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% Title.
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\title{Shift operators and complex systems}
\twoauthors
{C. Bertelle($^1$), G.H.E. Duchamp($^2$)}
{($^1$) LIH - University of Le Havre, \\
($^2$) LIPN - University of Paris XIII, \\ France}
{Lua\"\i Jaff}
{LIPN - University of Paris XIII, \\ France}
%
% {G. H. E. Duchamp}
% {LIPN - University of Paris XIII - France}
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% {K. Khatatneh, R. Ghneamat, S. Oqeili}
% {Al-Balqa' Applied University - Al-Salt, Jordan}

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%  {A. Author-one, B. Author-two\sthanks{Thanks to XYZ agency for funding.}}
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%  {C. Author-three, D. Author-four\sthanks{The fourth author performed the work
%   while at ...}}
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\begin{document}

\maketitle

%
\begin{abstract}
In a previous paper \cite{epnads}, some of the authors dealt with the use of automata with (various type of) 
multiplicities for modelizing agents with rational behaviour and how one could perform genetic operations on them. 
In this paper, we emphasize the role played by shift operators to identify automata with similar behaviour.   
\end{abstract}

\vspace*{0.5in}
\vspace*{-\baselineskip}


\section{Introduction: {\tt TODO}}

\section{Shift operators}

One of the most notaural operations on functions is the shifting of the argument. 
Using the denotation of\\ 
{\tt http://mathworld.wolfram.com/}\\
{\tt ShiftOperator.html},\\ 
one has 
\begin{equation}
	E^a(f)[x]:=f(x+a)
\end{equation}
What is little less known (but not less useful) is the use of sifting operations 
on the exponents of a taylor series (which amounts to the same realm, considering a series as a function of the monomials).\\
To each series, 
\begin{equation}
	S=\sum_{n\geq 0}\al_n z^n
\end{equation}
one make correspond the series 
\begin{equation}\label{gammadag}
	\gamma_z^\dag(S)=\sum_{n\geq 0}\al_{n+1} z^n
\end{equation}
In this manner one can deal not only with polynomials but with rational functions. This setting has a natural 
extension to the multivariate case allowing to dela with rational behaviours.
Rational behaviours are exactly the behaviours of automata with multiplicities and these automta are the type 
which is well fitted for the use of genetic algorithms over populations of agents with rational behaviours \cite{epnads}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Kinds of Automata}


Automata theory is at the edge of many areas of human skill
 (mathematics, physics, engeneering, ...) and roughly speaking, could be defined as 
 {\it all models dealing with transition rules}.\\
 Several domains have been developped within automata theory including: {\it cellular automata, 
 boolean automata (or finites-state automata) and multiplicity automata}.\\
 The first kind deals with with {\it local trasition rules} whereas the two last are defined by 
 {\it global rules}. As a matter of fact, the last one ({\it multiplicity automata theory \cite{Eil,BR}})
  includes the second by specilizing the scalars to the Boolean semiring. Indeed varying the scalars, 
  one can reach, with the same theory, models as diverse as {\it deterministic finite automata}, {\it finite Markov chains} and {\it transducers} (kinds of automta where transitions produce words and languages).\\  
This gives the general framework for {\it multiplicity automata} or {\it automata with multiplicities}. 
In this model, the output data of the automata with output belong to a specific 
algebraic structure, a semiring \cite{Ei,Go,St}. 
In that way, we will be able to build effective operations on such automata, 
using the power of the algebraic structures of the output data 
and we are also able to describe this automaton by means of a matrix 
representation with all the power of the new linear algebra (i.e. with semirings).\\
Moreover, the automaton can be observed by means of the function it generates. 
This function will be called the {\it behaviour} of the automaton.

