyberne cs. 6. Plesnevitch G.s., and M:.s.Saparov (1981). Algorithms in graph theory. Ilim, Ashkhabad.
ln td li g~ncc,
THE SYNTHESIS OF MINIMAL·DESCRIPTION MODELS FOR REPRESENTING KNOWLEDGE K. A. Pupkov and O. V. Evseev Mosco l\' fl w itllle of Electronic Mechanical E ngin eering ..Hosco l\ '. USS R
,
" v
.'
Abstract. We consider the problem of constructing minimaldescription models for totalities of objects, - optimal models containing only a minimum of necessary knowledge without any element of which the model as a whole becomes not adequate to the object domain. A mathematical technique is introduced to describe and analyse the structure of such models and characteristics expressing their structure are analytically exhibited. Precise and E. -approximated methods for the synthesis of optimal models having the exhibited structure are described. The method for minimal supplement of optimal models is given. Keywords. Optimal systems; models; approximation theory; graph theory; identification; tree(mathematical); linear programming. INTRODUCTION
FORMULATION OF THE PROBLEM
The so-called frameworks of minimal description have a number of useful characteristics. A model containing only a minimal necessary amount of description occupies less storage than any other model. Further, provided that the model contains redundant information about some element of enVironment, the analysis of data about the situation including this element will also be redundant. So in order to decrease the decision making time it is worth using minimal-description models that do not contain redundant information. But the available methods do not provide the means for a precise constructing of the minimal-description models and do not make it possible to estimate the difference between the heuristic solutions and optimal solutions.
Let the objects or situations {S1' ••• 'SnJ =S be adequately described by multigraphs {G 1 , ••• ,Gn l=G with coloured nodes and arcs. It is necessary to construct the minimal adequate model of knowledge (MK)n , containing information only about some subgraphs (subdescriptions) g1' ••• ~ of graphs GiEG. It is necessary to find such subdescriptions that: a) there exists MK describing S adequately only with the help of these subdescriptions, their correlationship and their connection with names of situations of S; b) the summary volume of subdescriptions {g1, ••• ,~J= G is minimal in the sense of estimation
If there is a certain initial description of objects a minimal description of each object can be obtained by removing from its description those elements without which it still remains distinguishable from other objects (Victorova, 1981). But in the general case the solution of such local problems does not provide the global minimum of the whole description of objects. The model to be found must contain the optimal system of correlated adequate minimal descriptions.
where aj,b j are the costs of nodes and arcs rj of GiEG. The costs are assigned due to a concrete problem and can represent a number of computer operations required to input information in to the system about the availability of basic
139
K.A. Pupkov a nd O. V. Evseev
140
units used to describe the situations in the given situation. One can take a j =b j =1 (V j).
MK is considered to be adequate if
there exists a procedure P for a correct identification of all situations on the basis of information stored P is a tree (Pig.1) whose nodes correspond to the checks of presence of su~s1tuations s(gi) described by gie G in the input situation; arcs correspond to positive and negative results of the checks, and leaves correspond to the situations of S. Such a procedure is a permissible procedural form of MK sought for and is given by a pair = (G,P).
inn .
n
Por fixed G a set of volume and adaquacy equivalent modelsn =(G,P) is given by hierarhical model n=(G,Q) where the tree Q is of the type shown in Pig.2. Nodes of Q correspond to subgraphs giE G and subclasses MS: G (they are rectangular in Fig.2). If successor of gi is M there is the subsituation s(gi) in all situations corresponding to graphs of M. Subclasses-successors of some class cover all this class. Permissible trees P differ one from another by permutation of checks corresponding to subgraphs-successors of a single class M.
n
In the model =(G,Q) subclasses M are subjected to the ratio of taxonomic hierarchy. Chains (from root to leaves)_point out generalized notions giEG describing the given situation and subclasses which this situation enters. Such a declarative formn =(G,Q) of MK is sought for The synthesis problem is to find ~ and Q providing minimum (1) such that modeln =(G, Q) would be adequate. is the model of minimal description as it does not permit removing a single element forming it without the loss of the model's adequacy_
n
THE STRUCTURE OF OPTIMAL MODELS Let.f(M,t)= lK 1 , ••• ,KJ, (1~2) be a covering of subset M~ G by classes K1 , ••• ,Kl where: a) for any Ki ,
-
(i=1,1-1) there exists a set
CKi~~
of subgraphs g of volume X (see (1) when G =(g)) belonging only to graphs of Ki ; b) each g does not belong to graphs of Kl • Let the so-called covering C(G) be a totality of coverings.f (M, t ) : ! (G,o) = {K1 ,··· 'IS.J is applied to G; .f (K i , Of) is applied to non one-element Ki' (1.6 i ~ 1) and so on, until all classes become oneelement (Fi~.). Any C(G) generates some MK n =C~,Q). It is sufficient to take arbitrary subgraph gi of each set CK . for obtaining the set ~ G. For formation Q it is sufficient to take a tree defining covering C(G) (Fig.) and "hang" nodes with names of chosen subgraphs to his edges.
