A universal symmetry detection algorithm

Transcription

1 DOI /s RESEARCH Open Access A universal symmetry detection algorithm Peter M Maurer * *Correspondence: Peter_Maurer@Baylor.edu Department o Computer Science, Baylor University, Waco, TX , USA Abstract Research on symmetry detection ocuses on identiying and detecting new types o symmetry. The paper presents an algorithm that is capable o detecting any type o permutation-based symmetry, including many types or which there are no existing algorithms. General symmetry detection is library-based, but symmetries that can be parameterized, (i.e. total, partial, rotational, and dihedral symmetry), can be detected without using libraries. In many cases it is aster than existing techniques. Furthermore, it is simpler than most existing techniques, and can easily be incorporated into existing sotware. The algorithm can also be used with virtually any type o matrix-based symmetry, including conjugate symmetry. Keywords: Symmetry detection, Symmetric Boolean unctions, Generalized symmetry Background A symmetric Boolean unction is a unction whose inputs can be rearranged in some ashion without changing the output o the unction. The importance such unctions was irst recognized by Shannon in (Shannon 1949), who characterized unction symmetries using permutations o the input variables. Since that time, the detection and exploitation o symmetric Boolean unctions has been o recurring interest in the ield o design automation (Abdollahi 2006; Biswas 1970; Born and Scidmore 1968; Butler et al. 2000; Chrzanowska-Jeske 2001; Chung and Liu 1998; Darga et al. 2008; Drechsler and Becker 1995; Hu and Marek-Sadowska 2001; Hu et al. 2008; Ke and Menon 1995; Kettle and King 2008; Kravets and Sakallah 2002; Maurer 2011; Mohnke et al. 2002; Moller et al. 1993; Muzio et al. 2008; Rice and Muzio 2002; Scholl et al. 1997; Tsai and Marek-Sadowska 1996; Wang and Chen 2004; Zhang et al. 2004). Virtually all o these algorithms are based on Shannon s Theorem (Shannon 1949) which detects symmetry by comparison o two-variable coactors. (See below.) Although comparison o two-variable coactors is powerul enough to detect all total and partial symmetries, there are many types o symmetries that cannot be detected in this manner. As the number o input variables grows, these types o symmetry become more common than partial and total symmetry. Some progress has been made in detecting symmetries beyond partial and total symmetry (Chrzanowska-Jeske 2001; Tsai and Marek-Sadowska 1994; Kravets and Sakallah 2000), but the problem o universal symmetry detection has remained open since Maurer. This article is distributed under the terms o the Creative Commons Attribution 4.0 International License ( creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate i changes were made.

2 Page 2 o 30 Figure 1 lists the number o symmetries and the number o symmetry types or each number o inputs rom 1 through 18. (Two symmetries are o the same type i they are the same but applied to dierent inputs.) Columns 2 and 3 are the total number o symmetry types and symmetries or each n. Columns 4 and 5 are the total number o symmetry types and symmetries or partial and total symmetry. Figure 2 gives the percentage o partial and total symmetries and symmetry types, compared to the total number o symmetries. The data rom these tables shows that partial and total symmetries account or less than hal o the available symmetry types when the number o inputs is more than 3, and only a tiny percentage o the available symmetries when the number o inputs is 8 or more. The experiments with standard benchmarks (Brglez et al. 1985) show that symmetries other than partial and total symmetries are common. Other researchers have noted the existence o such symmetries (Kravets and Sakallah 2002; Mohnke et al. 2002). Ignoring such symmetries can cause major ailures in layout veriication and regression (Maurer and Schapira 1988). These algorithms typically use graph-isomorphism algorithms, starting rom the known elements o the circuit (the primary inputs, or example). I a ailure occurs early in the process, such as not recognizing that the unction x 1 x 2 + x 3x 4 in one circuit is identical to the unction x 3 x 4 + x 1x 2 in the other, the ailure can cause a cascade o alse errors throughout a major portion o the circuit. When too many alse errors are reported, the real errors are extremely diicult to identiy. (The unction Input Count All Symmetry Types All Symmetries Partial/Total Symmetry Types Partial/Total Symmetries , , , ,694, , ,594, , ,126, , ,723 10,594,925, ,213, , ,238,308, ,644, ,154 5,651,774,693, ,899, , ,053,117,995, ,382,958, ,165 5,320,744,503,742, ,480,142, ,466, ,889,331,236,297, ,864,869, ,274,651 7,598,016,157,515,302, ,076,806,159 Fig. 1 Symmetries and symmetry types (Holt 2010).

3 Page 3 o 30 Inputs Partial/Total Symmetry Percentage Partial/Total Type Percentage % % % % % 75.00% % 45.45% % 36.84% % 19.64% % 15.63% % 7.43% % 5.42% % 2.64% % 1.81% % 0.72% % 0.48% % 0.18% % 0.11% % 0.03% % 0.02% % 0.01% Fig. 2 Percentage o partial/total symmetries. x 1 x 2 + x 3x 4 is neither totally nor partially symmetric, but it is obvious the set o inputs {x 1, x 2 } can be exchanged with the set {x 3, x 4 } without altering the output.) Correct handling o symmetry is also important when attempting to match design speciications to an existing library o unctions (Mohnke et al. 2002). Suppose the library contains a pre-laid-out implementation o the unction x 1 x 2 + x 3x 4, and the circuit being laid out contains the subcircuit x 3 x 4 + x 1x 2. I symmetry-detection ails, these two unctions will not be matched correctly, requiring a new layout (by hand or by machine) or x 3 x 4 + x 1x 2, when none is actually required. This can be costly, both in terms o time and o correctness. New implementations must be veriied and tested, whereas library implementations are already veriied and are much more likely to be correct. This paper presents an entirely new approach which, eectively, considers all inputs simultaneously instead o in pairs. This approach allows the algorithm to detect virtually any type o symmetry, including some types that go beyond permutations. (These types include matrix-based symmetry, auto-symmetry, Kronecker symmetry, anti-symmetry, and multi-phase symmetry.) For small numbers o inputs (less than 8) the USD is aster than using coactors. In addition, the coding is simpler. Pseudo code is presented in "The symmetry detection algorithm", which can easily be adapted or use in existing EDA algorithms. The USD algorithm also is somewhat easier to parallelize than the conventional algorithm, because it does not require the accumulation o results to completely characterize a unction. The conventional algorithm checks or symmetric variable pairs

