p-Adic Discrete Dynamical Systems and Collective Behaviour of Information States in Cognitive Models

We develop a model of functioning of complex information systems (in particular, cognitive systems) in that information states are coded by p-adic integers. An information state evolves by iterations of a discrete p-adic dynamical system. The padic utrametric on the space of information states (p-adic homogeneous tree) describes the ability of an information system to operate with associations. The main attention is paid to the collective dynamics of families of associations.


INTRODUCTION
The system of p-adic numbers Qp, constructed by Hensel in the 1890s, was the first example of an infinite number field (i.e., a system of numbers where the operations of addition, subtraction, multiplication and division are well defined) which was different from a subfield of the fields of real and complex numbers.During much of the last 100 years p-adic numbers were considered only in pure mathematics, but in recent years they have been extensively used in theoretical physics (see, for example, the books Vladimirov et al., 1994  and Khrennikov, 1994 and the pioneer papers Volovich, 1987; Freund and Olson, 1987; Manin,   1985), the theory of probability, Khrennikov  (1994) and investigations of chaos in dynamical sys- tems Khrennikov (1997), (1998) and Albeverio et al. (1998).In Khrennikov (1998); Albeverio et al.  (1999)  and Dubischar et al. (1999) p-adic dynami- cal systems were applied to the simulation of functioning of complex information systems (in particular, cognitive systems).In this paper we continue these investigations.We study the collec- tive dynamics of information states.We found that such a dynamics has some advantages compare to the dynamics of individual information states.First of all the use of collections of sets (instead of single points) as primary information (in particu- lar, cognitive) units extremely extends the ability of an information system to operate with large volumes of information.Another advantage is that (in the opposite to the dynamics of single states) the collective dynamics is essentially more regular.
As we have seen in Khrennikov (1997); Albeverio  et al. (1998), discrete dynamical systems over fields of p-adic numbers have the large spectrum of non- attracting behaviours.Starting with the initial point xoQp iterations need not arrive to an attractor.In particular, there are numerous cycles (and cyclic behaviour depends cruicially on the prime number p) as well as Siegel disks.In our information model attractors are considered as solutions of problems (coded by initial informa- tion states x0 Qp).lThe absence of an attractor implies that in such a model the problem x0 could not be solved.In the opposite to dynamics of single information states (p-adic numbers) collec- tive dynamics practically always have attractors (at least for the dynamical systems which have been studied in Khrennikov, 1999 and Albeverio   et al., 1998).So here each problem has the definite solution. 2 There are no physical reasons to use only prime numbers p > as the basis for the description of a physical or information model.Therefore, we use systems of so called m-adic numbers, where rn > is an arbitrary natural number, see, for example, Mahler (1980).
In our model each chain of neurons A/" has a hierarchic structure: neuron no is the most im- portant, neuron nl is less important than neuron n0,...,neuron nj is less important neurons than no,..., nj_ 1.This hierarchy is based on the possibility of a neuron to ignite other neurons in this chain: no can ignite all neurons nl,...,nk,...,nM, nl can ignite all neurons nz,...,nk,...,nM, and so on; but the neuron n cannot ignite any of the previous neurons no,...,nj_l.Moreover, the process of igniting has the following structure.If nj. has the highest level of excitement, a= m-1, then increasing of a to one unit induces the complete relaxation of the neuron nj, a a-0, and increasing to one unit of the level of excitement a+l of the next neuron in the chain,

m-ADIC HIERARCHIC CHAINS FOR CODING OF INFORMATION
The abbreviation "I" will be used for information.
The symbol 7-will be used to denote an/-system.Let 7-be an/-system (in particular, a cognitive system).We shall use neurophysiologic terminol- ogy: elementary units for /-processing are called neurons, a 'thinking device' of 7-is called brain.In our model it is supposed that each neuron n has Oj+l --+ Oj+ OZj+l --1. (2)  If neuron n+l already was maximally exited, ay+l=m-1, then transformation (2) will auto- matically imply the change to one unit of the state of neuron nj+2 (and the complete relaxation of 3 the neuron nj.+ l) and so on.
We shall use the abbreviation HCN for hier- archic chain of neurons.This hierarchy is called a horizontal hierarchy in the/-performance in brain.This is more or less the standard viewpoint for models based on neural networks, see, for example, Amit (1989).
Of course, the construction of this solution needs time and memory resources.An information system may have or may not have such resources.
