Corrected word_problem parsing and added algorithm option

This replaces the fragile string parsing in PermutationGroupElement.word_problem
with a stable  method using GAP's syllable functions (NumberSyllables,
GeneratorSyllable, ExponentSyllable). This resolves parsing errors
involving parentheses and generator name conflicts (e.g., x10 vs x1).

Introduces an 'algorithm' parameter:
- 'syllable' (Now default): Returns 1-based generator indices when as_list=True.
- 'legacy': Old implementation, preserved for testing but deprecated
  due to known bugs. Returns generator strings when as_list=True.

Updated docstrings and examples accordingly.

Fixes #36419
Related: #28556

Author: EigenVector22

src/sage/groups/perm_gps/permgroup_element.pyx

@@ -109,6 +109,7 @@ Check that :issue:`13569` is fixed::
 # ****************************************************************************
 
 import copy
+import warnings
 
 from libc.stdlib cimport qsort
 
@@ -1991,102 +1992,252 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             entries[i, self.perm[i]] = 1
         return M(entries)
 
-    def word_problem(self, words, display=True, as_list=False):
+    def word_problem(self, words, display=True, as_list=False, algorithm='syllable'):
         r"""
         Try to solve the word problem for ``self``.
 
         INPUT:
 
-        - ``words`` -- list of elements of the ambient group, generating
-          a subgroup
-
-        - ``display`` -- boolean (default: ``True``); whether to display
-          additional information
-
-        - ``as_list`` -- boolean (default: ``False``); whether to return
-          the result as a list of pairs (generator, exponent)
+        - ``words`` -- a list of elements of the ambient group, generating
+          a subgroup `H`.
+        - ``display`` -- boolean (default ``True``); whether to display
+          additional information (the raw GAP word and the list form).
+        - ``as_list`` -- boolean (default ``False``); whether to return
+          the result as a list. See Output section for format details.
+        - ``algorithm`` -- string (default ``'syllable'``); the algorithm to use:
+          - ``'syllable'``: (Recommended) Uses GAP's syllable functions
+            for robust parsing. Returns indices in list format.
+          - ``'legacy'``: Uses the old, broken string parsing method.
+            Preserved for backward compatibility testing but is deprecated
+            and will be removed. Returns strings in list format.
 
         OUTPUT:
 
-        - a pair of strings, both representing the same word
-
-        or
-
-        - a list of pairs representing the word, each pair being
-          (generator as a string, exponent as an integer)
-
-        Let `G` be the ambient permutation group, containing the given
-        element `g`. Let `H` be the subgroup of `G` generated by the list
-        ``words`` of elements of `G`. If `g` is in `H`, this function
-        returns an expression for `g` as a word in the elements of
-        ``words`` and their inverses.
-
-        This function does not solve the word problem in Sage. Rather it
-        pushes it over to GAP, which has optimized algorithms for the word
-        problem. Essentially, this function is a wrapper for the GAP
-        functions ``EpimorphismFromFreeGroup`` and
+        If `g = self` is in the subgroup `H` generated by ``words``, this
+        function returns an expression for `g` as a word in the elements
+        of ``words`` and their inverses.
+
+        - If ``as_list=False`` (default): Returns a pair of strings `(l1, l2)`.
+          - `l1`: The word in terms of abstract generators `x1, x2, ...`.
+          - `l2`: The word with the actual generator strings substituted.
+        - If ``as_list=True``:
+          - If ``algorithm='syllable'`` (default): Returns a list of pairs
+            `[generator_index, exponent]`, where `generator_index` is
+            the 1-based index into the input ``words`` list.
+          - If ``algorithm='legacy'``: Returns a list of pairs
+            `[generator_string, exponent]`. This algorithm is deprecated.
+
+        This function uses GAP's ``EpimorphismFromFreeGroup`` and
         ``PreImagesRepresentative``.
 
