Syril Corneille

Tchio Tchichelia

Hervé Talé Kalachi

Jules Waku

René Ndoundam
email: rene.ndoundam@facsciences-uy1.cm

Threshold Circuit based on Cyclic Codes

Keywords: Threshold Circuit, Words, Symmetric Difference, Functions, Cyclic code, Parity-check Matrix

We study the model of a threshold circuit. By using the parity-check matrix:

• Firstly, we build a threshold circuit that recognizes the words belonging to a cyclic code C.

• Secondly, we build a threshold circuit that recognizes the words belonging to the set C1 ∩ C2, where C1 and C2 are cyclic codes.

• Thirdly, we build a threshold circuit that recognizes the words belonging to a symmetric difference of two cyclic codes.

We use these functions to characterize the followings sets: T C 0 4 , T C 0 5 and T C 0 6 .

Introduction

Let B = {0,1}. A function f from B n to B is a boolean function on n variables and a single value, in other terms:

f : B n -→ B x = (x 1 , x 2 , • • • , x n ) -→ y
A circuit is a directed acyclic graph. The sources are called input nodes and are labeled with 0, 1, x 1 , x 2 , • • • , x n . Non-input nodes are called gates and are labeled by boolean functions, whose arity is the in-degree of the nodes. The in-degree (out-degree) of a gate is called the fan-in (fan-out). Sink nodes have fan-out 0 and are called output nodes [START_REF] Clote | Boolean Functions and Computation Models[END_REF].

A threshold circuit C is a boolean circuit where each gate computes a threshold function. Threshold circuits are studied in Complexity Theory and Neural Networks [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF].

1

A threshold function is a function that takes the value 1 if a specified function of the arguments exceeds or egals a given threshold and 0 otherwise.

The size of a circuit is the number of gates, while the depth is the length of the longest path from an input to an output node. Let us note α = (α 1 , • • • , α n ). The following basic functions (threshold functions) arise in the study of circuits:

T α k (x 1 , x 2 , • • • , x m ) =                1, if m i=1 α i x i ≥ k; 0, if m i=1 α i x i < k (1)
When α = (1, 1, • • • , 1), we also have the following threshold function:

T H n k (x 1 , x 2 , • • • , x n ) =                1, if n i=1 x i ≥ k; 0, if n i=1 x i < k (2)
Figure 1: A threshold circuit computing the function

T H n k (x 1 , x 2 , • • • , x n ) .
The equal function is defined as follows:

EQU AL(x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) =      1, if x i = y i ∀ 1 ≤ i ≤ n; 0, ∃ i 1 ≤ i ≤ n such that x i ̸ = y i (3) A boolean function f is symmetric if and only if f(x 1 , x 2 , • • • , x n ) = f(x α(1) , x α(2) , • • • , x α(n) )
for any permutation α on the set {1, 2, • • • , n}.

Hajnal et al. [START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] shown the following result: Proposition 1 [START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] Symmetric functions can be computed by depth-2 threshold circuits of linear size and weight 1.

Parberry [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] shown the following theorems:

Theorem 1 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Any symmetric function f : B n -→ B can be computed by a unit-weight threshold circuit with size 2n + 3 and depth 2.

Theorem 2 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Any threshold circuit of the weight w and depth 2 for IP:

B 2n -→ B must have size Ω(2 n/2 /w 2 ). Corollary 1 [2] IP / ∈ T C 0 2 Proof 1 [2]
By Theorem 2, any depth 2 circuit of weight n c for inner product must have size Ω(2 n/2 /n 2c ), which is larger than any polynomial. □ Let us consider the following symmetric function:

P ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x n = n i=1 x i
The complement of Parity is:

P ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x n
We easily observe that:

P ARIT Y (x 1 , x 2 , • • • , x n ) = P ARIT Y (x 1 , x 2 , • • • , x n-1 , x n )
Inner Product is defined in [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF][START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] as:

IP (x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) = n i=1 (x i ∧ y i ) (4)
P ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x n = n i=1
x i with depth 2 and size at most 2n + 3.

P ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x n with
IP (x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) = x 1 y 1 ⊕ x 2 y 2 ⊕ • • • ⊕ x i y i ⊕ • • • ⊕ x n y n
of depth 3 and of size at most 2n(2n + 3).

Theorem 3 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Any symmetric function f : B n -→ B can be computed by a unit-weight threshold circuit with 2n + 3 and depth 2.

Proof 2

Let f : {0, 1} n -→ {0, 1} be a symmetric function. f is fully(uniquely) defined by the set

S f = {m ∈ N | f (x ) = 1 f or all x ∈ B n with exactly m ones}. Suppose S f = {m 1 , • • • , m k }.
The circuit uses k pairs of gates on the first level. The ith pair has one gate active when the number of ones in the input is at least m i (this is a unit-weigth threshold-gate with threshold m i connected to the inputs x 1 , • • • , x n ), and the other gate active when the number of ones in the input is at most m i . When given an input x such that f (x ) = 0, exactly one of each pair is active, therefore, exactly k gates are active. When given an input x such that f (x ) = 1, one pair has both of its gates active, and all other pairs have exactly one of its gates active, therefore exactly k + 1 gates are active. The output gate therefore has threshold value k + 1 and inputs from all of the first level gates. This circuit has depth 2, and since k ≤ n + 1, size at most 2(n + 1) + 1. □ For example, Figure 9 shows a threshold circuit for computing PARITY in depth 2 and size 7.

Figure 10 shows a threshold circuit for computing P ARIT Y in depth 2 and size 9. Corollary 2 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Any symmetric function f : B n -→ B m can be computed by a unit-weight threshold circuit with size 2n + m + 2 and depth 2.

Proof 3 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Suppose f :B n -→ B m is a symmetric function. Computing each individual bit of the output of f is a symmetric function, and hence by Theorem 1 can be computed in-depth 2 and size 2n + 1. Thus, the obvious circuit for computing f uses m such circuits and has depth 2 and size 2nm + m. However, the first layer of this combined circuit can have at most 2(n + 1) different gates, giving the required size bound. □ Gates in the second layer of the threshold circuits constructed in Theorem 1 and Corollary 2 have an interesting property. They have unit weights, threshold k , and the number of ones in their input is guaranteed (by the rest of the circuit) to be either k or k -1 . Let us call this kind of Boolean linear threshold function a balanced one. The following result enables savings in depth whenever balanced threshold gates are used in any layer of a circuit but the last. This does not, of course, give savings in depth for the circuits constructed in Theorem 1 or Corollary 2, but it will enable a reduction in depth whenever these circuits are used as building blocks in the interior of another circuit [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF]. Figure 7: A threshold circuit computing the function

IP (x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) = P ARIT Y (x 1 ∧ y 1 , • • • , x i ∧ y i , • • • , x n ∧ y n ) of depth 3

and of size at most 2n(2n + 3).

A threshold gate g is said to be balanced iff the number or sum of the ones at the input of the threshold-gate g is either k or k-1, with k ∈ N, mathematically:

W H (x) ∈ {k, k -1}.
If the number or the sum of the ones at the input of the threshold-gate of g is different from k or k-1, then g is said to be unbalanced, mathematically:

W H (x) / ∈ {k, k -1}.
Lemma 1 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Let g 0 be a unit-weight threshold-gate that has inputs only from balanced threshold-gates g 1 , • • • , g m where for all 1 ≤ i < j ≤ m, gates g i and g j have distinct inputs. The gates g 0 , g 1 , • • • , g m can be replaced by a single thresholdgate.

Proof 4 [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] Let g 0 be a unit-weight threshold-gate that has inputs only from balanced threshold-gates

g 1 , • • • , g m . Suppose gates g 1 , • • • , g m collectively have inputs x 1 , • • • , x n ,
and that for all 1 ≤ i < j ≤ m, gates g i and g j have nonoverlapping input. Suppose g i has weight k i , for 0 ≤ i ≤ m. We claim that the entire circuit can be replaced by a threshold-gate g with threshold m i=0 k i -m (see Figure 11).

Suppose g 0 outputs 0. Then, at most k 0 -1 of the gates g 1 , • • • , g m output 1. Therefore, at most k 0 -1 of the gates g i for 1 ≤ i < j ≤ m see k i ones, and the rest see

k i -1 ones. Hence, x 1 , • • • , x n can have at most m i=1 (k i -1) + (k 0 -1) = m i=0 k i -(m + 1)
ones. Therefore, g outputs 0. Conversely, suppose g 0 outputs 1. Then, at least k 0 of the gates g 1 , • • • , g m output 1. Therefore, at least k 0 of the gates g i for 1 ≤ i ≤ m see k i ones, and the rest see

-1 + k i ones. Hence, x 1 , • • • , x n must have at least m i=1 (k i -1) + k 0 = m i=0 k i -m Figure 9: A threshold circuit computing the symmetric function x 1 ⊕ x 2 ⊕ x 3 ⊕ x 4 ⊕ x 5 .
ones. Therefore, g outputs 1.

We have shown that g outputs 1 iff g 0 outputs 1. Therefore, the circuit containing g 0 , g 1 , • • • , g m can be replaced by the threshold-gate g, as claimed. □ Definition 1 T C 0 is the class of languages accepted by threshold circuits of polynomial size and depth O(1).

T C 0

k is the class of languages accepted by threshold circuits of depth k and size O((log(n)) k ).

Hajnal et al. [START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] have shown the following result:

Lemma 2 [3]
Fix any ϵ > 0 and polynomial p. Assume that C is a depth-2 threshold circuit with weight ≤ p(n) computing INNER PRODUCT MOD 2 to two n-bit strings. Then if n is sufficiently large, the size of C is at least

2 (1/2-ϵ)n .
From the Lemma 2, Hajanal et al. [START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] deduce that:

Lemma 3 [3] INNER PROD MOD 2 is not in T C 0 2 , INNER PROD MOD 2 is in T C 0 3
Hajnal et al. [START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF] deduce the following strict inclusion: It was conjectured in [START_REF] Mix Barrington | On uniformity with in N C 1[END_REF][START_REF] Yao | Circuits and local computation[END_REF] that:

Theorem 4 [3] T C 0 1 ⊆ T C 0 2 ⊆ T C 0 3
T C 0 ̸ = T C 0 k , f or any k
Parberry stated in [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF] that:

It is an open problem as to whether the T C 0 hierarchy collapses, that is, whether more than three layers of threshold gates are needed to compute all functions in T C 0 in polynomial size.