-
Theorem. Por any adequate G and (G,Q) there is C(Q) generating adequate MK n'=(G' ,Q') not of larger volume than
n=
n•
Conclusion: optimal adequate MK is generated by covering of the form C(G); C(G) is a "carrier" of structure of optimal adequate 1~ in the first order approximation; ~ and Q can be searched only among G,Q generated by coverings of the form C(G). Let us study ) types of extremum of objective function (1): 1) conditional, 2) inconditionally local, ) global. Minimum (1) corresponds to conditionally optimal MK no=(~o,Qo) under the following restriction: the volume of subdescriptions gi does not vanish along chains of the tree Q from root to the leaves. We introduce a particular type E~(G) of covering C(G) a totality of partially ordered partions .f°(M, 0)= {K 1 , ••• ,Klj , (Kif1Kj=~' i~j). The partition,P°is
a particular type of covering.f : subgraphs gE CK1 , (1=1,1-1) are not a subgraphs of graphs of class Kq , (q~i, q=1,1-1). Partitions ~oare ordered in E~(G): the partition J~G, 00) (where min{o;j'p°(G,(n])
00=
~
is applied to G, the partition ,O(K,~) is applied to non one-element classes K of .p o(G, ~) and so on
The Synthesis of Minimal-Description Models
Figure 1.
Figure 2.
Figure 3.
141
K.A. Pupkov and O.V. Evseev
142
,
until we have classes K1, ••• ,K1
in-
divisible by no JO(K~,Oo) at given to; then
t1 >00) 0,=
min {a: '0
31° «,0),J
are chosen for non one-element Kj and J> o (Ki' , 'If,) is applied, and so on until all classes become one-element (Fig.3). Theorem. There is a partition E~(G) for any adequate G, and any partition E~(G) generates adequate conditionally optimal MK. Conclusion: the structure of conditionally optimal MK is o exact1y given by the properties of E*(G). The problem of synthesis congerges to constructing arbitrary E*(G) and taking arbi trary subgraphs gi E CKi where CKi correspond to pOeM, 0) in E~(G). The so-called inconditiona11y 10ca1o -0 0 1y optimal MK =(G ,Q ) is certain to be obtained from no=(Go,Qo) by permutation subgraphs gie Go in the tree Qo• The ordering of subdescriptions gi in Qo over decreasing their volume becomes not valid and the set Go serves as local domain of searChing the extremum.
n
The particular form of C(G) is a partition C~(G) which consists of partitions .f;(M,O) of subsets M~ G into two classes K1 ,K2, ~ is a minimal value for which there is the partition .P °01' set M into classes K1 ,K2 or into classes K2,K1 exactly. Theorem. There is a partition C~(G) generating adequate globally optimal model n*=(G*,Q*) for any adequate G. Conclusion: the structure of globally optimal MK is given the properties of the partition C~(G); the search domain converges to the set of omode1s generated by the partitions C*(G). OPTIMAL-MODEL SYNTHESIS ALGORITHMS To obtain the conditionally optimal
no=(Go,Qo) it is necessary to be able to construct partitionsfo(M,~)
3pD(M, o)J. But partition poeM, 0') if
where 0,,= min{o: '0
?f
there exists and only if in some graph of M there is a subgraph of volume '0 which is not a subgraph of all graphs of M. Therefore constructing ofr (M, if) converges to solving the problem: in graphs Gi E M to find subgraph go of minimal volume ~o =V(go) which is not a subgraph of all graphs of M. = min V(g)
(2)
gc G. -
l.
