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COMMENT ⊗ VALID 00042 PAGES C REC PAGE DESCRIPTION C00001 00001 VALID 00031 PAGES C00006 00002 .DEVICE TTY C00009 00003 Problems of structural isomerism in chemistry have received much C00011 00004 Acyclic molecules represent only a subset of molecular structures, C00014 00005 Central to the successful solution of this problem is the solution C00018 00006 .TITL METHOD C00022 00007 ∪Superatom. A chemical graph is separable into cyclic and acyclic C00025 00008 ∪Underlying ∪Graph. Underlying graphs (11) possess only tertiary or C00028 00009 ∪Multiple ∪Bonds. All multiple bonds (double, triple, ...) are C00031 00010 ∪Strategy. The strategy behind the cyclic structure generator is C00034 00011 The example chosen to illustrate each step of the C00037 00012 .BEGIN TABLE I,Partitioning Steps Performed by the Cyclic Structure Generator C00039 00013 PART A. Superatom Partitions. C00042 00014 If the unsaturation count is zero, the formula is passed immediately C00046 00015 .BEGIN INDENT 0,0 C00050 00016 .BEGIN TABLE II,Allowed Partitions of C↓6U↓3 Into Superatomparts and Remaining Atoms. C00052 00017 PART B. Ring-superatom Construction. C00056 00018 ↓_Degree List_↓. Each partition of free valences alters the effective C00059 00019 ∪Loops. As will be clarified in the subsequent discussion, there are C00063 00020 .BEGIN TABLE III,Partial Listing of Underlying Graphs in the CATALOG C00068 00021 With the reduced degree list of a superatompart, the program C00072 00022 .BEGIN TABLE IV,The Six Graph Labelling Steps Performed by the Labelling Algorithm C00077 00023 .<< there is something essentially different from the program (but not >> C00082 00024 ↓_Unciliated Skeletons_↓. The previous labelling steps specified the C00085 00025 ↓_Atom Name Labelling_↓. The nodes of a ciliated skeleton are then C00089 00026 ↓_Completeness and Irredundancy_↓. Although a mathematical proof of the C00095 00027 In summary, the verification procedures utilized have all C00099 00028 ∪Constraints. The structure generator is designed to produce a list of C00103 00029 .HD CONCLUSIONS C00105 00030 Appendix A. Equivalence Classes and Finite Permutation Groups. C00108 00031 Appendix B. Isomerism and Symmetry. C00113 00032 ↓_Isomers Under Total Molecular Symmetry_↓. If in addition to the above C00117 00033 Appendix C. Calculation of Loops. C00120 00034 ↓_Partitioning of Secondary Nodes_↓. For each of the possible numbers of C00125 00035 Appendix D C00127 00036 C1. METHOD FOR THE CASE WITH EVEN NUMBER OF TOTAL ATOMS. C00130 00037 There are two ways of partitioning the four items into two C00133 00038 Step 3. Form Molecules From Radicals. C00136 00039 Step 3. Generate Radicals. C00139 00040 C2. METHOD FOR ODD NUMBER OF TOTAL ATOMS C00143 00041 ∪General ∪Case. Radicals have uniquely defined centers as well, the C00146 00042 Step 5. Label Central Atom with Radicals. C00148 ENDMK C⊗; .DEVICE TTY .FOOTSEP←"------------------"; .BEGIN CENTER APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI Exhaustive Generation of Cyclic and Acyclic Isomers.(1,2) .END .GROUP SKIP 5 .BEGIN INDENT 6,6,6 ABSTRACT. A systematic method of identification of all possible molecular structures (isomers) fitting a given empirical formula is described. The method, embodied in a computer program, generates a complete list of isomers. Duplicate structures are avoided prospectively. .END .GROUP SKIP 20 .BEGIN VERBATIM ------------------ .END 1) For Part X see D.H. Smith, B.G. Buchanan, W.C. White, E.A. Feigenbaum, J. Lederberg, and C. Djerassi, Tetrahedron, submitted. 2) Financial support for this work was provided by the National Institutes of Health (RR00612-03) and the Advanced Research Projects Agency (SD-183). .EVERY FOOTING(,{PAGE},) .MACRO LMCOMMENT (REMARK) ⊂⊃ .MACRO HD (HEADING) ⊂ BEGIN TURNON "{";SKIP 1;NOFILL;BREAK; }HEADING{END;⊃ .TURNON "∪↑↓[]&→_" .EVERY HEADING(,APPLICATIONS OF ARTIFICIAL INTELLIGENCE FOR CHEMICAL INFERENCE - XI,) .SKIP TO COLUMN 1 .MACRO TITL (TITLE) ⊂ BEGIN TURNON "{"; SKIP 2; NOFILL; BREAK;CENTER;}TITLE{END;⊃ .MACRO TABLE (N,TITLE) ⊂ NOFILL;GROUP;BEGIN CENTER;TURNON "{"; SKIP 1;} -------------------------------------------------------------------- Table N TITLE{END;⊃ Problems of structural isomerism in chemistry have received much attention. But only occasional inroads have been made toward a systematic solution of the underlying mathematical problems of isomerism. Recently the "boundaries, scope and limits" (3) of the .SEND FOOT ⊂ 3) J. Lederberg, G.L. Sutherland, B.G. Buchanan, E.A. Feigenbaum, A.V. Robertson, A.M. Duffield, and C. Djerassi, J. Amer. Chem. Soc., 91, 2973 (1969). .⊃; subject of structural isomerism of acyclic molecules have been defined by the DENDRAL algorithm (3). This algorithm permits an enumeration and representation of all possible acyclic molecular structures with a given molecular formula. Acyclic molecules represent only a subset of molecular structures, however, and it may be argued that cyclic structures (including those posessing acyclic chains) are of more general interest and importance to modern chemistry. An approach to cyclic structure generation has appeared in a previous paper in this series (4). That .SEND FOOT ⊂ 4) Y.M. Sheikh, A. Buchs, A.B. Delfino, G. Schroll, A.M. Duffield, C. Djerassi, B.G. Buchanan, G.L. Sutherland, E.A. Feigenbaum, and J. Lederberg, Org. Mass Spectrom., 4, 493 (1970). .⊃; approach, which manipulates acyclic isomers by removing hydrogen atoms pairwise and replacing them with a new bond, has one serious drawback. Because the approach cannot make efficient use of the symmetry properties of cyclic graphs, an extensive amount of computer time (prohibitive for other than very simple cases) must be spent in retrospective checking of each candidate structure with existing structures to remove duplicates. For this reason, an alternative approach to construction of cyclic molecules has been developed. This approach is designed to take advantage of the underlying graph theoretic considerations, primarily symmetry, to arrive at a method for more efficient construction of a complete and irredundant list of isomers for a given empirical formula. Central to the successful solution of this problem is the solution of the subproblem of specification of all isomers having a specified common ring system. This topic has received attention for nearly 100 years, with limited success (5). Its more general ramifications go far .SEND FOOT ⊂ 5) See, for example, A.C. Lunn and J.K. Senior, J. Phys. Chem., 33, 1027 (1929) and references cited therein. .⊃; beyond organic chemistry. Graph theoreticians have considered various aspects of this topic, frequently, but not necessarily, in the context of organic molecules. Polya has presented a theorem (6) which .SEND FOOT ⊂ .BEGIN NOFILL 6) a) G. Polya, Compt. rend., 201, 1167 (1935); b) G. Polya, Helv. Chim. Acta, 19, 22 (1936); c) G. Polya, Z. Kryst. 92, 415 (1936); d) G. Polya, Acta Math., 68, 145 (1937). .END .⊃; permits calculation of the number of structural isomers for a given ring system. Hill (7) has applied this theorem to enumeration of .SEND FOOT ⊂ .BEGIN NOFILL 7) a) T.L. Hill, J. Phys. Chem., 47, 253 (1943); b) T.L. Hill, ibid., p. 413. .END .⊃; isomers of simple ring compounds and Hill (8) and Taylor (9) have .SEND FOOT ⊂ 8) T.L. Hill, J. Chem. Phys., 11, 294 (1943). 9) W.J. Taylor, ibid., p. 532. .⊃; pointed out that Polya's theorem permits enumeration of geometrical and optical isomers in addition to structural isomers. More recently, formulae for enumeration of isomers of monocyclic aromatic compounds based on graph theory, permutation groups and Polya's theorem have been presented (10). This history of interest and results provides .SEND FOOT ⊂ 10) A.T. Balaban and F. Harary, Rev. Rumaine de Chimie, 12, 1511 (1967). .