Krause and Wegener [START_REF] Krause | Circuit Complexity in Boolean Models and Methods in Mathematics[END_REF] have said that:

Indeed, no proof is known that a function is contained in N P \ T C 0 3 . Our aim is to show that the sets N P \ T C 0 3 , N P \ T C 0 4 and N P \ T C 0 5 are not empty.

The paper is organized as follows. Section 2 is concerned with the presentation of linear codes and cyclic codes, matrix generators, and parity-check matrix of linear codes. Section 3, we present our contribution. Concluding remarks are stated in Section 4.

Linear Codes and Cyclic Codes 2.1 Linear Codes

Definition 2 C is a linear code if C is a linear subspace of F n q . C is then called a [k , n]-code if dim(C)= k . If the minimal distance of C is d , we say that C is a [k , n, d ]-code
A linear code can be described by each of the following matrices : The generator matrix in systematic form have the following structure:

• A generator matrix G for an [k , n, d ]-code C is a k × n matrix
G = [I k |P ] ,
where I k is the k × k identity matrix and P ∈ F

k×(n-k) q . • A parity-check matrix H for an [k , n, d ] code C is an (n -k ) × n matrix of rank n -k sastisfying: ∀ x ∈ C, H t x = 0.
From a generator matrix in systematic form, one can compute the paritycheck matrix as follos :

H = t P |I n-k .
If G is a generator matrix and H a parity-check matrix of the same code, then G t H = 0.

Example 1

We construct a binary [START_REF] Krause | Circuit Complexity in Boolean Models and Methods in Mathematics[END_REF][START_REF] Hajnal | Threshold Circuits of Bounded Depth[END_REF]-code by choosing three vectors linearly independent of F 6 2 .

G =   1 0 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0  
We obtain all the codewords of C by calculating all the products mG with m ∈ F 3 2 .

The words of the code C is given by:

    1 0 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0     0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 1 0 1 0 1 0 0 1 1 1 1 0 1 1 1 0 0 0 1 1 1 0 0 1 1 1 0
We transform the matrix G in the systematic form(Gaussian elimination)

G =   1 0 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0   -→ • • • -→   1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 1 0   = (I 3 |P ) Example 2
Starting from the generator matrix defined as follow:

G =   1 0 0 1 0 1 0 1 0 0 1 1 0 0 1 1 1 0   = (I 3 |A)
We find the parity-check matrix H = ( t A|I 3 ) of C which yields:

H =   1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1   From H, we obtain the codewords of C ⊥     1 0 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1     0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 0 0 0 1 1 0 2.2 Cyclic Codes Definition 3 [11] A code C is cyclic if a) C is a linear code; b) any cyclic shift of a codeword is also a codeword, i.e. whenever v o v 1 • • • v n-1 ∈ C, then also v n-1 v 0 v 1 • • • v n-2 ∈ C and v 1 v 2 • • • v n-1 v 0 ∈ C Example 3
G =     1 0 0 0 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1    
is equivalent to a cyclic code.

Theorem 5 [START_REF] Adamek | Theory and Applications of Error-Correcting Codes with an Introduction to Cryptography and Information Theory[END_REF] Every nontrivial cyclic (n,k)-code contains a codeword g(x) of degree n-k. Then code words are precisely all the multiples q(x)g(x), where q(x) is any polynomial of degree smaller than k. The code has the following generator matrix:

G =      g(x) xg(x)
. . .

x k-1 g(x)      =       g 0 g 1 g 2 • • • g n-k 0 0 • • • 0 0 0 0 g 0 g 1 g 2 • • • g n-k 0 • • • 0 0 0 0 0 g 0 g 1 g 2 • • • g n-k • • • 0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 0 0 0 0 0 • • • g 0 g 1 g 2 • • • g n-k       Example 4
The even-parity code of length n is cyclic. The codeword of degree n-k = 1 is just one: 1 + x. The generator matrix is

G =     1 1 0 0 0 • • • 0 0 1 1 0 0 • • • 0 0 0 1 1 0 • • • 0 0 0 0 • • • 0 1 1     Proposition 2 [11]
A cyclic code with the parity check polynomial h(x

) = h 0 + h 1 x + • • • + h k-1 x k-1 + x k
has the following parity check matrix:

H =       0 0 • • • 0 0 0 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 1 h k-1 h 2 h 1 h 0 0 0 0 • • • 0 1 h k-1 h 2 h 1 h 0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 0       Example 5
The Hamming cyclic code of length 7 with the generator polynomial g(x) = 1 + x + x 3 has the parity check polynomial h(x) = (x 7 -1) ÷ (x 3 + x + 1) = x 4 + x 2 + x + 1. Thus, it has the following parity check matrix:

H =   0 0 1 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 0 0   Remark 1
If we denote H the control matrix of a cyclic code C

H =      h 1 * h 2 * . . . h (n-k) *      =       0 0 • • • 0 0 0 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 1 h k-1 h 2 h 1 h 0 0 0 0 • • • 0 1 h k-1 h 2 h 1 h 0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 0      
Based on Proposition 2 and on the structure of H, we easily observe that:

h i * = S(h (i+1) * ); 1 ≤ i ≤ n -k -1
where S is the right circular shift defined as: sometimes, right circular shift is noted as : ≫ It follows that

h (n-k-i) * = S i (h (n-k) * ); 1 ≤ i ≤ n -k -1
where

S i = SoSoSoSoSo • • • oS i times
we easily deduce that:

H =          h 1 * h 2 * . . . h (n-k-2) * h (n-k-1) * h (n-k) *          =           S (n-k-1) (h (n-k) * ) S (n-k-2) (h (n-k) * ) . . . S 2 (h (n-k) * ) S(h (n-k) * ) h (n-k) *          

Our Contribution

In the following subsection, we characterize a function that testing if a word belonging to a code by using a generator matrix.

Testing belonging of a word to a code by using Generator Matrix

We consider two cases of Generator Matrix: Generator Matrix in a systematic form and Generator Matrix in a general form.

Generator Matrix in a systematic form

Subsequently, without loss of generality we suppose that the generator matrix has the form:

G = (I k |B) =        1 0 0 • • • 0 | b 11 b 12 • • • b 1,n-k 0 1 0 • • • 0 | b 21 b 22 • • • b 2,n-k 0 0 1 • • • 0 | b 31 b 32 • • • b 3,n-k . . . . . . . . . . . . . . . | . . . . . . . . . . . . 0 0 0 • • • 1 | b k1 b k2 • • • b k,n-k        (7)
∀ V = (v 1 , v 2 , • • • , v n ) ∈ C, we have Coor(V ) =      v 1 v 2 . . . v k     
From the fact that G is a generator matrix of C, we can write:

V ∈ C ⇐⇒ Coor(V ).G = V
This is equivalent to:

V ∈ C ⇐⇒ (v 1 , v 2 , v 3 • • • , v k )        1 0 0 • • • 0 | b 11 b 12 • • • b 1,n-k 0 1 0 • • • 0 | b 21 b 22 • • • b 2,n-k 0 0 1 • • • 0 | b 31 b 32 • • • b 3,n-k . . . . . . . . . . . . . . . | . . . . . . . . . . . . 0 0 0 • • • 1 | b k1 b k2 • • • b k,n-k        = (v 1 , v 2 , v 3 • • • , v n )
This implies that:

V ∈ C ⇐⇒ (v 1 , v 2 , v 3 • • • , v k , k l=1 v l ×b l,1 , k l=1 v l ×b l,2 , • • • , k l=1 v l ×b l,n-k ) = (v 1 , v 2 , v 3 • • • , v n )
We conclude that:

V ∈ C ⇐⇒ k l=1 v l × b l,j = v k+j , 1 ≤ j ≤ n -k
It follows that:

V ∈ C ⇐⇒ IP (b * j , V ) = v k+j , 1 ≤ j ≤ n -k (8)
By using Equation ( 8), we define the following function

f 1 (V, G) = EQU AL((IP (b * 1 , V ), IP (b * 2 , V ), • • • , IP (b * n-k , V )), (v k+1 , v k+2 , • • • , v n ))
which testing if the word V belonging to a code C by using the generator matrix in the systematic form.

Generator Matrix in a general form

We consider the general case:

G =      g 11 g 12 g 13 • • • g 1n g 21 g 22 g 23 • • • g 2n . . . . . . . . . . . . . . . g k1 g k2 g k3 • • • g kn      where G is a k × n matrix such that g ij ∈ {0, 1}, f or 1 ≤ i ≤ k and 1 ≤ j ≤ n. We have to determine (z 1 , z 2 , z 3 • • • , z k ) ∈ {0, 1}, such that k i=1 z i × g i, * = V This is equivalent to: V ∈ C ⇐⇒ (z 1 , z 2 , z 3 • • • , z k )      g 11 g 12 g 13 • • • g 1n g 21 g 22 g 23 • • • g 2n . . . . . . . . . . . . . . . g k1 g k2 g k3 • • • g kn      = (v 1 , v 2 , v 3 • • • , v n )
This implies that:

V ∈ C ⇐⇒ ((z 1 ×g 11 + z 2 ×g 21 + z 3 ×g 31 +• • •+ z k ×g k1 ), (z 1 ×g 12 + z 2 ×g 22 + z 3 ×g 32 +• • •+ z k ×g k2 ), • • • , (z 1 × g 1n + z 2 × g 2n + z 3 × g 3n + • • • + z k × g kn )) = (v 1 , v 2 , • • • , v n ) V ∈ C ⇐⇒ ( k t=1 z t × g k1 , k t=1 z t × g k2 , • • • , k t=1 z t × g kn ) = (v 1 , v 2 , • • • , v n ) V ∈ C ⇐⇒ ( k t=1 z t ∧ g t,1 , k t=1 z t ∧ g t,2 , • • • , k t=1 z t ∧ g k,n ) = (v 1 , v 2 , • • • , v n )
We conclude that:

V ∈ C ⇐⇒ k j=1 z j ∧ g j,i = v i , 1 ≤ i ≤ n
It follows that:

V ∈ C ⇐⇒ IP (g * i , V ) = v i , 1 ≤ i ≤ n (9)
By using Equation ( 9), we define the following function

f 2 (V, G) = EQU AL((IP (g * 1 , V ), IP (g * 2 , V ), • • • , IP (g * n , V )), (v 1 , v 2 , • • • , v n ))
which testing if the word V belonging to a code C by using the generator matrix in the general form.

