JPH0636155B2 - Pattern comparison device - Google Patents

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a pattern comparison device that compares a plurality of types of registered patterns with an input pattern to identify the input pattern, and in particular to a continuously uttered word voice. The present invention relates to a pattern comparison device that can be applied to, for example, recognition.

2. Description of the Related Art Since voice, which is the most natural means of generating information for humans, is fully utilized as a human-machine input means, continuous normal speech recognition is possible without limiting the speaker. It is desirable.

FIG. 2 is a block diagram of a voice recognition device using a word unit as a recognition unit. (21) is an input terminal of a voice signal, (22) is an acoustic analysis unit that converts the input voice signal into a series of several numerical value groups (feature vectors) by frequency analysis, LPC analysis, PARCOR analysis, correlation analysis, etc. (23) is a standard pattern storage unit in which a word to be recognized is registered as a sequence of the feature vectors, (24) is a sequence of the feature vectors for the input speech signal to be recognized analyzed by the acoustic analysis unit (22). And a pattern matching unit that compares each of the standard patterns and calculates the distance or the similarity between the two, (25) is a standard pattern that is the closest to the input voice pattern based on the calculation result of the pattern matching unit (24). A determination unit that determines a corresponding word as a recognition result, and (26) is an output terminal that outputs this recognition result. In the speech recognition apparatus having such a configuration, as a pattern matching method, a method of performing matching (DP matching) by time-axis nonlinear expansion / contraction by dynamic programming is superior.

This DP matching is used in the continuous word recognition by the device of the present invention. Next, the DP matching algorithm will be briefly described. Now Are two voice patterns. That is, those voice patterns are represented by a series of feature vectors a _i and b _j for each.

When the distance between the vectors a _i and b _j is d (i, j), the weighted average of d (i, j) is calculated for various correspondences of the vectors forming the two series, and it is minimized. The optimum correlation between the two series is defined as follows, and the weighted average at that time is defined as the distance D (A, B) between the two series. This procedure is efficiently performed using dynamic programming. Is DP matching. Note that the Euclidean distance or city distance between the vectors a _i and b _j is usually used as d (i, j).

FIG. 3 illustrates this two-dimensionally. Correspondence of time between both patterns A and B, that is, the time conversion function j (i) is i−j.
Sequence of grid points c (K) = (i (K), j (K)) on the plane F = c (1) c (2) ... c (k) ... c (K) ... (2) (i ( K) = I, j (k) = J). At this time, D (A, B) is defined as follows.

Where w (K) is a non-negative constant whose value is the time conversion function j
It is determined by the method used when approximating (i) with a sequence of points.
Here, the denominator of equation (3) is a constant that does not depend on F. Then, D (A, B) can be efficiently obtained by dynamic programming. That is, Therefore, with g (c (1)) ＝ g (1,1) ＝ d (1,1), the recurrence formula
If (4) is solved and g (c (K)) ＝ g (I, J) is obtained, As D (A, B) is required.

As a method of making the denominator of the formula (3) constant, there are a method of setting M = I + J (symmetrical type) and a method of setting M = I or J (asymmetrical type). FIGS. 4 (a) to 4 (f) show examples of constraint conditions when selecting the point sequence F, and the route to the point (i, j) can be only the route indicated by the arrow in the figure. Also, the number shown on each line segment indicates the load w (K) when the line segment is selected as the route. (a) and (b) are M = I + J in the symmetric example, and (c) to (f) are M = I + J in the asymmetric type.
It becomes I.

The following is how to recognize a word voice using such a matching method. The word class to be recognized is represented by n (n = 1 to N), and its standard pattern is represented by B ⁿ . Distance between input A and each standard pattern B ⁿ D _n = D (A,
B ⁿ ) is calculated by the above method, The class n ₀ that gives is the recognition result for A.

If M = I is set in the asymmetric DP matching, M becomes a quantity related only to the input pattern length.
In (5), M is constant for any standard pattern, Can be defined as When the formula (6) is obtained under the constraint condition of FIG. 4 (c), the following recurrence formula (7) may be calculated.

Initial condition g (1,1) = d (1,1) Next, recognition of continuous word speech will be described. Continuous word speech recognition can be formulated as follows. Now X words q
A (1), q (2), ... Q (x) represents a voice pattern when it is uttered continuously.