4 Page 4 o 30 and then combines the results rom these tests to determine the symmetries o a unction. Even though each o the symmetric variable pair tests could be done in parallel, combining the results requires some sort o binary anin or pointer jumping, giving a parallel time bound o O(lg n). The USD tests an entire permutation group in one shot without needing to combine results. Each o these tests could be done in parallel, giving a parallel time bound o O(c). Despite the advantages o the new approach, the algorithm includes the conventional symmetric-variable-pair detection algorithms as a subset o the new detection algorithms. When no libraries exist or a particular number o inputs, this makes it possible to detect any partial, total, multi-phase, anti, and Kronecker symmetry using the conventional approach. Symmetries can be categorized into total symmetry, partial symmetry, and strong symmetry. Total symmetry permits the inputs o a unction to be rearranged arbitrarily without changing the output o the unction. Partial symmetry is similar to total symmetry in that it permits one or more subsets o inputs to be rearranged arbitrarily. Strong symmetry is a catch-all term that includes every type o symmetry that is neither total nor partial. The unction x 1 + x 2 + x 3 + x 4 is totally symmetric, the unction x 1 x 2 x 3 + x 4 is partially symmetric, while the unctions x 1 x 2 + x 3 x 4 and x 1 x 2 + x 3 x 4 are strongly symmetric. (Functions are speciied as expressions in which multiplication signiies AND, addition signiies OR, and the prime symbol speciies NOT.) In x 1 x 2 + x 3 x 4 no single variable can be exchanged with any other single variable, but the set {x 1, x 2 } can be exchanged with the set {x 3, x 4 }. The unction x 1 x 2 + x 3 x 4 is more problematical because most existing algorithms will detect two partial symmetries, but ignore the act that the set {x 1, x 2 } can be exchanged with the set {x 3, x 4 }. (The algorithm o (Kravets and Sakallah 2002) will detect the correct symmetry or this unction.) There are many more kinds o strong symmetry than partial and total symmetry (See Figs. 1, 2). Various sub-categories o strong o symmetry have been discovered, and algorithms have been created to detect and exploit some o these symmetries (Mohnke et al. 2002). Examples o such symmetries are hierarchical symmetry, rotational symmetry and dihedral symmetry (Kravets and Sakallah 2002). The primary tool or categorizing symmetry (in any ield) is the permutation group (Passman 1968). Let X be a inite set o objects. A permutation is a one-to-one unction rom X to itsel. In other words, a permutation rearranges the elements o X without creating or destroying any elements. Permutations can be multiplied using unction composition. I p and q are permutations, then so is pq where (pq)(x) = q(p(x)). The multiplication operation is associative, ( p(qr) = (pq)r), but not necessarily commutative, (pq need not equal qp). A set o permutations, G, that is closed under multiplication (or all a, b G, ab G) is called a Group. The set o all permutations o a set X is called the symmetric group on X and is written S X. Although it is possible to apply permutations to any inite set, the only thing that aects the structure o S X is the size o X. I X and Y are two sets o the same size, then S X and S Y are identical. I p S X, and the size o X is n then n is the degree o p. I X ={1, 2,..., n} S X is written as S n. When speaking o the input variables o a unction, the variables will be designated as x 1, x 2,..., x n. All permutations are assumed to be elements o S n, and permute the variables x 1, x 2,..., x n by operating on their indices.

5 Page 5 o 30 Every permutation group G has two important properties which are implied by G being closed under multiplication. First, the identity permutation, I, is a member o every group. (Ip = pi = p or all p.) Second, every permutation p G has an inverse permutation p 1 G such that pp 1 = p 1 p = I. Permutations can be speciied in many ways, but in this paper cycle notation will normally be used. Every permutation in S n can be characterized as one or more cyclic shits o some subset o the integers {1, 2,, n}. For example the permutation (1, 2, 3) S 3 maps 1 to 2, 2 to 3, and 3 to 1. A permutation may perorm several cyclic shits simultaneously, as in (1, 2, 3)(4, 5) S 5, which shits 1, 2, and 3 cyclically and also swaps 4 and 5. Elements that are not moved by the permutation are normally omitted rom the cycle notation, so the permutation (1, 2, 3)(4)(5)(6) S 6 would normally be written (1, 2, 3). In a cyclic shit, it doesn t matter which element comes irst, so the cycles (1, 2, 3), (2, 3, 1), and (3, 1, 2) all denote the same permutation. To avoid this ambiguity, the smallest element o a cycle is always listed irst. A similar ambiguity occurs when a permutation perorms two or more cyclic shits o the same size, as in (1, 2, 3)(4, 5, 6). In this case the cycles o the same length are written in ascending order by their smallest elements. Cycles o length two, such as (1, 2) and (2, 3) are called transpositions. Any cycle o length k can be actored into k 1 transpositions in the ollowing manner: (c 1, c 2, c 3..., c k ) = (c 1, c 2 )(c 1, c 3 )...(c 1, c k ). This implies that any permutation can be actored into a product o transpositions. This actorization is not unique, but i a permutation, p, can be actored into an even number o transpositions, then there is no way to actor it into an odd number o transpositions. By the same token i a permutation can be actored into an odd number o transpositions, then there is no way to actor it into an even number o transpositions. Thus it is possible to characterize each permutation as either even or odd. The product o two odd or two even permutations is even and the product o an odd permutation with an even permutation is odd. For every symmetric group S n, there is a subgroup A n, consisting o the even permutations o S n. The group A n is called the alternating group o degree n. Let p be a permutation o degree n and let be an n-input Boolean unction. The permutation p and the unction are compatible i using p to rearrange the variables o leaves the output o unchanged. The unction is also said to be invariant with respect to p. This terminology is extended in the obvious way to subgroups o S n. The symmetry group G is the set o all permutations that leave invariant. The group G is closed under multiplication because i p and q leave invariant, then so does pq. Thus G is a subgroup o S n. Because the identity element leaves every unction invariant, G is never empty. Most recognized types o symmetric unctions can be characterized using symmetry groups. For example, an n-input unction is totally symmetric, i and only i G = S n. A unction is non-symmetric i and only i G ={I}. Most existing symmetry-detection algorithms use symmetric variable pairs, which are detected by comparing the coactors o a unction (Chrzanowska-Jeske 2001). A coactor o is ound by setting one or more input variables to constant values. For example, let = x 1 x 2 + x 3 x 4. Two coactors o are xx1x = x 1 x 2 + x 4 and 0xxx = x 3 x 4. The our positions in the subscript correspond to the our input variables x 1, x 2, x 3, and x 4 respectively. The subscript indicates which variables have been set to constants and which are unaected. When the unaected variables are obvious, it is common to omit the x s.