Testing belonging of a word to a code by using Paritycheck Matrix

Let us denote C ⊥ the dual code of a code C. We note H the generator matrix of

C ⊥ . Let us note a word V = (v 1 , v 2 , • • • , v n ) ∈ C and let us express H t V in terms of IP : H t V =      h 11 h 12 • • • h 1n h 21 h 22 • • • h 2n . . . . . . . . . . . . h (n-k)1 h (n-k)2 • • • h (n-k)n           v 1 v 2 . . . v n      =                  n j=1 h 1j × v j . . . n j=1 h ij × v j . . . n j=1 h (n-k)j × v j                  =                  n j=1 h 1j ∧ v j . . . n j=1 h ij ∧ v j . . . n j=1 h (n-k)j ∧ v j                  =         IP (h 1 * , V ) . . . IP (h i * , V ) . . . IP (h (n-k) * , V )        
By definition, we have:

V ∈ C ⇐⇒ H t V = ⃗ 0 (10)
This implies that:

V ∈ C ⇐⇒ n-k i=1 IP (h i * , V ) = 0 (11) V / ∈ C ⇐⇒ n-k i=1 IP (h i * , V ) ≥ 1 (12)
It follows that:

V ∈ C ⇐⇒ n-k i=1 IP (h i * , V ) = (n -k) (13)
We deduce that:

V ∈ C ⇐⇒ T H (n-k) (n-k) (IP (h 1 * , V ), IP (h 2 * , V ), • • • , IP (h (n-k) * , V )) = 1 (14)
By using Equation ( 14), we define the following function:

f 3 (V, H) = T H (n-k) (n-k) (IP (h 1 * , V ), IP (h 2 * , V ), • • • , IP (h (n-k) * , V ))
where the vector

h i * = h i1 , h i2 , • • • , h in 1 ≤ i ≤ n -k (15)
are the generator of the dual space of C ⊥ (or the parity-check matrix of C).

The function f 3 tests if the word V belonging to a code C by using the parity-check matrix.

V ∈ C ⇐⇒ f 3 (V, H) = 1 (16)
Let us consider the following function:

f3 (V, H) = T H (n-k) 1 (IP (h 1 * , V ), IP (h 2 * , V ), • • • , IP (h (n-k) * , V )) (17)
It is easy to see that:

V / ∈ C ⇐⇒ f3 (V, H) = 1
Example 6

Let n = 3, k = 1 from where n -k = 2 construct the threshold circuit corresponding to the function

f 4 (V, H) = T H (2) (2) (IP (h 1 * , V ), IP (h 2 * , V ))
where the vector (2) (IP (h 1 * , V ), IP (h 2 * , V )) testing belonging of a word of length 3 to a code by using the parity-check-matrix of dimension 2 reduced by one level by using the Lemma 1 of Parberry [START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF]. This circuit is size 21 and depth 3.

h i * = h i1 , h i2 , h i3 1 ≤ i ≤ 2 (18) with IP (h 1 * , V ) = h 11 ∧ v 1 ⊕ h 12 ∧ v 2 ⊕ h 13 ∧ v 3 and IP (h 2 * , V ) = h 21 ∧ v 1 ⊕ h 22 ∧ v 2 ⊕ h 23 ∧ v 3

Element of T C 0 4

In this paragraph, we want to show that there exists a function which belongs to T C 0 4 .

We consider a cyclic code C.

Problem 1

Data : a cyclic code C, H its parity-check matrix , a word w.

Question : w ∈ C ?

Let us build a boolean function f 5 such that:

f 5 (w, h (n-k) * ) = 1 ⇐⇒ w ∈ C H =      h 1 * h 2 * . . . h (n-k) *      =      h 11 h 12 • • • h 1n h 21 h 22 • • • h 2n . . . . . . . . . . . . h (n-k)1 h (n-k)2 • • • h (n-k)n     
is the parity-check matrix of the cyclic code C.

f 5 (w, h (n-k) * ) = T H (n-k) (n-k) (IP (S (n-k-1) (h (n-k) * ), w), IP (S (n-k-2) (h (n-k) * ), w), • • • , 20
IP (S(h (n-k) * ), w), IP (h (n-k) * , w))
Theorem 6

f or k = n -⌈log 2 (n)⌉ f 5 ∈ T C 0 4 .

Proof 5

Based on the depth of evaluation of IP , we consider two cases:

First case: IP is evaluated by a circuit of depth 2.

From the fact that IP is evaluated by a circuit of depth 2 and from Lemma 2, we conclude that the size of IP is exponential in n, it follows that the size of the function f 5 is also exponential in n.

then f 5 / ∈ T C 0 3 .
Second case: IP is evaluated by a circuit of depth 3.

From the fact that the depth of IP is 3, we deduce that the depth of the function f 5 is 4.

From the Theorem 1, the size of the circuit which evaluates

IP (h i * , w) = IP (S (n-k-i) (h (n-k) * ), w) is O((log n) 3 ).
It follows that the size of

f 5 is O((log n) 4 ), because n -k = ⌈log 2 (n)⌉.
We easily conclude that f 5 ∈ T C 0 4 . □ Let us consider a cyclic code C, H is parity-check matrix where

H =      h 1 * h 2 * . . . h (n-k) *      =      h 11 h 12 • • • h 1n h 21 h 22 • • • h 2n . . . . . . . . . . . . h (n-k)1 h (n-k)2 • • • h (n-k)n      Problem 2
Data : a cyclic code C, H its parity-check matrix , a word w.

Question : w / ∈ C ?

Let us build a boolean function f5 such that:

f5 (w, h (n-k) * ) = 1 ⇐⇒ w / ∈ C f5 (w, h (n-k) * ) = T H (n-k) 1 (IP (S (n-k-1) (h (n-k) * ), w), IP (S (n-k-2) (h (n-k) * ), w), • • • , IP (S(h (n-k) * ), w), IP (h (n-k) * , w))
The following result characterizes the function f5 .

Theorem 7

For n sufficiently large, we define

k = n -⌈log 2 (n)⌉ f5 ∈ T C 0 4 .

Proof 6

Based on the depth of evaluation of IP , we consider two cases:

First case: IP is evaluated by a circuit of depth 2.

From the fact that IP is evaluated by a circuit of depth 2 and from Lemma 2, we conclude that the size of IP is exponential in n, it follows that the size of the function f5 is also exponential in n.

then f5 / ∈ T C 0 3 .
Second case: IP is evaluated by a circuit of depth 3.

From the fact that the depth of IP is 3, we deduce that the depth of the function f5 is 4.

From the Theorem 1, the size of the circuit which evaluates

IP (h i * , w) = IP (S (n-k-i) (h (n-k) * ), w) is O((log n) 3 ). It follows that the size of f5 is O((log n) 4 ), because n -k = ⌈log 2 (n)⌉.
We easily conclude that f5 ∈ T C 0 4 . □

• Two cyclic codes C 1 and C 2 , • H 1 parity-check matrix of C 1 , H 1 =      L 1 L 2 . . . L (n-k)      =       0 0 • • • 0 0 0 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 1 h k-1 h 2 h 1 h 0 0 0 0 • • • 0 1 h k-1 h 2 h 1 h 0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 0       • H 2 parity-check matrix of C 2 , H 2 =      L1 L2 . . . L(n-k)      =       0 0 • • • 0 0 0 1 hk-1 • • • h2 h1 h0 0 0 • • • 0 0 1 hk-1 h2 h1 h0 0 0 0 • • • 0 1 hk-1 h2 h1 h0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 hk-1 • • • h2 h1 h0 0 0 • • • 0 0 0      
• a word w. From the fact that IP and IP are evaluated by a circuit of depth 2 and from Lemma 2, we conclude that the size of IP or IP is exponential in n. It follows that the size of the function f 6 is also exponential in n.

Then f 6 / ∈ T C 0 4 . Second case: IP and IP are evaluated by a circuit of depth 3.

From the fact the depth of IP and IP is 3, we deduce that the depth of the function f 6 is 5.

From the Theorem 1 the size of the circuit which evaluates:

• IP (h (i * ) , w) = IP (S (n-k-i) (h (n-k) ), w) is O((log n) 3 ), • IP (h (i * ) , w) = IP (S (n-k-i) (h (n-k) ), w) is O((log n) 3 ).

It follows that the size of f 6 is O((log n) 5 ) , because n -k = (⌈log 2 (n)⌉) 2 . We easily deduce that f 6 ∈ T C 0 5 . □

The next subsection is denoted to the study of the set T C 0 • Two cyclic codes C 1 and C 2 ,

• H 1 parity-check matrix of C 1 ,

H 1 =      L (1) L (2)
. . .

L (n-k)      =       0 0 • • • 0 0 0 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 1 h k-1 h 2 h 1 h 0 0 0 0 • • • 0 1 h k-1 h 2 h 1 h 0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 h k-1 • • • h 2 h 1 h 0 0 0 • • • 0 0 0      
• H 2 parity-check matrix of C 2 ,

H 2 =      L(1) L (2)
. . .

L(n-k)      =       0 0 • • • 0 0 0 1 hk-1 • • • h2 h1 h0 0 0 • • • 0 0 1 hk-1 h2 h1 h0 0 0 0 • • • 0 1 hk-1 h2 h1 h0 0 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1 hk-1 • • • h2 h1 h0 0 0 • • • 0 0 0      
• a word w.

Question : w ∈ C 1 ∆ C 2 ?
Let us consider the function We characterize the function f 7 in the next result.

Theorem 9

For n sufficiently large, define

k = n -(⌈log 2 (n)⌉) 3 f 7 ∈ T C 0 6 .

  Figure 2: A threshold circuit computing the function EQU AL(x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) of depth 2 and of size at least 2n + 1.