A = a ₁ a ₂ … a _i … a _I …… (8) The standard pattern of the word q (x) is , The standard pattern obtained by connecting _x words B _{q (1)} , B _{q (2)} , ... B _{q (x)} is It is represented by. Here, the continuity of the pattern is represented.

Therefore, in continuous word voice recognition, DP matching is performed between this and the input voice pattern A, and D obtained at that time is obtained.
X and q (x) (x = 1,2, ..., so that (A,) is minimized
The problem is to decide x). Ie Is calculated, and the condition for minimizing T may be obtained. Formula (11)
A huge amount of calculation is required to properly execute the calculation of. That is, the maximum value of the number of continuously uttered words in the input voice pattern is K, and the number of standard word patterns is N.
Then, N ^K calculations will be executed. Therefore, we actually reduce this problem to the problem of solving the following recurrence formula.

In the input voice pattern A, a partial section from i = l + 1 to i = m is defined by a partial pattern A (l, m).

A (l, m) = a l + 1 a l + 2 ... a m ...... (12) In this case, the following can be said by defining the distance between the patterns by equation (6).

Using this fact, equation (11) can be solved as follows.

The meanings of the symbols used hereafter are summarized in Table 1.

If the solution of the recurrence formula is obtained, the recognition result is sequentially obtained from the last word name and segmentation result of the X word string to the first word name and segmentation result according to the flowchart shown in FIG.

ii) When the number of input words X is unknown The recognition result is obtained from the solution of the recurrence formula shown in the flowchart of FIG.

A two-stage DP method has been proposed to realize the above concept. An outline of the following two-stage DP method will be described.

The two-stage DP method is a method in which D ⁿ ₀ (s: t) is first obtained by DP for all combinations of s and t, and then D (i) is obtained by DP.
The feature is that the DP is in two stages. As the two-stage DP method, a forward algorithm and a backward algorithm have been proposed. Here, the backward algorithm will be described.

For frame i-1 of the input pattern, D (i-1), N (i
It is assumed that -1) and B (i-1) have been obtained.

DP matching is performed on the standard pattern of the word n (n = 1, 2 ... N) and the input pattern in the reverse time direction starting from i. Therefore, the constraint conditions of the route are shown in FIGS. 8 (a) (b) (c) (d) corresponding to FIGS. 4 (c) (d) (e) (f). The matching range may be set with the matching window width R, but here the slope is 1/2
It shall be performed in the range of 2 (within the inclination limit, the shaded portion in FIG. 7). This matching is performed without termination. As a result, D ⁿ ₀ (s: t) is obtained. However, i-2J ⁿ + 1 ≦ s ≦ i−
(1/2) J ⁿ .

Find D (i), N (i), and B (i) in equation (15).

Return to i = i + 1.

According to the above idea, the constraint condition of the matching route is as shown in FIG.
As shown in (c) to (f), it is an asymmetric type, and the normalized coefficient M is the number of frames of the input pattern (more generally, when weights are appropriately introduced into each frame of the input pattern). It depends on the total number of weights only), and in the case of a symmetrical path as shown in FIGS. 4 (a) and 4 (b), the normalization coefficient depends on the number of frames of the standard pattern. As a result, the recurrence formulas (14) and (15) are not established.

Next, the reason will be described. In this case, the cumulative matching distance also changes depending on the standard pattern length. Therefore, in order to evaluate which standard pattern best fits, the cumulative matching distance between the two patterns is calculated by summing the input pattern length and the standard pattern length as described above. It was necessary to divide (normalize) the distance.

Now, the standard pattern that best matches the partial pattern A (0, m) of the input pattern A is B ₁ , its length is b ₁ , other standard patterns are B ₂ , and its length is b _2. Then, the following equation holds.

Here, D (P, Q) represents the cumulative matching distance between the pattern P and the pattern Q before normalization.

Consider a case where the input is the i-th frame and the grasshopper pointer and the last word (monosyllabic) are searched based on equations (14) and (15) (assuming that the input pattern length and the standard pattern length are used for normalization). Assuming that the last word is X, its length is x, and the back pointer is m, the cumulative matching distance between the standard pattern combining B ₁ and X and the input partial pattern A (0, i) is the input pattern length. Normalized by the sum of standard pattern length It is represented by. In order to search m and X by the equations (14) and (15), α must be smaller than the following value.

