On Partial Differential Encodings, with Application to Boolean Circuits
aa r X i v : . [ c s . CC ] O c t ON PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS
EDINAH K. GNANG
In a epoch-making book titled “An Investigation of the Laws of Thought”, George Boole [Boo54] laid the founda-tion for what has come to be known as the Boolean algebra. This algebra undoubtedly, serves as the first pillar ofthe computing revolution. Interestingly, George Boole also initiated the branch of mathematics knowns as invarianttheory [Wol08]. There is a recognition [GIM +
19, GMQ16, Aar16, Gro20] that a rich interplay relates these seeminglydistinct branches both pioneered by Boole. Invariant theory emphasizes consequences which stem from symmetriesor lack thereof. The importance of symmetry in the analysis of Boolean function was well known to pioneers of thefield Shannon, Pólya and Redfield [Sha49, Pol40, Pol37, Red27]. The present work investigates a partial derivativeincarnation of Turing machines. Turing machines were introduced by Alan Turing [Tur36] as a mathematical modelof computation. Turing machines serve as the second pillar to the computing revolution. The use of Differentialdifferential operators in invariants theory is very old. Their origin can be traced back to the work of early pioneersof invariant theory. Most notably to the work of Arthur Cayley and James Joseph Sylvester [Cay89, Syl52] whoinstigated the analysis of differential operator used to construct invariants of group actions. This particular frame-work is known as Cayley’s Ω process. In complexity theory, differential operators were investigated in the contextof arithmetic complexity by Baur and Strassen [BS83] who analyzed the arithmetic complexity of computing partialderivative. More recently, Cornelius Brand and Kevin Pratt [BP20] were able to match the runtime of the fastestknown deterministic algorithm for detecting subgraphs of bounded pathwidth using a method of partial derivatives.We also refer the reader to the excellent survey of recent uses of partial derivatives in arithmetic complexity theory[CKW11, SY10]. The importance of investigating partial differential operators is also reinforced by the central rolethey play in Deep Learning. The present work ties together, aspects of arithmetic circuit complexity to Booleancircuit complexity. Fortunately, recent depth reduction results [VS81, AV08, Raz13, GKKS16, Hya79] reduce in-vestigations of arithmetic circuits lower bounds to investigations of depth 3 circuits lower bounds. These resultsjustify our emphasis on low depth arithmetic circuits. The present work argues that strong arithmetic circuit lowerbounds yield Boolean circuit lower bounds. In particular we show that Boolean formula complexity upper-boundsan algebraic variant of the Kolomogorov complexity of partial differential incarnations of Turing machines.1. Partial Differential Encodings.
Recall the “Needles in a Haystack” conundrum. The conundrum translates mathematically [Sha49], into theobservation that there are roughly speaking less than s ) Boolean circuits (expressed in the De Morgan basis)of size s among the (2 n ) possible Boolean functions on n -bit binary input strings. Consequently most circuits arelower bounded in size albeit crudely by n . Unfortunately, this particular incarnation of the conundrum suffersfrom a crippling deficiency. Namely, this pigeon hole argument fails to track any specific family of Boolean functionsacross varying input lengths. We circumvent this drawback by considering a different incarnation of the “Needlesin a Haystack” conundrum. This alternative incarnation is based upon the well known algebraic correspondence
1N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 2 introduced by George Boole [Boo54].(1.1)
True → , False → , ¬ x [ i ] → − x [ i ] , x [ i ] ∨ x [ j ] → x [ i ] + x [ j ] − x [ i ] · x [ j ] , x [ i ] ∧ x [ j ] → x [ i ] · x [ j ] . (1.2) → True , → False , x [ i ] · x [ j ] → x [ i ] ∧ x [ j ] , − x [ i ] → ¬ x [ i ] , ( x [ i ] + x [ j ]) mod → ( x [ i ] ∨ x [ j ]) ∧ ¬ ( x [ i ] ∧ x [ j ]) . Polynomials in Boolean variables { x [ i ] : i ∈ Z n } are taken modulo algebraic relations(1.3) (cid:26) ( x [ i ]) ≡ x [ i ] ∀ i ∈ Z n (cid:27) . Proposition 1.
The congruence class of polynomial interpolant for a Boolean function F ∈ { , } ( { , } n × ) suchas (1.4) F ( x ) = X b ∈ { , } n × F ( b ) = 1 Y d ∈{ , } n × \{ b } x [0] + P
Proof.