  Figure

Figure 4

 4 Figure 4: A threshold circuit computing the functionP ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x nwith depth 2 and size at most 2n + 3.

  Figure 4: A threshold circuit computing the functionP ARIT Y (x 1 , x 2 , • • • , x n ) = x 1 ⊕ x 2 ⊕ • • • ⊕ x i ⊕ • • • ⊕ x nwith depth 2 and size at most 2n + 3.

  Figure 5: A threshold circuit computing the functionIP (x 1 , x 2 , • • • , x n , y 1 , y 2 , • • • , y n ) = x 1 y 1 ⊕ x 2 y 2 ⊕ • • • ⊕ x i y i ⊕ • • • ⊕ x n y nof depth 3 and of size at most 2n(2n + 3).

Figure 6 :

 6 Figure 6: Example of a threshold circuit computing the function IP (x 1 , x 2 , x 3 , x 4 , x 5 , y 1 , y 2 , y 3 , y 4 , y 5 ) = x 1 y 1 ⊕ x 2 y 2 ⊕ x 3 y 3 ⊕ x 4 y 4 ⊕ x 5 y 5 of depth 3 and of size 17 .

Figure 8 :

 8 Figure 8: Example of a threshold circuit computing the function IP (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , y 1 , y 2 , y 3 , y 4 , y 5 , y 6 ) = P ARIT Y (x 1 ∧ y 1 , x 2 ∧ y 2 , x 3 ∧ y 3 , x 4 ∧ y 4 , x 5 ∧ y 5 , x 6 ∧ y 6 ) of depth 3 and of size 21.

Figure 10 :

 10 Figure 10: A threshold circuit computing the symmetric function P ARIT Y (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 ).Before:

Figure 11 :

 11 Figure11: Before and after pictures for Lemma[START_REF] Parberry | Circuit Complexity and Neural Networks[END_REF]

  whose rows form a basis for C. The lines of a generator matrix form a base for the code C.

a)

  {0} is a trivial cyclic code; b) Code C = {000, 101, 011, 110} is cyclic c) Hamming code Ham(3, 2): with the generator matrix

S(e p , e p- 1 ,

 1 • • • , e 2 , e 1 ) = (e 1 , e p , e p-1 , • • • , e 3 , e 2 )

  Figure12:A threshold circuit of the functionf 4 (V ) = T H (2) (2) (IP (h 1 * , V ), IP (h 2 * , V)) testing belonging of a word of length 3 to a code by using the parity-check-matrix of dimension 2. Here the function f 2 (v) is balanced, this circuit is size 23 and depth 4 .A corresponding threshold circuit the function f 2 is the following figure12It follows by application of the Lemma 1 of Parberry that the function f 4 belongs to T C 0 3 and it becomes unbalanced. A corresponding threshold circuit is the following figure13

Figure 13 :

 13 Figure 13: A threshold circuit of the function f 4 (v) = T H (2)

3. 4 5

 4 Element of T C 0 We tackle the existence of a function in the set T C 0 5 . Let us consider two cyclic codes C 1 and C 2 Problem 3

Question : w ∈ C 1 ∩ C 2 ? 8

 128 Let us consider the functionf 6 (w, L (n-k) , L(n-k) ) = T H 2 2 (f 5 (w, L (n-k) ), f5 (w, L(n-k) )) TheoremFor n sufficiently large, we definek = n -(⌈log 2 (n)⌉) 2 f 6 ∈ T C 0 5 . Proof 7Based on the depth of IP and IP we consider two cases:First case: IP and IP are evaluated by a circuit of depth 2.

6 3. 5 6

 65 Element of T C 0In this subsection, we are interested by the function who characterize the symmetric difference of two cyclic codes.Problem 4

f 7 (

 7 w, L (n-k) , L(n-k) ) = P ARIT Y (f 5 (w, L (n-k) ), f 5 (w, L(n-k) ))

We easily see that the complement of Inner Product is:

We easily deduce that:

Acknoweldgments

The authors thanks Professor Marcos KIWI(University of Chile) for useful suggestions.

Proof 8

Based on the depth of evaluation of IP and IP, we consider two cases: First case: IP and IP are evaluated by a circuit of depth 2.

From the fact that the size IP and IP is exponential in n, it follows that the size of the function f 7 is also exponential in n. We deduced that:

Second case: IP and IP are evaluated by a circuit of depth 3.

From the fact that the depth of IP or IP in 3, we deduce that the depth of the function f 7 is 6. a) From the Theorem 1, the size of the circuit which

b) It follows that the size of

We easily deduce that f 7 ∈ T C 0 6 .

□ 4 Conclusion

In this paper, by using cyclic codes and their parity check matrix, we have shown that the sets N P \ T C 0 3 , N P \ T C 0 4 and N P \ T C 0 5 are not empty.


Mario Talamo

Evolutions of Ethical Paradigms and Popular Fiction: The Case of Late Edo Tales of Vengeance

Keywords: katakiuchi mono, late Edo period, structuralism, morphology, functions, dramatis personae, intellectual paradigms, social context, nature, invention

Tales of vengeance were a popular Edo literary genre published in great numbers from the end of the eighteenth century. From the outset, tales of vengeance (katakiuchi mono) were informed by the neo-Confucian orthodoxy promoted by the military government in the immediate aftermath of the Kansei Reforms (1787-1793), and constituted ideological productions marked by consistent structures and dramatis personae with specific functions. With the passage of time, the intellectual paradigms-moral rules and sets of ideas governing everyday life-that underpinned these tales shifted. The present article explores the interaction between this broader ethical context and the tales of vengeance, arguing that the evolution of ethical paradigms from the immediate aftermath of the Kansei Reforms to the last part of the Edo period drove alterations to the tales.

The article sheds new light on the possibilities for philological and structural analysis to reveal fundamental structures and transformations in literature. The application of structuralism and morphological analysis to late Edo tales of vengeance provides us with information on the ideas, worries, and beliefs relevant to the people of the late Edo period. Changes over time-in the morphology of the tales, their structure, and their range of characters-reveal that their transformations reflect changes in reality. The article demonstrates the potential structuralism offers for the study of literature more broadly.

when Confucian virtues such as filial piety and loyalty were emphasized through the ideological propagation of neo-Confucianism by the bakufu 幕府. The Kansei edicts led to the silencing of the preceding generation of gesaku 戯作 (playful fiction) writers, the suppression of sharebon 洒落本 (frivolous books), and the disappearance of realistic elements and instances of contemporary life from the plots of the kibyōshi 黄表紙 (lit. "yellow cover" picture book) tales. Tales of vengeance remained popular, though, because the practice of blood revenge was justified as an act of filial piety and loyalty. Authors were thus free to write about a wide selection of material while respecting the new censorship protocols and "Vendetta fiction was a safe and appealing endeavor for popular writers as it presented moral acts that provided plenty of room for lurid and morally questionable scenes that sold well." 1 The result was a plethora of tales of adventure set in historical settings, with their plots dominated by ideology and doctrine. The article analyzes the connection between the changes that occurred in the form and structure of these tales and the broader social and intellectual environment.

While there is a wide variety of research on the theme of vendetta in theatrical genres like jōruri 浄瑠璃 (puppet theater) and kabuki 歌舞伎 (Japanese dance-drama), the narratives produced in the aftermath of the Kansei Reforms have never been studied in depth. Scholars such as Hirade Kōjirō 平出鏗二郎 and Ujiie Mikito 氏家幹人 offer exhaustive descriptions of the vengeful acts performed during the Edo period, but they have not paid sufficient attention to the social and ideological context within which these tales were produced. 2 The textual approach adopted here combines the precision of philological and paleographical work with a structural treatment of the selected materials to recover the social and political environment that influenced their production. Many of the texts examined here only exist as woodblock prints rather than in modern print editions, and require a supplementary process of transcription and translation, which makes this kind of sociocritical research on the literary domain of the Edo period particularly rare.

This article adopts a multidisciplinary approach based on the analysis of specific literary productions. It draws on the structuralist theories of Vladimir Propp (1895-1970), and on the models elaborated by Mikhail Bakhtin (1895-1975), Algirdas Greimas (1917-1992), and Denise Paulme (1909-1998) relating to the interpretation of tales. 3 These constitute a methodological tool for dividing the plot and the range of characters in the texts of the corpus into minimal actions and basic roles, and for connecting structural transformations and changes in the range of actions of the dramatis personae to the evolving social milieu. As Propp states, "We need to take into account the tale in relation with its milieu, the situation in which it is created. Here, real life and religion-in the broad sense of the term-play a determinant role." 4 The first section of this article provides details of the corpus of texts analyzed and explains the methodology utilized. The rest of the article is divided into two parts: the first, consisting of three sections, is purely structuralist, and outlines the morphological 1 Atherton 2013, p. 185. 2 See Hirade 1909; Ujiie 2007. 3 With Vladimir Propp's work, literary studies began shifting from formalism, which set aside subjective interpretation, to structuralism, which studies tales as a series of indivisible narrative units belonging to reality. Structuralism is appropriate for analyzing tales of vengeance due to their schematic yet formulaic structure. 4 Propp 1965, p. 176. and functional structure of the tales of vengeance, and the changes they undergo. The second half of the analysis, the remaining four sections, reads these changes against a series of intellectual evolutions in the late Edo period. This approach is inspired by Simondon's theory of transductivité that recognizes the mutual relation existing between reality and writing, with a circuit of interactions resulting in an exchange between objects and mental reproductions. Reality is thus part of narrative material, which is, in turn, a part of reality, reflecting ideas and mentalities.5 I assert that the transformations in the tales are external to the narratives themselves, and argue that it was new socioeconomic values, along with the loss of authority of the edicts imposed during the Kansei (1789-1801) and Tenpō 天保 (1830-1844) eras, which caused the reconfiguration of the morphological structure and dramatis personae of the tales of vengeance.