⊃; only marginal benefit to the organic chemist. Although the number of isomers may be interesting, these methods (5-9) do not indicate the structure of each isomer. Also, these methods do not provide information on the more general case where the ring system is embedded in a more complex structure. For even simple cases the task of specifying each structure by hand, without duplication, is an onerous one. .TITL METHOD .HD OVERVIEW ∪Framework. The framework for this method is that chemical structures consist of some combination of acyclic chains and rings or ring systems (4). The problem of construction of acyclic isomers has been solved previously (3). If, therefore, all possible ring systems can .LMCOMMENT this still does not solve the problem of embeding the rings; .LMCOMMENT that is still a labelling problem; be generated from all or part of the atoms in the empirical formula, the rings can be linked together or be attached to acyclic radicals using the acyclic generator. This yields the complete list of isomers. The method for construction of ring systems is described below. This description employs some terms which require definition. The definitions also serve to illustrate the taxonomic principles which underlie the operation of the cyclic structure generator. The generator's view of molecular structure differs in some respects from the chemist's. A chemist, for example, may view structures possessing the same functional group or ring as related. The generator works at the more fundamental level of the underlying graph (4,11), as described below. .LMCOMMENT it might be noted early that dbl bonds make rings ∪Chemical ∪Graph. A molecular structure may be viewed as a graph, termed the "chemical graph." A chemical graph consists of "nodes," with associated atom names, connected to one another by "edges." Consider as an example the substituted piperazine, 1, whose chemical graph is illustrated in Scheme 1 as structure 2. Note that hydrogen atoms are ignored by convention, while the symbol "U" in used to specify the unsaturation. The degree (primary, secondary, ...) of a node in the chemical graph has its usual meaning, i.e., the number of (non-hydrogen) edges connected to it. The valence of each atom determines its maximum degree in the graph. ∪Superatom. A chemical graph is separable into cyclic and acyclic parts, each of which may be termed a "superatom." A superatom is a structural sub-unit possessing at least one "free valence" (12). The .SEND FOOT ⊂ 12) A free valence is a bond on an atom in the superatom to which another atom (which is not part of the superatom) may be connected. .⊃; chemical graph 2 arises from a combination of three superatoms. Two are carbon atoms with one free valence (methyl radicals). The third is superatom 3, which possesses the indicated free valences to which the remaining hydrogen and two methyl radicals will be attached (Scheme 1). In the subsequent discussion, those superatoms which consist of rings are termed "ring-superatoms." ∪Ciliated ∪Skeletons. A "ciliated skeleton" is a graph in which nodes (but not atom names) and their interconnections are specified, together with their free valences. Superatom 3 arises from the ciliated skeleton 4 by associating the atom names of eight carbon and two nitrogen atoms with the skeleton (Scheme 1). ∪Unciliated ∪Skeleton. Skeletons where nodes and edges are specified, but free valences are not, are termed "unciliated skeletons." The ciliated skeleton 4 arises from one way of associating sixteen free valences with the nodes on the unciliated skeleton 5 (Scheme 1). .BEGIN NOFILL ---------------------------------- Scheme 1 .GROUP SKIP 6 ---------------------------------- .END ∪Underlying ∪Graph. Underlying graphs (11) possess only tertiary or higher degree nodes. The underlying graph of the unciliated skeleton 5 is the regular trivalent graph (11) of two nodes, 6. Note that the remaining nodes of the unciliated skeleton 5 are of degree two. Removal of these secondary nodes from 5 while retaining the interconnections of the two tertiary nodes yields 6. As an illustration of the variety of structures which may be constructed from a given underlying graph and empirical formula, for example, C↓1↓0H↓2↓0N↓2, consider that graph 6 is the underlying graph for all bicyclic ring systems (excluding spiro forms). Unciliated skeletons 7 and 8 (Scheme 2), for example, may be constructed from eight secondary nodes and 6. There are many ways of associating sixteen free valences with each unciliated skeleton, resulting in a larger number of ciliated skeletons. For example, 9 and 10 arise from different allocations of sixteen free valences to 5 (Scheme 2). There is only one way to associate eight carbon atoms and two nitrogen atoms with each ciliated skeleton to yield superatoms (e.g. 11 and 12, Scheme 2). However, several structures are obtained by associating the remaining two carbon atoms (in this example) with each superatom, as an ethyl or two methyl groups. Chemical graphs 13 and 14, for example, arise from two alternative ways of associating two methyl groups with superatom 3. .BEGIN NOFILL ---------------------------------- Scheme 2 .GROUP SKIP 6 ---------------------------------- .END ∪Multiple ∪Bonds. All multiple bonds (double, triple, ...) are considered to be small rings by the program. Although chemists normally maintain strong distinctions between multiple bonds and rings, (for good reason) it was easier when implementing the algorithm in a computer program to generate multiple bonds, whether isolated or within ring systems, along with the rings. As will be mentioned subsequently, from the program's standpoint acyclic portions of structures cannot possess multiple bonds. .<< ... it was found easier ... >> ∪Aims. The cyclic structure generator must produce a complete list of structures without duplication. By duplicate structures we mean structures which are equivalent in some well-defined sense. The concept of equivalence is discussed in more detail in Appendix A and in the accompanying paper (13). .SEND FOOT ⊂ 13) Accompanying description of the labelling algorithm. .⊃; The class of isomers generated by the program includes only connectivity isomers (see Appendix B). Transformations (see Appendix A) allowed under connectivity symmetry preserve the valence and bond distribution of every atom. Connectivity symmetry does not consider bond lengths or bond angles. This choice of symmetry results in construction of a set of topologically unique isomers. A more detailed discussion of isomerism and symmetry is presented in Appendix B. ∪Strategy. The strategy behind the cyclic structure generator is strongly tied to the framework described above. The strategy is summarized in greatly simplified form in Figure 1. The underlying graphs from which structures are constructed can be specified for a given problem by a series of calculations. Thus Part A of the program (Figure 1) assigns atoms in all possible ways to superatom partitions. Each partition consists of atoms assigned to one or more "superatom- parts" and "remaining atoms." Each superatompart is a collection of atoms from which all possible, unique ring-superatoms (see above definition) can be constructed based on the appropriate underlying graphs (Part B, Fig. 1). Each ring-superatom will be a ring system in completed structure. Remaining atoms will form acyclic parts of the final structures when combined in all possible, unique ways with the ring-superatoms from each superatompart in the initial partition (Part C, Fig. 1). .HD DESCRIPTION The subsequent detailed description parallels the operation of the computer program (called the cyclic structure generator). Programming details as well as complete descriptions of the underlying mathematics are omitted for the sake of clarity and because they are available from other sources (13-16). .SEND FOOT ⊂ 14a) H. Brown, L. Masinter, and L.Hjelmelend, Discrete Mathematics, submitted; b) H. Brown, L. Masinter, and L. Hjelmelend, Stanford Computer Science Memo 318. 15) H. Brown and L. Masinter 16) Additional information will be available from the authors on request. .