\begin{definition}
{\bf (Automaton with multiplicities)}\\
 An automaton with multiplicities over an alphabet  $A$  and a semiring $K$ is 
the 5-uple $(A,Q,I,T,F)$ where
\begin{itemize}
\item  $Q=\{S_1,S_2\cdots S_n\}$ is the (finite) set of states;
\item   $I: Q\mapsto K$  is a function over the set of states, which 
associates to each initial state a value in $K$ (for instance probability, word or boolean), 
called entry cost, and to non-initial state a zero value ;
\item	$F: Q\mapsto K$  is a function over the set states, which 
associates to each final state a value in $K$, called final cost, and to non-final 
state a zero value;
\item $T$ is the transition function, that is\\
 $T: Q\times A\times Q\mapsto K$ 
which to a state $S_i$, a letter $a$ and a state $S_j$ associates a value $z$ in 
$K$ (the cost of the transition) 
if it exist a transition labelled with $a$ from the state $S_i$  to the 
state $S_j$ and and zero otherwise.\\   
\end{itemize}
\end{definition}

\begin{remark}{\tt to redo} 
i) As told above, automata with multiplicities are a generalisation of finite automata. In fact, 
finite automata can be considered as automata with multiplicities in the 
semiring $K$, the boolan set $B=\{0,1\}$ (endowed with the logical ``or/and'').
 To each transition, we affect 1 if it exists and 0 if not.\\
ii) We have not yet, on purpose, defined what a semiring is. Roughly it is the least 
structure which allows the matrix ``calculus'' with unit (one can think of a ring without 
the "minus" operation).
\end{remark}

With such a setting, the automaton can be seen as a machine which accepts a word and provides a scalar 
(for instance a probability, a word or a boolean). Let us describe now how the output is computed.

We associate to the support of $T$ the weighted graph $\Gamma_T$ with
edges $edges(\Gamma_T) =$
\begin{equation}
\{(q,a,\alpha,r) \in Q\times A\times k\times Q\}_{\ T(q,a,r) \neq 0}
\end{equation}

\noindent
This means that every edge $(q \rightarrow r)$ is superscripted by the 
pair $(a|\alpha)$ such that $T(q,a,r) = \alpha\ne 0$. 
Classically call the {\bfseries{path}}, a sequence of edges
$(p_j, a_j,\alpha_j, q_j)_{1\leq j \leq n}$ such that, for all $j<n$, 
one has $ q_j = p_{j+1}$.

\noindent
Now, the cost of the path is $\prod_{1 \leq j \leq n} \alpha_i$ (product in K) 
and its label is $a_1 a_2 \,...\, a_n$ (concatenation)
\begin{equation} 
A(w) = \sum_{q,r \in Q} I(q) \big(\sum_{c\ path\ q\rightarrow r\atop label(c)=w}\textrm{cost}(c)\big) F(r)
\end{equation}

\begin{remark}
The previous automata with multiplicities can be, equivalently, expressed by its 
matrix representation which is a triplet 
\begin{itemize}
\item $\lambda\in K^{1\times Q}$ which is a row-vector which coefficients are 
$\lambda_i=I(S_i)$,
\item $\gamma\in K^{Q\times 1}$ is a column-vector which coefficients are 
$\gamma_i=F(S_i)$,
\item $\mu: A\mapsto K^{Q\times Q}$ is the mapping such that 
the coefficient on the $q_i$th row and $q_j$th column of $\mu(a)$ is 
$T(q_i,a,q_j)$ 
\end{itemize} 
\end{remark}

\noindent
Due to the effect of matrix computation we have

\medskip
\noindent
\begin{theorem} \cite{BR,Ei} Still denoting $\mu$ the extension of $\mu$ to words, 
one has
$$A(w) = I \mu(w) T $$
\end{theorem}

As the behaviour is a function on the words, we have will have advantage to 
consider operators on such functions.\\ 
In this paper we will concentrate on shifts which are generalizations the 
univariate case considered above \mref{gammadag}. 

The behaviour of an automaton $\A$ is, by definition, 
\begin{equation}
	Behaviour(\A)=\sum_{w\in A^*}\A(w)w
\end{equation}
where $w\rightarrow \A(w)$, is the output function\\
{\tt la d\'efinir en termes de chemins, puis de matrices et montrer que c'est la m\^eme chose}