Sources and Methodology

Vengeance, of course, has a long history in the Japanese literary tradition. The oldest vendetta tales date back to the Kojiki 古事記 (712) and Nihon shoki 日本書紀 (720), when Prince Mayowa 眉輪 avenged his father by murdering the twentieth emperor Ankō 安康; other famous episodes include the vengeance exacted by the Soga 曽我 brothers, who killed Kudō Suketsune 工藤祐経 in a famous story from the Muromachi 室町 period (1336-1573). While many works dealt with revenge, though, these tales of vengeance were not considered a genre: the revenge fiction by Ihara Saikaku 井原西鶴 (1642-1693), for example, was considered as a branch of ukiyozōshi 浮世草子 (tales of the floating world), while Chikamatsu Monzaemon 近松門左衛門 (1653-1724) wrote jōruri. It was only in the late Edo period (1603-1868) that stories about this practice diffused widely, and the number of tales expanded rapidly down to the Meiji 明治 era (1868-1912).6 As Saeki Shin'ichi 佐伯真一 argues, it is at this point the term katakiuchi mono comes to designate a specific literary category. 7 Moral values, virtues, and ideology play a pivotal role in their narrative, and like Bakhtin's tales of challenges (prüfungsromans), tales of vengeance recount a series of tasks performed by the hero, which test them through suffering and temptation in order to cultivate loyalty, courage, virtues, and sanctity. 8 The sources for this article are drawn from two subsets of the macro-genre of gesaku published in the city of Edo: the shorter kibyōshi and the longer gōkan 合巻 (bound volumes). These are literary works in which illustrations occupy most of the page, with narration and dialogue written around them (see figures 1 and 2).

My corpus spans the Kansei to Ansei 安政 (1855-1860) eras and comprises seven tales of vengeance (see Appendix 1). These texts were all written in the aftermath of the reforms instituted by Matsudaira Sadanobu 松平定信 (1758-1829), from 1787 the chief senior councilor of the Tokugawa 徳川 shogunal administration. The Kansei Reforms sought to respond to uprisings during the Tenmei 天明 famine (1782-1788) by mandating both government and personal austerity-samurai were to abandon frivolous hobbies like poetry composition and return to serious literature (bun 文) and martial arts (bu 武). Criticism of these measures published in 1788-1789 resulted in the punishment of a number of popular writers, and restrictions were applied to the publishing sector from Kansei 2 (1790) onward. Sadanobu decreed that "all scabrous contents must be banned from books, that new publications must be registered with the city magistrate, and that all anonymous works should be withdrawn from the market." 9 Following these reforms, Confucian virtues such as filial piety and loyalty were emphasized through the bakufu's ideological propagation of neo-Confucianism. 10 The texts in my corpus demonstrate how the tales evolved after the Kansei Reforms. The work of Nansenshō Somahito 南仙笑楚満人 (1749-1807) shows the morphological peculiarities of the first tale of vengeance issued after the Kansei Reforms. The texts by Jippensha Ikku 十返舎一九 (1765-1831) and Santō Kyōzan 山東京山 (1769-1858), issued after the two reforms of Kansei and Tenpō (1841-1843) respectively, illustrate the influence of political edicts on the functional evolution of these productions. The remaining texts, those belonging to the first decade of the nineteenth century by Santō Kyōden 山東京伝 (1761-1816) and Shikitei Sanba 式亭三馬 (1776-1822), and the one dating to the beginning of the Tenpō era signed by Goryūtei Tokushō 五柳亭徳升 (1793-1853), show the ethical evolutions occurring in Japanese society.

The texts examine various stages in the production of revenge fiction. When researching on ethical paradigms and their influence on literary productions, it is necessary to allow for sufficient time to pass between the different texts selected to observe how new ideas circulated widely among the populace. These helped to shape new perceptions of reality that the authors, voluntarily or involuntarily, conveyed in their works. The collation of primary sources dated from Kansei to the Bunka 文化 (1804-1818) period was simple due to the popularity enjoyed by tales of vengeance, but from the third decade of the nineteenth century onward, when vendetta fiction was supplanted by other productions, minor works had to be examined.

I adopt a twofold methodology: paleographical and philological, as most of the corpus is still written in hentaigana 変体仮名 (non-standardized variants of hiragana) and needs to be reconstructed, and structuralist, based on the "functional reduction" proposed by Vladimir Propp in Morfologija skazki (1928), and on other theories adapted to tales of vengeance. Edo fictional genres have formulaic structures that suit the application of Proppian studies. Propp was the first narratologist to reconstruct a common code concerning functions-basic actions constituting indivisible units that form the plot of tales-and characters, and aimed to find historical developments reflected in folklore and literary production.11 This methodology enables me to outline a recurring list of functions in the plots of tales of vengeance and to identify a fixed range of characters. The results of my analysis, in turn, enabled me to understand the evolution of functional structures and the influence of social developments and changes in the intellectual environment.

Two previous studies, by Koike Masatane 小池正胤 and Konita Seiji 小二田誠二, have drawn on Propp's theories to develop structural classifications of tales of vengeance based on Proppian binaries/oppositions (see Appendix 2). Koike examined Ikku's kibyōshi and gōkan to develop a very simple classification of the plot of tales of vengeance, showing how these changed from the beginning to the end of the Bunka period. Konita sought to break down the tales of the Soga and Ishii 石井 brothers into their constituent elements based on their phases of tribulation. I accept Konita's claims for the importance of the Soga monogatari 曽我物語 in the Japanese vendetta tradition, but in contrast to Koike assert that the tales were constituted by a common code, one that does not change over the period. I therefore draw on their models while establishing a link between the structural transformations of the tales and the social and intellectual milieu within which that transformation occurs.

The article's primary contribution is the adoption of a philological and paleographical approach that, through the examination of characters and morphological structure, reveals the broader context shaping these literary productions.

Functional Structure and Dramatis Personae of the Early Productions

Tales of vengeance are fundamentally melodramatic productions. They begin and end in the same idyllic or social space; the succession of events which spans the tales are the attempts of the protagonists to return to the original "world of innocence."12 This return to origins is the main aim of the tales.

The plots of the tales of vengeance in my corpus can be divided into ten recurring functions:

1. The idyll; 2. The end of the idyll; 3. The death of the father-figure; 4. The identification of the culprit; 5. The decision to avenge; 6. The tribulations and financial struggles of the hero; 7. The meeting with the auxiliary; 8. The journey to the final battle; 9. The final vendetta; 10. The return of the hero.

The initial situation, the idyll, represents the life of the protagonists before the loss of their social status. Uncontrolled passion results in murder and brings the first function to an end. The identification of the culprit is a pivotal phase of the plot. When it cannot take place through a rational event (a search for evidence), authors often resort to magic. In Kyōden's Katakiuchi Magotarōmushi 敵討孫太郎虫, the victim Kaname, killed by Hitokaku, returns into the world of the living as a ghost to reveal the identity of the murderer to his wife Sakurado. 13 While the functions may vary in order, generally the first four and the last three are fixed. The central three functions, constituting the tribulations of the avenger, are freer, depending on author and audience. Indeed, this central set may be repeated more than once. Magotarōmushi's morphological structure, for instance, comprises more than one cycle of privations. After the death of her father Kuranoshin, the protagonist Sakurado becomes a beggar, loses her husband Kaname, and risks being the victim of sexual violence. The duplication and triplication of misadventures is a "recognizable trait which heightened the suspense"14 and emotional engagement of readers. These struggles end through a meeting with an auxiliary, who gives strength to the avenger.

Initially, the characters are part of a Manichean conflict between good and evil: the hero is a positive character while the antagonist represents pure evil. 15 In his work on folktales, Propp points to seven different categories of characters: the antagonist, the donor, the helper, the princess or the king, the dispatcher, the hero, and the false hero. 16 The texts of my corpus provide four categories of characters as recurring typologies of agents: the hero (or the avenger), the antagonist, and the auxiliaries (the helper and the donor).17 In addition, while the victim does not figure among the dramatis personae identified by Propp, the actions of Propp's princess and king comprise the same functions as that of agent in the tales of vengeance. These five types-avenger (hero), antagonist, helper, donor, and victimconstitute the dramatis personae of my study.

The avenger is key, as their quest to regain the initial social idyll provides the central motor for the plot. Their trajectory-initially losing status before returning to a position of strength-is made possible by meeting the auxiliary. Male avengers in late Edo tales of vengeance are always aged fifteen, on the cusp of becoming an adult samurai, when they can be useful to the nation. Their age remains constant throughout. 18 Another peculiarity of the avenger is their celibacy; love stories find very limited space in the plots. 19 Heroes in tales of vengeance are paragons of virtue, but also liminal figures positioned between two ages-childhood and adulthood-and two states-human and divine. Late Edo tales of vengeance aimed to instruct readers by presenting examples of virtue.

It is difficult to outline the peculiarities of the antagonist, as their numbers and personalities vary. Most of the villains in this article's corpus are rōnin 浪人, samurai without master or dwelling. Antagonists are stronger than avengers, and, often accompanied by a group, initially overcome them. This victimization of the avenger has two main purposes: to demonstrate the avenger's moral integrity through their ability to withstand pain and suffering, and to cause the reader to identify with the avenger. The antagonist is named, but after the murder they sometimes live in obscurity. 20 The auxiliaries-helpers and donors-have similar functions. The helper assists the avenger, while the donor endows him with a magical weapon or skill, acquired after extensive physical training and a binary series of questions to test the avenger's state of virtue; this Propp defines as "the sequence of the donor."21 Avengers and helpers are linked by blood ties and working relations, but the encounter with the donor depends on chance and virtuous behavior. Donors appear as part of the Buddhist pantheon, like the Jizōbosatsu 地蔵菩薩 (Jizo Bodhisattva) in Kyōden's Magotarōmushi; as an animal, like Koyuki the cat in Sanba's Katakiuchi Yomeodoshidani 復讐娶嚇谷 or even as a creature of local folklore, such as the tengu 天狗 in Katakiuchi aiyadobanashi 復讐相宿噺 by Tokushō.

The final agents are the victims. Their death drives the entire plot, from innocence to absence. The main peculiarity of the victim is their moral integrity and capacity to foresee and understand future events.

Each of these dramatis personae are defined by specific, formulaic descriptions. The victim, for instance, is generally presented as an honest man (monogatai 物堅い). Standard expressions reoccur, such as "clenching fist and grinding teeth" (kobushi o nigiri, hagami o nashi 拳を握り、歯嚙みをなし), which shows an agent's sense of impotence. These descriptive formulae usually refer to Confucianism. Avengers are loyal, gentle, possessed of filial piety, and chaste (in the case of female avengers); while villains are evil, false, and lacking in any sense of justice or loyalty. Extrapolating the common code structuring the tales of vengeance allows for transformations in the morphology of these texts to be analyzed.