         EXAMPLES::
 
             sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]], canonicalize=False)
             sage: g1, g2 = G.gens()
             sage: h = g1^2*g2*g1
-            sage: h.word_problem([g1,g2], False)
+            sage: h.word_problem([g1,g2], display=False, as_list=False) # Default algorithm
             ('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')
 
-            sage: h.word_problem([g1,g2])
-               x1^2*x2^-1*x1
-               [['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]
+        Displaying output (using the default 'syllable' algorithm)::
+
+            sage: h.word_problem([g1, g2])
+            x1^2*x2^-1*x1
+            [[1, 2], [2, -1], [1, 1]]
             ('x1^2*x2^-1*x1', '(1,2,3)(4,5)^2*(3,4)^-1*(1,2,3)(4,5)')
 
-            sage: h.word_problem([g1,g2], False, as_list=True)
+        Getting the list output (default algorithm, uses indices)::
+
+            sage: h.word_problem([g1, g2], display=False, as_list=True)
+            [[1, 2], [2, -1], [1, 1]]
+
+        Using the legacy algorithm (deprecated, returns strings in list)::
+
+            sage: h.word_problem([g1, g2], display=False, as_list=True, algorithm='legacy')
+            doctest:warning...:
+            DeprecationWarning: The 'legacy' algorithm for word_problem is deprecated...
             [['(1,2,3)(4,5)', 2], ['(3,4)', -1], ['(1,2,3)(4,5)', 1]]
 
-        TESTS:
+        Fixing :issue:`36419` (parentheses issue)::
+
+            sage: P = PermutationGroup([[(1, 2)], [(1, 2, 3, 4)]])
+            sage: p = P([(1, 4)])
+            sage: p.word_problem(P.gens(), display=True, as_list=True) # Default algorithm
+            x1^-1*x2^-1*(x2^-1*x1^-1)^2*x2
+            [[1, -1], [2, -2], [1, -1], [2, -1], [1, -1], [2, 1]]
+            [[1, -1], [2, -2], [1, -1], [2, -1], [1, -1], [2, 1]]
+            sage: x1, x2 = P.gens()
+            sage: word_new = x1^-1*x2^-1*(x2^-1*x1^-1)^2*x2 
+            sage: word_new == p
+            True
+
+        Compare with legacy algorithm (shows the parsing error)::
+        
+            sage: p.word_problem(P.gens(), display=True, as_list=True, algorithm='legacy')
+            x2^-1*(x2^-1*x1^-1)^2*x2*x1^-1*x2^-2*x1^-1*x2
+            [['(1,2,3,4)', -1], ['((1,2,3,4)', -1], ['(1,2)', 1], ['(1,2,3,4)', 1], ['(1,2)', -1], ['(1,2,3,4)', -2], ['(1,2)', -1], ['(1,2,3,4)', 1]]
+            [['(1,2,3,4)', -1],
+             ['((1,2,3,4)', -1],
+             ['(1,2)', 1],
+             ['(1,2,3,4)', 1],
+             ['(1,2)', -1],
+             ['(1,2,3,4)', -2],
+             ['(1,2)', -1],
+             ['(1,2,3,4)', 1]]
+
+        Fixing :issue:`36419` (x10 vs x1 issue)::
+
+            sage: P = PermutationGroup([[(1, 2)]] * 9 + [[(3, 4, 5)]], canonicalize=False)
+            sage: p = P([(3, 4, 5)])
+            sage: gens = P.gens()
+            sage: p.word_problem(gens, display=False, as_list=False) # Default algorithm
+            ('x10', '(3,4,5)')
+            sage: p.word_problem(gens, display=False, as_list=True) # Default algorithm
+            [[10, 1]]
+
+        Compare with legacy algorithm (shows the replacement error)::
+
+            sage: p.word_problem(gens, display=False, as_list=False, algorithm='legacy')
+            ('x10', '(1,2)0')
+
+        Compare with legacy algorithm (shows the replacement error)::
+
+            sage: p.word_problem(gens, display=False, as_list=True, algorithm='legacy')
+            [['(1,2)0', 1]]
 
         Check for :issue:`28556`::
 
             sage: G = SymmetricGroup(6)
             sage: g = G('(1,2,3)')
-            sage: g.word_problem([g], False)
+            sage: g.word_problem([g], display=False, as_list=False) # Default algorithm
             ('x1', '(1,2,3)')
+            sage: g.word_problem([g], display=False, as_list=True) # Default algorithm
+            [[1, 1]]
         """
         if not self._parent._has_natural_domain():
-            raise NotImplementedError
-
-        def convert_back(string):
-            L = copy.copy(string)
-            for i, w_i in enumerate(words):
-                L = L.replace("x" + str(i + 1), str(w_i))
-            return L
+            raise NotImplementedError("Word problem only implemented for groups acting on {1, ..., n}")
 
-        g = words[0].parent()(self)
-        H = libgap.Group(words)
-        ans = H.EpimorphismFromFreeGroup().PreImagesRepresentative(g)
-
-        l1 = str(ans)
-        l2 = convert_back(l1)
+        if not words:
+            if self == self.parent().one():
+                if as_list: return []
+                else: return ('', '')
+            else:
+                raise ValueError("Cannot express non-identity element with zero generators")
 