That is, if β <α holds, D in Eq. (15)
This is because it is not possible to use D (m) obtained in the m-th frame as (m).

However, α <β does not generally hold. For example, if D (A (0, m), B ₁ ) = 10, D (A (0, m), B ₂ ) = 20 m = 20, b ₁ = 10, b ₂ = 20, then equation (16) In, the left side = 10 / (20 + 10) = 1/3 and the right side = 20 / (20 + 20) = 1/2, and the above numerical values satisfy equation (16). However, if i = 40, x = 10, D (A (m + 1, i), X) = 60, α = (10 + 60) / (40 + 10 + 10) = 7/6 β = (20 Since +60) / (40 + 20 + 10) = 8/7, α> β, and equation (16) is no longer satisfied.

However, the asymmetric type that depends only on the input pattern length
In case of DP method If Since it is clear, equations (14) and (15) can be used without contradiction.

As described above, when the symmetrical DP path is used, continuous word speech recognition based on the above formulation cannot be performed as it is.

Means for Solving the Problems In order to solve the above problems, the present invention uses an input signal as a sequence of feature vectors. Feature extraction means for converting to A standard pattern B ⁿ (where n = 1, 2, ..., N)
Is a sequence of feature vectors with a fixed number of frames J A standard pattern normalizing means for converting into a standard pattern, a standardized standard pattern storing means for storing the standardized standard pattern, a partial pattern A (0, m) from the first frame to the m-th frame of the input pattern, and Normalized standard pattern x-1
The minimum cumulative matching distance D _x-1 (m) with the concatenated pattern obtained by optimally concatenating the
Partial pattern A (m, i) from the 1st frame to the i-th frame
And the normalized standard pattern And the minimum cumulative matching distance D ⁿ _x (m + 1: i) for n, m are minimized to obtain the minimum cumulative matching distance D _x (i from the first frame to the i-th frame of the input pattern. ) Is calculated and the m, n at that time is calculated as B _x (i), N
_When the back pointer storage means and the last word storage means to be stored as _x (i) and the final frame of the input when the input is completed are I, and the input consists of x words, the final word is N. _x (I), the second to last word is N
_x-1 (B _x (I)), the third to last word is N _x-2 (B _x-1 (B
_x (I))) ... and the like, which has means for recognizing consecutively input word sounds in reverse order, and when the number of input words is specified, the input is the specified number of words. It outputs the optimum word string under the assumption.

Action With this configuration, the series of feature vectors A standard pattern B ⁿ (where n = 1, 2, ..., N)
A standardized standard pattern with a fixed number of frames J And the minimum cumulative matching distance D _x-1 between the partial pattern from the first frame to the m-th frame of the input pattern and the concatenated pattern obtained by optimally concatenating the normalized standard patterns _x-1 (m) and the (m + 1) th input pattern
Minimum cumulative matching distance D ⁿ _x (m + 1: i) between the partial pattern from the frame to the i-th frame and the normalized standard pattern ⁿ
The minimum cumulative matching distance D _x (i) from the 1st frame to the i-th frame of the input turn is calculated assuming that the sum of and is minimized for n and m, and m at that time is calculated as the back pointer D.
_{Let x} (i) be the last word name N _x (i) when n is the end of the input at the i-th frame, and i = 1, 2 ... I, x = 1, 2,
...... For each X, B _x (i) and N _x (i) are sequentially obtained, and when the input is completed (when i = I), if the input consists of x words, the final word is N _x ( i), the second last word is N _x-1 (B _x (I)), the third last word is N _x-2 (B _x-1 (B
_x (I))) ... Recognizes word speech continuously input in the reverse order.