gftr
Here r is a maximal isomorpha1 subgraph for graphs of M. To obtain..? 0 one can take J'" (M, 00) ={K1 Kd where
f
K1 = graphs of M, ~ontaining gol, K2=M' K1 • To form Go one can take an arbitrary subgraph from CK 1 , for instance, the subgraph go. The problem (2) can be po1ynomia11y converged to a problem of linear 0,1programming. But we consider a more efficient method using a modificated model proposed by V.G.Vizing when estimating the chromatic a1sss of graph (P1esnevitch,Saparov,1981; Vizing, 1965). Let us describe the polynomial desmant1ing procedure for graphs Gi , r without using index i of graph Gi • Let i,j be the nodes of
r ;
K,l - nodes of G; r(i,j),r(K,l) - colours of arcs connecting i,j in rand K,l in G (if i,j are not connected then r(i,j)=O). Let us define a set of pairs P1=(i,K), ••• ,Pm=(j,1) where i,K form the pair if their colours are identical. Let us assosiate every pair with node of graph B. If i=j or K=l then ps=(i,K),Pq=(j,l) are connected in B by an arc called a rigid one. If i~j, K~l, r(i,j)~r(K,l) ~O then Ps,Pq are connected with an arc of colour r(K,l) called a soft one. If i~j, K~l, r(i,j)=r(K,l)~ then Ps,Pq are connected with an arc of
Th e Synthe sis o f Minimal-Description Models
colour r(K,I) called a coloured one. Let us ~rk nodes of G presented in pairs P1, ••• ,Pm. Let us remove coloured arcs r(K,I) and isolated marked nodes from G. Let G' be the graph obtained. Let us remove arcs and non-marked nodes of G' from G. Let G" be the graph obtained. Let us remove from graph B soft arcs of colour r connecting nodes (i,K),(j,l) where nodes K,l of G' are connected by an arc of colour r. Let B(r,G) be the graph obtained. Lemma. The subgraph ~o being the sorutron of problem (2) is among those and only those graphs that are of~be following form: 1) a non~rked node of G'; 2) a subgraph of "node-arc-node" form of G' where both nodes are marked; 3) t+1 nodes of the same colour of G" where there are only t nodes in 4) a subgraph f~ G", rro with single jointed arc r of G" connecting nodes of f that is absent in Here r"., is one of subgraphs r of G.
r;
rm.
The subgraphs of 1,2,3 type are the simple solutions of the problem (2) that can be found quickly in polynomial number of steps. Here it is not necessary to construct B,B(r,G) it is sufficient to construct G',G". The graph B(r,G) is used for searching solutions of type 4. Theorem. An arbitrary coloured graphs rand G contain a subgraph f 5; r, G if and only if there is a set of nodes I={(i1,K1),(i2'~)' ••• ' (it,Kt )} in graph B(r,G) which are not connected with each other by rigid arcs - the so-called rigid independent set (RIS). Here f contains nodes K1,~, ••• ,Kt and nodes Kq,K s are connected by arc of colour r in f only if (i q ,K q ),(i s ,Ks )E! are connected by coloured arc of colour r in B( r ,G). Let us consider that RIS I generates f S;r,G. Let nodes (i1'K 1 ), (i2'~)~ I be connected by soft arc of colour r=r(K1'~). If we add r connecting K1 ,K2 to f, the subgraph g of type 4 will be constructed. If g $. r , it can be a solution of problem (2).
14 3
Lemma. If I consists of 2 elements, g=(K 1rK2),"r. If RIS I={(i1'K 1 ), ••• , (it,Kt )], (t ~ J) generates a subgraph fS; G with set R of arcs,then g!/: r if and only if an adjoint RIS I= { (i 1,K 1 ), ••• , (it ,Kt) generating subgraph f', G with arcs R,r does not exist.
J
An algorithm for searching 4-type solutions of problem (2) for each soft arc rE B( r , G) constructs sets I (RIS with nodes connected by r) increasing number of nodes in I. A set R of arcs of f c r,G is constructed according to the last theorem. If the conditions of the last lemma are satisfied the subgraph f is constructed, r is added to it and the 4-type g is obtained. The simplest algorithm for synthesis of the conditionally optimal MK =(~,Qo) searches the subgraph go solving problem (2) for graphs of M~ G(M=G on the first step), includes go in Go' Ki={grafhS of M, containing gol, K2 =M' K1 , constructs the part of the tree Qo (M is a root; go'~ - successors of M; K1 successor of go). Further these operations are applied to non one-element classes obtained if any. The upper bound for a number of problems (2) to be solved here is
no
A
N
=n
([log2n] + 1).
