# Common Transducers (Finite State Machines Generators)¶

Transducers in Sage can be built through the transducers object. It contains generators for common finite state machines. For example,

sage: I = transducers.Identity([0, 1, 2])


generates an identity transducer on the alphabet $$\{0, 1, 2\}$$.

To construct transducers manually, you can use the class Transducer. See finite_state_machine for more details and a lot of examples.

Transducers

 Identity() Returns a transducer realizing the identity map. abs() Returns a transducer realizing absolute value. operator() Returns a transducer realizing a binary operation. all() Returns a transducer realizing logical and. any() Returns a transducer realizing logical or. add() Returns a transducer realizing addition. sub() Returns a transducer realizing subtraction. CountSubblockOccurrences() Returns a transducer counting the occurrences of a subblock. Wait() Returns a transducer writing False until first (or k-th) true input is read. weight() Returns a transducer realizing the Hamming weight GrayCode() Returns a transducer realizing binary Gray code.

AUTHORS:

• Clemens Heuberger (2014-04-07): initial version
• Sara Kropf (2014-04-10): some changes in TransducerGenerator
• Daniel Krenn (2014-04-15): improved common docstring during review
• Clemens Heuberger, Daniel Krenn, Sara Kropf (2014-04-16–2014-05-02): A couple of improvements. Details see #16141, #16142, #16143, #16186.
• Sara Kropf (2014-04-29): weight transducer
• Clemens Heuberger, Daniel Krenn (2014-07-18): transducers Wait, all, any

ACKNOWLEDGEMENT:

• Clemens Heuberger, Daniel Krenn and Sara Kropf are supported by the Austrian Science Fund (FWF): P 24644-N26.

## Functions and methods¶

class sage.combinat.finite_state_machine_generators.TransducerGenerators

Bases: object

A class consisting of constructors for several common transducers.

A list of all transducers in this database is available via tab completion. Type “transducers.” and then hit tab to see which transducers are available.

The transducers currently in this class include:

CountSubblockOccurrences(block, input_alphabet)

Returns a transducer counting the number of (possibly overlapping) occurrences of a block in the input.

INPUT:

• block – a list (or other iterable) of letters.
• input_alphabet – a list or other iterable.

OUTPUT:

A transducer counting (in unary) the number of occurrences of the given block in the input. Overlapping occurrences are counted several times.

Denoting the block by $$b_0\ldots b_{k-1}$$, the input word by $$i_0\ldots i_L$$ and the output word by $$o_0\ldots o_L$$, we have $$o_j = 1$$ if and only if $$i_{j-k+1}\ldots i_{j} = b_0\ldots b_{k-1}$$. Otherwise, $$o_j = 0$$.

EXAMPLES:

1. Counting the number of 10 blocks over the alphabet [0, 1]:

sage: T = transducers.CountSubblockOccurrences(
....:     [1, 0],
....:     [0, 1])
sage: sorted(T.transitions())
[Transition from () to (): 0|0,
Transition from () to (1,): 1|0,
Transition from (1,) to (): 0|1,
Transition from (1,) to (1,): 1|0]
sage: T.input_alphabet
[0, 1]
sage: T.output_alphabet
[0, 1]
sage: T.initial_states()
[()]
sage: T.final_states()
[(), (1,)]


Check some sequence:

sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: T([0, 1, 0, 1, 1, 0])
[0, 0, 1, 0, 0, 1]

2. Counting the number of 11 blocks over the alphabet [0, 1]:

sage: T = transducers.CountSubblockOccurrences(
....:     [1, 1],
....:     [0, 1])
sage: sorted(T.transitions())
[Transition from () to (): 0|0,
Transition from () to (1,): 1|0,
Transition from (1,) to (): 0|0,
Transition from (1,) to (1,): 1|1]


Check some sequence:

sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: T([0, 1, 0, 1, 1, 0])
[0, 0, 0, 0, 1, 0]