Embodiment An embodiment of the present invention will be described below with reference to the drawings. First
FIG. 1 is a block diagram showing an embodiment of a continuous voice recognition device of the present invention. In Figure 1, an input terminal of the I _n audio signal, (1) is a feature extraction composed of a filter bank or the like, wherein the input speech signal vector It is converted into the series A. (2) is a word standard pattern storage unit in which N words, which are recognition vocabularies, are standard patterns (N = 1, 2, ..., N) is registered in advance in the form of a series of feature vectors. (3) is a standard pattern normalization unit that linearly expands and contracts the j-th frame of the pattern B ⁿ held in the word standard pattern storage unit (2). Corresponding to the frames, the time axis normalization is performed to convert the pattern of the frame number J ^{n into} the pattern of the frame number J. Here, J is a constant. FIG. 9 is a diagram for explaining this conversion method. FIG. 9 (a) shows that the frame j of the standard word pattern B ⁿ is It is shown to be converted into. (100) where x is the horizontal axis and y is the vertical axis Is a straight line given by. Actually, j can take only integer values. By j and Correspond to. Here, [Z] represents an integer which is less than or equal to Z and is closest to Z. FIG. 9 (b) shows the vector of the j-th frame in the pattern B ⁿ represented by the series of feature vectors. And the vector before time axis normalization And vector after time axis normalization To explain the relationship. According to equation (20)) And j corresponding to Which is below the intersection (102) of the straight line and the straight line (100),
Closest to the intersection It is mapped as the y coordinate of the upper grid point (103), where Is A vector at the y coordinate of the lattice point (101) which is above and is closest to the intersection (102) above It is obtained by linear interpolation from and. That is, if the intersection point (102) divides the grid point (101) and the grid point (103) into 1-S vs. S, Given in. S is in equation (19) It is obtained by subtracting j given by equation (20) from y obtained by substituting I.e. Becomes

(4) is a normalized standard pattern storage unit, which stores a series of feature vectors whose time axis is normalized by the standard pattern normalization unit (3) as a standard pattern. In this way, if the number of frames in the standard pattern is made constant regardless of words,
Equation (14) holds even when the symmetrical DP path is used. That is, in the above example, if b ₁ = b ₂ = c = J (const), then corresponding to equations (17) and (18) Therefore, according to equation (16) This is because α <β can always be said if is satisfied. However, these denominators are equal when the number of words is the same, and the above can be said, but the number of words is unknown as in the case of Equation (15), which requires comparison for different numbers of words. It cannot be used in some cases. Therefore, the present invention is applicable when the number of words is known.

(5) is an inter-vector distance calculation unit, which is a feature vector in the i-th frame of the input pattern And the nth word standard pattern Feature vector The distance d ⁿ (i, j) with respect to j = 1, 2, ..., J is obtained. In the present embodiment, ^assuming that the matching route shown in FIG. 4 (b) is used, d ⁿ (i, j) is set to 1 in the next calculation each time.
Since it is used only once, according to the method of the present embodiment, it is not necessary to store it once it has been used once. Therefore, it is possible to set d = d ⁿ (i, j) here. d ⁿ (i,
j) is for example It can be defined as the city distance between and. That is, the vector dimension is l When Etc.

(6) is a cumulative distance calculation unit, which is a part for obtaining the solution of the recurrence formula (14) in the case where the route constraint condition is as shown in FIG. 4 (b). Intermediate cumulative distance D ⁿ _x (i, j) for the i-th frame,
The terminal cumulative distance D _x (i), the intermediate back pointer D ⁿ _x (i, j), and the back pointer B _x (i) are obtained for j = 1,2, ... J; n = 1,2, ... N, and i Find N _x (i) indicating the last word when is the end frame. According to the constraint condition of the matching path in this embodiment, D ⁿ _x (i, j) and D _x (i) are j = 1,2, ...
... J is obtained by sequentially calculating the following recurrence formulas.

Initial condition D ⁿ _x (i-1,0) ＝ D _x-1 (i-1) D ⁿ _x (i, j) and B ⁿ _x (i) are obtained from the following equations.

Initial condition D ⁿ _x (i-1,0) ＝ i-1 Also Is.

Equations (24) to (28) are calculated for each of x = 1, 2, ... X, and finally for each frame, D _x (i), B _x (i), N _x
Get (i). Here, since D _x (i) is necessary only in the calculation of the next frame, it is possible to set D _x = D _x (i). That is, only D ₁ , D ₂ , ... D _x need be stored.

The terminal cumulative distance D _x , the back pointer B _x (i), and the last word name N _x (i) obtained as described above are respectively x = 1,2,
, X are stored in the cumulative end distance storage unit (7), the back pointer storage unit (8), and the last word name storage unit (9).