Let us construct the unconditionally locally optimal MK n°. Let there be I ~ 2 identical nodes g of volume (f in Qo • Moving in Qo from nodes g along the chains to the root, let's define class K where these chains converge. Let Q(K) be a subtree with the root K, 'lf1 < 1/ be the volume of successors of K. Let us describe a polynomial transformation 0. of MK J no... (Go,Qo)· a) Construct K1={ graphs of K, containing g} , K2 =K' K1 and a tree T: K is a root; g1K2 are successors of K; K1 is successor of g. b) Construct trees Q(Ki ) giving conditionally optimal models for Ki ,(i=1,2). For
K.A. Pupkov and O.V. Evse ev
144
this purpose it is necessary to remove graphs being absent in Ki from the nodes-classes of Q(K). c) Substitute for subtree obtained by attaching roots Ki of Q(Ki ) to the leaves of T the subtree Q(K) in Qo • If Q(K) gives a conditionally optimal model for K, K is referred to as a dominant class. Then, the application of to the tree Qo results in the tree Q' = Qo and the volume of MK is decreased. An unconditionally local minimum of (1) corresponds to a sequence of transformations tSj" (5 gi ' • • • , 6" gin giving maximal vo-
Cl,
og
2
lume compression of
-V(~;11
C·
(c5ji2
MK
A ~V( Q
(63'1 Q o ) ... )).
o
)-
n
Here V(Q) is the volume of MK = (G,Q). The problem is solved by E- approximated method of "branches and bounds" ( E. ~ 0). Branching OCCUl!'S by all possible sequences, bounds are calculated in polynomial number of steps. Let us denote Qi ••• jq= 639"C6jj
(...
. = ••• Jq ... V(Q )-V(Q . . ). Let in the tree o l:"J q Qi ••• jq there is some sets of Is
(03i Qo ) ... )) ,
identical nodes gs'
A V.
~
(s~i, •••
,j,q).
Let U be a set of all different gs which have a dominant class in Qi ••• j q' be the volume of suc-
rS
cessor of K , (f'$ be a volume of g • Let us marksnodes-classes whose suScessors are our gs in the tree with the root Ks. Let us find a node-subgraph of minimal volume ~$ with marked su~cessor. Lemma. If a sought for sequence begrns-with transformations ()g, , ••• 03;, Ot9" the upper and lower bounds of the A value A V* are: f i ••• jq=.AVi ••• Jq . +
These bounds also make it possible to evaluate.tl V* and to neglect obtaining too little compression of the model's volume. Let us consider the E -approximated method based on the "branches and bounds" technique for the synthesis of globally optimal MK n~=(~*,Q*). The method consists in constructing tree B~(G) which c~tains as a root subtree the tree Q* corresponding globally optimal MK exactly to given t~ 0; then Q! is selected in Be< G) by means of a linear algorithm. n~
is generated by the partition
C~(G). Therefore, the subgraphs
gic G* are among those subgraphs which can belong to the sets CKi of parti tions .1'". 0 (M, ~) in C~ (G) (it is sufficient to single any subgraph in every CKi ). Let us denote Z(G) the set of subgraphs which can form G*. Z(G) can be found when constructing conditionally optimal MK if all the solutions of each problem (2) are found. Let us construct a tree B(G) with a root G. The successor of non one-element node-class M f: G (M=G on the first step) are nodes-subgraphs gjcZ(G) for which M~M Nj~~ where
J;
Nj = [graphs of G, containing gj gj has 2 successors M(1N j , M'(MlrN j ). The leaves of B(G) are one-element classes. Let QCB(G) be a subtree with the root G where any class has only one successor. There is a one-to-one correspondence between subrees of the form Q and partitions of the form C~(G). Hence the sought for Q* is among the subtrees QCB(G). Let us assign oosts to nodes of B(G): a) cost of any leaf is 0; b) cost of node gj is the volume of gj plus costs of its successors-classes; c) the cost of a class which is not a leaf is the least of costs of its successors gj. For any olass in B(G) let us leave only one successor with the least cost. Then in B(G) a sought for tree Q* will be the single tree connected with the root G.
Th e Synthesis of Minima l-Des c ription Mode ls
The prqcess of directed constructing tree Bi(G)C B(G) provides the availability of the sought for tree Q* in it. We continue to construct successors only for those leaves K of B(G) which are the leaves of subtree Q'cB(G) pos~sing the property: the lower bound V of value v*=V(n*) is minimal for Q' under condition that the root part of the tree Q* is Q', ~/' and E =V-V is minimal for Q' where ",. V is the upper bound of V* under the same condition. Bounds V,, are obtained by the volume of nodessubgraphs in Q' and values of bounds '" A V(K)~ V(K) of volume V*(K) of globa!lY optimal models for leavescl~~es K of Q'. Bounds '" V(K), -A V(K) are calculated in polynomial number of steps according to the results of synthesis of conditionally optimal MK. Let no(K)= =(G~,Q~) be conditionally optimal MK for graphs of K. Let there are subgraphs of volume ~ ~ in G~. Let us find in Q~ for each
to, , ...,
1E W(K)=b'f'~t··· ~m J the class K such that there is the greatest number m ( 6) of subgraphs of volume just below it. Let ~o (K) be a volume of successors of K in Q', and I K I be a number of elements 8f K. K'~
t
~.