⊃; The example chosen to illustrate each step of the method is C↓6H↓8. This example, however, does not contain bivalent or trivalent atoms (e.g., oxygen and nitrogen, respectively) or atoms of valence greater than four, nor any univalent atoms other than hydrogen (e.g., chlorine, fluorine). The description indicates how these additional atoms are considered by the program. ↓_Partitioning and Labelling_↓. The mechanism for structure generation involves a series of "partitioning" steps followed by a series of "labelling" steps. Partitions are made of items which must be assigned to objects (usually graph structures or parts thereof) as the molecular structures are built up from the underlying graphs. The process by which items are assigned to the graphs is termed labelling (13,14). Examination of Scheme 1 reveals the different types of items involved. For example, nodes are partitioned among and labelled upon the edges of the underlying graphs to yield the unciliated skeletons. Free valences are partitioned among and labelled upon the nodes of unciliated skeletons to yield ciliated skeletons, and so forth. Partitioning steps in the subsequent discussion are carried out assuming that objects among which items are partitioned are indist- inguishable. Distinguishability of objects (edges, nodes, ...) is specified during labelling and will be discussed in a subsequent section. The partitioning steps performed by the program are outlined in Table I. Each step is described in more detail below. .BEGIN TABLE I,Partitioning Steps Performed by the Cyclic Structure Generator Step # Partition Among 1 Atoms and Unsaturations Superatomparts and in Empirical Formula Remaining Atoms 2 Free Valence Atoms in Superatompart 3 Secondary Nodes Loops 4 Non-looped Secondary Edges of Graph Nodes 5 Looped Secondary Nodes Loops 6 Ring-superatoms and Efferent Links Remaining Atoms (see Appendix D) ------------------------------- .END PART A. Superatom Partitions. Ring-superatoms are "two-connected" structures, i.e., the ring-superatom cannot be split into two parts by scission of a single bond. The atoms in an empirical formula may be distributed among from one to several such two-connected ring-superatoms. A distribution which allots atoms to two or more superatomparts will yield (respectively) structures containing two or more ring-superatoms linked together by single bonds (or acyclic chains) (17). .SEND FOOT ⊂ 17) Chemists are more familiar with terms such as rings or ring systems. The term two-connected is used here in conjunction with ring-superatoms for a more precise description. For example, biphenyl may be viewed as a single ring system or two rings depending on the chemical context. In this work, however, biphenyl consists of two ring-superatoms (two phenyl rings) linked by a single bond. .⊃; The initial formation of superatom partitions is linked intimately with chemistry unlike several of the subsequent operations within the cyclic structure generator. Thus, there are several well-known concepts which are used during partitioning, dealing primarily with valence and unsaturation. .<<really doesn't have anything to do with chemistry- only valence>> The first step is to replace the hydrogen count with the degree of unsaturation. The number of unsaturations (rings plus double bonds) is determined from the empirical formula in the normal way, as given in equation 1. .BEGIN NOFILL U = 1/2 (2+↑n&↓[i=1]&∃(i-2)a↓i)→(1) U = unsaturation i = valence n = maximum valence in composition a↓i= number of atoms with valence i .END If the unsaturation count is zero, the formula is passed immediately to the acyclic generator. Specifying the unsaturations as U's, the example C↓6H↓8 becomes C↓6U↓3 (hydrogen atoms are omitted by convention). There are several rules which are used during the partitioning scheme, as follows: .BEGIN INDENT 0,5 I. The resulting formula is stripped of other univalent atoms (e.g., chlorine) as such atoms cannot be part of two-connected ring-superatoms. These univalent atoms are assigned to the category of remaining atoms. II. The remaining atoms in a given partition (those not allocated to superatomparts) can contain NO unsaturations. Thus ALL rings and/or double bonds will be generated from the superatomparts. III. It follows from I and II that every superatompart in the partition must contain at least two atoms of valence two or higher plus at least one unsaturation. If there are no unsaturations the atoms belong in the remaining atoms category of another partition. In addition, an unsaturation cannot be placed on a single atom. This rule defines the minimum number of atoms and unsaturations in a superatompart. IV. The maximum number of unsaturations in a superatompart is given by Equation 2. Superatoms must possess at least one free valence (12), so that superatomparts with no free valences, e.g., O↓2U↓1 or C↓2U↓3, are not allowed. .BEGIN NOFILL U↓[max]= 1/2 (↑n&↓[i=3]&∃(i-2)a↓i)→(2) U↓[max]= maximum unsaturation of a superatompart i = valence n = maximum valence in composition a↓i = number of atoms with valence i .END V. The maximum number of superatomparts for a given formula is defined by equation 3. .BEGIN NOFILL S↓[max]= 1/2↑n&↓[i=2]&∃a↓i→(3) n = maximum valence in composition S↓[max]= maximum number of superatomparts in a superatom partition note: the summation is over all atoms of valence >2; univalents are not considered. .END .BEGIN INDENT 0,0 Rules I-V define the allowed partitions of a group of atoms into superatomparts. These rules do not, however, prevent generation of equivalent partitions, which would eventually result in duplicate structures. Defining a canonical ordering scheme to govern partitioning prevents equivalent partitions. (There may be alternative schemes): .<<is this necessary? the partitioning is nice, but this isn't complete>> .<<anyway, and gets into too much detail>> .END VI. Canonical Ordering: .BEGIN INDENT 6,6 a. Partition in order of increasing number of superatomparts. b. For each entry in each part of (a), partition in order of decreasing size of superatompart by allocation of atoms one at a time to remaining atoms. c. Each individual partition containing two or more superatomparts must be in order of equal or decreasing size of the superatompart. In other words, the number of atoms and unsaturations in superatompart n+1 must be equal to or less than the number in superatompart n. The program notes the equality of superatomparts in a partition to avoid repetition. .END .END The application of rules I-VI is best illustrated through reference to the example of C↓6U↓3. The maximum number of superatomparts for this example is three (Equation 3). There is one way to partition *C6U3 into one superatompart with no remaining atoms, partition 1, Table II. Subsequent assignment of carbon atoms to the category of remaining atoms results in partitions 2-4, Table II. The next partition following the sequence 1-4 would be C↓2U↓3 with C↓4 assigned to remaining atoms. This partition is forbidden as C↓2U↓3 has no free valences. The three ways to partition C↓6U↓3 into two superatomparts are indicated along with the corresponding partitions following assignment of atoms to remaining atoms, as partitions 5-10, Table II. Note that the next logical steps are partitions C↓3U↓1/C↓3U↓2, C↓2U↓2/C↓4U↓1, and so forth, which are not allowed by the canonical ordering rule VIc, above. There is only one unique way of partitioning C↓6U↓3 into three superatomparts, partition 11, Table II. .BEGIN TABLE II,Allowed Partitions of C↓6U↓3 Into Superatomparts and Remaining Atoms. Superatom Partition Number of Superatompart Number Remaining Number Superatomparts 1 2 3 Atoms 1 1 C↓6U↓3 - - - 2 1 C↓5U↓3 - - C↓1 3 1 C↓4U↓3 - - C↓2 4 1 C↓3U↓3 - - C↓3 5 2 C↓4U↓2 C↓2U↓1 - - 6 2 C↓3U↓2 C↓2U↓1 - C↓1 7 2 C↓2U↓2 C↓2U↓1 - C↓2 8 2 C↓4U↓1 C↓2U↓2 - - 9 2 C↓3U↓1 C↓2U↓2 - C↓1 10 2 C↓3U↓2 C↓3U↓1 - - 11 3 C↓2U↓1 C↓2U↓1 C↓2U↓1 - .END PART B. Ring-superatom Construction. Each partition (Table II) must now