In fact, transformation (2) is the addition with respect to mod m.
p-ADIC DISCRETE DYNAMIC SYSTEMS 61 HCNs provide the basis for forming asso- ciations.Of course, a single HCN is not able to form associations.Such an ability is a feature of an ensemble B of HCNs of 7-.Let s E {0, 1,...,m-1}.A set As {x (00,..., cM) XI co s} C XI is called an association of the order 1.This association is represented by a collection B of all HCNs A/'-(no, n,..., nt) which have the state Co-S for neuron no.Thus any association As of the order is represented in the brain of 7-by some set B of HCNs.Of course, if the set B is empty the association As does not present in the brain (at this instance of time).Associations of higher orders are defined in the same way.
Let So,...,sl_ {0, 1,...,m-1}, l< M. The set Aso...Sl {x (o0,... ,OM) Xl" is called an association of the order l.Such an association is represented by a set Bs0...st c B of HCN.We remark that associations of the order M coincide with/-states for HCN.We shall demon- strate that an/-system 7-obtains large advantages by working with associations of orders l<<M.
Denote the set of all associations of order by the symbol XA,I.We set XA U x,.This is the set of all possible associations which can be formed on the basis of the/-space XI.
Sets of associations J c Xa also have a coginitive meaning.Such sets of associations will be called ideas of 7-(of the order 1).Denote the set of all ideas by the symbol XIz. 4 Homogeneous ideas are ideas which are formed by associations of the same order.For example, ideas J= {As,... ,Aq}, s,..., q {0, 1,... ,m-}, or J {ass2,... ,aq,q2}, si,...,qi6 {0, 1,...,m-1} are homogeneous.An idea J {As,Aslsz,...,Aqlq2 qt} is not homoge- neous.Denote the space of all ideas formed by associations of the fixed order by the symbol Xiz,z (these ideas are homogeneous).Denote the space of all ideas formed by associations of orders less or equal to L (where L is the fixed number) by the symbol Xo.
The hierarchy /-state association idea is called a vertical hierarchy in the/-performance in the brain.
Remark 2.1 (Associations, ideas and complexity of cognitive behaviour) One of the main features of our model is that not only/-states x XI, but also associations A X and ideas JXID have a cognitive meaning.One of the reasons to use such a model is that complex cognitive behaviour can be demonstrated not only by living organisms 7- which are able to perform in 'brains' large amounts of 'pure information' (/-states), but also by some living organisms with negligibly small 'brains'.It is well known that some primitive organisms 7-pr having (approximately) N=300 nervous cells can demonstrate rather complex cognitive behaviour: ability for learning, complex sexual (even homosexual) behaviour.Suppose, for example, that the basis m of the coding system of 7-pr is equal to 2.Here each nervous cell n can yield two states: 0, nonfiring, and 1, firing.Nonhier- archic coding of information gives the possibility to perform in the brain (at each instance of time) 300 bits of information.In the process of 'think- ing' (see Section 3) 7-pr transforms these 300 bits into another 300 bits.It seems that such 300-bits /-dynamics could not give a complex cognitive behaviour.We now suppose that 7-pr has the ability to create hierarchic chains of nervous cells (horisontal hierarchy).Let, for example, such HCNs have the length L 5. Thus 7-pr has N 60 HCNs (so the set B -pr has 60 elements).The total number of/-states, x (c0, Cl, c2, c3, c4), cj 0, 1, which can be performed by HCNs of the length L 5 is equal to NI= 2 s= 32.Thus brain's hard- ware B 7"pr can perform all/-states (since NI < N).
We assume that all/-states are performed by the brain at each instant of time.We suppose that 7-pr 4in principle, it is possible to consider sets of ideas of the order as new cognitive objects (ideas of the order 2) and so on.However, we restrict our attention to dynamics of ideas of order 1. is able to use the vertical hierarchy in the /- performance.The 7-pr have Na-2 ' associations of order k--1, 2,..., 5.The number of homogeneous ideas of 7-pr is equal NIp  (22 1) + (222 1) + (2 23 1) + (2 24 1) + (22s 1) 4295033103 > > 300 (each term contains -l, because empty sets of associations are not considered as ideas).Hence 7-pr works with more than 4295033103 'ideas' (having at the same time only NI= 32/-strings in his brain).
In our model 'hardware' of the brain ofis given by an ensemble B of HCNs.For an HCN A/" E B-, we set i(A/')= x, where x is the/-state of A/'.The map i: B-Xz gives the correspondence between states of brain and states of mind. 5In general it may be that i(A/')=i (A/'2) for A/' -A/'2.It is natural to assume that in general the map depends on the time parameter t: i=it.In particular, if is discrete, we obtain a sequence of maps it: t=0, 1, 2,....