3. Counting the number of 1010 blocks over the alphabet [0, 1, 2]:

sage: T = transducers.CountSubblockOccurrences(
....:     [1, 0, 1, 0],
....:     [0, 1, 2])
sage: sorted(T.transitions())
[Transition from () to (): 0|0,
Transition from () to (1,): 1|0,
Transition from () to (): 2|0,
Transition from (1,) to (1, 0): 0|0,
Transition from (1,) to (1,): 1|0,
Transition from (1,) to (): 2|0,
Transition from (1, 0) to (): 0|0,
Transition from (1, 0) to (1, 0, 1): 1|0,
Transition from (1, 0) to (): 2|0,
Transition from (1, 0, 1) to (1, 0): 0|1,
Transition from (1, 0, 1) to (1,): 1|0,
Transition from (1, 0, 1) to (): 2|0]
sage: input =  [0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 2]
sage: output = [0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0]
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: T(input) == output
True

GrayCode()

Returns a transducer converting the standard binary expansion to Gray code.

INPUT:

Nothing.

OUTPUT:

A transducer.

Cf. the Wikipedia article Gray_code for a description of the Gray code.

EXAMPLE:

sage: G = transducers.GrayCode()
sage: G
Transducer with 3 states
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: for v in srange(0, 10):
....:     print v, G(v.digits(base=2))
0 []
1 [1]
2 [1, 1]
3 [0, 1]
4 [0, 1, 1]
5 [1, 1, 1]
6 [1, 0, 1]
7 [0, 0, 1]
8 [0, 0, 1, 1]
9 [1, 0, 1, 1]


In the example Gray Code in the documentation of the finite_state_machine module, the Gray code transducer is derived from the algorithm converting the binary expansion to the Gray code. The result is the same as the one given here.

Identity(input_alphabet)

Returns the identity transducer realizing the identity map.

INPUT:

• input_alphabet – a list or other iterable.

OUTPUT:

A transducer mapping each word over input_alphabet to itself.

EXAMPLES:

sage: T = transducers.Identity([0, 1])
sage: sorted(T.transitions())
[Transition from 0 to 0: 0|0,
Transition from 0 to 0: 1|1]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T.input_alphabet
[0, 1]
sage: T.output_alphabet
[0, 1]
sage: sage.combinat.finite_state_machine.FSMOldProcessOutput = False
sage: T([0, 1, 0, 1, 1])
[0, 1, 0, 1, 1]

Wait(input_alphabet, threshold=1)

Writes False until reading the threshold-th occurrence of a true input letter; then writes True.

INPUT:

• input_alphabet – a list or other iterable.
• threshold – a positive integer specifying how many occurrences of True inputs are waited for.

OUTPUT:

A transducer writing False until the threshold-th true (Python’s standard conversion to boolean is used to convert the actual input to boolean) input is read. Subsequently, the transducer writes True.

EXAMPLES:

sage: T = transducers.Wait([0, 1])
sage: T([0, 0, 1, 0, 1, 0])
[False, False, True, True, True, True]
sage: T2 = transducers.Wait([0, 1], threshold=2)
sage: T2([0, 0, 1, 0, 1, 0])
[False, False, False, False, True, True]

abs(input_alphabet)

Returns a transducer which realizes the letter-wise absolute value of an input word over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.

OUTPUT:

A transducer mapping $$i_0\ldots i_k$$ to $$|i_0|\ldots |i_k|$$.

EXAMPLE:

The following transducer realizes letter-wise absolute value:

sage: T = transducers.abs([-1, 0, 1])
sage: T.transitions()
[Transition from 0 to 0: -1|1,
Transition from 0 to 0: 0|0,
Transition from 0 to 0: 1|1]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([-1, -1, 0, 1])
[1, 1, 0, 1]


Returns a transducer which realizes addition on pairs over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.
• number_of_operands – (default: $$2$$) it specifies the number of input arguments the operator takes.

OUTPUT:

A transducer mapping an input word $$(i_{01}, \ldots, i_{0d})\ldots (i_{k1}, \ldots, i_{kd})$$ to the word $$(i_{01} + \cdots + i_{0d})\ldots (i_{k1} + \cdots + i_{kd})$$.