6 Page 6 o 30 Symmetric variable pairs are pairs o variables that can be exchanged without aecting the output o the unction. Shannon s theorem (Shannon 1949) states that (x 1, x 2 ) is a symmetric variable pair i and only i 01 = 10. Symmetric variable pairs are transitive, which means that i (x i, x j ) and (x j, x k ) are symmetric variable pairs, then so is (x i, x k ). Because o this, all partial and total symmetries can be detected using symmetric variable pairs. However, symmetric variable pairs cannot be used to detect strong symmetries. This paper introduces a new method o categorizing permutation groups called Boolean Orbits. Boolean orbits are used as the basis o a new symmetry detection algorithm that can determine a unction s compatibility with any permutation group. The concept o Boolean orbits can be extended to virtually all types o known symmetry, allowing these types o symmetry to be detected by the algorithm described here. Results and discussion Boolean orbits Orbits have been used by mathematicians or many years to analyze and categorize permutation groups (Mohnke et al. 2002; Passman 1968). They have also been used to some extent to analyze symmetric Boolean unctions (Mohnke et al. 2002). Orbits are computed as ollows. Let G be a permutation group that is compatible with a Boolean unction, and let X be the set o input variables o. Two variables x i, x j X are said to be in the same orbit o G i there is a permutation p G, such that p(x i ) = x j. Intuitively, an orbit contains all the variables that can be exchanged with one another, so the unction x 1 x 2 x 3 + x 4 has two orbits {x 1, x 2, x 3 } and {x 4 }. Belonging to the same orbit is an equivalence relation, so it breaks the set o input variables into a collection o disjoint subsets. Orbits can be used to distinguish total and partial symmetries, but are not particularly eective with strong symmetries. Consider the unction x 1 x 2 + x 3 x 4, which possesses dihedral symmetry. At irst it may appear that this unction has two orbits, but in act it has only one, {x 1, x 2, x 3, x 4 }. By the same token, the totally symmetric unction x 1 + x 2 + x 3 + x 4 has a single orbit, {x 1, x 2, x 3, x 4 }. Thus the unctions x 1 + x 2 + x 3 + x 4 and x 1 x 2 + x 3 x 4 have the same orbits even though their symmetries are quite dierent. (Strictly speaking, orbits are properties o permutation groups, not o unctions. Thus, the orbit {x 1, x 2, x 3, x 4 } is the orbit o the permutation group o x 1 x 2 + x 3 x 4, not o the unction itsel.) This paper presents a new type o orbits, called Boolean Orbits, that permit one to deal eectively with strong symmetries as well as partial and total symmetries. Boolean orbits are computed with respect to the Boolean input vectors o a unction rather than with respect to the variables. (Again, it is important to note that Boolean orbits are properties o permutation groups, not o Boolean unctions.) Permutations o degree n can operate on n-element vectors by permuting the indices o the elements. For example, one can apply the permutation (1, 2, 3) to the vector (v 1, v 2, v 3 ) to obtain (v 3, v 1, v 2 ). Applying this permutation to the speciic vector (1, 1, 0) yields the vector (0, 1, 1). The concept o Boolean orbits is ormalized in the ollowing deinition. Deinition 1 Given a permutation group G o degree n, two n - input vectors v and w are in the same Boolean orbit o G i there is a permutation p G such that p(v) = w.