Let O be some subset of XI.The space of associations which are composed by /-states x belonging to the set O is denoted by the symbol XA(O).The corresponding space of ideas is denoted by the symbol XD(O).
In the spatial domain model each HCN A/" is a chain of physical neurons which are connected by axons and dendrites, see, for example, Eccles   (1974).In principle, such a chain of neurons can be observed (as a spatial structure in the Euclidean space R3), compare with Cohen et al. (1997) and Courtney et al. (1997).In the frequency domain model (see Hoppensteadt, 1997) digits ajE{0, 1,...,m-1} can be considered as (discretized) frequencies of oscillations for neurons nj., j 0, 1, 2,...,which form a 'frequency HCN' A/'.Here A/" need not be imagine as a connected spatial structure.It may have a dust-like structure in R3.

DYNAMICAL EVOLUTION OF INFORMATION
In this section we shall study the simplest dynamics of/-states, associations and ideas.Such /-dynamics is "ruled" by a function 35 XI XI which does not depend on time and random fluctuations.This "process of thinking" has no memory: the previous/-state x determines a new /-state y via the transformation y--fix).In this model time is discrete, t--0, 1, 2,..., n,..., K. Set U io(B"-), U il(B"-),..., U'(B'), A set U of/-states is called an/-universe of -.This is the set of all/-states which are generated by the ensemble B of HCNs ofat the instant of the time t-n.We suppose that sets {U}_ 0 of /-states can be obtained by iterations of one fixed map f'.XI XI.Thus dynamics (3) of/-universe ofis induced by pointwise iterations: Xn+l f(Xn).
If X U, then y =fix) U+1.Suppose that, for each association A, its image B=f(A) {y =fix): x A} is again an association.
Denote the class of all such maps f by the symbol ,A(XI).IffE A(XI), then dynamics (4) of/-states of induces dynamics of associations An+l =f(An). (5) Starting with an association A0 (which is a subset of/-universe U)obtain a sequence of associations: A0, A1 =f(A0),..., An + =f(An),.... Dynamics of associations (5) induces dynamics of ideas: jt =f(j)= {B-=f(A) A J}. Thus each idea evolves by iterations: Jn+a --f(Jn). (6) In fact, the map provides the connection between the material and mental worlds.
We are interested in attractors of dynamical system (6) (these are ideas-solutions).To define attractors in the space of ideas XID, we have to define a convergence in this space.This will be done in Section 5.

m-ADIC REPRESENTATION FOR INFORMATION STATES
It is surprising that number systems which provide the adequate mathematical description of HCN were developed long time ago by purely number theoretical reasons.These are systems of so called m-adic numbers, rn > is a natural numbers.First we note that in mathematical model it would be useful to consider infinite/-states: aj 0, 1,...,m-1. (7) Such an /-state x can be generated by an ideal infinite HCN 32.Denote the set of all vectors (7) by the symbol Zm.This is an ideal /-space, XI Zm.On this space we introduce a metric Pm corresponding to the hierarchic structure between neurons in chain A/" having an /-state x: two holds true.This inequality has the simple /- meaning.Let JV',Ad,Z; be HCNs having /-states x, y, z respectively.Denote by k(.A/', Ad) (k(N', E) and k(Ad,/2)) length of an initial segment in chains N" and .AA (A/" and/2, AA and 12) such that neurons in A/" and AA have the same level of exiting.Then is evident that k(A/', A//) _> min[k(A/', ), k(, AA)].
of balls.Geometrically Z can be represented as a homogeneous tree.Associations are represented as bundles of branches on the m-adic tree.Ideas are represented as sets of bundles.So dynamics (4), ( 5), ( 6) are, respectively, dynamics of branches, bundles and sets of bundles on the m-adic tree.
/-dynamics on Zm is generated by maps ]2 Zm--+ Zm.We are interested in maps which belong to the class ,A(O), where (.9 is some subset of Zm. They map a ball onto a ball: f(Ur(a))= Ur, (a').To   give examples of such maps, we use the standard algebraic structure on Zm.
Each sequence x (c0, c1, ct,... ) is identi- fied with an m-adic number X--Z ojmJ-Ozo-+-01 m -t--o2 m 2 + + On m n +.... j=O (10) It is possible to work with such series as with ordinary numbers.Addition, subtraction and multiplication are well defined.Analysis is much simpler for prime numbers m-p > 1. Therefore we study mathematical models for p-adic numbers.
Hence here associations are transformed into associations.m-adic balls U (a) have an interesting property which will be used in our cognitive model.Each point b E Ur(a) can be chosen as a center of this ball: U(a) =_ U(b).Thus each/-state x belonging to an association A can be chosen as a base of this association, m-adic balls have another interesting property which will by used in our cognitive model.Let Ur(a) and Us(b) be two balls.If the intersection of these balls is not empty, then one of the balls is contained in another.