Note that D ⁿ (i, j), B ⁿ (i, j) (j = 1,2, ... J; n = 1,2, ... N)
Are temporarily stored in the internal memory of the cumulative distance calculation unit (6) until they are no longer needed. In this embodiment, the values of the previous frame and the current frame are used for those calculations of the i-th frame. It is necessary to store only two frames because it is necessary only. That is, for k = 0.1, D ⁿ (k, j), B ⁿ (k, j)
For example, in an input odd-numbered frame, k = 0 is calculated as the previous frame, and k = 1 is calculated as the current frame. In an even-numbered input frame, k = 1 is calculated as the previous frame and k = 0 is calculated as the current frame. It can be calculated as a frame.

Reference numeral (10) is a voice section detection unit that determines the voice section from the magnitude of the input signal and the like. Voice section detector (10)
However, when it is detected that voice input is started, the frame number counting unit (11) starts counting for each frame. The above-mentioned processing was processing for the i-th frame, but the count value of the frame number counting section (11) sets the i. Therefore, the same processing as described above is performed each time the frame advances. The frame number counting unit (11) starts counting when a voice section is detected, and is reset when the voice section ends. Therefore, N _x (i) and B _x (i) are i = 1, 2, ... I in the last word storage section (9) and the back pointer storage section (8).
x = 1, 2, ... X will be stored.

The segmentation unit (12) is a back point storage unit (8)
In response, a command to read a predetermined back pointer is issued. That is, the segmentation unit (12)
When the value B _x (i) is issued to the back pointer storage unit (8), the back pointer B _x-1 (B
_x (i)) is read. When the segmentation unit (12) receives a value of B _{x -1} (B _x (i)) from the back pointer storage unit (8), the segmentation unit (12) issues the same value to the back pointer storage unit (8).
Therefore, when the voice section detecting unit (10) detects the end of voice input, the final position I and the word number X of the frame number counting unit (11) are supplied to the segmentation unit (12), and the segmentation unit (12) first The values I and X are issued to the back pointer storage unit (8). Thereafter, according to the operation described above, B _x (I), B _x-1 (B _x (I)), B _x-2 (B _x-1 (B
_x (I))), ... Outputs are sequentially obtained. These values are the end frame of the penultimate word, ibid.
4th ending frame, 4th ending frame, ...
Since N _x (i) is a word ending in the i frame, if this value is given to the last word storage unit (9) as it is, the recognition result is obtained from the last word in the reverse order. In order to prevent the recognition results from being obtained in the reverse order, this order conversion is performed on the output of the back pointer storage unit (8) or on the output of the last word storage unit (9). Just go.

FIG. 10 is a flow chart in the case where the function of the apparatus of the above-described embodiment is realized by software, and the operation of each unit of the apparatus of the above-described embodiment will be described below.

Steps (100) to (105) are cumulative distance D _x , intermediate cumulative distance D ⁿ _x
(i, j), back pointer B _x (i), intermediate back pointer D ⁿ _x
This is the part that initializes (i, j).

Steps (107) to (117) are processing executed in the i-th frame. Step (111) is a part for giving the intermediate cumulative distance and the initial value of the intermediate back pointer in the i-th frame. The processes of steps (114) to (117) are processes mainly performed by the cumulative distance calculation unit (6). Notation in step (116) Is the ⁿ that minimizes D ⁿ _x (k, J ⁿ ). It means to put. In step (114), the intermediate cumulative distance D ⁿ _x (k, j) and the intermediate back pointer B ⁿ _x (k, j) are obtained. Step (116) calculates n = 1,2,
, N when the i-th frame of the input input is the end of the input, the last word with which the cumulative distance D ⁿ _x (k, J ⁿ ) is minimized Is a process for obtaining. Step (117) is the best word found in step (116) As opposed to Are stored in the memory respectively, and those memories are the last word storage unit (9) and the cumulative distance storage unit.
(7) corresponds to the back pointer storage unit (8).

When i is an odd number by steps (106) and (108), k =
When 1 and i are even numbers, k = 0, and as described above, the frame is switched in the memory for storing the intermediate cumulative distance D ⁿ _x (k, j) and the intermediate back pointer B ⁿ _x (k, j). Each time it is switched, the storage for the immediately preceding frame and the storage for the current frame are switched alternately. That is, step (11
In 1), (114), (116) and (117), k means the current frame and k means the immediately preceding frame.