be treated in turn. The complete set of ring-superatoms for each superatompart in a given partition must be constructed. The major steps in the procedure are outlined in Figure 2. ∪Valence ∪List. The first step in part B is to strip the superatompart of atom names, while retaining the valence of each atom. The numbers of each type of atom are saved for later labelling of the skeleton. A valence list may then be specified, giving in order the number of bi-, tri- , tetra- and n-valent nodes which will be incorporated in the superatom. Thus the superatompart C↓6U↓3 is transformed into the valence list 0 bivalents, 0 trivalents, 6 tetravalents (0, 0, 6), and C↓4U↓2 becomes (0, 0, 4) (Figure 2). ↓_Calculation of Free Valence_↓. From the valence list and the associated unsaturation count the number of free valences of each superatompart is determined using equation 4. .BEGIN NOFILL FV = (2 +↑n&↓[i=3]&∃(i-2)a↓i)-2U→(4) U = unsaturation of superatompart i = valence n = maximum valence in composition a↓i= number of atoms with valence i FV = free valence .END ↓_Partitioning of Free Valence_↓. The free valences are then partitioned among the nodes in the valence list in all possible, unique ways. As in the earlier partitioning problem, there are some simple rules which must be followed. Because ring-superatoms are two-connected structures two valences of each atom of a superatompart must be used to connect the atom to the ring-superatom. Thus no free valences can be assigned to bivalent nodes in the valence list, a maximum of one to each trivalent, a maximum of two to each tetravalent, and so forth. The example (Fig. 2) is further simplified in that there are only tetravalent nodes in the valence list. Inclusion of trivalent nodes (e.g., nitrogen atoms) merely extends the number of possible partitions. The free valences are partitioned among the tetravalent nodes in all ways, as illustrated in Figure 2. It is important to note that removal of atom names makes all n-valent (n=2 or 3 or ...) nodes in the valence list equivalent at this stage. Thus the partitions (of eight free valences among six tetravalent nodes) 222200, 222020, 222002, ......, 002222 are all equivalent. Only one of these partitions is considered to avoid eventual duplication of structures. .BREAK ↓_Degree List_↓. Each partition of free valences alters the effective valence of the nodes in the original valence list with respect to the ring-superatom. In the example, assignment of one or two free valences to a tetravalent node transforms this node into a tri- or bivalent node respectively. As the ring-superatom is constructed, those tetravalent nodes which have been assigned, say, two free valences, have then only two valences remaining for attachment to the ring-superatom. These nodes are then of degree (18) two and may be termed secondary nodes (18). Thus the partition of free valences 2,2,2,2,0,0 .SEND FOOT ⊂ 18) The term "valence with respect to the ring-superatom" is confusing as valence has a clear chemical meaning. Rather, we speak of the degree of a node in the chemical graph (see Overview). Standard use of the terms primary, secondary, ... when speaking of degree refers to bonding to atoms which are not hydrogen atoms. Use of degree in this report refers to the number of bonds other than free valences. A free valence may or may not eventually be attached to a hydrogen atom in the final structure. .⊃; on six tetravalent nodes yields the degree list (4,0,2) (Fig. 2) as four of the tetravalent nodes receive two free valences each yielding four nodes of degree two (secondary) and leaving two nodes of degree four (quaternary). The program keeps track of the number of free valences assigned to all nodes for use in a subsequent step. ∪Loops. As will be clarified in the subsequent discussion, there are several general types of ring-superatoms which cannot be constructed from the underlying graphs available in the CATALOG (described below). These are all cases of multiple extended unsaturations either in the form of double bonds or rings. Examples are the following: .BEGIN NOFILL 1) bi-, tri-, ... n-cyclics with exocyclic double bonds; 2) some types of SPIRO ring systems; 3) allenes extended by additional double bonds, e.g., c=c=c=c . .END The concept of a loop, consisting of a single unsaturation and at least one bivalent node must be specified for these cases. Examples of loops containing one, two and three bivalent nodes are shown in Scheme 3. Note that the two remaining "ends" of the unsaturation will yield a "looped structure" when attached to a node in a graph (shown as X, Scheme 1). .BEGIN NOFILL --------------------- Scheme 3 bivalents = 1 2 3 .GROUP SKIP 4 --------------------- .END The method for specification of loops is discussed in Appendix C. This procedure yields the reduced degree list which contains none of the secondary nodes originally present in the degree list (see Appendix C). ↓_Underlying Graphs_↓. The reduced degree lists are used to specify a set of underlying graphs for the eventual ring-superatoms. All two- connected structures can be described by their underlying graphs, which are, for most structures, regular trivalent graphs. This concept has been described in detail by Lederberg (11), who has also presented a generation and classification scheme for such graphs. Given a set of all underlying graphs, the set of all ring-superatoms may be specified. The underlying graphs are maintained by the program in the CATALOG. Catalog entries for regular trivalent graphs possessing two, four and six nodes are presented in Table III. This list must be supplemented by additional underlying graphs to cover several special cases. Several of these additional graphs, which are sufficient for generation of all structures in the example, are also presented in Table III. .BEGIN TABLE III,Partial Listing of Underlying Graphs in the CATALOG .APART Reduced Structure Name* Degree List Remarks 2A (k,2,0,0) Regular trivalent graph of 2 nodes 4AA (k,4,0,0...) Regular trivalent graph of 4 nodes 4BB (k,4,0,0...) Regular trivalent graph of 4 nodes 6AAA (k,6,0,0...) Regular trivalent graph of 6 nodes 6ABB (k,6,0,0...) Regular trivalent graph of 6 nodes 6ACA (k,6,0,0...) Regular trivalent graph of 6 nodes 6BCB (k,6,0,0...) Regular trivalent graph of 6 nodes 6CCC (k,6,0,0...) Regular trivalent graph of 6 nodes "Singlering e.g. M" (4,0,0...) A single ring composed of M secondary nodes. "Daisy" (k,0,1,0...) A single 2n-degree node (n=2,3...) e.g. $1AA "T03" (k,0,3,0...) Graph composed of three quaternary nodes $3BCB (k,2,1,0...) Graph composed of two tertiary and one quaternary node "T22" (k,2,2,0...) Graph composed of two tertiary and two quaternary nodes "T22" (k,2,2,0...) Graph composed of two tertiary and two quaternary nodes. "T22" (k,2,2,0...) Graph composed of two tertiary and two quaternary nodes "T22" (k,2,2,0...) Graph composed of two tertiary and two quaternary nodes $5CECC (k,4,1,0...) Graph composed of four tertiary and one quaternary nodes $BCCB (k,4,1,0...) Graph composed of four tertiary and one quaternary nodes $B:5AECA (k,4,1,0...) Graph composed of four tertiary and one quaternary nodes $5ABCB (k,4,1,0...) Graph composed of four tertiary and one quaternary nodes $5ACDB (k,4,1,0...) Graph composed of four tertiary and one quaternary nodes e.g.