7 Page 7 o 30 Like ordinary orbits, belonging to the same Boolean Orbit is an equivalence relation, so this relation breaks the set o n - input Boolean vectors into a collection o disjoint sets. I G is the symmetry group o a Boolean unction,, then the Boolean orbits o G will partition the truth-table o into disjoint sets. In act, the symmetry o a Boolean unction is completely determined by the Boolean orbits o its permutation group. Figure 3 shows the symmetry groups and the Boolean orbits o the two unctions x 1 + x 2 + x 3 + x 4 and x 1 x 2 + x 3 x 4. The irst two lines o Fig. 3 give the unction and the conventional orbits o the unction. The inal part o Fig. 3 contains the Boolean orbits o the unction with one Boolean orbit per line. The Boolean orbits o the two unctions are quite dierent, even though the conventional orbits are the same. (When listing Boolean orbits, all orbits o size 1 are omitted, since these orbits do not aect the symmetry o the unction.) The important properties o Boolean orbits are summarized in the ollowing theorems. Theorem 1 states that a symmetric Boolean unction must map the elements o each o the Boolean orbits o its symmetry group to a unique value. Theorem 1 Let be a Boolean unction, and G be the symmetry group o. I K is a Boolean orbit o G and u, v K, then (u) = (v). Proo I K is a Boolean orbit o G and u, v K, then there is a permutation p G such that p(u) = v. Since every element o G must leave invariant, (u) = (p(u)) = (v). Theorem 2 is the converse o Theorem 1. It states that i the Boolean unction,, maps the orbits o a symmetry group, G, to unique values, then is compatible with G. Theorem 2 Let be an n-input Boolean unction and let G S n be a group such that or every Boolean orbit, K, o G, and or every pair o elements u, v K, (u) = (v) then is invariant with respect to G, and G G. Proo Let p be an element o G, and let u be any input o. The vectors u and p(u) are in the same Boolean orbit, K, o G. Since (u) = (v) whenever u, v K, (u) = (p(u)), and is invariant with respect to p. Since p was arbitrary, every element o G is compatible with. The group G contains every permutation that leaves invariant, so i p G then p G, and G G. x1+ x2 + x3+ x4 xx 1 2+ xx 3 4 { x1, x2, x3, x4} { x1, x2, x3, x4} Fig. 3 Orbits, and Boolean orbits

8 Page 8 o 30 Obviously, the singleton orbits (those containing a single vector) do not aect the symmetry o a unction, so when testing a unction or compatibility with a group G, the singleton orbits can be ignored. Theorem 3 deals with the problem o unctions that have more than one type o symmetry. As this theorem shows, i a Boolean unction has two dierent types o symmetry A, and B, then also possesses an overarching symmetry that includes both A and B. Thus i one can identiy the largest symmetry group that is compatible with, then all o the symmetries possessed by have been discovered. Theorem 3 Let be a Boolean unction and let G and H be two permutation groups that are compatible with. Then there is a permutation group, K compatible with such that G and H are both subgroups o K. Proo Let K be the smallest subgroup o S n containing G H. Since S n contains both G and H, K must exist. From group theory it is known that that every element o p K is o the orm p = q 1 q 2...q k where q i G or q i H. Since every element o either G or H is compatible with, p must also be compatible with, and K is the required group. The remainder o the paper will make extensive use o the characteristic unction o an orbit. Let S be any set o n-element Boolean vectors. The characteristic unction o S, C s is an n-input Boolean unction which is equal to 1 on every element o S, and zero elsewhere. Figure 4 gives a set o orbits along with their characteristic unctions in truthtable orm. The characteristic unctions o the Boolean Orbits can be used to determine the symmetries o a Boolean unction. Given a permutation group, G, computing the Boolean orbits o G is straightorward. The algorithm is given in Fig. 5. Shannon s theorem, which is the basis o virtually all other symmetry detection algorithms, is a special case o Theorem 2. Shannon s theorem deals with symmetric variable pairs o a unction, (x i, x j ). Formally, symmetric variable pairs are deined as ollows. Deinition 2 Let be an n-input Boolean unction and let x i and x j be two input variables o. The pair (x i, x j ) is a symmetric variable pair i and only i is compatible with the permutation group {I, (i, j)}. The proo o Theorem 4 shows that Shannon s theorem is a special case o Theorem 2 above. (The truth o this theorem is, o course, well known.) Theorem 4 Let be a Boolean unction, and let 00, 01, 10 and 11 be the coactors o with respect to the pair o variables (x i, x j ). The pair (x i, x j ) is a symmetric variable pair o, i and only i 01 = Fig. 4 Boolean orbits and characteristic unctions.

9 Page 9 o 30 Set Orbit-Collection B to Empty For Each n-element Boolean vector v I v has not been assigned to an orbit Create a new orbit Z Assign v to Z Mark v as assigned to an orbit For Each permutation p in group G Apply p to v to obtain w I w is not in Z Assign w to Z Mark w as assigned to an orbit End I End For Add Z to B End I End For Fig. 5 Generating Boolean orbits. Proo I (x i, x j ) is a symmetric variable pair o, then must be invariant with respect to the permutation group {I, (i, j)}. The Boolean orbits o this group are either singletons o the orm { 0 0 }, or { 1 1 }, or pairs o the orm { 0 1, 1 0 }, where the unspeciied parts are identical or each vector. The pairs run through all Boolean combinations o the unspeciied parts. The unction must map the two elements o each Boolean orbit o the orm { 0 1, 1 0 } to the same value. I all vectors o the orm 0 1 are combined into a single set, the 0, and the 1 are ignored, the result is the truth table o the coactor 01. The truth table o the coactor 10 is obtained in the same manner. Because each element o each Boolean orbit must be mapped to a single value, the truth tables must be the identical, implying that 01 = 10. For the converse, i 01 = 10, it is possible to expand each o the coactors into a truth table. These truth tables must be identical. It is then possible to insert 0 and 1 into the appropriate positions o each truth table to obtain the values o or each o the inputs o the orm 0 1 and 1 0. Now consider two vectors o the orm 0 1 and 1 0 where the unspeciied parts are identical. These vectors were obtained rom a single entry in a single truth table, thereore they must be mapped to the same value by. Thereore must be invariant with respect to the permutation group {I, (i, j)}, and (x i, x j ) must be a symmetric variable pair. The implication relation Let and g be two n-input Boolean unctions. The unction is said to imply g i g(v) = 1 whenever (v) = 1. The concept o implication to deines the undamental relationship on which the USD algorithm is based. This result is given in Theorem 5. Theorem 5 Let G S n be a permutation group, and let be an n-input Boolean unction. Let K ={K 1, K 2,..., K n } be the collection o Boolean orbits o G with characteristic