Steps (118) to (120) are N _x obtained as above.
(i), B _x (i) is the part that obtains the word recognition result in the reverse order. The segmentation part (12), back pointer storage part
(8) corresponds to the processing performed between the last word storage unit (9).

Various values can be set as x in step (118).
For example, if x = 5 is calculated, the number of input words is 1
Since D _x (I) has been obtained for each of the cases of And the steps (118) to (200) are calculated from x =, recognition is possible even when the number of input words is unknown for x = 1 to 5.

In the present embodiment, the case has been described in which consecutively uttered words are recognized, but a single syllable or the like may be used instead of the word, and the present invention can be applied to the recognition of other continuous patterns.

Although the constraint condition of the route shown in FIG. 4 (b) is used, other constraint conditions such as the constraint condition shown in FIG. 4 (a) may be used, and the physical condition of the input pattern It can be set appropriately according to the physical properties.

Further, although the present embodiment has been described by using the distance, the similarity between vectors can be defined by the similarity by using the correlation or the like, and the method of maximizing the similarity instead of minimizing the distance. Can be achieved with exactly the same treatment.

EFFECTS OF THE INVENTION As described above, according to the present invention, continuously uttered voices can be recognized using a symmetrical DP path, and a continuous voice recognition device with high recognition accuracy can be realized.

[Brief description of drawings]

FIG. 1 is a block diagram showing an embodiment of the present invention, FIG. 2 is a block diagram explaining the principle of a voice recognition device, FIG. 3 is a diagram explaining non-linear expansion / contraction matching in isolated word voice recognition, and FIG. FIG. 6 is a diagram showing an example of a route restriction condition in DP matching, FIG. 5 is a flow chart showing a flow of processing in a word determination stage when continuous word speech recognition is made known, and FIG. 6 is also a word number. FIG. 7 is a flowchart showing an unknown condition, FIG. 7 is a diagram showing a range of matching when performing continuous word speech recognition, and FIG. 8 is a diagram showing an example of a route limiting condition when performing reverse time DP matching, and FIG. Is a diagram for explaining a method of exchanging the length of the standard pattern with a constant value to create a normalized standard pattern, and FIG. 10 is an NS chart for explaining a method of realizing the above-mentioned embodiment by software. (1) …… Feature extraction unit, (2) …… Word standard pattern storage unit,
(3) …… Standard pattern normalization unit, (4) Normalized standard pattern storage unit, (5) …… Vector distance calculation unit, (6) …… Cumulative distance calculation unit, (7) …… Terminal cumulative distance Storage unit, (8) …… Pack pointer storage unit, (9) …… End word storage unit, (10) ……
Voice section detector, (11) …… Frame number counter, (12) ……
Segmentation department

Claims (1)

[Claims] 1. A sequence of feature vectors a ₁ a ₂ ...
Feature extraction means for converting to a _I , and feature vector series A standard pattern B ⁿ (where n = 1, 2, ..., N)
Is a sequence of feature vectors with a fixed number of frames J A standard pattern normalizing means for converting into a standard pattern, a standardized standard pattern storing means for storing the standardized standard pattern, a partial pattern A (0, m) from the first frame to the m-th frame of the input pattern, and The normalized standard pattern is x-
Minimum cumulative matching distance (or maximum cumulative matching similarity) D with the connection pattern obtained by optimally connecting one
_x-1 (m), the partial pattern A (m, i) from the (m + 1) th frame to the i-th frame of the input pattern, and the normalized standard pattern Minimum cumulative matching distance (or maximum cumulative matching similarity) D
The sum of ⁿ _x (m + 1: i) and n, m are minimized (maximized) to obtain the partial pattern A (0, i) from the first frame to the i-th frame of the input pattern and the normalization. Minimum (maximum) distance (similarity) between x standard patterns and combined patterns
Minimum cumulative matching distance (or maximum cumulative matching similarity)
Cumulative matching distance (or cumulative matching similarity) calculating means for calculating D _x (i), back pointer storing means for storing m at that time as B _x (i), and n at that time are N _x (i) ) And the last frame of the input when the input is completed, and the final frame of the input is I, if the input consists of x words, the final word is N _x (I), Two
The third word is N _x-1 (B _x (I)), the third last word is N
A pattern comparison device having means for recognizing continuously input word sounds in the reverse order such as _x-2 (B _x-1 (B _x (I))).