(k,0,2,0,...) Two nodes of equal valence (>3) * The graphs are named according to Lederberg (11). ------------------------------------------------------------------ .END With the reduced degree list of a superatompart, the program requests the appropriate CATALOG entries. In the example (Fig. 2), the degree list (4,0,2) specifies underlying graphs containing two quaternary nodes. The degree list (2,4,0) specifies regular trivalent graphs of four nodes, of which there are two, 4AA and 4BB (Table T2). Note that secondary nodes in the degree list are ignored since underlying graphs have only trivalent or higher nodes. The exception is when only secondary nodes are present in the degree list, when the underlying graph "Singlering" (Table III) is utilized. ∪Interlude. Up to this point in the method, the program has effectively decomposed the problem into a series of subproblems, working down from the empirical formula through a set of partitions and subpartitions to the set of possible underlying graphs. Subsequent steps now work back up, in a sense, as the underlying graphs are expanded in a series of steps to the final structures employing the results of the previous partitions. ↓_Labelling Edges of Underlying Graphs with Special Secondary Nodes_↓. The first step in building up structures is to attach the secondary nodes in the reduced degree list to the underlying graphs. It will be recalled that these secondary nodes are those that will .<<ONLY POSSIBLY, WHEN THERE WERE LOOP PARTITIONS>> have loops attached, thus the term "special secondary nodes" is used. Because a graph structure is involved, the procedure by which the possible attachments of the nodes to the graph are specified may be viewed as a "labelling" of the graph. This is the first of six such graph labelling steps required by the program. The six steps are listed in Table IV. All steps involve the same labelling problem, that of associating a set of n not necessarily distinct labels with a set of n not necessarily distinct objects. This problem is solved by a single algorithm, the "labelling algorithm", for each of the six labelling steps (Table III). This algorithm is described in some detail in the subsequent publication (11). A description of the underlying mathematics and proof of thoroughness and irredundancy will appear separately (12). .BEGIN TABLE IV,The Six Graph Labelling Steps Performed by the Labelling Algorithm Labelling Step Function 1 Label Edges of Underlying Graphs with Special Secondary Nodes. 2 Label Edges of Resulting Graphs with Non-Looped Secondary Nodes. 3 Label Loops of Resulting Graphs with Looped Secondary Nodes. 4 Label Nodes of Skeletons with Free Valences. 5 Label Nodes of Skeletons with Atom Names. 6 Label Free Valences of Superatoms with Radicals (see Appendix D). .END It is important in the context of this discussion, however, to discuss some aspects of the first labelling step to indicate how equivalent labellings (which would eventually yield duplicate structures) may be avoided prospectively. This discussion is valid for each of the subsequent labelling steps. Equivalent labellings are prospectively avoided by recognition of the symmetry properties of, in the first labelling, the underlying graph. These symmetry properties are expressed in terms of the permutation group (see Appendix A and refs. 13 and 14), in the first labelling, on the edges of the underlying graph. This permutation group, which defines the equivalence of the edges, may be specified in the CATALOG or, alternatively, calculated as needed by a separate part of the cyclic structure generator. As subsequent labelling steps are executed, a new permutation group (e.g., on the nodes for labelling step four) is calculated, again as necessary. Thus, only labellings .<<I want it to be clear that the group is not recalculated from scratch>> which result in unique expansions of the structure are permitted. In the simple example, (Fig. 2), the symmetries of the underlying graphs and subsequent skeletons can be discerned easily by eye. In the general case this is not true, however. For example, all edges of the underlying graph containing two tetravalent nodes are equivalent, as are the edges of the regular trivalent graphs 2A and also 4BB. The $3BCB graph (Table II, Fig. 2) has four equivalent edges and one other edge, and so forth. In the general case, however, the symmetries of the underlying graphs and subsequent expansions thereof are not always obvious. .<< there is something essentially different from the program (but not >> .<< really wrong) here - that is, the loop-secondary nodes are partitioned>> .<< among the loops before hand; and it is the loops WITH NODES which are>> .<< labelled on at first >> With distinguishability specified by the group on the edges, labelling of the underlying graphs with special secondary nodes is carried out. The results of this procedure for partitions containing loops are indeicated in Figure 2. ↓_Partition of and Labelling With Non-Loop Secondary Nodes_↓. The secondary nodes which were not assigned to loops ("non-looped secondary nodes") are partitioned among the edges of the graphs which arise from the previous labelling step. As in previous partitionings, the edges are initially assumed to be indistinguishable. In addition, loops are not .<<what does "... indistinguishable" mean???>> counted as edges. There are, for example, five ways to partition four non-loop secondary nodes among the edges of the underlying graph possessing two quaternary nodes (Fig. 2). Labelling of the graphs with non-loop secondary nodes is then carried out. Each of the five partitions mentioned immediately above results in a single labelling, as all edges of the graph are equivalent. When edges are distinguishable there may be several ways to label a graph with a single partition. There are, for example, for the $3BCB graph, two ways to label with the partition 3,0,0,0,0, four ways with the partition 2,1,0,0,0 and three ways with the partition 1,1,1,0,0 (Fig. 2). ↓_Partitioning of and Labelling with Loop Seconday Nodes_↓. There remain unassigned to the graphs at this point only secondary nodes which were assigned to loops. These nodes are first partitioned among the loops assuming indistinguish- ability and remembering that each loop must receive at least one secondary node. For example, following the path from the degree list (4,0,2) through the specification of two loops (Fig. 2), there are two ways of labelling the two equivalent loops with four secondary nodes. There is one way to label the two loops of the adjacent graph with three secondary nodes and one way of labelling the two loops of each of the two remaining graphs in this section of Figure 2 with two secondary nodes. In this example (*C6U3) the loops in every case are equivalent or there is only one loop to be labelled. In the general case loops may not be equivalent, resulting in a greater number of ways to label loops with a given partition of secondary nodes. ↓_Unciliated Skeletons_↓. The previous labelling steps specified the number of secondary nodes on each edge of and loop attached to the underlying graphs. All atoms in the original superatompart are thus accounted for. A representation of the result is the unciliated skeleton, where nodes and their connections to one another are specified. These skeletons begin to resemble the final structures. ↓_Free Valence Labelling_↓. Each of the nodes in an unciliated skeleton is then labelled with free valences. This labelling is trivial in the example, as all atoms are of the same valence (four). This labelling is illustrated in some cases in Figure 2, yielding the indicated ciliated skeletons. This labelling is performed independent of the IDENTITIES of the atoms, but with knowledge of how many atoms of each valence type were present in the original superatompart. A simple example illustrates the potential complexity of this labelling problem when atoms of more than one valence type are present. If there were, say, n nitrogen atoms in the superatompart, in addition to m carbon atoms, then n nodes in the ciliated skeleton must be trivalent, and m must be tetravalent, in all unique ways. The free valence labelling would be done accordingly, yielding bivalent nodes with one or two free valences and trivalent nodes with zero or one free valences. In the general case there may be several ways to perform this labelling on a single unciliated skeleton, whereas in the C↓6U↓3 example there is only one way. ↓_Atom Name Labelling_↓. The nodes of a ciliated skeleton are then labelled with atom names to yield the ring-superatom(s). Again this labelling is trivial in the