SEMI-METRIC SPACES OF SETS
Let X be a set.A function p X X R+ is said to be a metric 6 if it has the following properties: (1) p(x, y)=0 iff x=y; (2) p (x, y)= p(y, x); (3)   p(x, y) <_ p(x, z) + p(z, y) (the triangle inequality).
The pair (X, p) is called a metric space.The set Ur(a) {x X: p(x,a) _< r}, a X, r > 0, is a ball in X.This set is closed in the metric space (X, p).
A metric p on X is called an ultra-metric if the so called strong triangle inequality p(a, b) <_ max[p(a, c), p(c, b)], a, b, c X, holds true; in such a case (X, p) is called an ultra- metric space.
A distance between a point a X and a subset B of X is defined as p(a,B)-inf p(a,b) bB (if B is a finite set, then p(a, B)= minb z p(a, b)).
Denote by Sub(X) the system of all subsets of X.
The Hausdorf distance between two sets A and B belonging to Sub(X) is defined as p(A, B) sup p(a, B) sup inf p(a, b).aEA aEA bB (11) If A and B are finite sets, then p(A, B) max p(a, B) max min p(a, b).
The Hausdorf distance p is not a metric on the set Y=Sub(X).In particular, p(A,B)=O does not imply that A B. For instance, let A be a subset of B. Then, for each a A, p(a, B)= infb z e p(a, b)= p(a, a)=0.So p(A,B)=O.However, in general p(A, B)=0 does not imply A cB. 7 Moreover, the Hausdorf distance is not symmetric: in general p(A, B) 7 p(B, A). s 6The symbol R+ denotes the set of non-negative real numbers.
Let B be a non-closed subset in the metric space X and let A be the closure of B. Thus B is a proper subset of A. Here, for each a E A, p(a, B) inf6 B p(a, b) 0. Hence p(A, B) O.
Let A C B and let p(b, A) # 0 at least for one point b E B. Then p(A, B) 0. But p(B, A) > p(b, A) > O.
p-ADIC DISCRETE DYNAMIC SYSTEMS   65   We shall use the following simple fact.Let B be a closed subset in the metric space X. 9 Then p(A, B) 0 iff A c B. In particular, this holds true for finite sets.
The triangle inequality p(A,B) <_ p(A, C) + p(C,B), A,B, C E Y, holds true for the Hausdorf distance.
Let T be a set.A function p: Tx TR+ for that the triangle inequality holds true is called a pseudometric on T; (T,p) is called a pseudometric space.So the Hausdorf distance is a pseudometric on the space Y of all subsets of the metric space X; (Y, p) is a pseudometric space.
The strong triangle inequality p(A,B) <_ max[p(a, C), p(C,B)] A,B, C Y, holds true for the Hausdorf distance corresponding to an ultra-metric p on X.In this case the Hausdorf distance p is a ultra-pseudometric on the set Y= Sub(X).
We can repeat the previous considerations starting with the Hausdorf pseudometric on .
We set Z=Sub(Y) and define the Hausdorf pseudometric on Z via (11).As p Yx YR+ is not a metric (and only a pseudometric) the Hausdorf pseudometric p: Z x Z R+ does not have the same properties as p: Yx YR+.In particular, even if A, B Z-Sub(Y) are finite sets, p(A, B)--0 does not imply that A is a subset of B. For example, let A {u} and B= {v} are single- point sets (u,v Y=Sub(X)) and let u cv (as subsets of X).Then p(u, v)=0.If u is a proper subset of v, then A is not a subset of B (in the space Y).
PROPOSITION 5.1 Let A, B Z Sub(Y) be finite sets and let elements of B be closed subsets of X.If #(A, B)-O, then, for each u A, there exists v B such that ucv.
there exists v B such that p(u, v)=0.As v is a closed subset of X, this implies that u c v.
Let A, B Z and let, for each u A, there exists v B such that u c v. Such a relation between sets A and B is denoted by the symbol A C CB (in particular, A C B implies that A c c B).We remark that AcCB and B c cA do not imply A=B.For instance, let A--{Ul,U2} and let B= where ul cue.We also remark that A1 c c B1 and AeCC Be implies that A1 t_J A2 Cc B U B2.
Let f: T T, where T= Y or T-Z, be a map.