Narrative Structures and Stylistic Remarks

In addition to the functional structure and character types of late Edo tales of vengeance, my analysis concerns their narrative structure and style. Denise Paulme proposes that there are seven main plot structures in tales from sub-Saharan Africa, typically based on two fundamental movements: ascending and descending. The former starts with a lack and, through a series of improvements, leads to a satisfactory condition; the latter is the opposite as it starts from an idyllic situation but ends in poverty. These two movements are defined by Paulme as the "Simple Forms" of far more complex structures. 22 This kind of morphological analysis is suitable for both oral and written texts, since both verbal and textual narratives can be divided into many functions, each with determined types of agents. 23 The texts of the corpus offer two narrative structures: the first resembles an hourglass while the second is cyclical. In the first, avenger and antagonist enter the scene together. After the murder, the avenger experiences various vicissitudes while the antagonist ameliorates working and financial problems; the situation reverses when the avenger meets the helper, which gives the hero the strength to fight while jeopardizing the future of the villain (see figure 3). In the second, the future avenger joins the narrative from a privileged position-as a member of the military aristocracy-and goes through a descending phase of tribulations and financial struggles. These vicissitudes end with their meeting the auxiliary, which enables the hero to ascend to the final vendetta (see figure 4). The hourglass structure characterizes the tales in the early part of the corpus, those issued from the Kansei to the Bunka period, while the tales from the Bunsei 文政 (1818-1830) era onward adopt a cyclical structure.

Late Edo tales of vengeance are chronocentric productions. Heroes, warriors, and enemies appear at full strength, endowed with incomparable skills. However, readers never witness a real learning process, or the physical growth of the protagonists. The tales alternate between static moments, in which the protagonists lead their lives, and dynamic periods, marked by the pursuit of avengers and villains. The dynamic tempo of the narrative is enhanced by the "syntax of enchainment," in which sentences end with verbs in the continuative form (ren'yōkei+te 連用形+て) and adjectives in the suspensive form (ren'yōkei), sometimes spaced out with izenkei+ba 已然形+ば (causal nexus) that directs the action. 24 This style appears mostly in passages narrating fights and movements, instilling a rhythm to the sentences. The tempo of the vendetta is extraordinary: daily life is unexpectedly interrupted by an adversative preposition, introduced by "ga" (but), followed by the verb 24 Jacqueline Pigeot showed the syntax of the enchainment in the Heike monogatari 平家物語. According to her, this endows the discourse with an inner dynamism. See Pigeot 2009, p. 252. in the rentaikei 連体形 form (attributive form) and tokoroni (right when) or other temporal expressions such as orifushi (in that moment), aruhi (one day), arutoki (once), and orishimo (right then). Their occurrence marks the beginning of an extraordinary time, signaling vicissitudes, misadventures, natural disasters, and divine epiphanies.

The tales are also vectorial: the author supplies information enabling the reader to understand the roles of dramatis personae and crucial passages of the plot. 25 The vectors in Edo tales of vengeance usually concern characters and are extremely helpful for understanding aspects of the intrigue. Initially, readers identified agential roles by when they join the narrative. The first tales of vengeance provided a fixed and recurring scheme, as Koike Masatane demonstrated in his functional model: first we meet the antagonist, then the victim, and after the murder we get to know the avenger(s). 26 That tripartite structure was fundamental, for it enabled readers to understand the role of every single character. From Kyōden's Magotarōmushi onward, however, the audience cannot necessarily distinguish between the various agents who join the narrative in the prologue; the additional information provided by the vectors substitutes for Koike's tripartite structure. Also, tales of vengeance immediately reveal the age of the characters, pivotal to understanding who could aspire to become the male hero: fifteen years marked the official coming of age for a samurai, and for avengers-former samurai excluded from their social class due to an injustice-represents the terminus ante quem by which privileges should be reacquired through vengeance.

The structural analysis of tales of vengeance allows for their plot structures to be elucidated. In examining the evolution of these tales through the nineteenth century, I distinguish structural particularities (functions and characters that appear throughout the corpus) from temporary characteristics, the latter of which reflect literary fashion and authorial choices. These functions allow me to study the influence of the ideological and social context. Functions constitute the standard pattern of a production. Changes and new elements show the original scheme altered by new ideas. Dialogues (particularly between avengers and auxiliaries) reveal ideology and social problems. The evolution of dramatis personae and their functions is connected to ideas in society, and structural changes in the narrative demonstrate shifts in the surrounding social milieu reflected in the production of these tales of vengeance.

Morphological Evolution of Tales of Vengeance

The tales of vengeance early in the corpus have a very simple structure and limited ensemble of characters, as Koike shows. The five types of agents typical of tales of vengeance appear in one iteration in the narrative. The narratives themselves are linear functional structures, lacking any kind of subplot. For instance, Somahito's Katakiuchi gijo no hanabusa 敵討義女 英 offers three victims, one hero, one antagonist, a donor, and a helper, with a narrative structure that comprises the punctual performance of the ten basic functions. From the Bunsei era onwards, though, the morphology of the tales of vengeance changes: the authors prolong the central "phase of tribulations" and add several subplots. Tales of vengeance in this period show a greater range of characters, with new types of dramatis personae, including a mediator and Propp's dispatcher. We also observe single characters fulfilling multiple roles, which, following Greimas, we will refer to here as the archi-agent.27 This is true of Gōsuke in Magotarōmushi, for instance, who initially acts as helper and subsequently as donor, or Hayanoshin in Kyōzan's Katakiuchi teijo kagami 敵討貞女鑑, a donor undertaking a role usually performed by the mother (or the dying victim) in conferring the role of avenger on Osayo. This expansion in the numbers and roles of characters is also reflected in the morphological structure through multiple rounds of "meeting with the auxiliary" and an extended "movements and phases of tribulations."

Change also occurs through the disappearance of elements formally central to the tales. One peculiarity of the morphology of late Edo tales of vengeance is a function referred to by Propp as "the lack filled," which is ever-present in the tales issued from the Kansei to Bunka periods. After the murder, the antagonist steals something precious (usually a sword or money), the restitution of which is crucial for the avenger to restore their lost privileges: Ōyanagi Hitokaku, antihero in Magotarōmushi by Kyōden, steals a precious sword after the assassination of Kuranoshin; in Yomeodoshidani by Sanba, Kensawa Gonpei steals one thousand ryō 両; Tetsudō, in Aiyadobanashi by Tokushō, does the same with the victim's sword; and Yokoshima Akuzō, the villain in Teijokagami by Kyōzan, steals the money from the victim Jūsuke. However, this function subsequently disappears-in the tales of vengeance issued from the Bunsei era onward, we no longer read of the restoration of a precious object as being crucial to the storyline.

There is another pivotal function of the morphology that loses its centrality over time. Initially, after the murder, the avenger had to go through an extensive phase of tribulations and financial struggles, but in later tales, the hero's suffering disappears from the story. By the first decade of the nineteenth century, his troubles and pains are no longer deemed to be important, and several authors narrate how the avengers, although deprived of their guarantor of prosperity (their father-figure), continue to lead stable lives. While the heroine Sakurado in Magotarōmushi is forced to live as a beggar after the death of Kuranoshin, in later tales we see that Kichinosuke, the avenger in Tokushō's Aiyadobanashi, moves into the house of his swordmaster, while Shizuma, the hero in Iga no adauchi 伊賀之仇討 by Dontei Robun 鈍亭魯文 (1829-1894), continues to act as a vassal. Tales of vengeance from the Bunsei era onwards relate a history of (psychological) suffering in which financial and other troubles do not affect the protagonists.

The application of structuralist theories allows for the classification and division of the texts into minimal units. They reveal the stable elements which constitute the tales of vengeance of my corpus and highlight transformations in the structure and dramatis personae of the tales. As I will show below, these morphological transformations are expressions of the ideological paradigms underpinning the production of late Edo tales of vengeance. These tenets and their alterations will constitute the core of the analysis below.

Ideology and Tales of Vengeance

The remainder of this article highlights evolutions that occurred in late Edo thought and connects them with transformations in the morphological structure of the tales. Tales of vengeance of the late Edo period are commercial productions, but ones with a strong ideological and philosophical background. In the texts of my corpus, the ethical structures introduced by the tales largely belong to a neo-Confucian orthodoxy that sought for individuals to order society and to accept the system in several ways, particularly through exhorting people to respect the preestablished social order, and promoting the Confucian virtues mentioned in the titles of the tales (teijo 貞女 chaste woman; gijo 義女 righteous woman). Vengeance was an expression of two of the most important Confucian virtuesloyalty and filial piety-and the names of many characters explicitly reference proper Confucian attitudes (such as Misao 操 for chastity). 28 Murders in tales of vengeance are always a consequence of uncontrolled feelings; a negative view of human emotions directly reflecting neo-Confucian orthodoxy. The tales also emphasize the social relations of each character (kyōdai 兄弟 brothers; fūfu 夫婦 husband and wife), reflecting a social order in which "every individual had an identity certifying his status, lifestyle, behavior and relations."29 Finally, the presentation of the natural environment-the frame for every murder and destructive event-comes straight from the Confucian understanding of nature. 30 Late Edo tales of vengeance also reference Buddhism: the Jizo Bodhisattva serves as donor in Kyōden's Magotarōmushi, protecting and healing the future avenger Magotarō, while zense no mukui 前世の報い (punishment for sins committed in earlier lives) allows the characters in my corpus to justify events.

It is the paradigm of natural creation (shizen 自然), reflected in the Kansei Reforms, which provided for a society based on rules, relationships, and productive activities rooted in nature. This upheld the immutability of the social and feudal order and invoked the five relational virtues and four social groups to claim all should respect the social position they inherited by fulfilling the obligations that came together with their mibun 身分 (social status). The Zhu Xi 朱熹 (1130-1200) neo-Confucianism of the Kansei Reforms sought to respond to social crises by fostering these "natural" Confucian virtues, which had supposedly characterized Tokugawa society in its earlier years. 31 Nevertheless, this effort to return to nature would prove unsuccessful, as it failed to account for the shift towards a proto-capitalistic economy, characterized by new employment relations and an increasingly monetized society.