\section{Populations of automata}

For complexity theory, it is important to deal with {\it populations of automata with 
multiplicities}. The first step, before considering neibourhood of {\it close behaviours} is
\begin{enumerate}
	\item to be able to test whether two automata have the same behaviour
	\item to be able to reduce the number of states of automata (the behaviour being preserved)
\end{enumerate}
In fact, given an automaton $\A$ and if the scalars are taken in a field 
(also certain rings can do, see \cite{Lu}) 
there is a minimal automaton having the same behaviour as $\A$. This automaton is the automaton of 
the {\it shifts of $Behaviour(\A)$}.\\
For a general function $f: A^*\mapsto k$,denoted as a sum $f=\sum_{w\in A^*}f(w)w$ one considers the 
{\it positive shifts} $(\gamma_u)_{u\in A^*}$
\begin{equation}
	\gamma_u(f)=\sum{w\in A^*}f(w)uw
\end{equation}
and the {\it negative shifts}
\begin{equation}
	\gamma_u^\dag(f)=\sum_{w\in A^*}f(uw)w
\end{equation} 
 

\section{Topological considerations}


If $K$ is a field, one sees that the space $\A_{(n)}$ of automata of dimension 
$n$ (with multiplicities in $K$) is a $K$-vector space of dimension 
$k.n^2+2n$ ($k$ is here the number of letters). So, in case the ground field is 
the field of real or complex numbers \cite{Bo1}, one 
can take any vector norm (usually one takes one of the H\"older norms 
$||(x_i)_{i\in I}||_\alpha := \big(\sum_{i\in I} | x_i 
|^\alpha\big)^{\frac{1}{\alpha}}$ 
for $\alpha\geq 1$,
 but any norm will do) and the distance is derived, in the classical way, by 
\begin{equation}
d(\A_1,\A_2)=norm(V(\A_1)- V(\A_2))
\end{equation}
where $V(\A)$ stands for the vector of all coefficients of 
$\A=(\lambda,\mu,\gamma)$ 
arranged in some order. One has then the result of Theorem \ref{th1}.
 Assuming that $K$ is the field of 
real or complex numbers, we endow the 
space of series/behaviours with the topology of  pointwise convergence (Topology 
of F. Treves \cite{Tr}).  

\begin{theorem}\label{th1}
Let  $(\A_n)$ be a sequence of automata with limit $\L$ ($\L$ is an automaton), 
then one has
\begin{equation}
Behaviour(\L)=\lim_{n\rightarrow \infty} Behaviour(\A_n)
\end{equation}
where the limit is computed in the topology of Treves.
\end{theorem}


\section{Genetic automata as efficient operators}


We define the chromosome for each automata with multiplicities as the sequence 
of all the matrices 
 associated to each letter from the (linearly ordered) alphabet. 
The chromosomes are composed with alleles 
which are here the lines of the matrix \cite{BFJOP2}.\\

In the following, genetic algorithms are going to generate new automata 
containing possibly new transitions 
from the ones included in the initial automata.\\

The genetic algorithm over the population of automata with multiplicities 
follows a reproduction iteration 
broken up in three steps \cite{Gol,Mi,Ko}:
\begin{itemize}
\item 	{\it Duplication}: where each automaton generates a clone of itself; 
\item	{\it Crossing-over}: concerns a couple of automata. Over this couple, we 
consider a sequence of lines of each 
matrix  for all. For each of these matrices, a permutation on the lines of the 
chosen 
sequence is made between the analogue matrices of this couple of automata; 
\item	{\it Mutation}: where a line of each matrix  is randomly chosen and a 
sequence 
of new values is given for 
this line.
\end{itemize}

Finally the whole genetic algorithm scheduling for a full process of 
reproduction over all the population of 
automata is the evolutionary algorithm:

\begin{enumerate}
\item For all couple of automata, two children are created by duplication, 
crossover and mutation 
mechanisms;
\item The fitness for each automaton is computed;
\item For all 4-uple composed of parents and children, the performless automata, 
in term of fitness 
computed in previous step, are suppressed. The two automata, still living, 
are the result of the 
evolution of the two initial parents.
\end{enumerate}

\begin{remark}
The fitness is not defined at this level of abstract formulation, but it is 
defined 
corresponding to the context for which the automaton is a model, as we will do 
in 
the next section.
\end{remark}

\section{Applications to competition-cooperation modeling using prisoner 
dilemma}

We develop in this section how we can modelize competition-cooperation processes 
in a same automata-based representation. The genetic computation allows to make 
automatic transitions from competition to cooperation or from cooperation to 
competition. The basic problem used for this purpose is the well-known prisoner 
dilemma \cite{Ax}.

