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corresponding to edge    Note that we omit edges labeled    " By our definition KK
an edge whose transition condition is   does not exist  We can alternatively denote as x vertex  j  For example  in Figure      k  TRANSMITTING  PAUSING   is  I receive L pause Figures and illustrate these FSA definitions There are FSA plans for three agents F I and L with vertices  edges  and transition conditions  An incoming arrow to a state not from any other state signifies that this is an initial state A multiagent plan is formed from single agent plans by taking the tensor product also called the  synchronous product" or simply  product"  of the FSAs corresponding to the individual plans  Formally  the tensor product is defined as n 

$$\dot{i}\qquad\quad\dot{i}\qquad\quad\dot{i}$$
$$\begin{array}{cccc}n&\mbox{of}&\mbox{trans}\\ &&1&1&1^{\prime}\\ &&1&1&1\end{array}$$

where is the Cartesian product  and the tensor product sing the state of the control of the control.

n n non's for a 1 1' FSA is formed by taking the Cartesian product of the vertices and the intersection of the transition conditions Initial states of the product FSA are tuples formed from the initial states of the individual FSAs The product FSA models a set of synchronous FSAs  The Boolean algebra correspond ing to the product FSA is the product algebra  For Figures    and    to formulate the FSA
S modeling the entire multiagent plan  we take the tensor product S    F    I    L of the three FSAs  For this tensor product   S      COLLECTING RECEIVING TRANS
MITTING  COLLECTING RECEIVING PAUSING  COLLECTING RECEIVING RECEIVING Part of the tensor product FSA is shown in Figure Next we define the by the FSA plan To do this we first define a Γ 
Formally a string x is an infinite dimensional vector    0 infinite     length sequence of actions  where    is the Boolean algebra used by A
run v of string x is a sequence o of vertices such that one is x i K  i  i+1  because the  i are atoms  In other words  a run of a string is the i e ssequence of vertices visited in an FSA when the string satisfies the transition conditions along the edges