In his work on Tokugawa intellectual history, Maruyama Masao 丸山真男 (1914-1996) argued that modernity happens when humanity overcomes nature. 32 Indeed, it is often thought that the denaturalization of society corresponds to its modernization. 33 Modernity here is defined by three points: a positivist metaphysics, the doctrine of freewill and personal responsibility, and a vision of society as contractual in its relationships. For Maruyama, the development of a modern society coincides with the passage from the paradigm of natural creation to the idea of artificial invention (sakui 作意). The two paradigms of nature and invention describe two sets of social relations in tales of vengeance: natural and voluntary. 34 In natural relations, "typically governed by pre-determined sets of traditions, where individuals are glued to these structuring habits which discriminately assign goods, positions, and identities," the clan and the family occupy the center of social life. 35 These ties characterize the first group of tales of vengeance, those published between the Kansei and Bunka periods. Here, auxiliaries have social ties to the hero, as long-standing vassals or faithful servants, and are not randomly encountered during the hero's long journey. This is the case with the Adachi brothers, the helpers in the texts by Jippensha Ikku, or the faithful servant Nadeshiko in Magotarōmushi by Kyōden, who tries to soothe the pain of the heroine Sakurado. In these early texts, there is no seventh function, that of "meeting with the auxiliary," as the auxiliary role is naturally constituted. This contrasts with later texts, in which this relationship is a voluntary one.

Sakui and Its Manifestations

The Kansei Reforms attempted to restore a traditional vision of society, economy, and productive activities, and asserted a concept of natural creation that shaped the early part of the corpus of tales of vengeance. Between the Kansei and Bunka periods, natural relations were central to the tales of vengeance, with hierarchies (and respect for them) clear in relations between the agents. These tales maintained social harmony with even vengeance itself only carried out once the necessary bureaucratic procedure-the authorization to undertake revenge-is completed.

However, the structure of the tales comes to reflect the idea of artificial invention, namely that society and its institutions are not natural, but stem from wise men and sage kings. 36 As with nature, this idea of sakui is expressed in the tales of vengeance through their dramatis personae. Early in the corpus, the helpers all have natural ties to the hero. From the Bunsei era onward, this gives way to voluntary relations, in accordance with the 32 Thomas 2001b, p. 16. 33 Thomas 2001a, p. 20. 34 The first is constituted before the birth of each individual (family, clan); the second is associations that individuals form according to their personal will and interests (syndicate, political party), see Maruyama 1974, p. 224. Weber refers to this dichotomy of human relations as traditional and rational authority, Durkheim to mechanical and organic solidarity, while Louis Dumont argues only two kinds of human societies exist, the holistic and the individualistic. 35 Ansart 2014a, p. 185. 36 The idea of artificial invention was based on the hypothesis that institutions, and even the five elements and the five virtues, had been created by the sages (the legendary kings of the first Chinese dynasties). They were thus products of political leadership. The opposition between natural order and artificial invention was understood to date back to the age of Xunzi 荀子 (ca. 313 BC-238 BC) and the Legists, the Chinese philosophical school whose speculation focused on how to maintain social order, security, and political stability.

paradigm of artificial invention. Voluntary relations are not blood or preexisting social ties but are based on a contract reflecting convenience and personal interests: avengers come to meet individuals useful to their plan of revenge, thus enlarging the range of characters they interact with. Atherton argues that this proliferation of characters and actions are matters of genre. Kibyōshi were shorter, with a limited page count, and could not provide a greater range of characters and plot developments, which was why the longer gōkan were created. 37 However, an increase in the number of characters and functions is already evident in the morphology of the last two kibyōshi of my corpus, Magotarōmushi and Yomeodoshidani. The former lists a larger range of agents-thirteen, including three auxiliaries, two donors, one dispatcher-and a far more elaborate functional structure, while the latter offers the same number of characters divided into those who support the avenger and those who side with the antagonist. It is therefore argued here that the number of characters does not increase solely due to the appearance of new genres, for it reflects the fact that interpersonal relations between the characters become more functional and contractual. Santō Kyōden's Magotarōmushi marks the point at which relations change. This work can be divided into two sections, comprising different relations between auxiliaries and avenger. The first part of the tale is dominated by natural ties to the helper Nadeshiko, who belongs to the protagonist's family. In the second part, there are two other auxiliaries: Gōsuke and his wife, and the Jizo Bodhisattva. The former are natural relations, for the couple previously worked for Kaname's family, but that with the Jizo is a voluntary relationship resulting from Sakurado's virtuous behavior and desire for repentance. In the texts of the corpus from Magotarōmushi onward, auxiliaries are no longer connected with the family of the avenger.

The effect of sakui on my corpus is particularly clear in the appearance of two new dramatis personae: the dispatcher and the mediator. The dispatcher-a character that sends individuals and objects from one place to another-is a perfect expression of the idea of invention in the Edo sociocultural milieu. Dispatchers appear in Magotarōmushi and Yomeodoshidani, where two characters, Masauji-kō and Midori no mae, take part in the vicissitudes of the protagonists at the outset and conclusion of the tale. Their function is to dispatch vassals to the place of vengeance in order to recognize the virtuous exploits of the avengers. In Tokushō's Aiyadobanashi, dispatchers play an influential role in the narrative and are now actively involved throughout the text. The prime example is Horikoshi Minbu, an archi-agent serving as both auxiliary and dispatcher, who sends the antagonist to Imagawa, exploiting his ambition to become a vassal, and enabling the avenger to locate the villain in the process. Mediators-agents who seek to foster consensus between charactersalso increasingly come to lead the action and drive events. They first appear in the Iga no adauchi by Robun after a specific function of "meeting with the mediator." Dispatchers and mediators show the impact that sakui exerted on Japanese society; they are a product of political leadership that directly acts to solve problems.

The loss of authority of the Kansei Reforms reflected the decline in the natural paradigm, and the diminution of the role of agriculture in economy. With the advancement of the idea of sakui, entrepreneurship came to be celebrated, together with activities of creation and invention through a vision of individuals as "animals producing tools." 38 This is visible in the tales of vengeance. In the early texts, avengers-after losing their social status-live humbly and make their living through agriculture, but later tales tend to have merchants as the main actors. This is clear in Kyōzan's Teijokagami, where the auxiliaries of the protagonist Osayo are all merchants. The example of Osayo's brother, Tōsaku, who lives in the countryside, emphasizes that now farmers are associated with poverty rather than serving as the quintessence of virtue:

In the province of Musashi, in the district of Usu, near the village of Kine, lived Tōsaku, a peasant who owned only one horse. . . . His life was sad and he lived eating taros. 39 In their valorization of entrepreneurship, and the emergence of collaboration through personal skills and contracts, the tales increasingly reflected a more positivist vision, together with a contractual attitude depending on freewill and self-interest. 40

Social Utilitarianism

A vision of society as contractual meant that social ideals no longer ref lected the presumption that society was natural. This made Confucian social and economic theories anachronistic and led many intellectuals to question the values imposed by the bakufu.

One key thinker here is Kaiho Seiryō 海保青陵 (1755-1817), who envisioned a contractual society in which every social agent played a productive role. 41 Seiryō's theories reveal the new mindset in late Edo society, and reinterpret Confucian virtue (toku 徳) as a means to achieve specific aims. 42 The changes captured by Seiryō are also reflected in the tales of vengeance; and particularly through meetings between archi-agents and avengers. Early in the corpus, auxiliaries were part of natural groups (family), and avengers did not need to demonstrate virtue to be aided. In later texts, though, with helpers and donors being drawn from outside the avengers' families, virtue becomes the currency that buys their help. The role of the archi-agent in these late-period tales of vengeance is patterned: they meet the protagonist as a helper, but after testing the hero's virtue through a series of questions and answers, they become a donor. The virtue of the protagonist is thus the means to facilitate the passage of the archi-agent from the role of helper to donor. The first multi-role character appears in Magotarōmushi, where Gōsuke debuts as a helper but, due to the future avenger Magotarō's filial devotion, becomes a donor teaching Magotarō the secrets of martial arts. In Teijokagami by Santō Kyōzan, the character of Sonowaza Hayanoshin, who joins the plot as 38 Maruyama 1974, p. 302. 39 KTK, p. 931. 40 Ansart 2014a, p. 200. 41 An acute observer of Tokugawa society, Seiryō was born in Edo, where he spent the first thirty years of his life, before traveling around the country advising lords and vassals on economy and politics. Seiryō's speculations revaluated productivity and the active contribution to society of every single member. Chief among his works are two discourses: Keizaiwa 経済話 (Discourse on the pacification and administration of the country, 1806), and Keikodan 稽古談 (Lessons of the past, 1813). 42 According to Seiryō, who employs the term utsuwa 器 (instrument) borrowed from the Analects by Confucius, all virtues are instruments of prosperity. See Ansart 2014b, pp. 118-119. helper, becomes a donor thanks to the healing of his blind daughter Otsuyu by the heroine Osayo. The transition takes place after recognition of Osayo's loyalty (chūsetsu 忠節), with Hayanoshin exclaiming that "This time, thanks to your loyalty, I got my daughter back."43 Virtue is the basis for avengers to receive help, unlike in the first part of the corpus, where natural relations guaranteed the support of auxiliaries belonging to the familial milieu.

Another consequence, already highlighted, of the structural changes brought about by this new mentality is the lack of suffering and financial troubles of the hero. In the first part of the corpus, after the murder, avengers go through hardships and tribulations. The "phase of tribulations and financial troubles" begins with the death of the head of the family and ends after vengeance has been realized with the reacquisition of old social privileges. The emphasis on financial vicissitudes follows the pattern established by the Soga monogatari, where the Soga brothers, after the assassination of their father, spend the rest of their childhood in poverty. 44 This pattern was retained in the early tales of my corpus, but then alters, as noted in the first section. Towards the end of the Tokugawa period, the value of a single person becomes measurable through success, and the wisdom (chie 知恵) of an individual is reduced to an ability to acquire good things and avoid the bad. 45 Poverty and difficult financial conditions were blamed on those who suffer them, for "if the poor people are poor, it is their fault. Laziness, lack of skills and stupidity bring them to this condition."46 This negative vision of poverty is reflected in the morphology of the tales of vengeance. Avengers, members of the higher social strata, no longer suffer hardship, as this would suggest a lack of skill. This removes the tribulations of the avengers from the plots of the tales. This transition from tears of pain to the disappearance of suffering is already clear in Yomeodoshidani by Sanba, where the avenger Kazuma, even after the loss of his status (he was a samurai), lives in a hut but gives no indication of economic hardship. He gets help from Yukino-the archi-agent performing the role of helper and donor-and together they live as lovers in an extra-temporal dimension.