\subsection{From adaptive strategies to probabilistic automata}

The prisoner dilemma is a two-players game where each player has two possible 
actions: 
cooperate ($C$) with its adversary or betray him ($\overline{C}$). So, four 
outputs are possible for the global 
actions of the two players. A payoff is defined relatively to these 
possible outputs, as 
described in the following table where the rows correspond to one player 
behaviour and the 
columns to the other player one.\\

\begin{table}[htp]
        \begin{center}
        \begin{tabular}{|l|c|c|} \hline
                                & $C$           & $\overline{C}$ \\ \hline
                $C$             & (3,3)     & (0,5)       \\ \hline
                $\overline{C}$  & (5,0)     & (1,1)       \\ \hline
        \end{tabular}
        \caption{Prisoner dilemma payoff}
        \label{prisonerDilemmaPayoff}
\end{center}
\end{table}

In the iterative version of the prisoner's dilemma, successive steps can be 
defined.  
Each player do not know the action of its adversary during the current step but 
he knows it for the preceding step. 
So, different strategies can be defined for a player behaviour, the goal of each 
one is to 
obtain a maximal payoff for himself.\\
 
In Figures \ref{titfortat} and \ref{vindictive}, we describe two strategies 
with transducers. 
Each transition is labeled by the input corresponding to the player perception 
which is the precedent adversary action and the output corresponding to the 
present player 
action.  
The only inital state is the state 1, recognizable by the incoming arrow labeled 
only by the output. 
The final states are the states 1 and 2, recognizable with the double circles.\\
 
In the strategy of Figure \ref{titfortat}, the player has systematically the 
same behaviour as its adversary at the previous step. 
In the strategy of Figure \ref{vindictive}, the player chooses definitively to 
betray as soon as his adversary does it.  
The previous automaton represents static strategies and so they are not well 
adapted for the 
modelization of evolutive strategies. 
For this purpose, we propose a model based on a probabilistic automaton 
described by Figure \ref{probaDilemma} \cite{BFJOP1}.\\


This automaton represents all the two-states strategies for cooperation  and 
competitive   
behaviour of one agent against another in prisoner's dilemma.\\

The transitions are labeled in output by the probabilities $p_i$ of their 
realization.  The first state is the state reached after cooperation action and 
the second state is reached after betrayal. \\

For this automaton, the associated matrix representation, as described 
previously, is:

\begin{eqnarray} 
I &=& \pmatrix{p_1 & 1-p_1}; \\
F &=& \pmatrix{p_6\cr 1-p_6};\\
T(C) &=& \pmatrix{p_2 & 1-p_2\cr p_3 & 1- p_3};\\ 
T(\overline{C}) &=& \pmatrix{p_4 & 1-p_4\cr p_5 & 1- p_5}
\end{eqnarray}


\subsection{From probabilistic automata to genetic automata}

From the matrix representations of the automata, we can compute genetic automata 
as described 
in previous sections. Here the chromosomes are the sequences of all the matrices 
associated to 
each letter. We have to define the fitness in the context of the use of these 
automata. The 
fitness here is the value of the payoff. 


\subsection{General Genetic Algorithm Process for Genetic Automata}


A population of automata is initially generated. These automata are playing 
against a predefined 
strategy, named $S_0$.\\

Each automaton makes a set of plays. At each play, we run the probabilistic 
automaton 
which gives one of the two outputs: ($C$) or ($\overline{C}$). With this output 
and the $S_0$'s output, we compute the payoff of the automaton, according with 
the payoff table.\\

At the end of the set of plays, the automaton payoff is the sum of all the 
payoffs of each play. 
This sum is the fitness of the automaton. At the end of this set of plays, each 
automaton has 
its own fitness and so the selection process can select the best automata. At 
the end of these 
selection process, we obtain a new generation of automata.\\

This new generation of automata is the basis of a new computation of the 3 
genetic operators.\\

This processus allows to make evolve the player's behavior which is modelized by 
the probabilistic multi-stra\-te\-gies two-states automaton from cooperation to 
competition or from competition to cooperation. The evolution of the strategy is 
the expression of an adaptive computation. This leads us to use this formalism 
to implement some self-organisation processes which occurs in complex systems.  