Consider the polynomial interpolant for a Boolean function F given by(1.5) F ( x ) = X b ∈ { , } n × F ( b ) = 1 Y d ∈{ , } n × \{ b } x [0] + P
0] + X ≤ w 0] + X ≤ w Polynomials whose expanded form are expressed as (1.26) X R ⊆ S ⊆ Z nk τ R Y j ∈ R x [ j ] and X Z nk ⊇ R ⊇ S τ R Y j ∈ R x [ j ] , s.t. ( τ R ) m = 1 ∀ R ⊆ Z n k , are optimally expressed as sums of products of linear forms given by (1.27) µ P ⊆ S ( diag ( u ) · x ) and µ P ⊇ S ( diag ( u ) · x ) , where u ◦ m = n k × , µ m = 1 .Proof. The prime factors in the factorization of the integer count for the number of terms in the expanded form of P ⊆ S ( x ) and P ⊇ S ( x ) yields a lower bound for the number of terms which make up each linear form in a productwhose expanded form has | S | and n k −| S | monomial terms respectively. (cid:3) Consequently, the smallest hypermatrix which underlies P ⊆ S ( x ) must be of size × | S | × (cid:0) n k (cid:1) while thesmallest hypermatrix which underlies P ⊇ S ( x ) must be of size × n k × (cid:0) n k (cid:1) . Furthermore, there are only m | S | +1 out of a total of m | S | optimal PDEs for F ⊆ S and similarly there are only m n k −| S | +1 out of a total of m nk −| S | optimal PDEs for F ⊇ S . There are (cid:18) | S | (cid:19) · m | S | +1 − − m | S | +1 distinct second best PDEs for F ⊆ S specified via multilinear polynomials of the form µ (cid:0) γ ∅ + γ { u } x [ u ] + γ { v } x [ v ] + γ { u,v } x [ u ] x [ v ] (cid:1) Y i ∈ S \{ u,v } (1 + ( diag ( u ) · x ) [ i ]) = X R ⊆ S ⊆ Z nk τ R Y j ∈ R x [ j ] . The corresponding underlying hypermatrix is of size × | S | × (cid:0) n k (cid:1) . Quite similarly, there are (cid:18) n k − | S | (cid:19) · m n k −| S | +1 − − m n k −| S | +1 distinct second best PDEs for F ⊇ S specified via multilinear polynomials of the form µ (cid:0) γ ∅ + γ { u } x [ u ] + γ { v } x [ v ] + γ { u,v } x [ u ] x [ v ] (cid:1) Y i ∈ S x [ i ] ! Y i ∈ S \{ u,v } (1 + ( diag ( u ) · x ) [ i ]) = X Z nk ⊇ R ⊇ S τ R Y j ∈ R x [ j ] . The corresponding underlying hypermatrix is of size × n k × (cid:0) n k (cid:1) . Both counts are obtained by accountingfor all choices of the roots of unity coefficients appearing in the expansion of the product of two of the binomialfactors. Since these expressions necessarily include the optimal ones, we must subtract from the total the number ofoptimal ones. The argument sketched above can be used to enumerate the number of PDEs for F ⊆ S and F ⊇ S whosemultilinear polynomial is expressed via hypermatrices of sizes at least t × | S | × (cid:0) n k (cid:1) and t × n k × (cid:0) n k (cid:1) respectively. For instance we see that there are m | S | − | S | · m ( ( | S |− ) , distinct expressions which expand into a polynomial of the form X R ⊆ S ⊆ Z nk τ R Y j ∈ R x [ j ] , s.t. ( τ R ) m = 1 ∀ R ⊆ Z n k , N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 7 whose underlying hypermatrices are of size at least | S | × | S | × (cid:0) n k (cid:1) . Similarly there are m nk −| S | − (cid:0) n k − | S | (cid:1) · m (cid:18) ( nk −| S |− ) (cid:19) , distinct expressions whose expanded form are given by X Z nk ⊇ R ⊇ S τ R Y j ∈ R x [ j ] , s.t. ( τ R ) m = 1 ∀ R ⊆ Z n k , with underlying hypermatrices of size at least n k −| S | × n k × (cid:0) n k (cid:1) . The latter count therefore determines thecount for the worst possible PDEs for F ⊆ S and F ⊇ S and we see that asymptotically these are the most numerousones.It serves our purpose to think of the n k entries of the vector x to be associated with distinct directed hyperedgesof a complete directed n -vertex hypergraph of order k . Optimal PDEs such as the one devised for F ⊆ S and F ⊇ S , epitomize the membership of Boolean functions into the class P is the class of Boolean functions which admitsefficient PDEs ( i.e. PDEs whose size is upper bounded by some polynomial in n k ). The class P is taken to beclosed under negation, and closed under any polynomial ( in n k ) number of conjunctions, disjunctions.2. PDEs of cardinality variants of F ⊆ S and F ⊇ S . We now turn our attention to a slightly more intricate family of Boolean functions associated with cardinalityvariants of F ⊆ S and F ⊇ S defined by(2.1) F ≤| S | ( T ) = if | T | ≤ | S | otherwise and F ≥| S | ( T ) = if | T | ≥ | S | otherwise . With the goal in mind to express optimal PDEs for F ≤| S | and F ≥| S | , consider the n k × orbital vector O Z whoseentries depict the action of the symmetric group S n k on the hyperedge set. The entries of O Z are monomials inentries of a symbolic matrix Z of size n k × (cid:0) n k ! (cid:1) .