10 Page 10 o 30 unctions {C 1, C 2,..., C k }. The group G leaves invariant i and only i C i implies either or or every i, 1 i k. Proo Suppose C i implies, and v K i. Then C i (v) = 1, and because C i implies, (v) = 1 or all v K i. Now suppose C i implies. I v K i then C i (v) = 1 and since C i implies, (v) = 1, implying that (v) = 0 or all v K i. By Theorem 2, must be invariant with respect to G. Now suppose is invariant with respect to G. By Theorem 1, i u, v K i then (u) = (v). But C i (u) = C i (v) = 1. I (u) = (v) = 1, then C i implies. I (u) = (v) = 0, then C i implies. Theorem 5 gives a principle that can be used to detect symmetry with respect to any permutation group. Given a permutation group G it is straightorward to compute the characteristic unctions o its Boolean orbits. Given it is straightorward to compute. Once these unctions have been computed, one only need check the characteristic unction o each orbit to determine whether is symmetric with respect to G. Although symmetry detection is normally done on single-output unctions, the principle can be extended easily to multiple-output unctions. First it is necessary to deine the unction orbits o multiple output unctions. Deinition 3 Let be a Boolean unction with n inputs and m outputs. For each m-element vector v, S [v] is the set o all n-element input vectors on which takes the value v. The sets S [v] are known as the unction orbits o. The unction [v] is the characteristic unction o S [v]. Theorem 6 extends the implication principle to Boolean unctions with multiple outputs. Theorem 6 Let G S n be a permutation group, and let be an n-input Boolean unction with m outputs. Let K ={K 1, K 2,..., K n } be the collection o Boolean orbits o G with characteristic unctions {C 1, C 2,..., C k }. The group G leaves invariant i and only i or every i, 1 i k, C i implies [v] or some m-element vector v. The proo is essentially identical to that o Theorem 5. The symmetry detection algorithm Figure 6 gives the pseudo code or the universal symmetry detection (USD) algorithm. In this igure, it is assumed that the algorithm is being applied to a collection o unctions, and that a library o symmetries is being used. Each library entry contains a set o characteristic unctions that correspond to the Boolean orbits o a symmetry group. In most cases, the library will contain all subgroups o S n or some integer, n, although there are a number o other more specialized libraries. Complete libraries or S 2 through S 8 have been created. Subgroup libraries or S 9 through S 18 exist on the web (Holt 2010; Peier 2005), but these have not yet been adapted or use with the USD. When used with a complete library or S n symmetry detection begins with the largest group so the algorithm may stop as soon as a compatible group is ound.

11 Page 11 o 30 As Fig. 6 shows, the algorithm reads each unction, and compares it to each library entry until a compatible entry is ound. Most libraries contain a non-symmetric entry which permits each unction to be associated with at least one library entry. However, such an entry is not required. I no compatible subgroup can be ound, the unction is marked as non-symmetric. Comparison between a unction and a subgroup is done by enumerating the Boolean orbits o the subgroup. Each Boolean orbit is tested against and seeking an implication. I a particular subgroup orbit does not imply either unction orbit, the comparison with the group is terminated. However, i all subgroup orbits imply a unction orbit, the subgroup is assigned to the unction as its symmetry group, and testing o the unction terminates. Libraries are not necessary or symmetries that can be parameterized or an arbitrary number o inputs. As yet, only a ew symmetries have been so categorized, the most well-known o which are total, symmetry, partial symmetry, rotational symmetry, dihedral symmetry, and various types o hierarchical symmetry. The USD algorithm has special generators or total, partial, dihedral and rotational symmetry, which permits these Load Library Sort Library into descending order by subgroup size. For each unction For each subgroup G in Library GroupCompatible = True; For each orbit K o G While GroupCompatible OrbitCompatible = False; For each orbit P o I C K implies CP OrbitCompatible = True; Break; EndI EndFor I Not OrbitCompatible GroupCompatible = False; Break; EndI EndFor I GroupCompatible Assign G as the symmetry group o ; Break; EndI; EndFor; I Not GroupCompatible Mark as nonsymmetric EndI; EndFor; Fig. 6 The universal symmetry detection algorithm.