%(to be completed ... we can add a discussion about prisoner dilemma and how 
% that can be used in economical modeling : firm concurence, cold war, ...)

\section{Extension to Emergent Systems Modeling}

In this section, we study how evolutive automata-based modeling can be used to 
compute automatic emergent systems. The emergent systems have to be understood 
in the meaning of complex system paradigm that we recall in the next section. We 
have previously defined some way to compute the distance between automata and we 
use these principles to define distance between agents behaviours that are 
modeled 
with automata. Finally, we defined a specific fitness that allows to use genetic 
algorithms as a kind of reinforcement method which leads to emergent system 
computation \cite{Ho}.


\subsection{Complex System Description Using Automata-Ba\-sed Agent Model}


According to General System Theory \cite{gst, Mo}, a complex system is composed 
of entities in mutual interaction and interacting with the outside environment. 
A 
system has some characteristic properties which confer its structural aspects, 
as schematically described in part (a) of Figure \ref{sys2beh}: 
\begin{itemize}
\item The set elements or entities are in interactive dependance. The alteration 
of only one entity or one interaction reverberates on the whole system. 
\item A global organization emerges from interacting constitutive elements. This 
organization can be identified and carries its own autonomous behavior while it 
is in relation and dependance with its environment. The emergent organization 
possesses new properties that its own constitutive entities do not have. "The 
whole is more than the sum of its parts". 
\item The global organization retro-acts over its constitutive components. "The 
whole is less than the sum of its parts" after E. Morin.\\
\end{itemize}

The interacting entities network as described in part (b) of Figure 
\ref{sys2beh} leads each entity to perceive informations or actions from other 
entities or from the whole system and to act itself.\\

A well-adapted modeling consists of using an agent-based representation which is 
composed of the entity called agent as an entity which perceives and acts on an 
environment, using an autonomous behaviour as described in part (c) of 
Figure \ref{sys2beh}.\\

To compute a simulation composed of such entities, we need to describe the 
behaviour of each agent. This one can be schematically described using internal 
states and transition processes between these states, as described in part 
(d) of Figure \ref{sys2beh}.\\  

There are several definitions of ``agents'' or ``intelligent agents'' according 
to their behaviour specificities~\cite{Fe, We}.
Their autonomy  means that the agents try to satisfy a goal
and execute actions, optimizing a satisfaction function to reach it.\\

For agents with high level of autonomy, specific actions are realized even when no 
perception are detected from the environment. 
To represent the process of this deliberation, different formalisms can be 
used and a behaviour decomposed in internal states is an effective approach.
Finally, when many agents operate, the social aspects must also be taken into 
account. 
These aspects are expressed as communications through agent organisation with 
message
passing processes. 
Sending a message is an agent action and receiving a message is an agent 
perception. The previous description based on the couple: perception and action, 
is well adapted to this.

\subsection{Agent Behavior Semi-Distance}

We describe in this section the bases of the genetic algorithm used on the 
probabilistic automata allowing to manage emergent self-organizations in the 
multi-agent simulation.\\

For each agent, we define $e$ an evaluation function of its own
behaviour returning the matrix $M$ of values such that $M_{i,j}$ is 
the output series from all possible successive perceptions when
starting from the initial state $i$ and ending at the final state $j$,
without cycle. It will clearly be $0$ if either $i$ is not an initial
state or $j$ is not a final one and the matrix $M_{i,j}$ is indeed a matrix of 
evaluations \cite{BR} of subseries of 
\begin{equation}
M^*:=(\sum_{a\in A} \mu(a)a)^*
\end{equation}

%equal to the value of the sum of the multipicities of all
%the successful paths between $i$ and $j$, without cycle, if $i$ is an
%initial state and $j$ is a final state, and is equal to $0$ otherwise. 
Notice that the
coefficients of this matrix, as defined, are computed whatever the
value of the perception in the alphabet $A$ on each transition on the successful
path\footnote{A {\it succesful path} is a path from an initial state to a final 
state}. This means that the contribution of the agent behaviour for
collective organization formation is only based, here, on
probabilities to reach a final state from an initial one.     
This allows to preserve individual characteristics in each agent
behaviour even if the agent belongs to an organization.\\