Let H be a fixed point off, f(H)= H.A basin of attraction of H is the set AT(H)-{J E T: limn p(f"(J), H) 0}.We remark that J A T(H) means that iterations f(J) of the set J are (approxi- mately) absorbed by the set H. The H is said to be an attractor for the point J Z if, for any fixed point H off such that lim,_ p(fn(j), H) 0 (so J A T(H)), we have H c H. Thus an attractor for the set J( 6 Sub(Y)) is the minimal set that attracts J.The attractor is uniquely defined.
Let T Z Sub(Y), Y Sub(X).The H is said to be an c c-attractor for the point J E Z if, for any fixed point H off such that lim p(f(J), H)--0 (so JAT(H)), we have H CC H. c C- attractor is not uniquely defined.For example, let J= {u}, u Y, flu)= u.Here the set J is an attractor (for itself) as well as any refinement of J: A {u,v, ,Vu}, where vj C u.
All previous considerations can be repeated if, instead of the spaces Y= Sub(X) and Z---Sub(Y) of all subsets, we consider some families of subsets: U c Sub(X) and V= Sub(U).We obtain pseudo- metric spaces (U, p) and (V, p).Let J2 U-U be a map.For u U, we set O+,k(u) {fl(u): _> k},k 0, 1,2,..., and 0 k=O Proof As p(A, B)--0, then, for each uA, p(u, B) minb B p(U, b) 0. Thus, for each u E A, For a set J V, we set O +,k(J) [Ju J O +,k(u) and O (J) [,Ju j O (u).
9A closed set B can be defined as a set having the property: for each x E X, p(x, B)---0 implies that x E B. LEMMA 5.1 Let the space U be finite.Then, for each J E V, Y is attracted by the set Oo (J).Proof First we remark that, as O(u)c...c 0 +,k + l(U) C 0 +,k(u).C 0 +,o(U), and 0 +,o(U) is finite, we get that O(u) =O+,k(u) for k >_ N(u) (where N(u) is sufficiently large).
We prove that, for each u E J, the set O(u) is f-invariant and lim p(fn (u) O (u)) O.
THEOREM 5.1 Let the space U be finite and let the Hausdorf distance on the space U be a metric which is bounded from below.Then each set J V has an attractor, namely the set O(J).
Proof By Lemma 5.1 we have that J EAT (O(J)).We need to prove that if, for some set A V, lim p(fk(u) A)-0, then O(u) c A. Let p(ft(u),A) < 5 for > k >_ N(u) (here 5 is defined by condition ( 12)).As A is a finite set (so p(d, A)-mina A p(d, a)), we obtain that p(fl(u), a) 0 ( 14) for some a-a(u, 1) A. Hence fl(u) a(u, 1) A, l >_ k. (15) Thus Oo(u) 0 +,k(U) C A. Let lim p(fk(j),A) 0 ( 16) As U is finite (and so J is also finite), ( 16) holds true iff (13) holds true for all uJ.Thus Oo(u) c A for each u E J.So O(J) c A.
If the Hausdorf distance is not a metric on U (and only a pseudometric), then (in general) the set O(J) is not an attractor for the set J. However, we have the following result: THEOREM 5.2 Let the space U be finite and let all elements of the space U C Sub(X) be closed subsets of the metric space (X, p).Let the Hausdorf pseudometric on the space U be bounded from below.Then each set J G V has an c C-attractor, namely the set O(J).
Proof By Lemma 5.1 we again have that J AT(O(J)).We need to prove that if, for some set A V, (13) holds true, then O(u)cc A. We again obtain condition (14).However, as p is just a pseudometric, this condition does not imply (15).We apply Proposition 5.1 and obtain that ft(u) c a(u, 1).As O(u)= 0 +,k(U) for sufficently large k, we obtain that, for each wOo(u) (w =ft(u), > k), there exists a E A such that w c a. Thus Oo(u) c c A.
In applications to the/-processing we shall use the following construction.
Let (X, p) be an ultrametric space.We choose U cSub(X) as the set of all balls Ur(a).The Hausdorf distance is an ultra-pseudometric on the space of balls U.As balls are closed, p(Ur(a), U(b))=O implies Ur(a) C U(b).In particular, p(Ur(a), U(b))=O implies U(a) U,,(b).performed via mod pm arithmetics for natural numbers.In particular, the attractor O(J): {U1/pm(t)'tEO(D)}.Therefore, the solution O(J) of the problem J can constructed purely mod pro-arithmetically.CONECTt3RE The process of thinking (at least its essential part) is based on mod pm arithmetics.
The same considerations can be used for non- homogeneous ideas J EX(O).Here J-{Jm}, where JmXz),m(O).Due to properties of the map fn all homogeneous ideas Jm proceed independently.