(2.2) O Z [ i ] = Y σ ∈ S nk Z [ σ ( i ) , lex ( σ )] , ∀ i ∈ Z n k . Let the canonical representative of the congruence class(2.3) P ⊆ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Y j ∈ R Y [ j, lex ( R )] : | R | ≤ | S | σ ∈ S n k , denote the unique member of the congruence class which is a polynomial depending only on entries of Y , notdepending on any entry of Z . Proposition 3. The canonical representative of the congruence class (2.4) P ⊆ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Y j ∈ R Y [ j, lex ( R )] : | R | ≤ | S | σ ∈ S n k , N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 8 expresses the orbit list generating polynomial (2.5) X ≤ t ≤| S | (cid:18) | S | t (cid:19) Y R ⊆ Z n k | R | = t Y i ∈ R Y [ i, lex ( R )] ! ( n k −| R | ) ! ·| R | ! . Proof. The canonical representative of(2.6) P ⊆ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Y j ∈ R Y [ j, lex ( R )] : R ⊆ Z n k σ ∈ S n k is the polynomial obtained by replacing into the expanded form of P ⊆ S ( O Z ) every occurrence of monomials in theentries of Z given by(2.7) Y i ∈ R Z [ i, lex ( σ )] ∀ R ⊆ Z n k σ ∈ S n k , with a monomial in the entries of Y given by(2.8) Y j ∈ R Y [ j, lex ( R )] . It therefore follows that the canonical representative is(2.9) X R ⊆ Z n k | R | ≤ | S | Y T ⊆ Z n k | T | = | R | Y i ∈ T Y [ i, lex ( T )] ! ( n k −| R | ) ! ·| R | ! , re-expressed as(2.10) X ≤ t ≤| S | (cid:18) | S | t (cid:19) Y R ⊆ Z n k | R | = t Y i ∈ R Y [ i, lex ( R )] ! ( n k −| R | ) ! ·| R | ! . (cid:3) An identical argument to the one used to prove Prop. (3) establishes that the canonical representative of thecongruence class(2.11) P ⊇ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Y j ∈ R Y [ j, lex ( R )] : | R | ≥ | S | σ ∈ S n k , yields the orbit list generating polynomial(2.12) X R ⊆ Z n k | R | ≥ | S | Y T ⊆ Z n k | T | = | R | Y i ∈ R Y [ i, lex ( T )] ! ( n k −| R | ) ! ·| R | ! , N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 9 (2.13) = X | S |≤ t ≤ n k (cid:18) n k − | S | t (cid:19) Y R ⊆ Z n k | R | = t Y i ∈ R Y [ i, lex ( R )] ! ( n k −| R | ) ! ·| R | ! . We modify slightly the orbit list generating function construction above to devise PDEs for F ≤| S | and F ≥| S | . Letcanonical representative of congruences classes(2.14) P ≤| S | ( x ) = P ⊆ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Q i ∈ R x [ i ]( n k −| R | )! ·| R | ! · ( | S || R | )0 1 : R ⊆ Z n k , | R | ≤ | S | σ ∈ S n k [0 , , (2.15) P ≥| S | ( x ) = P ⊇ S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Q i ∈ R x [ i ]( n k −| R | )! ·| R | ! · ( nk −| S || R | )0 1 : R ⊆ Z n k , | R | ≥ | S | σ ∈ S n k [0 , , and(2.16) P = | S | ( x ) = P = S ( O Z ) mod Y i ∈ R Z [ i, lex ( σ )] − Q i ∈ R x [ i ]( n k −| R | )! ·| R | ! : R ⊆ Z n k , | R | = | S | σ ∈ S n k [0 , , be polynomials in the respective congruence classes depending only on entries of the vector x , not depending onany entries of Z . Proposition 4. The multivariate multilinear polynomials P ≤| S | ( x ) and P ≥| S | ( x ) are used to specify PDEs (2.17) F ≤| S | ( T ) = ∂ | T | µP ≤| S | ( diag ( u ) · x ) Q i ∈ T ∂ x [ i ] x = nk × m and F ≥| S | ( T ) = ∂ | T | µP ≥| S | ( diag ( u ) · x ) Q i ∈ T ∂ x [ i ] x = nk × m , where u ◦ m = n k × and µ m = 1 .Proof. Similarly to the proof of Prop. (3), canonical representatives for the first of these two congruence classes isobtained by replacing into the expanded form of P ⊆ S ( O Z ) every occurrence of monomials in the entries of Z givenby(2.18) Y i ∈ R Z [ i, lex ( σ )] ∀ R ⊆ Z n k σ ∈ S n k by the × matrix(2.19) Q i ∈ R x [ i ]( n k −| R | )! ·| R | ! · ( | S || R | )0 1 canonical representatives for the second of the two congruence classes is obtained by replacing into the expandedform of P ⊇ S ( O Z ) every occurrence of monomials in the entries of Z given by(2.20) Y i ∈ R Z [ i, lex ( σ )] ∀ R ⊆ Z n k σ ∈ S n k N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 10 by the × matrix(2.21) Q i ∈ R x [ i ]( n k −| R | )! ·| R | ! · ( nk −| S || R | )0 1 The desired multilinear polynomial appears as the [0 , entry of the × matrix resulting from the substitutions.The matrix is so constructed to result in a multilinear polynomial whose non-vanishing coefficients are all equal toone and whose terms list all subsets of Z n k with the desired properties. This follows from the observation that ifthere were some other PDEs whose expanded multilinear polynomials had fewer terms in its expanded form thanthe number of terms in P ≤| S | ( x ) and P ≤| S | ( x ) respectively, then this would contradict the optimality assertion for P ⊆ S ( x ) and P ⊇ S ( x ) . (cid:3) The construction above accounts only for m | S | +1 of the m P ≤ t ≤| S | ( nkt ) possible PDEs for F ≤| S | . Similarly, theconstruction above only accounts for m n k −| S | +1 of the m P | S |≤ t ≤ nk ( nkt ) possible PDEs for F ≥| S | . Incidentally, itfollows that(2.22) F = | S | ( T ) = (cid:0) F ≤| S | ( T ) ∧ F ≥| S | ( T ) (cid:1) = if | T | = | S | otherwise , (2.23) = ⇒ F = | S | ( T ) = ∂ | T | µP = | S | ( diag ( u ) · x ) Q i ∈ T ∂ x [ i ] x = nk × m , where(2.24) P = | S | ( x ) = X R ⊆ Z n k | R | = | S | Y j ∈ R x [ j ] . Consider the equality(2.25) P ≤| S | ( x ) = X ≤ u<ρ Y ≤ v 0] + X ≤ w We introduce here a variant of PDEs called Partial Differential Programs ( or PDPs for short). By contrast toPDEs, in a PDP, the multilinear polynomial used to specify it is only implicitly prescribed. In fact multivariatepolynomial interpolant such as the ones expressed in Eq. (1.6) illustrate such implicit descriptions. More concretely,we seek to reduce the size of expressions describing multilinear polynomials such as P ≤| S | ( x ) and P ≥| S | ( x ) used tospecify PDEs for F ≤| S | and F ≥| S | . We do this by prescribing such multilinear polynomials only up to congruencemodulo(3.1) (cid:26) ( x [ i ]) − x [ i ] ∀ i ∈ Z n k (cid:27) Exploiting the reduction in sizes of the description of the polynomials incurred by reducing modulo these relationsentails a broadening of the proposed model of computation. This broadening hinges upon the fact we implicitlydescribe multilinear polynomials used to specify PDEs. A multilinear polynomial used to specify a PDE is implicitlydescribed by supplying a member of its congruence class. We refer to such implicit encodings as a Partial DifferentialPrograms. Consequently, there is an essential distinction separating PDEs from PDPs. For a simple illustrationconsider a PDE associated with the Boolean function whose truth table is x [1] x [0] F ( x )0 0 10 1 11 0 01 1 1 N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 12 (3.2) = ⇒ F ( T ) = (cid:18) ∂∂ x [0] (cid:19) T [0] (cid:18) ∂∂ x [1] (cid:19) T [1] P ( x ) !