example, as only one type of atom is present (carbon), yielding in each case only a single superatom (Fig. 2). If there is more than one type of atom with the same valence (e.g., silicon and carbon), the labelling problem is more complex. Each node of appropriate valence may be labelled with either type of atom. Duplicate structures are avoided by calculating the group on the set of nodes of equal valence and labelling accordingly. .HD PART C. ACYCLIC GENERATOR. The superatom partition expanded in the example had no atoms assigned to acyclic chains (remaining atoms). The set of superatoms on completion of Part B, above, thus yields the set of 36 structures on placement of a hydrogen atom on each free valence (Fig. 2). If the superatom partition (partitions 2-11, Table II) contained more than one superatompart or remaining atoms, the acyclic generator must be used to connect the segments of the structure in all ways. This procedure is described in detail in Appendix D. .TITL DISCUSSION ↓_Completion of C↓6U↓3_↓. The example (Fig. 2) has considered only expansion of a single superatom partition. The total number of isomers of C↓6U↓3 and their structures considering all partitions has been determined by the program to be 159. It may be instructive for the reader to attempt to generate, by hand, structures from another partition, following the method outlined above and in Figure 2. The initial superatom partitions yield some indication of the types of structures which will result from each partition. For example, partition 4 (Table II), C↓3U↓3 in a single superatompart, plus three carbons in the remaining atoms, should yield all structures containing a three-membered ring possessing two double bonds or a triple bond. As there are only two free valences, the remaining atoms can be in a single chain (as a propyl or iso-propyl radical) or as a methyl and an ethyl group, but not as three methyl groups. ↓_Completeness and Irredundancy_↓. Although a mathematical proof of the completeness and irredundancy of the method exists (14 15), there is no guarantee that the implementation of the algorithm in a computer program maintains these desired characteristics. Confidence in the thoroughness and irredundancy of the algorithm can be engendered in the following ways: .BEGIN INDENT 6 1) Duplication of the program's performance by another, completely independent approach. If such an approach exists (to our knowledge it does not), it must be proven mathematically. An independent algorithm has been developed which is capable of enumeration, but not construction, of all isomers of a composition containing C,H,N, and O. Implementation of that algorithm is limited at the present time, unfortunately, to compositions containing only 5 or fewer carbon atoms and various numbers of hydrogen atoms. The limitation is due to computer memory constraints. For these cases, however, the numbers of isomers obtained by both programs agree. 2) Testing by manual generation of structures. Several chemists, all without knowledge of the algorithm described above, have been given several test cases, including C↓6U↓3, from which structures were generated by hand. Familiarity with chemistry is no guarantee of success, as evidenced by the performance of three chemists for the superficially simple case of C↓6U↓3 (C↓6H↓8, Table V). This example indicates that for more than very trivial cases, it is extremely difficult to avoid duplicates (tricyclics, for example, are difficult to visualize when testing for duplicates) and omissions. Omissions appear to result from both being careless and forgetting ring systems that are implausible or unfamiliar. The program seems better at testing the chemist than vice versa. In every instance of manual structure generation, no one has been able to construct a legal structure that the program failed to construct. No one has been able to detect an instance of duplication by the program. This performance builds some confidence, but manual verification of more complicated cases is extremely tedious and difficult. Isomers for many empirical formulae have been generated, and some results are tabulated in Table VI. The choice of examples has been motivated by a desire to test all parts of the program where errors may exist while keeping the number of isomers small enough to allow verification. In this manner all obvious sources of error have been checked, for example, construction of loops on loops, multiple types of atoms of the same valence (e.g. Cl, Br, I) and partitions containing atoms of several different valences including penta- and hexavalent atoms. 3) Varying the order of generation. The structure of the program permits additional tests by doing some operations in a different order. For example, one variation allowed is to leave hydrogens associated with the atoms in each partition rather than to strip them away initially and place them on the remaining free valences in the last step. Each such test has resulted in the same set of isomers. .END In summary, the verification procedures utilized have all indicated absence of errors in the computer implementation of the algorithm. Also, there is no clear reason why generation of larger sets of isomers should not also proceed correctly. The final verdict however, must await development of new mathematical tools for verification by enumeration (see above) or an alternative algorithm. .BEGIN TABLE V,|Performance of Three↑* Chemists in Manual Generation of Isomers of C↓6H↓8 (C↓6U↓3). There are 159 Isomers| Number Generated Type of Error Chemist 1 161 4 duplicates; 4 omissions; 2 with 7 carbon atoms. Chemist 2 168 16 duplicates; 7 omissions Chemist 3 160 2 duplicates; 1 omission * One PhD and two graduate students. ----------------- .END .BEGIN TABLE VI,The Number of Isomers for Several Empirical Formulae Empirical Example Number of Isomers Manually Verified? Formula Compound C↓6H↓6 benzene 217 yes C↓6H↓8 1,3-cyclohexadiene 159 yes C↓6H↓1↓0 cyclohexene 77 yes C↓6H↓1↓2 cyclohexane 25 yes C↓6H↓1↓4 hexane 5 yes C↓6H↓6O phenol 2237 C↓6H↓1↓0O cyclohexanone 747 no C↓6H↓1↓2O 2-hexanone 211 yes C↓3H↓4N↓2 pyrazole 155 no C↓3H↓6N↓2 2-pyrazoline 136 yes C↓3H↓8N↓2 tetrahydropyrazole 62 no C↓3H↓1↓0N↓2 propylenediamine 14 yes C↓4H↓9P↓1 (pentavalent P) 110 no ----------------- .END ∪Constraints. The structure generator is designed to produce a list of all possible connectivity isomers (Appendix B). This list contains many structures whose existence seems unlikely based on present chemical knowledge. In addition, the program may be called on to generate possible structures for an unknown in the presence of a body of data on the unknown which specify various features, (e.g., functional groups) of the molecule. In such instances mechanisms are required for constraining the generator to produce only structures conforming to specified rules. The implementation of the acyclic generator possessed such a mechanism in the form of GOODLIST (desired features) and BADLIST (unwanted features) (3) which could be utilized during the course of structure generation. The cyclic generator is less tractable. As in prospective avoidance of duplicate structures, it is important that unwanted structures, or portions thereof, be filtered out as early in the generation process as possible. It is relatively easy to specify certain general types of constraints in chemical terms, for example, the number of each of various types of rings or ring systems in the final structure, ring fusions, functional groups, sub-structures and so forth. It is not always so easy to devise an efficient scheme for utilizing a constraint in the algorithm, however. As seen in the above example (Fig. 2) the expanded superatom partition results in what would be viewed by the chemist as several very different ring systems. The design of the program facilitates some types of constraints. For example, the program may be entered at the level of combining superatoms to generate structures from a set of known sub-structures. If