Values ~(K)=
Lm(!)·(o(K»).
L m(~)'1+(IKI-1-
~EW(K)
~
, V(K)=
oe:W(K)
= V(no(K» are upper and lower bounds of value V*(K) of volume of globally optimal-model for graphs of K: V(K)~ V*(K)~ V(K). These bounds also make it possible to evaluate the difference V( no)-V( n1fo ) in polynomial number of steps and to abandon obtaining too little reducing of the MK volume in constructing n~. ,,~ Besides, if V(G)=v(G), no =(Go,Qo) is
-
globally optimal MK. If Go consists of subgraphs of identical volume, is globally optimal MK.
nD
145
SUPPLEMENT OF OPTTIlIAL KNOWLEDGE MODEL of the formn(G)=(G,P) converges to inputing of data about the new situation Sn+1 f= S= { S1' ••• 'Sn} described by graph Gn+1~ G= {G 1 , ••• ,Gn ) into the model. If we input Sn+1 to the entry of procedure P, Sn+1 is certain to be erroneously assosiated with situation Sib S. Let us use the result of this erroneous identification for the correction of n (G). It is necessary to distinguish Sn+1 from Si for P to provide a correct identification of all S1, ••• ,Sn+1. For this purpose let us add in set a subgraph go of one of the graphs Gi , Gn +1 such that another of these graphs does not contain go' and let us substitute for node go with successors Si,Sn+1 the leaf Si in the tree P. The resultant new tree P' provides correct identification of all S1, ••• ,Sn+1. For obtaining a minimal supplement of the model's volume it is necessary to take a minimal in volume go' i.e. to solve problem (2). As a result we have a new , adequate model n =(GUgo'P') for situations S1' ••• Sn+1. There is a minimal difference in the volume between this procedural model and the ini tial procedural model (G). Ap¥lications. The main application of me hods eleborated is the synthesis of mjnimal-description models for robots with microprocessor control. The methods were used to construct a subsystem for planning a path connection of a mobile robot (Evseev,Solonin, 1982). Two minimal-description models were used. The first model contained information about applicability conditions of local plan operators - operators of constructing paths going round obstacles. The second model contained knowledge about the form of typical obstacles and estimations of complexity of overcoming them by robot. It helped us to
~
-
n
K. A. Pupk ov a nd O.V. Evs eev
146
construct a plan of the robot's movement with minimal executing difficulty in concrete situations. The models were synthesised in declara ti ve form (the volume cutting down obtained in comparison with the initial description was about 6 times). Each model was transformed to equivalent procedural form ~= (G*,P*) where tree P* represents the procedure for the identification of situations with a minimal summary cost of elementary operations in checks. These procedures formed the basis of the software of a robot's decision-making system. As a result, the very strict requirements of storage and speed of decision-making system were met. Experience of using the presented methods shows that in a number of practical problems optimal solutions may be found manually without a computer. Apparently, this is connected with the fact that objects usually have such descriptions that problem (2) has the simplest solutions for them i.e. 1,2,3 (given in the lemma) that are easily found. REFERENCES Vizing, V.G. (1965). Chromatic class of a multigraph. Cybernetics.3. Victorova, N.P. (1981). Generalization and classification methods of level structures of special kind. proceed~s of Academf of Sciences. Tec~cal Cyberne ics,
2·
Evseev, a.v., and A.V.Solonin (1982). Branches and boundy method in the problem of planning of optimal ronte of a robot motion. In D.E.Okhotsimsky (Ed.), Advances in Information and Control Systems of robots. Aplied mathematics Institute of Academy of Sciences, Moscow. Evseev, a.v., and K.A.Pupkov (1981). Minimum resentation of external worls model in decision problems of stimUlus-reaction kind. Proceedings of Acadeil of Sciences, Technical Cyberne cs. 6. Plesnevitch G.s., and M:.s.Saparov (1981). Algorithms in graph theory. Ilim, Ashkhabad.