The input alphabet of the generated transducer is the cartesian product of number_of_operands copies of input_alphabet.

EXAMPLE:

The following transducer realizes letter-wise addition:

sage: T = transducers.add([0, 1])
sage: T.transitions()
[Transition from 0 to 0: (0, 0)|0,
Transition from 0 to 0: (0, 1)|1,
Transition from 0 to 0: (1, 0)|1,
Transition from 0 to 0: (1, 1)|2]
sage: T.input_alphabet
[(0, 0), (0, 1), (1, 0), (1, 1)]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([(0, 0), (0, 1), (1, 0), (1, 1)])
[0, 1, 1, 2]


More than two operands can also be handled:

sage: T3 = transducers.add([0, 1], number_of_operands=3)
sage: T3.input_alphabet
[(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1),
(1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)]
sage: T3([(0, 0, 0), (0, 1, 0), (0, 1, 1), (1, 1, 1)])
[0, 1, 2, 3]

all(input_alphabet, number_of_operands=2)

Returns a transducer which realizes logical and over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.
• number_of_operands – (default: $$2$$) specifies the number of input arguments for the and operation.

OUTPUT:

A transducer mapping an input word $$(i_{01}, \ldots, i_{0d})\ldots (i_{k1}, \ldots, i_{kd})$$ to the word $$(i_{01} \land \cdots \land i_{0d})\ldots (i_{k1} \land \cdots \land i_{kd})$$.

The input alphabet of the generated transducer is the cartesian product of number_of_operands copies of input_alphabet.

EXAMPLE:

The following transducer realizes letter-wise logical and:

sage: T = transducers.all([False, True])
sage: T.transitions()
[Transition from 0 to 0: (False, False)|False,
Transition from 0 to 0: (False, True)|False,
Transition from 0 to 0: (True, False)|False,
Transition from 0 to 0: (True, True)|True]
sage: T.input_alphabet
[(False, False), (False, True), (True, False), (True, True)]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([(False, False), (False, True), (True, False), (True, True)])
[False, False, False, True]


More than two operands and other input alphabets (with conversion to boolean) are also possible:

sage: T3 = transducers.all([0, 1], number_of_operands=3)
sage: T3([(0, 0, 0), (1, 0, 0), (1, 1, 1)])
[False, False, True]

any(input_alphabet, number_of_operands=2)

Returns a transducer which realizes logical or over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.
• number_of_operands – (default: $$2$$) specifies the number of input arguments for the or operation.

OUTPUT:

A transducer mapping an input word $$(i_{01}, \ldots, i_{0d})\ldots (i_{k1}, \ldots, i_{kd})$$ to the word $$(i_{01} \lor \cdots \lor i_{0d})\ldots (i_{k1} \lor \cdots \lor i_{kd})$$.

The input alphabet of the generated transducer is the cartesian product of number_of_operands copies of input_alphabet.

EXAMPLE:

The following transducer realizes letter-wise logical or:

sage: T = transducers.any([False, True])
sage: T.transitions()
[Transition from 0 to 0: (False, False)|False,
Transition from 0 to 0: (False, True)|True,
Transition from 0 to 0: (True, False)|True,
Transition from 0 to 0: (True, True)|True]
sage: T.input_alphabet
[(False, False), (False, True), (True, False), (True, True)]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([(False, False), (False, True), (True, False), (True, True)])
[False, True, True, True]


More than two operands and other input alphabets (with conversion to boolean) are also possible:

sage: T3 = transducers.any([0, 1], number_of_operands=3)
sage: T3([(0, 0, 0), (1, 0, 0), (1, 1, 1)])
[False, True, True]

operator(operator, input_alphabet, number_of_operands=2)

Returns a transducer which realizes an operation on tuples over the given input alphabet.

INPUT:

• operator – operator to realize. It is a function which takes number_of_operands input arguments (each out of input_alphabet).
• input_alphabet – a list or other iterable.
• number_of_operands – (default: $$2$$) it specifies the number of input arguments the operator takes.

OUTPUT:

A transducer mapping an input letter $$(i_1, \dots, i_n)$$ to $$\mathrm{operator}(i_1, \dots, i_n)$$. Here, $$n$$ equals number_of_operands.