additional atoms are present in an unknown configuration, they can be treated as a separate generation problem, the results of which are finally combined in all ways with the known superatoms. This approach will not form additional two-connected structures, however. Constraints which disallow an entire partition may be easily included. For example, it is possible to generate only open chain isomers by "turning off" the appropriate initial superatom partitions. Much additional work remains, however, before a reasonably complete set of constraints can be included. The implementation of each type of constraint must be examined and tested in detail to ensure that the generator remains thorough and irredundant. .HD CONCLUSIONS The boundaries, scope and limitations of chemical structure can now be specified. .HD ACKNOWLEDGEMENTS .NEXT PAGE Appendix A. Equivalence Classes and Finite Permutation Groups. Two members of a set of possible isomers may be defined to be equivalent if a specified transformation of one member causes it to be superimposable upon another member of the set. For example, there are fifteen possible ways of attaching two chlorine and four hydrogen atoms to a benzene ring (Scheme 4). .BEGIN NOFILL ----------------------------------------------------------------- Scheme 4 .GROUP SKIP 8 ----------------------------------------------------------------- .END If rotations by multiples of 60 degrees are specified as allowed transformations, the fifteen structures fall logically into three classes, termed "equivalence classes" (Scheme Sa). Within each equivalence class structures may be made superimposable by the rotational transformation. If one element (in this case a molecular structure) is chosen from each equivalence class, the complete set of possible structures is determined, without duplication. It is the task of the labelling algorithm to produce one and only one graph labelling corresponding to one member of each equivalence class. The set of transformations which define an equivalence class is termed a "finite permutation group." This permutation group may be calculated based on the symmetry properties of a graph (or chemical structure in the example of Scheme Sa). This calculation provides the mechanism for prospective avoidance of duplication. These procedures are described more fully in the accompanying paper (13). .NEXT PAGE Appendix B. Isomerism and Symmetry. Appendix A introduced the concept of equivalence classes and finite permutation groups. The selection of transformation (Appendix A) directs the calculation of the permutation group and thus defines the equivalence classes. Different types of transformation may be allowed depending on the symmetry properties of the class of isomers considered. This Appendix discusses several of the possible types of isomerism, most of which are familiar to chemists. The reader seeking a more thorough discussion of some types of isomerism discussed below is referred to an exposition of molecular symmetry in the context of chemistry and mathematics (19). .SEND FOOT ⊂ 19) I. Ugi, D. Marquarding, H. Klusacek, G. Gokel, and P. Gillespie, Angew. Chem. internat. Edit., 9, 703 (1970). .⊃; Isomers are most often defined as chemical structures possessing the same empirical formula. Different concepts of symmetry give rise to different classes of isomers, some of which are described below. ∪Permutational ∪Isomers. Permutational isomers are isomers which have in common the same skeleton and set of ligands. They differ in the distribution of ligands about the skeleton. Gillespie et al. (20) .SEND FOOT ⊂ 20) P. Gillespie, P. Hoffman, H. Klusacek, D. Marquarding, S. Pfohl, F. Ramirez, E. A. Tsolis, and I. Ugi, Angew. Chem. internat. Edit., 10, 687 (1971). .⊃; and Klemperer (21) .SEND FOOT ⊂ .BEGIN NOFILL 21) a) W. G. Klemperer, J. Amer. Chem. Soc., 94, 6940 (1972); b) W. G. Klemperer, ibid, p. 8360. .END .⊃; have used the concept of permutational isomers to probe into unimolecular rearrangement or isomerization reactions. ∪Stereoisomers. Ugi et al. (19) have defined the "chemical constitution" of an atom to be its bonds and bonded neighbors. Those permutational isomers which differ only by permutations of ligands at constitutionally equivalent positions form the class of stereoisomers. ↓_Isomers Under Rigid Molecular Symmetry_↓. If one conceives of molecular structures as having rigid skeletons, the physical rotational (three dimensional) symmetries and transformations may be readily defined. Each transformation causes each atom (and bond) to occupy the position of another or same atom (and bond) so that the rotated structure can physically occupy its former position and at the same time be indistinguishable from it in any way. This is the most familiar form of symmetry. Under this type of symmetry conformers are distinguishable and belong in distinct equivalence classes. Every transformation is orthogonal and preserves bond angles and bond lengths as well as maintaining true chirality. If one allows other orthogonal transformations that alter chiral properties of structures, equivalence classes result that treat both the left-handed and right-handed forms of chiral molecules to be the "same". Thus a "mirror image" transformation when suitably defined permits the left-handed form to exactly superimpose the right-handed form and vice-versa. ↓_Isomers Under Total Molecular Symmetry_↓. If in addition to the above mentioned rigid molecular transformations one recognizes the flexional movements of a nonrigid skeleton, a dynamic symmetry group may be defined. Under this definition, different conformers now are grouped together. Thus the "chair" and "boat" conformations of cyclohexane belong to the same equivalence class under dynamic symmetry. The permutation group of skeletal flexibility is computable separately and independently of rigid molecular symmetry. One can then view total molecular symmetry as the product of the two finite permutation groups. ↓_Isomers Under Connectivity Symmetry_↓. The concept of connectivity symmetry was introduced previously (METHOD section). Every permutation of atoms and bonds onto themselves is a symmetry transformation for connectivity symmetry if, .BEGIN INDENT 6,6 a) each atom is mapped into another of like species, e.g., N to N, C to C, O to O, and b) for every pair of atoms, the connectivity (none, single, double , triple, ...) is preserved in the mapping, i.e. the the connectivity of the two atoms is identical to the connectivity of the atoms they are mapped into. .END One can readily recognize that transformations as defined automatically preserve the valence and bond distribution of every atom. It is very probable that readers accustomed to three dimensional rotational and reflectional symmetries will tend to equate them with the symmetries of connectivity. It is emphasized again that connectivity symmetry does not consider bond lengths or bond angles, and it includes certain transformations that are conceivable but have no physical interpretation save that of permuting the atoms and bonds. .NEXT PAGE 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. .BEGIN NOFILL MINLOOPS = max{0,a↓2+1/2(2j↓[max]-↑n&↓[i=2]&∃ja↓j)}→(5) MAXLOOPS = min{a↓2,↑n&↓[j=4]&∃((i-2)/2)a↓j}→(6) MINLOOPS = minimum number of loops MAXLOOPS = maximum number of loops a↓2 = number of secondary nodes in degree list j↓[max] = degree of highest degree item in degree list j = degree n = highest degree in list a↓j = number of nodes with degree j. .END The form of the equations results from the following considerations: .BEGIN INDENT 6,6 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. .END ↓_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 1). 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 .<<the word is level>> 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↓6U↓5 is 15. .BEGIN NOFILL c=c=c=c=c=c 15 .END 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 .<<that it is generated is more of a bug in the program (harmless bug) >> .