% x = × . The corresponding multilinear polynomial P ( x ) used to specify the PDE is(3.3) P ( x ) = 1 + x [0] + x [0] · x [1] . Trivially, the hypermatrix which underlies P ( x ) can be taken to be of size × × . However, the optimalhypermatrix which underlies P ( x ) can be taken to be of size × × as seen from the equality(3.4) P ( x ) = ( B [0 , , 0] + B [0 , , · x [0] + B [0 , , · x [1]) ( B [0 , , 0] + B [0 , , · x [0] + B [0 , , · x [1]) +( B [1 , , 0] + B [1 , , · x [0] + B [1 , , · x [1]) ( B [1 , , 0] + B [1 , , · x [0] + B [1 , , · x [1]) where non-zero entries of B are(3.5) B [0 , , 0] = B [0 , , 0] = B [1 , , 1] = B [1 , , 0] = B [1 , , 2] = 1 . This observation stems from the equality(3.6) P ( x ) = 1 + x [0] + x [0] · x [1] = 1 + x [0] · (1 + x [1]) . The optimal PDE for F is thus given by(3.7) F ( T ) = (cid:18) ∂∂ x [0] (cid:19) T [0] (cid:18) ∂∂ x [1] (cid:19) T [1] (cid:16) x [0] · (1 + x [1]) ⌋ x = × (cid:17) , On the other hand, we may instead prescribe P ( x ) implicitly up to congruence modulo relations(3.8) ( x [0]) − x [0] and ( x [1]) − x [1] . We then specify a PDP for the Boolean function F using a polynomial Q ( x ) not necessarily multilinear. A PDPfor F is thus expressed as(3.9) F ( T ) = (cid:18) ∂∂ x [0] (cid:19) T [0] (cid:18) ∂∂ x [1] (cid:19) T [1] Q ( x ) mod (cid:26) ( x [0]) − x [0]( x [1]) − x [1] (cid:27)(cid:23) x = × ! . In this setting, we can find an optimal hypermatrix B ′ ∈ C × × which underlies the polynomial such that(3.10) Q ( x ) = ( B ′ [0 , , 0] + B ′ [0 , , · x [0] + B ′ [0 , , · x [1]) ( B ′ [0 , , 0] + B ′ [0 , , · x [0] + B ′ [0 , , · x [1]) One particular solution is completely determined by the partial assignment(3.11) B ′ [0 , , 1] = − √ √ qp √ √ (cid:0) √ √ (cid:1) + 8 √ √ √ √ p √ √ (cid:16) √ √ p √ √ (cid:17) (3.12) B ′ [0 , , 1] = − , B ′ [0 , , 1] = − q √ 15 + 17 − , B ′ [0 , , 2] = 1 . Unfortunately the exact expression of B ′ [0 , , and B ′ [0 , , although easily obtainable are too prohibitivelylarge to be displayed here. The example illustrates a concrete instance where the optimal PDP is “smaller” thanthe optimal PDE.By definition PDEs form a proper subset of PDPs, since polynomials used to specify PDPs are not necessarilymultilinear. Note that there are finitely many PDEs for any given Boolean functions on n -bit input strings ( when N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 13 the exponent parameter m is fixed). By contrast there are infinitely many distinct PDPs for a given Booleanfunctions on n -bit input strings ( even when the exponent parameter m is fixed ).Note that Boolean functions F ≤| S | , F ≥| S | and F = | S | are all symmetric with respect to permutations of the inputvariables, and so too are the multilinear polynomials P ≤| S | ( x ) , P ≥| S | ( x ) and P = | S | ( x ) used to specify their PDEs.Recall the well known Newton identity. These identities relate the densest (in their monomial support) set ofgenerators of the ring of symmetric polynomials given by(3.13) e t ( x ) = X R ⊆ Z n k | R | = t Y j ∈ R x [ j ] , ∀ t ∈ Z n k +1 to the sparsest set of generators ( for the same ring ) given by(3.14) p t ( x ) = X ≤ i For all t ∈ Z n k +1 \ { } , we have (3.15) e t = ( − t X m m ··· + tmt = tm ≥ ,...,mt ≥ Y ≤ i ≤ t ( − p i ) m i m i ! i m i . Proof. Consider the polynomial(3.16) n k − X σ ∈ S nk Y γ ∈ S nk ( x [ σ (0)] − y [ γ (1)]) ( nk − ) ! = X i ∈ Z nk Y j ∈ Z nk ( x [ i ] − y [ j ]) (3.17) = ⇒ X ≤ t ≤ n k p t ( x ) · e n k − t ( x ) , solving the resulting triangular systems of linear equations in e t ( x ) via back-substitution yields the desired Newton’sidentity(3.18) e t = ( − t X m m ··· + tmt = tm ≥ ,...,mt ≥ Y ≤ i ≤ t ( − p i ) m i m i ! i m i . (cid:3) The abundance of non-vanishing terms in multilinear polynomials used to specify PDEs for F ≤| S | , F ≥| S | con-stitutes the main obstacle to certifying membership of F ≤| S | , F ≥| S | into the class P . We use Newton’s identityto eliminate as many cross terms as possible from the multilinear polynomial used specify PDEs of of F ≤| S | and F ≥| S | . Crucially, our proposed elimination procedure must done in such a fashion as to leave the congruence classunchanged and thereby converts PDEs into PDPs. The fact that this is at all possible is a direct consequence ofthe Boolean congruence identity(3.19) p t ≡ p mod (cid:26) ( x [ i ]) − x [ i ] ∀ i ∈ Z n k (cid:27) , ∀ t ∈ Z n k \ { } . Consequently(3.20) e t ≡ ( − t X m m ··· + tmt = tm ≥ ,...,mt ≥ Y ≤ i ≤ t ( − p ) m i m i ! i m i mod (cid:26) ( x [ i ]) − x [ i ] ∀ i ∈ Z n k (cid:27) . N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 14 It therefore follows that within the congruence classes(3.21) (cid:18) P ≤| S | ( x ) mod (cid:26) ( x [ i ]) − x [ i ] ∀ i ∈ Z n k (cid:27)(cid:19) and (cid:18) P ≥| S | ( x ) mod (cid:26) ( x [ i ]) − x [ i ] ∀ i ∈ Z n k (cid:27)(cid:19) lies respectively univariate polynomials in P ≤ i Experience suggests that it is very probably not the case. Certainly one would wish for a proof of this claim; Wehave meanwhile temporarily put aside the search for such a proof after some fleeting futile attempts, as it appearsunnecessary for the next objective of our investigation. The subtle complexity separation between PDPs and PDEsappears to capture the separation between the permanent and the determinant as suggested by the following result.Consider(3.27) H n ( A ) = Y i ∈ Z n (cid:16) ( x [ i ]) n +1 − x [ i ] (cid:17) · ( x [ i ] − − mod n ( x [ i ]) j +1 − A [ i, j ] θ i : i, j ∈ Z n o (3.28) mod θ m − O ≤ k The family of formulas H n in entries of the n × n matrix A evaluates to a family of polynomials inthe entries of A which include as their two extreme members det ( A ) and Per ( A ) depending only on how expandedwe express the canonical representative ( i.e. the member of the congruence class which does not depend on entriesof x ) (3.29) Y ≤ i The factored form of the canonical representative of the congruence class(3.31) Y ≤ i 0] : 0 ≤ i < n < j < n (cid:27) , Once again it is crucial that the division be performed by decreasing order of the exponent j where the hypermatrixwhich underlies the polynomial Y ≤ i There exist a non-trivial circuit in the proposed model whose complexity reduces to the complexity ofencoding the integer d such that the circuit expresses a function whose real roots coincides with the roots of Y Starting with(3.39) Y S, T ⊆ Z n k we define the isomorphism equivalence relation between S and T as(4.5) T ≃ R ⇐⇒ ∃ σ ∈ S nk / Aut T s.t. R = σT. N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 19 Similarly we define the sub-isomorphism relation relation between S and T as(4.6) T ⊂ ∼ S ⇐⇒ ∃ R ⊆ S s.t. T ≃ R. Isomorphism variant of F ⊆ S and F ⊇ S are Boolean functions prescribed by(4.7) F ⊂ ∼ S ( T ) = ( if T ⊂ ∼ S otherwise and F ⊃ ∼ S ( T ) = ( if T ⊃ ∼ S otherwise . Recall that the entries of the orbital n k × vector O Z are monomials in entries of a symbolic matrix Z of size n k × (cid:0) n k ! (cid:1) such that(4.8) O Z [ i ] = Y σ ∈ S nk Z [ σ ( i ) , lex ( σ )] , ∀ i ∈ Z n k . (4.9) = ⇒ O Z [ i ] = Y Λ ∈ G n Y σ ∈ S nk / G n Z [ σ Λ ( i ) , lex ( σ Λ)] , ∀ i ∈ Z n k . Consider the sub-orbital n k × vector P Z whose entries are divisors of the entries of O Z as follows(4.10) P Z [ i ] = O Z [ i ] Q Λ ∈ G n Q σ ∈ ( S nk / G n ) \{ id } Z [ σ Λ ( i ) , lex ( σ Λ)] ! , ∀ i ∈ Z n k . (4.11) = ⇒ P Z [ i ] = Y Λ ∈ G n Z [Λ ( i ) , lex (Λ)] , ∀ i ∈ Z n k . For simplicity and with minimal lost of generality in our discussion we set k to and typically disregard loop edges.Let the canonical representative of the congruence class(4.12) P ⊆ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Y j ∈ R Y [ j, lex ( R )] : R ⊂ ∼ S Λ ∈ G n , denote the unique member of the congruence class which is a polynomials in the entries of Y and most importantlydoes not depend upon any entry of Z . Proposition 8. The canonical representative of the congruence class (4.13) P ⊆ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Y j ∈ R Y [ j, lex ( R )] : R ⊂ ∼ S Λ ∈ G n expresses the orbit list generating polynomial (4.14) X R ⊂ ∼ S Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 20 (4.15) = X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S γ S,R Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , where ℘ ( Z n ) / Iso denotes the set of distinct conjugacy classes of n -vertex directed graphs and γ S,R denotes the numberof distinct subgraphs of S isomorphic to R .Proof. The canonical representative of(4.16) P ⊆ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Y j ∈ R Y [ i, lex ( R )] : R ⊂ ∼ S Λ ∈ G n is the polynomial obtained by replacing in the expanded form of P ⊆ S ( P Z ) every occurrence of monomials in theentries of Z given by(4.17) Y i ∈ R Z [ i, lex (Λ)] , ∀ R ⊂ ∼ S Λ ∈ G n by a monomial in the entries of Y given by(4.18) Y i ∈ R Y [ i, lex ( R )] It therefore follows that the canonical representative is(4.19) X R ⊂ ∼ S Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , (4.20) = X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S γ S,R Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , as claimed. (cid:3) Similarly, the canonical representative of the congruence class(4.21) P ⊇ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Y j ∈ R Y [ i, lex ( R )] : R ⊃ ∼ S Λ ∈ G n yields the orbit generating polynomial for supergraphs of S (4.22) X R ⊃ ∼ S Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 21 (4.23) = X R ∈ ℘ ( Z n ) / Iso R ⊃ ∼ S Γ S,R Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | , where ℘ ( Z n ) / Iso denotes the set of distinct conjugacy classes of n -vertex graphs and Γ S,R denotes the number ofdistinct supergraphs of S isomorphic to R . The canonical representatives above list respectively unlabeled subgraphand supergraphs of S . Let canonical representative of congruences classes(4.24) P ⊂ ∼ S ( x ) ≡ P ⊆ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Q i ∈ R x [ i ] | G n / Aut R || Aut R | γ S,R : R ⊆ Z n Λ ∈ G n [0 , , (4.25) P ⊃ ∼ S ( x ) ≡ P ⊆ S ( P Z ) mod Y i ∈ R Z [ i, lex (Λ)] − Q i ∈ R x [ i ] | G n / Aut R || Aut R | Γ S,R : R ⊆ Z n Λ ∈ G n [0 , . be respective polynomials in the entries of x and do not depend upon any entry of Z . Proposition 9. Multivariate multilinear polynomials P ⊂ ∼ S ( x ) and P ⊃ ∼ S ( x ) are multilinear polynomials used toexpress PDEs (4.26) F ⊂ ∼ S ( T ) = ∂ | T | µ P ⊂ ∼ S ( diag ( u ) · x ) Q i ∈ T ∂ x [ i ] x = ( n ) × m , F ⊃ ∼ S ( T ) = ∂ | T | µ P ⊃ ∼ S ( diag ( u ) · x ) Q i ∈ T ∂ x [ i ] x = ( n ) × m , where u ◦ m = n k × , µ m = 1 .Proof. Similarly to the proof of the Prop. (6), canonical representatives are obtained by replacing in P ⊆ S ( P Z ) every occurrence of monomials in the entries of Z given by(4.27) Y i ∈ R Z [ i, lex (Λ)] , ∀ R ⊂ ∼ S Λ ∈ G n by the following × matrix(4.28) Q i ∈ R x [ i ] | S n / Aut R || Aut R | γ S,R for the first of the two congruence classes and we replace every occurrence of monomials in the entries of Z givenby(4.29) Y i ∈ R Z [ i, lex (Λ)] ∀ R ⊃ ∼ S Λ ∈ G n by the × matrix(4.30) Q i ∈ R x [ i ] | S n / Aut R || Aut R | Γ S,R N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 22 for the second congruence class. The desired polynomial appears as the [0 , entry of the × matrix resulting fromthe matrix substitutions. The matrix is so constructed to result in a multilinear polynomial whose non-vanishingcoefficients are all equal to one and whose terms list all subsets of Z n with the desired properties. This followsfrom the observation that if there were some other PDEs whose expanded multilinear polynomials had fewer termsthan the number of terms in P ⊂ ∼ S ( x ) and P ⊃ ∼ S ( x ) respectively then this would contradict the optimality assertionfor P ⊆ S ( x ) and P ⊇ S ( x ) . (cid:3) Incidentally(4.31) F ≃ S ( T ) = (cid:16) F ⊂ ∼ S ( T ) ∧ F ⊃ S ( T ) (cid:17) = if T ≃ S otherwise , (4.32) = ⇒ F ≃ S ( T ) = Y i ∈ Z n (cid:18) ∂∂ x [ i ] (cid:19) T [ i ] µ P ≃ S ( diag ( u ) · x ) x = ( n ) × m . where(4.33) P ≃ S ( x ) = X R ≃ S Y i ∈ R x [ i ] ! = X σ ∈ G n / Aut S Y i ∈ S x [ σ ( i )] , whose underlying hypermatrix is of size | G n / Aut S | × | S | × n k The PDE for F ≃ S accounts for at most m | S | +1 of thetotal m | G n / Aut R | . Note that alternative PDEs for F ⊂ ∼ S , F ⊃ ∼ S can be devised directly from list generating functionsas was done for F ≤| S | , F ≥| S | .The following mapping ensues when comparing the cardinality variant to isomorphism variants(4.34) (cid:0) n − | R | (cid:1) ! · | R | ! → | Aut R | , (cid:0) | S || R | (cid:1) → | G n / Aut R | · γ S,R , (cid:0) n −| S || R | (cid:1) → | G n / Aut R | · Γ S,R . In stark contrast to polynomials used to specify PDPs of F ≤| S | and F ≥| S | , whose polynomials are expressed by aunivariate polynomial in P σ ∈ S n x [ σ (0)]( n − , we now show that any univariate polynomial in P σ ∈ S n x [ σ (0)]( n − which lies in the same congruence class of either P ⊂ ∼ S ( x ) , P ⊃ ∼ S ( x ) or P ≃ S must have exponentially large degree. N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 23 Theorem 10. The congruence class of polynomials used to specify PDEs for F ≃ S , F ⊂ ∼ S and F ⊃ ∼ S include symbolicroots to univariate polynomial in a symbolic variables ζ of degree (cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12) whose coefficients each express a productof linear forms in which each linear factor is symmetric with respect to permutations of the edges variables.Proof. We start by discussing P ≃ S ( x ) . For simplicity assume that S ⊂ Z ( n ) , is associated with a rigid graph ( i.e. Aut ( S ) is trivial ). For all σ ∈ S ( n ) / G n , let(4.35) z [ lex ( σ )] = P Λ ∈ G n Q i ∈ S x [ σ Λ ( i )] Aut ( S ) . Consider the densest set of generators for the ring of symmetric polynomials in the entries of the vector z (4.36) e ( z ) = X σ ∈ S ( n ) / G n P Λ ∈ G n Q i ∈ S x [ σ Λ ( i )] Aut ( S ) . By symmetry,(4.37) P = | S | ( x ) = e ( z ) ≡ γ Y ≤ j< | S | γ ,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) . The remaining generators are associated with the subscript parameter < t ≤ (cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12) such that(4.38) e t ( z ) ≡ γ t Y ≤ j< | S | t γ t,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) , and in particular the last generator is given by(4.39) e (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( z ) = Y σ ∈ S ( n ) / G n P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] Aut ( S ) (4.40) ≡ γ (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) Y ≤ j< | S | t γ (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) . It thus follows that(4.41) P ≃ S ( x ) ∈ Roots in ζ of X ≤ t ≤ (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( − t e t ( z ) · ζ t N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 24 A similar derivation expresses P ⊂ ∼ S ( x ) and P ⊃ ∼ S ( x ) for arbitrary S as follows. For all σ ∈ S ( n ) / G n , let(4.42) z [ lex ( σ )] = X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . Consequently(4.43) e ( z ) = X σ ∈ S ( n ) / G n