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α|ααMCGRK25;αM1FIX25;α;
 APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI


Appendix C.  Calculation of Loops.

There are several  rules which must  be followed in  consideration of
loop  assignment  to  ring-superatoms.   The  minimum  (MINLOOPS) and
maximum (MAXLOOPS)  numbers of  loops for  a given  valence list  are
designated by equations 5 and 6.
				    n
     MINLOOPS = max{0,a +1/2(2j  -  αFCSαF1ja α←.α→p)}                     (5)
		       2       max  i=2α←pα→.j

		        n
     MAXLOOPS = min{a , αFCSαF1((i-2)/2)aα←.α→p }                          (6)
		     2  j=4α←pα→.

     MINLOOPS = minimum number of loops

     MAXLOOPS = maximum number of loops

           a  = number of secondary nodes in degree list
	    2

        j     = degree of highest degree item in degree list
	 max

            j = degree

            n = highest degree in list

           a  = number of nodes with degree j.
	    j

The form of the equations results from the following considerations:

1) Only secondary  nodes may be assigned  to loops.  Nodes  of higher
degree will always be in the non-loop portion of the ring-superatom.

2) A  loop, by definition, must be attached by  two bonds to a single
node in the  resulting ring-superatom.  The  loop cannot be  attached
through  the free  valences.   Thus the  degree list  must possess  a
sufficient  number of  quaternary or  higher degree nodes  to support
the loops(s).

3) Each  loop must have  at least  one secondary  node, which is  the
reason  MAXLOOPS  is restricted  to  at most  the  maximum number  of
secondary nodes in the degree list (Equation 6).

4) There must be  available one unsaturation for  each loop (this  is
implicit in the  calculation of MINLOOPS  and MAXLOOPS) as  each loop
effectively forms a new ring.

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 APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI

Partitioning_of_Secondary_Nodes.   For  each of the  possible numbers
of loops  (0,1, ...) the secondary nodes  are removed from the degree
list and partitioned among the loops, remembering that the  loops are
at present indistinguishable and each  loop must receive at least one
secondary  node.  In the  example (Fig. 2),  starting with the degree
list  (4,0,2),  there  are  three  ways  of   partitioning  the  four
secondary  nodes  among   two  loops  and  the  remaining  non-looped
portion.  Removal of  the four secondary nodes  from the degree  list
and assignment of two, three or four of them  to two loops results in
the  list  specified  in  Figure  2  as  the "reduced  degree  list".
Specification of  two loops transforms  the two  quaternary nodes  in
the degree  list into  two secondary  nodes.   This results  from the
fact that  two valences of a quaternary or higher degree node must be
used to  support  each loop.  These  are "special"  secondary  nodes,
however,  as  these  particular nodes  are  the  ones  which will  be
connected to  loops as  the structure  is  built up.   Thus,  in  the
example, any secondary  nodes which are  found in the  reduced degree
list  will have  a loop attached  in a  subsequent step.   The degree
list (4,0,2)  thus becomes  the reduced  degree list  (2,0,0) in  the
partition specifying two loops (Fig.  2). Similarly, the partition of
one  loop for  the degree  list (3,2,1)  results in a  reduced degree
list of (1,2,0) with  the three original secondary  nodes partitioned
among loop and non-loop portions (Figure 2).

If, after the  first, second,   ... nth loop partition,  there remain
one or  more quaternary or higher degree  nodes in the reduced degree
list,   the  list  must  be  tested  again  for  the  possibility  of
additional loops.   Each loop  partition will  generate a new  set of
structures.   The second pass will  yield those structures possessing
loops on loops,   and so  forth.  One  such superatom which would  be
generated in this manner from a composition of (at least) C U is 15.
                                                           6 5


          c=c=c=c=c=c                                              15


The partition of (4,0,2)  including one loop results in  each case in
a  reduced  degree list  (1,0,1).   This  list is  disallowed  in the
subsequent step, as  the vertex-graph  for one quaternary  node is  a
daisy (Table  II), which  requires a minimum  of two  secondary nodes
with which  to label the daisy loops (a minimum of one secondary node
in the reduced degree list for each loop of the daisy).




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 APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI

Appendix D

A method of construction  of structures similar to the  generation of
acyclic  molecules is utilized  to join  multiple ring-superatoms and
remaining atoms.  The  DENDRAL algorithm for construction of  acyclic
molecules (3,24)  relied on  the existence of  a unique  central atom
(or bond)  to every molecule.  The present acyclic generator uses the
same idea.   The present algorithm, though  simpler in not having  to
to  treat   interconnection  of  atoms   or  ring-superatoms  through
multiple bonds,  is more  complex because  of the  necessity to  deal
with the symmetries of the ring-superatoms.

D1. Method for the case with even number of total atoms.

The superatom  partition C U /C U /-/C   (partition  7, Table II  and
			  2 2  2 1    2
Figure  2)  will be  used here  to  illustrate this  procedure.   The
superatomparts   C U    and   C U    have    exactly   one   possible
		  2 2          2 1
ring-superatom for each (see Table VII).
---------------------------------------------------------
		Table VII

     Superatompart        Superatom

       C U                 -C≡≡C-
	2 2
 
       C U                >C==C<
	2 1

----------------------------------------

Thus acyclic structures are to be built with -C≡≡C- , >C==C< and
two C's.

There are an even number atoms and ring-superatoms.  The structures
to be generated fall into two categories:(a) those with a central bond;
(b) those with a central atom.


-----------------------
24) A more  complete description of  the algorithm is  available; see
B.  G.  Buchanan, A.  M.  Duffield,  and A.  V.  Robertson,  in "Mass
spectrometry, Techniques  and  Applications," G.  W. A.  Milne,  ed.,
John Wiley and Sons, Inc., 1971, p. 121.



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 APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI

Category A.  CENTRAL BOND (see Fig. 3).

Step 1.  Partition into Two Parts.

The atoms  and ring-superatoms are  partitioned into two  parts, with
each part  having exactly half the total number  of items.  Each atom
or ring-superatom  is  a  single item.    Each part  has  to  satisfy
equation 7, called the Restriction on Univalents.

Restriction on Univalents:

		n
              1+αFCSαF1(i-2)aα←.α→p ≥ 0                                   (7)
		i=1 α←pα→.i

     i = valence
    a  = number of atoms or superatoms of valence i
      i
     n = maximum valence in composition

There are  two ways  of partitioning  the four  items into two  parts
(Fig. 3).   The restriction on univalents  is satisfied in each case.
The restriction  will  disallow  certain partitions  that  have  "too
many"  univalents other  than  hydrogens and  therefore is  essential
only   in  partitioning  compositions  that  contain  any  number  of
non-hydrogen univalents.

Step 2.  Generate Radicals from Each Part.

Using a procedure  described in Section C3, radicals  are constructed
from  each part in  each partition.   Table VIII shows  the result of
applying this procedure to the example.




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