<<than a reflection on the algorithm - since the formal def of the algo->> .<<rithm specifies to check as part of the loop partitioner for invalid >> .<<things >> subsequent step, as the underlying graph (vide infra) 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). 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) .SEND FOOT ⊂ 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. .⊃; 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. C1. METHOD FOR THE CASE WITH EVEN NUMBER OF TOTAL ATOMS. The superatom partition C↓2U↓2/C↓2U↓1/-/C↓2 (partition 7, Table II and Figure 2) will be used here to illustrate this procedure. The superatomparts C↓2U↓2 and C↓2U↓1 have exactly one possible ring-superatom for each (see Table VII). .BEGIN TABLE VII Superatompart Superatom C↓2U↓2 -C≡≡C- C↓2U↓1 >C==C< ---------------------------------------- .END 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. 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. .BEGIN NOFILL Restriction on Univalents: 1+↑n&↓[i=1]&∃(i-2)a↓i≥ 0→(7) i = valence a↓i = number of atoms or superatoms of valence i n = maximum valence in composition .END 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. .BEGIN TABLE VIII Part Radicals ---------------------+------------------------------ -C≡≡C- , >C==C< -> -C≡≡C-CH=CH2 -CH=CH-C≡≡CH -C-C≡CH " CH2 C2 -> -CH2-CH3 -C≡≡C- , C -> -C≡≡C-CH3 -CH2-C≡≡CH >C==C< , C -> -CH==CH-CH3 -C-CH3 " CH2 -CH2-CH==CH2 ==================================================== .END Step 3. Form Molecules From Radicals. The radicals are combined in unique pairs, within each initial partition. Each pair gives rise to a unique molecule, for each of which is the center is a bond. There are nine such molecules that have a central bond, for the example chosen (Fig. 3). Category B. CENTRAL ATOM (see Fig. 4). Step 1. Selection of Central Atom. One must consider every unique atom or ring-superatom that has a free valence of three or higher as a central atom. In the example, of three candidates available: -C≡≡C- , >C==C< and C, the first is not chosen for it has a free valence of only two. Step 2. Partition the Rest of the Atoms. The atom or ring-superatom chosen for the center is removed from the set and the rest are partitioned into a number of parts less than or equal to the valence of the central atom. Each part must have less than half the total number of items being partitioned (again a ring-superatom is a single item). Each part must satisfy the restriction on univalents (equation 7). Thus, for the case where a carbon is the center, from one to four partitions are generated with the condition that each part has at most (4-1)/2 or 1 atom. No valid partitions are possible for one, two or four parts. There is exactly one partition for three parts, i.e., one atom in each. The partitions are shown in Figure 4. Step 3. Generate Radicals. Once again, using the procedure described in Section C3, radicals are generated for each part in each partition. For example, the partition -C≡≡C- gives rise to exactly one possible radical -C≡≡CH (Fig. 4). Step 4. Combine Radicals. Although in the example shown every part generates only one radical, in the general case there will be many radicals for each part. If so, the radicals must be combined to give all unique combinations of radicals within each partition. Step 5. Form Molecules from Central Atom and Radicals. If the central atom is not a ring-superatom but is a simple atom, then each combination of radicals derived in Step 4 defines a single molecule that is unique. Thus for example when C is chosen as the center, step 4 gives one combination of radicals which determines a single molecule when connected to the central C (see Figure 4). If the central atom is a ring-superatom and the valences of the ring-superatom are not identical then different ways of distributing the radicals around the center may yield different molecules. Labelling of the free valences of the central ring-superatom with radicals treated as labels (supplemented with adequate number of hydrogens to make up the total free valence of the ring-superatom) generates a complete and irredundant list of molecules. Thus >C==C< is labelled with the label set: .BEGIN NOFILL one of -C≡≡CH , two of -CH3 , and one of -H. .END There are two unique labellings as shown in Figure 4. C2. METHOD FOR ODD NUMBER OF TOTAL ATOMS With an odd number of total atoms, no structures can be generated with a central bond. Only the case of a central atom is to be considered. However, it is possible for structures to be built with a bivalent atom at the center. Thus the procedure outlined in Category B above is followed, in this case also allowing a bivalent atom to be the central atom. C3. GENERATION OF RADICALS The goal of this procedure is to generate all radicals from a list of atoms and ring-superatoms. A radical is defined to be an atom or superatom with a single free valence. When a composition of atoms and ring-superatoms is presented, from which radicals are to be generated, two special cases are recognized. Case 1. Onle One Atom in Composition. When only one atom which is not a ring-superatom makes up the composition only one radical is possible. As for example, with one C, the radical -CH3 is the only possibility. Case 2. Only One Item (a Ring-superatom) in Composition. In this case, depending upon the symmetry of the ring-superatom, several radicals are possible. This is determined by labelling the free valences of the ring-superatom with one label of a special type, a "radical-valence". .BEGIN NOFILL GROUP Example: A composition consists of one ring-superatom, 16. C- /" >C< " \" C- 16 .END Two radicals result from labelling with one radical valence. .BEGIN NOFILL GROUP CH C- /" /" -CH< " CH2< " \" \" CH CH 17 18 .END ∪General ∪Case. Radicals have uniquely defined centers as well, the center always being an atom of valence two or higher. The steps for generation of radicals are as follows. Step 1. Selection of Central Atom. Any bivalent or higher valent atom or ring-superatom is a valid candidate to be the center of a radical. Thus, for example, for the composition -C≡C-, >C=C< (see part 1 in Figure 3) both are valid central atoms (Figure 5). Step 2. Partition the Rest of the Atoms. The atom chosen for the central atom is removed from the composition. One of the valences of the central atom is to remain free. Therefore, the rest of the atoms in the composition are partitioned into less than or equal to (valence of central atom - 1) parts. Of course, each part should satisfy the restriction on univalents (equation 7) but for constructing radicals there is no restriction on the size of the parts. Step 3. Form Radicals from Each Part. The procedure to generate radicals is freshly invoked on each part thus generating radicals. Each part in Figure 5 gives rise to only one radical each arising from special case 2. Step 4. Combine Radicals in Each Part. For the example in Figure 5, each part yields only one radical. In a more general situation, where the rest of the composition after selection of center, is partitioned into several parts, and where each part yields several radicals, the radicals are combined to determine all unique combinations of radicals. Step 5. Label Central Atom with Radicals. If the center is an atom (not a ring-superatom) then each unique combination defines a single unique radical. If the center is a ring-superatom, the radicals are determined by labelling the center with a set of labels which includes;I) the radicals; ii) a leading radical-valence; iii) an adequate number of hydrogens to make up the remaining free valences of the ring-superatom. One selection of center gives one radical and the other gives two more, to complete a list of three radicals for the example chosen (Fig. 5). Summary For the example chosen to illustrate the operation of the acyclic generator, twelve isomers are generated, nine shown in Figure 3 and three shown in Figure 4.