12 Page 12 o 30 types o symmetries to be detected without having a precomputed library. O course it is possible to cache the output o these generators or uture use. Most o the existing libraries contain one entry per symmetry group. However, or large numbers o inputs it is not easible to store libraries in this ashion. (See Fig. 1.) The conjugacy relation can be used to reduce the size o the library or large numbers o inputs. Certain types o symmetry are undamentally the same, but applied to dierent inputs, and certain types o symmetry are undamentally dierent. For example, a 3-input partial symmetry on the irst three inputs o a unction is not undamentally dierent rom a three-input partial symmetry on the last three variables. But a partial symmetry in the irst three variables is undamentally dierent rom a partial symmetry in the irst two variables. The conjugacy relation is used to distinguish symmetries that are essentially the same rom symmetries that are undamentally dierent. Deinition 4 permits the ormalization this idea. Deinition 4 Two permutations p and q are conjugate to one another i there is another permutation s such that p = s 1 qs. Conjugacy can be best understood by visualizing it in this way: to permute the last three variables o a unction, move them to the irst three variables using s, then apply q to the irst three variables, and then use s 1 to move the variables back where they were. This relationship can be extended to permutation groups in the ollowing way: s 1 Gs ={s 1 ps p G}. I two symmetries are undamentally the same then their permutation groups will be conjugate to one another. For example, all partial symmetries on three inputs have conjugate symmetry groups. Conjugacy is an equivalence relation, so the subgroups o a group can be partitioned into a set o conjugacy classes. The Column 2 o Fig. 1 shows the number o conjugacy classes and Column 3 shows the number o subgroups or the symmetric groups rom S 1 through S 18. The ull libraries store each subgroup o the symmetric group. Reduced libraries store only one member o each conjugacy class. To regenerate a conjugacy class, it is necessary to compute the conjugates o each library entry. However or each subgroup G o S n there are many pairs o permutations (p, q) such that p = q, but p 1 Gp = q 1 Gq. To avoid duplicated work a set o permutations is stored with the library entry. There is one permutation or each conjugate, so applying each permutation to the entry will restore the entire class. The permutations are computed when creating the library. A group theoretic result states that the number o conjugates o a group is equal to the index (i.e. number o right cosets) o its normalizer (Robinson 1995). This is a simple idea, but requires some background discussion. The normalizer o a group, G, written N(G), is the set o permutations that leave G unchanged with respect to conjugacy. That is, the set N(G) ={p p 1 Gp = G}. (N(G) is always a subgroup o S n.) Since G is closed under multiplication, every element o G leaves G invariant under conjugation, and G is a subgroup o N(G). However, N(G) oten contains other elements as well. I p / N(G), then p 1 Gp = G, but or every element q N(G), (qp) 1 Gqp = p 1 q 1 Gqp = p 1 Gp. So, i it is desired to include only a minimal set o permutations in the library entry, it is necessary to avoid including both

13 Page 13 o 30 p and qp. In act, a theorem o group theory states that i two permutations r and s produce the same conjugate o G, (i.e. r 1 Gr = s 1 Gs), then there must be some q N(G) such that qr = s. This means that the entire set o permutations that produce the same conjugate as permutation r is the set N(G)r ={qr q N(G)}. The set N(G)r is called a right coset o N(G). Another theorem o group theory states that given two right cosets o N(G), N(G)r and N(G)s, either N(G)r = N(G)s or N(G)r N(G)s = φ. So i r and s belong to the same right coset o N(G), then r 1 Gr = s 1 Gs. I r and s belong to dierent right cosets o N(G), then r 1 Gr = s 1 Gs. This means that i one can generate all o the right cosets o G, and then select one permutation rom each, one will have a minimal set o permutations. Such a set is called a set o coset representatives. In summary: 1. There is set o permutations that don t change G under conjugation. 2. This set can be used to generate a collection o sets, each one o which contains permutations that all produce the same conjugate o G. 3. Choosing one representative rom each o the sets will give a minimal set o permutations. Figure 7 gives the algorithm or creating a reduced library entry. Figure 8 gives the library entry that is used to detect 2-variable partial symmetry in 3-input unctions. The permutations are coded in the orm o a list o numbers rom the set {0, 1, 2}. The irst permutation is the identity, I. The irst line o Fig. 8 is the name o the symmetry, the second is the number o inputs, the third is the list o coset representatives, and the remainder is the Boolean orbits, one orbit per line. For the larger symmetric groups, reconstructing all conjugacy classes is a physical impossibility. So instead, the set o stored permutations is used to alter the unction under test. Let s suppose that g is invariant with respect to p 1 Gp. Then, there is an which is invariant with respect to G such that p 1 = g. I g is invariant with respect to p 1 Gp, then pg is invariant with respect to G. Figure 9 gives the pseudocode or detecting symmetry with reduced library entries. The test or compatibility in Fig. 9 is identical to that in Fig. 6. In most cases, this code will be slower than using a ully expanded library, so the algorithm o Fig. 9 is used only when necessary. Forbidden groups There are certain permutation groups, and certain matrix groups that cannot be the symmetry group o any unction. For example, suppose G is a permutation group with Boolean orbits B, and suppose there is another group H such that G H(G = H) such that H also has Boolean orbits B. In this case, G cannot be the symmetry group o any unction, because any unction, that is compatible with G will also be compatible with H, but G does not contain every permutation that is compatible with, so G cannot be the symmetry group o. Groups o this nature are called orbidden groups. Forbidden groups can arise in two dierent ways. Some orbidden groups are output limited and some are input limited.

14 Page 14 o 30 Read group G Normalizer = {}; For each permutation p in Sn 1 I p Gp = G Add p to Normalizer Endi EndFor CosetCollection = {} For each permutation p in Sn I Gp is not in CosetCollection Add Gp to CosetCollection EndI EndFor CosetRepresentatives = {} For each Coset c in CosetCollection Select one element p rom c; Add p to CosetRepresentatives EndFor Write G with CosetRepresentatives Fig. 7 Generating coset representatives. Fig. 8 A reduced library entry. S3.S2 3 CSR: (0,1,2), (0,2,1), (2,1,0) Read unction ; FoundSymmetry = False; For each Group G while Not FoundSymmetry For each coset representative p in G Compute p i p is compatible with G 1 Assign p Gp as the symmetry group o G FoundSymmetry = True; Break; EndI EndFor EndFor Fig. 9 The reduced-library detection algorithm. More ormally, an output limited group, G, has three or more Boolean orbits o the same weight that must be distinguished rom one another to keep a unction rom being compatible with larger groups containing G. However, because the unction has only