The characters of the tales, and the conceptions of traditional Confucian virtue underpinning them, were influenced by the contractual society represented in Seiryō's writings and redefined by ideas that placed profit and exchange at their foundation. These were no longer compatible with accounts of the poverty of the protagonists, which consequently disappeared from the tales of vengeance.

Understanding Good and Evil

Initially, tales of vengeance were primarily informed by Confucian and neo-Confucian traditions. These tenets reflected right and wrong through a strict division between hero and antihero. As time passed, the loss of influence of the edicts of the Kansei Reforms was reflected in the texts of the corpus by the increasingly ambiguous relation between good and evil. This affects the narrative structures of the tales: the hourglass structure of the early tales shifted towards a cyclical structure.

In the early tales, revenge fiction exhibits an hourglass structure. The narrative follows the vicissitudes of both the avenger and the antagonist, and their contrasting destinies. After the murder, the avenger goes through hardships, while the antagonist enjoys good fortune. Subsequently, however, their trajectories invert, leading to the moment of vengeance. The hourglass structure is the result of a clear opposition between good and evil; its didactic role is to show that no matter how rich and strong an individual (the antagonist) becomes, if his behavior is not ethically sound, his success will prove illusory and turn to dust. Conversely, those who conform to notions of virtue will achieve their fortune.

From the Bunsei era onwards, however, the tales adopt a cyclical structure that focuses solely on the actions of the avenger; the antagonist is depicted only at the very beginning, when he kills the victim, and at the end, when he is killed by the avenger. This indicates the shift from a society strongly influenced by the Confucian tenets imposed by the Kansei Reforms-kanzen chōaku 勧善懲悪 (incitement of virtue and chastisement of vices) and a clear vision of good and evil-to a more fluid ethical environment. In addition, another shift marking this ethical transformation is the disappearance of "restitution of the lack" from the tales. Although the entire corpus has the antagonists stealing an object of value from the hero's family, only the early texts emphasize the object's restitution as a sign of what is right, and through which old privileges are reacquired. Later, this becomes a nonstructural optional element, and some texts end without the restitution of the precious object (for example, Aiyadobanashi, Teijokagami, and Iga no adauchi).

The distribution of the characters also alters. Tales of vengeance initially featured helpers and donors solely on the side of the hero; readers never encountered a meeting between villains and these agents. However, this changes from the beginning of the Bunka period in Sanba's Yomeodoshidani, where there are villains with helpers, blurring the boundaries between avenger and antagonist. The absence of Manichean opposition in the structure of the tales is maintained through the rest of the corpus. By the final text, Robun's Iga no adauchi, the Sakurae brothers aid both the avenger and the antagonists, in a role that I term the "involuntary donor." Over time, ethical remarks also issue from the mouths of the antagonists, not as pedagogical remarks, but instead combining humor and Propp's oduračivanije (deception). 47 In Yomeodoshidani, the antagonist Takashina provides the first example of subjective morality by misjudging the victim Hanasaki. In Tokushō's Aiyadobanashi, Horikoshi Minbu tricks the antagonist, enabling the avenger to find him easily. 48 In Iga no adauchi, Kanda Kanbei admits that the antagonist Sawai Matagorō is disloyal (fujitsu 不実), feigning sincere repentance before a pact to exchange hostages. 49 These transformations in narrative structure, the disappearance of "restitution" from the plots of tales of vengeance, the appearance of helpers and donors on the side of the antagonists, and the deceptive use of morality are all new elements in the tales. Their emergence shows how the invocation of Confucianism in these plays, which had been socially central in the period immediately after the Kansei Reforms, subsequently altered the distinction between actions praised and condemned.

Conclusion

The above demonstrates how a corpus of tales of vengeance evolved in the wake of a series of ideas and events that had a profound impact on late Edo society. Tales of vengeance published after the two periods of reform show two common features: a morphological linear structure, without subplots, and the partial loss of the functions of the dispatcher. These changes resulted from a perception of political stability among the populace because of the reforms. The subsequent affirmation of the idea of invention was reflected in the development of dramatis personae. The impact of social utilitarianism and of new socioeconomic structures was clear in the role of Confucian virtue and in the new conception of poverty as a fault of individuals. The transformation of narrative structure, the diffusion of auxiliaries among villains, and the deceptive use of morality should be interpreted as consequences of the ambiguous relationship between good and evil in the later tales. What remained unchanged, though, was the restorative function of vendetta and the role of the avenger, who was still a model and whose existence was "based on action." 50 This article shows through a combination of descriptive practice and a speculative approach the potential of philological and paleographical analysis. This survey of tales of vengeance clearly demonstrates that the entire literary tradition inspired by vengeful practice shows the same morphological structure and the same range of characters, regardless of the historical era in which it was written and the literary genre to which it belongs. It was therefore the theme of vengeance that shaped literary productions with plots containing the ten typical structural functions and the five typologies of agents I highlight here. The article opens new avenues for scholars interested in a structural approach, which may determine the morphological classification of literary genres and single texts alike.

APPENDIX 1

Figure 1 .

 1 Figure 1. The Sumiyoshimōde, as the villain Taema kills the head of the Usami clan. In this kibyōshi tale of vengeance, illustrated by the author, most of the page consists of illustrations, with the text of the tale above. Dialogue would sometimes appear below the illustrations. Jippensha Ikku, Sumiyoshimōde, pp. 4 ura, 5 omote. Courtesy of the Waseda University Library "Kotenseki Sogo Database."

Figure 3 .

 3 Figure 3. Hourglass Structure of the tales of vengeance, designed by the author. Based on the early texts of the corpus, this highlights the actions of the two main characters, avenger and antagonist, in relation to one another.

Figure 4 .

 4 Figure 4. Cyclical Structure of the tales of vengeance, designed by the author. This schematizes the journey of the avenger on the basis of tales from the latter half of the corpus.

[START_REF] Simondon | Du mode d'existence des objets techniques[END_REF]

Yamaki 2002, p. 24.

Saeki 2004, p. 7.

Bakhtin 1988, pp. 197-198.

See Ooms 1975, pp. 129-141.

See Ooms 1975, p. 55.

[START_REF] Propp | [END_REF]

Greimas 1974, p. 219.

The names of most of the characters were originally written in kana.

Dundes 1964, p. 81.

Konita 1989, p. 77.

Propp 1965, pp. 96-97.

The terms "agent," "dramatis persona," and "character" are nearly synonymous. However, "dramatis persona" refers to the characters or actors of a play or novel; the term "agent" represents the role a character plays, determined by their actions (the avenger avenges, the helper helps).

Females function as avengers in some of the tales, but are always accompanied by a boy able to carry out the actual act of vengeance and regain lost privileges. After their heroic exploits, female avengers often enter religion.

In his essay on the theory of the mythic tale, Greimas proposes another model to decompose the narrative. Among the peculiarities of the hero, he lists iron celibacy; see Greimas 1969. In tales aimed at women, however, relations between avengers and courtesans are the main attraction.

In Ikku's tale, Iwafuchi Motoemon, after raping Osechi, stops using his surname; see KSM 2, p. 7 ura.

Etienne Souriau argues that the difference between the two roles is based on the contribution of each agent to the final plan. SeeSouriau 1950, p. 79.

Paulme 1976, pp. 26-44.

In addition, one of the main differences between formalism and structuralism concerns the applicability of their theories to form and content. In formalism, the two parts should be separated, as only the forma (the oral element) is intelligible-ergo analyzable-while the content is devoid of any significance. According to the structuralist perspective adopted here, this opposition does not exist because forma and content have the same nature and therefore can become object of the same analysis. See LéviStrauss 1966, pp. 163-199.

Gérard Genette, in his essay titled "Auteur impliqué, lecteur impliqué," deals very briefly with the vectoriality of the literary production, the information that each author shares with his model reader. SeeGenette 1983, p. 417.

According to Greimas, the archi-agent represents "two roles covered by one single agent." For example, the helper and the donor occurring in one character, or the dispatcher and the donor.Greimas 1968, pp. 210-215.

During this period, Matsudaira Sadanobu stressed filial piety (kō 孝) and loyalty (chū 忠) as the central virtues, and called filial piety "the basis of all virtue." SeeOoms 1975, p. 33. After the Kansei Reforms, in Kyōwa 享和 1 (1801), the government published Kōgiroku 孝義録 (Records of filial piety and righteousness), whose aim was to educate the populace by collecting all the meritorious deeds that took place in the country. The Kōgiroku comprised fifty volumes and detailed the good deeds of almost eight thousand six hundred individuals, without distinguishing their class. Deeds of filial piety represented 60 percent of the total. For further information, seeSugano 1999, pp. 3-5.

This short excerpt from the Dialogues by Confucius comes from Ansart 2014b, p. 162.

In addition to that, several tales of vengeance present characters appeasing wild horses. This animal appears also in a number of Confucian texts as a symbol of human passions. The act of pacifying the horse would thus represent the capacity of individuals to overcome violent sentiments.

According to Zhu Xi, the real order is feudal and hierarchical; seeThomas 2001b, p. 23.

Original text: 扨此度、其方が忠節にて一人の娘を拾ひたり; see KTK, p. 961.

This work influences late Edo tales of vengeance in many ways. A couple of examples: the mother, who always reminds the two boys of the name of their enemy and their final mission, is recapitulated in many katakiuchi mono of the corpus; the adjective monogatai, applied to victims, describes the virtuous attitude of the father in the Soga monogatari.

This excerpt from the Keikodan comes from Ansart 2014b, p. 203.

Jolles 1972, p. 34.