The input alphabet of the generated transducer is the cartesian product of number_of_operands copies of input_alphabet.

EXAMPLE:

The following binary transducer realizes component-wise addition (this transducer is also available as add()):

sage: import operator
sage: T = transducers.operator(operator.add, [0, 1])
sage: T.transitions()
[Transition from 0 to 0: (0, 0)|0,
Transition from 0 to 0: (0, 1)|1,
Transition from 0 to 0: (1, 0)|1,
Transition from 0 to 0: (1, 1)|2]
sage: T.input_alphabet
[(0, 0), (0, 1), (1, 0), (1, 1)]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([(0, 0), (0, 1), (1, 0), (1, 1)])
[0, 1, 1, 2]


Note that for a unary operator the input letters of the new transducer are tuples of length $$1$$:

sage: T = transducers.operator(abs,
....:                          [-1, 0, 1],
....:                          number_of_operands=1)
sage: T([-1, 1, 0])
Traceback (most recent call last):
...
ValueError: Invalid input sequence.
sage: T([(-1,), (1,), (0,)])
[1, 1, 0]


Compare this with the transducer generated by abs():

sage: T = transducers.abs([-1, 0, 1])
sage: T([-1, 1, 0])
[1, 1, 0]

sub(input_alphabet)

Returns a transducer which realizes subtraction on pairs over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.

OUTPUT:

A transducer mapping an input word $$(i_0, i'_0)\ldots (i_k, i'_k)$$ to the word $$(i_0 - i'_0)\ldots (i_k - i'_k)$$.

The input alphabet of the generated transducer is the cartesian product of two copies of input_alphabet.

EXAMPLE:

The following transducer realizes letter-wise subtraction:

sage: T = transducers.sub([0, 1])
sage: T.transitions()
[Transition from 0 to 0: (0, 0)|0,
Transition from 0 to 0: (0, 1)|-1,
Transition from 0 to 0: (1, 0)|1,
Transition from 0 to 0: (1, 1)|0]
sage: T.input_alphabet
[(0, 0), (0, 1), (1, 0), (1, 1)]
sage: T.initial_states()
[0]
sage: T.final_states()
[0]
sage: T([(0, 0), (0, 1), (1, 0), (1, 1)])
[0, -1, 1, 0]

weight(input_alphabet, zero=0)

Returns a transducer which realizes the Hamming weight of the input over the given input alphabet.

INPUT:

• input_alphabet – a list or other iterable.
• zero – the zero symbol in the alphabet used

OUTPUT:

A transducer mapping $$i_0\ldots i_k$$ to $$(i_0\neq 0)\ldots(i_k\neq 0)$$.

The Hamming weight is defined as the number of non-zero digits in the input sequence over the alphabet input_alphabet (see Wikipedia article Hamming_weight). The output sequence of the transducer is a unary encoding of the Hamming weight. Thus the sum of the output sequence is the Hamming weight of the input.

EXAMPLES:

sage: W = transducers.weight([-1, 0, 2])
sage: W.transitions()
[Transition from 0 to 0: -1|1,
Transition from 0 to 0: 0|0,
Transition from 0 to 0: 2|1]
sage: unary_weight = W([-1, 0, 0, 2, -1])
sage: unary_weight
[1, 0, 0, 1, 1]
sage: weight = add(unary_weight)
sage: weight
3


Also the joint Hamming weight can be computed:

sage: v1 = vector([-1, 0])
sage: v0 = vector([0, 0])
sage: W = transducers.weight([v1, v0])
sage: unary_weight = W([v1, v0, v1, v0])
2


For the input alphabet [-1, 0, 1] the weight transducer is the same as the absolute value transducer abs():

sage: W = transducers.weight([-1, 0, 1])
sage: A = transducers.abs([-1, 0, 1])
sage: W == A
True


For other input alphabets, we can specify the zero symbol:

sage: W = transducers.weight(['a', 'b'], zero='a')
sage: add(W(['a', 'b', 'b']))
2