X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . Hence(4.44) P ≤| S | ( x ) = e ( z ) ≡ α Y ≤ j< | S | α ,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) . The remaining generators for the corresponding ring of symmetric polynomials are associated with the index pa-rameter < t ≤ (cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12) we have(4.45) e t ( z ) ≡ α t Y ≤ j< | S | t α t,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) , and in particular(4.46) e (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( z ) = Y σ ∈ S ( n ) / G n X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . It thus follows that(4.47) P ⊂ ∼ S ( x ) ∈ Roots in ζ of X ≤ t ≤ (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( − t · e t ( z ( x )) · ζ t . N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 25 Similarly, for all σ ∈ S ( n ) / G n , let(4.48) z [ lex ( σ )] = X R ∈ ℘ ( Z n ) / Iso R ⊃ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . Consequently(4.49) e ( z ) = X σ ∈ S ( n ) / G n X R ∈ ℘ ( Z n ) / Iso R ⊃ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . Hence(4.50) P ≥| S | ( x ) = e ( z ) ≡ β Y ≤ j< | S | β ,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) . The remaining generators for the corresponding ring of symmetric polynomials are associated with the index pa-rameter < t ≤ (cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12) we have(4.51) e t ( z ) ≡ β t Y ≤ j< | S | t ( n −| S | ) t β t,j + P σ ∈ S ( n ) x [ σ (0)] (cid:0) (cid:0) n (cid:1) − (cid:1) ! mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) , and in particular(4.52) e (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( z ) = Y σ ∈ S ( n ) / G n X R ∈ ℘ ( Z n ) / Iso R ⊃ ∼ S P Λ ∈ G n Q i ∈ R x [ σ Λ ( i )] | R | ! (cid:0) (cid:0) n (cid:1) − | R | (cid:1) ! | Aut R | . It thus follows that(4.53) P ⊃ ∼ S ( x ) ∈ Roots in ζ of X ≤ t ≤ (cid:12)(cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12)(cid:12) ( − t · e t ( z ) · ζ t . (cid:3) It follows as corollary of Thrm. 18 and well known properties of symbolic roots of univariate algebraic equations[Stu00, May36, Bir27] that polynomials used to specify PDPs for F ≃ S , F ⊂ ∼ S and F ⊃ ∼ S include in their congruenceclass a hypergeometric functions of products of linear form which are symmetric with respect to permutations of theedge variables. In particular it follows that neither polynomials P ≃ S ( x ) , P ⊂ ∼ S ( x ) and P ⊃ ∼ S ( x ) can not be expressed N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 26 as a sum over fewer then (cid:12)(cid:12)(cid:12) S ( n ) / G n (cid:12)(cid:12)(cid:12) product of linear forms symmetric with respect to permutation of the edgevariables.Thus far we have obtained canonical PDPs constructions for Boolean functions and for F ≃ S , F ⊂ ∼ S and F ⊃ ∼ S via Lagrange interpolation construction and elimination by symmetrization in Thrm. (18). All of which typicallyresult in exponentially large PDPs. Note that the lower bound for optimal PDP/PDE is always determined by theprime factorization of the integer count of number of non vanishing terms occurring in the expanded form of thepolynomial used to specify the PDE. For illustration purposes suppose that for either Boolean functions F ≃ S , F ⊂ ∼ S and F ⊃ ∼ S we knew the prime factorization of the count of the number of terms to be Y ≤ i Let S ⊆ Z ( n ) such that (4.54) F ⊂ ∼ S ( T ) = ∂ | T | Q i ∈ T ∂ x [ i ] µ Q ⊂ ∼ S ( diag ( u ) · x ) mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) x = ( n ) × m (4.55) Q ⊂ ∼ S ( x ) = X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · x [ w ] , where u ◦ m = n k × , µ m = 1 . Then a bound on the size of the hypermatrix B ∈ C ρ × d × ( ( n )) which underlies thepolynomial Q ⊂ ∼ S ( x ) used to specify the PDP is the smallest bound derived between the equalities (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊂ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊂ S | R | ≤ d )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ρ · d · (cid:18) (cid:18) n (cid:19)(cid:19) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊃ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊃ S | R | ≤ d )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ρ · d · (cid:18) (cid:18) n (cid:19)(cid:19) . N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 27 Proof. This implies that(4.56) X R ⊂ ∼ S Y i ∈ R x [ i ] ≡ X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · x [ w ] mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) , We substitute on both sides of the equal x [ i ] by P Z [ i ] . Recall that(4.57) P Z [ i ] = Y Λ ∈ G n Z [Λ ( i ) , lex (Λ)] , ∀ i ∈ Z ( n ) . (4.58) = ⇒ X R ⊂ ∼ S Y i ∈ R P Z [ i ] ≡ X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · P Z [ w ] mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) )X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · P Z [ w ] mod Q i ∈ R Z [ i, lex (Λ)] − Q j ∈ R Y [ i, lex ( R )] R ⊂ ∼ S, Λ ∈ G n ≡ (4.59) X T ⊂ ∼ S Y i ∈ T P Z [ i ] mod Q i ∈ R Z [ i, lex (Λ)] − Q j ∈ R Y [ i, lex ( R )] R ⊂ ∼ S, Λ ∈ G n mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) . X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · P Z [ w ] mod Q i ∈ R Z [ i, lex (Λ)] − Q j ∈ R Y [ i, lex ( R )] R ⊂ ∼ S, Λ ∈ G n ≡ (4.60) X T ⊂ ∼ S Y i ∈ T P Z [ i ] mod Q i ∈ R Z [ i, lex (Λ)] − Q j ∈ R Y [ i, lex ( R )] R ⊂ ∼ S, Λ ∈ G n mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) )X ≤ u<ρ Y ≤ v 0] + X ≤ w< ( n ) B [ u, v, w ] · P Z [ w ] mod Q i ∈ R Z [ i, lex (Λ)] − Q j ∈ R Y [ i, lex ( R )] R ⊂ ∼ S, Λ ∈ G n ≡ (4.61) X R ∈ ℘ ( Z n ) / Iso R ⊂ ∼ S | G n / Aut R | Y σ ∈ G n / Aut R Y i ∈ σR Y [ i, lex ( σR )] ! | Aut R | mod ( ( x [ i ]) − x [ i ] ∀ i ∈ Z ( n ) ) The equation above yields a system of (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊂ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊂ S | R | ≤ d )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) equationsin the ρ · d · (cid:0) (cid:0) n (cid:1)(cid:1) unknowns entries for B ∈ C ρ × d × ( ( n )) where clearly d ≥ | S | . We know from elimination via N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 28 resultant that the corresponding system has finitely many solutions if there are as many algebraically independentequations as unknowns, from which we derive the bound(4.62) (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊂ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊂ S | R | ≤ d )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ρ · d · (cid:18) (cid:18) n (cid:19)(cid:19) . Taking d = | S | suggest the lower bound(4.63) ρ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊂ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊂ S | R | ≤ | S | )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | S | · (cid:0) (cid:0) n (cid:1)(cid:1) . As similar argument for F ⊃ ∼ S yields the bound, where we take d = 2 (cid:0) n (cid:1) (4.64) ρ ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊃ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊃ S | R | ≤ | S | )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:0) n (cid:1) · (cid:0) (cid:0) n (cid:1)(cid:1) , derived from the exact same argument applied to F ⊃ ∼ S . So the bound on the hypermatrix size that we get is theminimum between the two expressions above. (cid:3) Note that we know from Polya Enumeration Theorem [Pol37, Pol40] that (cid:12)(cid:12)(cid:12)(cid:12)(cid:26) R ∈ ℘ Z ( n ) ! / Iso : R ⊂ ∼ S (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( R ∈ ℘ Z ( n ) ! / Iso : R ⊂ S | R | ≤ | S | )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | S | · (cid:0) (cid:0) n (cid:1)(cid:1) ∼ (cid:0) ( n ) | S | (cid:1) n ! . Application The proposed PDP construction for F ≃ S and F S differ in their weighted count sizes by exactly by | S | . Theproposed PDP construction for F ⊃ ∼ S and F ⊃ S differ in their weighted count sizes by at most (cid:0) n +12 (cid:1) · | S | . Weexplicitly certify membership to the class NP [Wig19, Aar16] by modifying PDE and PDP construction to outputNP certificates instead of the binary 0,1 output. Without lost of generality, assume that NP certificate are specifiedby binary strings which may be viewed as decimal expansions in base 2. Consequently, as a result establishingthat in general F ⊃ ∼ S has no polynomial size formulas, we extrapolates this result to establish that F ⊃ ∼ S admitsno polynomial size PDPs construction which outputs NP certificate strings ( such as a binary string describinga permutation matrix ) instead of the binary , outputs. The extrapolation follows by symmetry breaking ofconjugacy classes of graphs associated with non-vanishing terms. In fact we can think of a PDP which output NPcertificates as a tuple of ordinary PDPs with binary , outputs. For instance, a PDE for F ≃ S which outputs NPcertificates is expressed in terms of the expanded multilinear polynomials of the form(5.1) P ≃ S ( x ) = X σ ∈ S n / Aut S τ σ m s X i ∈ Z n n · i + σ ( i ) Y i ∈ S x [ σ ( i )] . N PARTIAL DIFFERENTIAL ENCODINGS OF BOOLEAN FUNCTIONS 29 (5.2) F ≃ S ( T ) = Y i ∈ Z ( n ) (cid:18) ∂∂ x [ i ] (cid:19) T [ i ] P ≃ S ( x ) x = ( n ) × m = P ≤ i Concluding remarks The discussion suggests a natural hierarchy gradually increasing the complexity of variants of F ⊆ S and F ⊇ S .It is natural to inquire about the next level in this hierarchy of variants. For this purpose we think of a point in ( Z n ) n × as describing a member of the transformation monoid Z Z n n . For all T ⊆ ( Z n ) n × (6.1) σT σ ( − = n lex (cid:16) σf σ ( − (0) , · · · , σf σ ( − ( n − (cid:17) : lex ( f (0) , · · · , f ( n − ∈ T o . (6.2) ConjAut T = n σ ∈ S n : T = σT σ ( − o . For all T, R ⊆ ( Z n ) n × ,(6.3) T ≃ R ⇐⇒ ∃ σ ∈ S n / ConjAut T such that R = σT σ ( − and for T, S ⊆ ( Z n ) n × ,(6.4) T ⊂ ∼ S ⇐⇒ ∃ R ⊆ S s.t. T ≃ R. Let S (1) := S and(6.5) S ( k +1) = n x [ lex ( f g (0) , · · · , f g ( i ) , · · · , f g ( n − f, g ∈ S ( k ) o and(6.6) Span ( S ) = lim k →∞ S ( k ) The next variant in the hierarchy correspond to span variants of F ⊆ S ( x ) and F ⊇ S ( x ) prescribed by followingBoolean functions(6.7) F ⊆ Span ( S ) ( T ) = if Span ( T ) ⊆ Span ( S )0 otherwise , and(6.8) F ⊇ Span ( S ) ( T ) = if Span ( T ) ⊇ Span ( S )0 otherwise . As well as their isomorphism span variants of F ⊆ S ( x ) and F ⊇ S ( x ) prescribed by following Boolean functions(6.9) F ⊂ ∼ Span ( S ) ( T ) = if Span ( T ) ⊂ ∼ Span ( S )0 otherwise , and(6.10) F ⊃ ∼ Span ( S ) ( T ) = if Span ( T ) ⊃ ∼ Span ( S )0 otherwise . 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