Let $x$ and $y$ be two agents and $e(x)$ and $e(y)$ their respective
evaluations as described above.
We define $d(x,y)$ a semi-distance (or pseudometrics, see \cite{Bo1} ch IX) 
between the two agents $x$ and $y$ as 
$||e(x)-e(y)||$, a matrix norm of the difference of their
evaluations. Let ${\cal{V}}_x$ a
neighbourhood of the agent $x$, relatively to a specific criterium, for
example a spatial distance or linkage network.  
We define $f(x)$ the agent fitness of the agent $x$ as~:
$$
f(x) = 
\left\lbrace
\begin{array}{ll}
\frac{ {\displaystyle card({\cal{V}}_x) } }
     { {\displaystyle \sum\limits_{y_i \in {\cal{V}}_{x}} d(x, y_i)^2} } 
     \ \ \ \ &\mbox{if} \sum\limits_{y_i \in {\cal{V}}_{x}} d(x,
y_i)^2 \neq 0 \\
\infty &\mbox{otherwise}
\end{array}
\right.
$$

\subsection{Evolutive Automata for Automatic Emergence of Self-Organized Agent-
Based Systems}
 
In the previous computation, we defined a semi-distance between two agents. This 
semi-distance is computed using the matrix representation 
of the automaton with multiplicities associated to the agent behaviour. This 
semi-distance is based on the computation of successful paths which 
needs to define initial and final states on the behaviour automata. For specific 
purposes, we can choose to define in some specific way, the 
initial and final states. This means that we try to compute some specific action 
sequences which are chararacterized by the way of going from some specific 
states (defined here as initial ones) to some specific states (defined here as 
final ones).\\

Based on this specific purpose which leads to define some initial and final 
states, we compute a behaviour semi-distance and then the fitness function 
defined 
previously. This fitness function is an indicator which returns high value when 
the evaluated agent is near, in the sense of the behaviour semi-distance defined 
previously, to all the other agents belonging to a predefined neighbouring.\\

Genetic algorithms will compute in such a way to make evolve an agent population 
in a selective process. So during the computation, the genetic algorithm will 
make 
evolve  the population towards a newer one with agents more and more adapted to 
the 
fitness. The new population will contain agents with better fitness, so the 
agents of a population will become nearer to each others in order to improve their 
fitness. 
In that way, the genetic algorithm reinforces the creation of a system which 
aggregates agents with similar behaviors, in the specific way of the definition 
of initial and final states defined on the automata.\\

The genetic algorithm proposed here can be considered as a modelization of the 
feed-back of emergent systems which leads to gather agents of similar behaviour, 
but these formations are dynamical and we cannot predict what will be the 
set of these aggregations which depends of the reaction of agents during the 
simulation. Moreover the genetic process has the effect of generating a feed-
back of the emergent systems on their own contitutive elements in the way that 
the fitness improvement leads to bring closer the agents which are picked up 
inside the emergent aggregations.\\

For specific problem solving, we can consider that the previous fitness 
function can be composed with another specific one which is able to measure the 
capability of the agent to solve one problem. This composition of fitness 
functions leads to create emergent systems only for the ones of interest, that 
is, these systems are able to be developed only if the aggregated agents are 
able to satisfy some problem solving evaluation. 

\section{Conclusion}

The aim of this study is to develop a powerful algebraic structure to represent 
behaviors concerning cooperation-competition processes and on which we can 
add genetic operators. We have explained how we can use these structures for 
modeling adaptive behaviors needed in game theory. More than for this 
application, we have described how we can use such adaptive computations to 
automatically detect emergent systems inside interacting networks of entities 
represented by agents in a simulation. 


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%
\bibitem{Bo1} Bourbaki N., {\em Elements of Mathematics: General Topology}, 
Chapters 5-10, Springer-Verlag Telos (dec. 1998).
%
\bibitem{gst} L. von Bertalanffy, {\em General System Theory}, Georges Braziller 
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%
\bibitem{BFJOP2}C. Bertelle, M. Flouret, V. Jay, D. Olivier, and J.-L. Ponty 
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\end{document}