15 Page 15 o 30 two output values, this is impossible. Output limitations disappear when there are more than two output values. An input limited group, G, contains a set o permutations that must be distinguished rom those o a larger group containing G to keep a unction rom being compatible with the larger group. However, the permutations o the larger group cannot be so distinguished, because the only thing the larger group adds is a permutations that move the identical elements o each input vector. Input limitations disappear when inputs are allowed to have more than two values. Input limited groups are easy to detect because they will have the same Boolean orbits as a larger group. Output limited groups are more complicated. For example, the group K4 = {I, (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3)} is output limited. (This group is known as the Klein 4-group, hence the name K4.) The Boolean orbits o K4 are given in Fig. 10. These orbits are quite similar to the Boolean orbits o the three conjugates o D 8, the dihedral group o order 8. These orbits are given in Fig. 11. The problem with K4 lies with the three orbits {0011, 1100}, {0101, 1010} and {0110, 1010}. I a unction maps the irst two orbits to the same value, the result will be dihedral symmetry o type D8.2. I it maps the last two orbits to the same value, the result will be a dihedral symmetry o type D8.3. I it maps the irst and last orbits to the same value, the result will be a dihedral symmetry o type D8.1. Because a singlevalued unction has only two possible output values, at least two o these orbits must be mapped to the same value. Thus no single-valued unction can have symmetry K4. On the other hand, it is possible or multiple-valued unctions to have K4 symmetry. The unction o Fig. 12 is an example. K4 also appears as a sub-symmetry in unctions with ive or more inputs. A similar phenomenon occurs with input limited groups. For example, the two symmetry groups S 3 and A 3 have identical Boolean orbits, as shown in Fig. 13. Thus Fig. 10 The Boolean orbits o K D8.1 D8.2 D Fig. 11 The three conjugates o D8.

16 Page 16 o 30 Fig. 12 A unction with K4 symmetry. { S3 = I,(1, 2),(1, 3),(2,3),(1, 2,3),(1, 3, 2) { I } A3 =,(1, 2,3),(1,3, 2) Fig. 13 The Boolean orbits o S 3 and A 3. } A 3 is orbidden. All o the alternating groups or n > 2 are orbidden, as is shown by Theorem 8. To show that all alternating groups are orbidden, it is necessary to start with the ollowing lemma. Lemma 1 Let n be greater than 2, v be an n-element vector, and p be a permutation o degree n. I p(v) = w, then there is an even permutation q such that q(v) = w. Proo Suppose p is odd. Because n is greater than 2, there must be two elements o v that are identical. Suppose these are positions i and j, with i = j. Thus (i, j) is a 2-cycle, and q = (i, j)p is an even permutation. But (i, j)(v) = v, and q(v) = p((i, j)(v)) = p(v) = w. Theorem 7 For all n > 2 A n is orbidden. Proo Suppose u and v are in the same Boolean orbit o S n. Then there must be a p S n such that p(u) = v. By Lemma 1, there must be a q A n such that q(u) = v. Thereore u and v must be in the same Boolean orbit o A n. Thus every Boolean orbit o S n must be contained in a Boolean orbit o A n. Because the Boolean orbits o S n are as large as possible, the Boolean orbits o S n and A n must be identical, and because A n S n, A n must be orbidden. The groups A n are examples o input limited groups. They are orbidden because there are only two distinct values or each input. For unctions with multiple-valued inputs,

17 Page 17 o 30 not all o the alternating groups are be orbidden. (Extended versions o Lemma 1 and Theorem 7, would still be true or suiciently large n.) As an example, the subgroup o S 6, S3a = {I, (1, 2, 3)(4, 5, 6), (1, 3, 2)(4, 6, 5)}, is isomorphic to A 3 (but not conjugate to A 3 ). S3a is not orbidden, in act the unction x 1 x 4 x 5 + x 2x 5 x 6 + x 3x 4 x 6 possesses S3a symmetry. This unction is derived rom the three cubes o Fig. 14, which make it easy to see that a rotation o the irst three variables must be accompanied by a rotation o the last three variables. The reason that S3a is not orbidden is that the inputs operate in pairs: {x 1, x 4 }, {x 2, x 5 } and {x 3, x 6 }. Each pair o inputs has our possible values, so the argument o Theorem 7 does not apply. The elimination o orbidden groups rom symmetry libraries is an important optimization step, because it eliminates symmetry tests that can never succeed. Matrix-based symmetry also displays the phenomenon o orbidden groups. Sub symmetries For unctions with many inputs, it may be more useul to detect smaller, more manageable symmetries on a subset o inputs. Such symmetries are called Sub-Symmetries. The USD algorithm is capable o detecting sub-symmetries using two dierent techniques. The irst technique is to promote existing symmetry rules to a collection o rules or a larger number o inputs. Although this process is technically easible, it is cumbersome, and can be extremely slow or reduced library entries. The second procedure alters the unctions under test rather than the libraries. It is based on the ollowing Theorem 8. Theorem 8 Let R be a symmetry group o degree k, and let be a unction o n > k inputs. Let S be a subset o k inputs taken rom the n inputs o. I possesses R symmetry in the set o k variables, then every coactor obtained by ixing the n k variables to constant values must possess R symmetry. The Proo is obvious. There are 2 n k such coactors or each set o k inputs. When testing an n-input unction using a symmetry rule o degree k < n, the USD algorithm begins by generating all combinations o n inputs taken k at a time. For each combination, the USD algorithm generates all 2 n k coactors, and tests each one or R symmetry. Figure 15 gives the algorithm or detecting sub-symmetries. I n k is extremely large, this algorithm may be unacceptably slow, but or small n k, the perormance is reasonable. The algorithm then tests or a sub-symmetry in the k variables selected by the combination. Each such test requires computing 2 n k coactors. These coactors are computed by setting the variables not selected by the permutation to every possible combination o zeros and ones. The procedure continues until a sub-symmetry is ound, or until all combinations have been exhausted. The C-Bar return value is a combination o n things xx10x1 x1xx10 Fig. 14 The cubes o x 1 x 4 x 5 + x 2x 5 x 6 + x 3x 4 x 6. 1xx10x

