add alternating groups; add a parent class 'PermutationSampler' for symmetric and alternating groups

automata.py

@@ -18,6 +18,7 @@ import csv
 import json
 import os
 import itertools
+from sympy.combinatorics.permutations import Permutation
 
 import datasets
 import numpy as np
@@ -29,9 +30,6 @@ major, minor = sys.version_info[:2]
 version = major + 0.1*minor
 OLD_PY_VERSION = 1 if version < 3.8 else 0
 
-# Local imports
-# from symmetric import SymmetricSampler
-
 _CITATION = """\
 """
 
@@ -315,19 +313,16 @@ class FlipFlopSampler(AutomatonSampler):
     return x, self.f(x)
 
 
-class SymmetricSampler(AutomatonSampler):
+
+class PermutationSampler(AutomatonSampler):
   """
-  TODO: add options for labels as functions of states
-  - parity (whether a state is even): this may need packages (e.g. Permutation from sympy)
-  - position / toggle: for S3 ~ D6, we can add labels for substructures as in Dihedral groups.
+  Subclasses: SymmetricSampler, AlternatingSampler
   """
   def __init__(self, data_config):
     super().__init__(data_config)
 
     if 'n' not in data_config:
-      data_config['n'] = 5 # Default to S5
-    if 'n_actions' not in data_config:
-      data_config['n_actions'] = 3
+      data_config['n'] = 5
     if 'label_type' not in data_config:
       # Options: 'state', 'first_chair'
       data_config['label_type'] = 'state'
@@ -335,6 +330,46 @@ class SymmetricSampler(AutomatonSampler):
     self.n = data_config['n'] # the symmetric group Sn
     self.label_type = data_config['label_type']
 
+    self.__info__ = \
+          "  - label_type (str): choosing from the following options:\n" \
+        + "    - 'state' (default): the state id.\n" \
+        + "    - 'first_chair': the element in the first position of the permutation.\n" \
+        + "          e.g. if the current permutation is [2,1,4,3], then 'first_chair' is 2.\n" \
+        + self.__info__
+
+  def get_state_label(self, state):
+    enc = self.state_encode(state)
+    return self.state_label_map[enc]
+
+  def f(self, x):
+    curr_state = np.arange(self.n)
+    labels = []
+    for action in x:
+      curr_state = self.actions[action].dot(curr_state)
+
+      if self.label_type == 'state':
+        labels += self.get_state_label(curr_state),
+      elif self.label_type == 'first_chair':
+        labels += curr_state[0],
+
+    return np.array(labels)
+
+  def sample(self):
+    T = self.sample_length()
+    x = np.random.choice(range(self.n_actions), replace=True, size=T)
+
+    return x, self.f(x)
+
+
+class SymmetricSampler(PermutationSampler):
+  """
+  TODO: add options for labels as functions of states
+  - parity (whether a state is even): this may need packages (e.g. Permutation from sympy)
+  - position / toggle: for S3 ~ D6, we can add labels for substructures as in Dihedral groups.
+  """
+  def __init__(self, data_config):
+    super().__init__(data_config)
+
     self.name = f'S{self.n}'
 
     """
@@ -349,6 +384,8 @@ class SymmetricSampler(AutomatonSampler):
     """
     Get actions (3 defaults: id, shift-by-1, swap-first-two)
     """
+    if 'n_actions' not in data_config:
+      data_config['n_actions'] = 3
     self.n_actions = data_config['n_actions']
     self.actions = {0: np.eye(self.n)}
     # shift all elements to the right by 1
@@ -377,39 +414,58 @@ class SymmetricSampler(AutomatonSampler):
         + "- Labels: depending on 'label_type'.\n" \
         + "- Config:\n" \
         + "  - n (int): number of objects, i.e. there are n! states.\n" \
-        + "  - label_type (str): choosing from the following options:\n" \
-        + "    - 'state' (default): the state id.\n" \
-        + "    - 'first_chair': the element in the first position of the permutation.\n" \
-        + "          e.g. if the current permutation is [2,3,1,4], then 'first_chair' is 2.\n" \
+        + "  - n_actions (int): number of permutations to include in the generator set;\n" \
+        + "        the ordering is given by itertools.permutations, and the first 'n_actions' permutations will be included.\n" \
         + self.__info__ 
 
 
-  def get_state_label(self, state):
-    enc = self.state_encode(state)
-    return self.state_label_map[enc]
+class AlternatingSampler(PermutationSampler):
+  """
+  TODO: other choices of generators (currently using (12x))?
+  """
+  def __init__(self, data_config):
+    super().__init__(data_config)
 
-  def f(self, x):
-    curr_state = np.arange(self.n)
-    labels = []
-    for action in x:
-      curr_state = self.actions[action].dot(curr_state)
+    self.name = f'A{self.n}'
 
-      if self.label_type == 'state':
-        labels += self.get_state_label(curr_state),
-      elif self.label_type == 'first_chair':
-        labels += curr_state[0],
+    """
+    Get states
+    """
+    self.state_label_map = {}
+    self.state_encode = lambda state: ''.join([str(int(each)) for each in state])
+    cnt = 0
+    for si, state in enumerate(itertools.permutations(range(self.n))):
+      if not Permutation(state).is_even:
+        continue
+      enc = self.state_encode(state)
+      self.state_label_map[enc] = cnt
+      cnt += 1
+
+    """
+    Get actions: all 3 cycles of the form (12x)
+    """
+    self.actions = {0: np.eye(self.n)}
+    for idx in range(2, self.n):
+      # (1, 2, idx) 
+      shift_idx = list(range(self.n))
+      shift_idx[0],shift_idx[1], shift_idx[idx] = shift_idx[1], shift_idx[idx], shift_idx[0]
+      self.actions[idx-1] = np.eye(self.n)[shift_idx]
+    self.n_actions = len(self.actions)
+
+    self.__info__ = f"Alternating group on n={self.n} objects:\n" \
+        +f"- Inputs: tokens from 0 to n-3, corresponding to all 3-cycles of the form (12x).\n" \
+        + "- Labels: depending on 'label_type'.\n" \
+        + "- Config:\n" \
+        + "  - n (int): number of objects, i.e. there are n!/2 states.\n" \
+        + self.__info__ 
 
-    return np.array(labels)
 
-  def sample(self):
-    T = self.sample_length()
-    x = np.random.choice(range(self.n_actions), replace=True, size=T)
 
-    return x, self.f(x)
 
 
 dataset_map = {
   'abab': ABABSampler,
+  'alternating': AlternatingSampler,
   'gridworld': GridworldSampler,
   'flipflop': FlipFlopSampler,
   'parity': ParitySampler,