18 Page 18 o 30 Read the Boolean Orbits o R For each combination C o n things taken n-k at a time For each n-k-element vector V // Obtain the coactor o E = Set the variables o speciied by C to the values o V I R is not compatible with E Return No Sub-Symmetry EndI EndFor EndFor Return C-Bar Fig. 15 The sub-symmetry detection algorithm. taken k at a time that speciies the variables containing the sub-symmetry. It is easily computed rom C. The algorithm o Fig. 15 can be easily modiied to ind multiple sub-symmetries o the same type, however to avoid detecting overlapping symmetries, it is better to run the algorithm o Fig. 15 on coactors o that do not include the variables speciied by C-Bar. Partial symmetries In some cases, it may be desirable to restrict the ocus to partial and total symmetries. These symmetries are well understood, and can be exploited by several existing algorithms. The universal symmetry detection algorithm can be used to detect these types o symmetries either by testing or symmetric variable pairs or by using a library o partial symmetry orbits. Although partial symmetries are easy to understand, enumerating all o them is a diicult task. For n inputs, the number o partial symmetries is equal to the number o partitions o a set o size n. A partition o a set S is a collection o non-empty sets P ={Q 1, Q 2,..., Q k }, such that Q 1 Q 2... Q k = S and Q i Q j = φ whenever i = j. The number o partitions o a set o size n is given by the Bell number B n, where B 1 = 1, and B n+1 = n ( ) n B i i. This ormula was used to compute the numbers o partial symmetries listed in Fig. 1. i=0 The number o conjugacy classes o partial symmetries is equal to the number o integer partitions o the integer n, p(n). Exact ormulas or p(n), are known, but they are so complex that, or small n, it is easier to generate all integer partitions o n and count them. The results or all integers less than or equal to 18 are given in Fig. 1. To generate a library o all partial symmetries or a speciic number o inputs, n, it is necessary to generate all partitions o the input variables. Because the Bell numbers provide only a count o the partitions, not the partitions themselves, it is necessary to begin with the integer partitions o n. Each integer partition generates a number o partitions. Each partition generates a set o Boolean orbits. The Boolean orbits are stored in a library in unction orm. When the number o inputs is seven or less, using libraries is aster. For larger numbers o inputs, the USD uses the conventional approach o generating and comparing coactors to detect symmetric variable pairs.

19 Page 19 o 30 Extended symmetric variable pairs Researchers have identiied various types o symmetric variable pairs that go beyond those discussed in Background. These extended types are deined in terms o the coactors o a unction, with respect to a pair o variables {x i, x j }. As beore, these are designated 00, 01, 10, and 11. The extended types o symmetry are deined by relations between these our coactors. Figure 16 lists the six possible relations, and the types o symmetry deined by each. It is possible to create Boolean orbits or each o these six relations. Assume that it is desired to detect these symmetries in a our-input unction, with respect to the variables x 1 and x 2. The Boolean orbits or these six relations are given in Fig. 17. As pointed out in (Maurer 2011), the single-variable relations are a special case o conjugate symmetry, and are best handled through matrix-based symmetry. Multi-phase symmetry is discussed more thoroughly in the ollowing section. Multi phase symmetry Multi-phase symmetry is the same as ordinary symmetry with one or more inputs inverted with respect to the others. In other words, the inputs o the unction are assumed to be a mixture o active-high and active-low inputs. Multi-phase symmetry is normally deined in terms o pairs o symmetric variables, but like ordinary symmetry, it can be deined entirely by Boolean orbits. Suppose it is desired to detect a multi-phase symmetry between the irst two variables o a our-input unction,. In terms o coactors, it is necessary to veriy that 00 = 11. This is just the Shannon relation, 10 = 01, with one o the inputs inverted (it doesn t matter which one). The Boolean orbits o this relation are given in Fig. 18, along with the Boolean orbits o an ordinary symmetry in the same two variables. (These are the same orbits given in Fig. 17.) Figure 17 shows that one can obtain the multi-phase Boolean orbits by inverting one o the inputs o the ordinary Boolean orbits. In act, it is possible invert any subset o the inputs, but not all such inversions will change the orbits. For example, inverting the last two bits o the orbits o Fig. 18 will leave them unchanged. In terms o vectors, the multi-phase orbits o Fig. 18 were obtained by XORing a constant vector, 0100, to each vector o each orbit as ollows. I S is an orbit and v is a vector, then S v ={w v w S}, where denotes the XOR operation. This principle can be applied to any set o orbits, not just those representing symmetric variable pairs. Consider the symmetry group {I, (1, 2)(3, 4)}. This type o symmetry cannot be deined in Fig. 16 The extended symmetry relations. Relation Type = Ordinary = Multi-Phase = Single Variable = Single Variable = Single Variable = Single Variable 10 11

20 Page 20 o 30 = = Fig. 17 Boolean orbits or extended relations. = = = = Multi-Phase Ordinary Fig. 18 The Boolean orbits o a multi-phase symmetry. terms o symmetric variable pairs. (The unction x 1 x 2 + x 3x 4 rom Background possesses such symmetry.) It is possible detect multi-phase symmetry in an arbitrary number o variables, by XORing an arbitrary vector with each o the orbits o the original symmetry. Assume that it is desired to detect symmetry o type {I, (1, 2)(3, 4)} in a unction whose irst and last variables are active high. To do this, one must add the vector 1001 to all orbits o the original symmetry. Figure 19 shows how this is done. Computation o multi-phase orbits can be perormed on the ly or loaded rom a library. I v w = , the vector o all 1 s, then adding v to a set o orbits will produce the same result as adding w to the same orbits. In practice, only those vectors whose irst element is zero are used. Conventional multi-phase symmetries (i.e. those related to partial and total symmetry) can also be detected by using coactor relations. Original Multi-Phase Fig. 19 Complex multi-phase symmetry.