-        if display or as_list:
-            l3 = l1.split("*")
-            l4 = []
-            for m in l3:  # parsing the word for display
-                m_split = m.split("^")
-                if len(m_split) == 2:
-                    l4.append([m_split[0], int(m_split[1])])
+        # Ensure self is in the parent group of the generators
+        # Coerce self into the parent group of the first generator
+        # This handles cases where self might be from a different but compatible group (e.g., S_n)
+        try:
+            g = words[0].parent()(self)
+        except (TypeError, ValueError) as e:
+            raise ValueError(f"Element {self} cannot be coerced into the parent group of the generators: {e}")
+
+        # --- Legacy Algorithm ---
+        if algorithm == 'legacy':
+            warnings.warn(
+                "The 'legacy' algorithm for word_problem is deprecated due to known bugs "
+                "in string parsing and will be removed in a future Sage version. "
+                "Use the default 'syllable' algorithm instead.",
+                DeprecationWarning, stacklevel=2)
+
+            # Original flawed convert_back function (only used for legacy)
+            def convert_back_legacy(string_in):
+                L = copy.copy(string_in) # Copy needed here? Strings immutable. Keep for safety.
+                for i, w_i in enumerate(words):
+                    # This replace is the source of the x10 vs x1 bug
+                    L = L.replace("x" + str(i + 1), str(w_i))
+                return L
+
+            H_legacy = libgap.Group(words)
+            try:
+                ans_legacy = H_legacy.EpimorphismFromFreeGroup().PreImagesRepresentative(g)
+            except RuntimeError as e:
+                 # Check if element is not in subgroup
+                 if "is not in" in str(e) and "generated by" in str(e):
+                     raise ValueError(f"Element {g} is not in the subgroup generated by the given words.")
+                 else:
+                     raise # Re-raise other GAP errors
+
+            l1_legacy = str(ans_legacy)
+            l2_legacy = convert_back_legacy(l1_legacy)
+
+            if display or as_list:
+                l3_legacy = l1_legacy.split("*")
+                l4_legacy = []
+                # Original flawed parsing logic
+                for m in l3_legacy:
+                    m_split = m.split("^")
+                    # This parsing fails on nested parentheses like (x1*x2)^2
+                    if len(m_split) == 2:
+                        # Need to handle potential errors if exponent isn't int
+                        try:
+                            exponent = int(m_split[1])
+                        except ValueError:
+                            # Handle cases where parsing might fail unexpectedly
+                            # This part of legacy is inherently broken for complex cases
+                            exponent = 1 # Or raise error? Legacy behavior unclear.
+                        l4_legacy.append([m_split[0], exponent])
+                    else:
+                        l4_legacy.append([m_split[0], 1])
+                # Apply flawed conversion
+                l5_legacy = [[convert_back_legacy(w), e] for w, e in l4_legacy]
+
+            if display:
+                print(l1_legacy)
+                print(l5_legacy)
+
+            if as_list:
+                return l5_legacy
+            else:
+                return l1_legacy, l2_legacy
+
+        # --- Syllable Algorithm (Default) ---
+        elif algorithm == 'syllable':
+            H = libgap.Group(words)
+            try:
+                ans = H.EpimorphismFromFreeGroup().PreImagesRepresentative(g)
+            except RuntimeError as e:
+                 # Check if element is not in subgroup
+                 if "is not in" in str(e) and "generated by" in str(e):
+                     raise ValueError(f"Element {g} is not in the subgroup generated by the given words.")
+                 else:
+                     raise # Re-raise other GAP errors
+
+            l1 = str(ans) # Get the raw GAP string representation (x1*x2^...)
+
+            # Build the list representation using syllable functions
+            num_syllables = ans.NumberSyllables()
+            syllable_list_indices = [] # List of [index, exponent]
+            if num_syllables > 0 : # Handle identity element case
+                for i in range(1, num_syllables + 1):
+                    # GeneratorSyllable returns 1-based index
+                    generator_index = int(ans.GeneratorSyllable(i))
+                    exponent = int(ans.ExponentSyllable(i))
+                    syllable_list_indices.append([generator_index, exponent])
+
+            # Build the substituted string representation (l2) correctly
+            l2_parts = []
+            for idx, exp in syllable_list_indices:
+                gen_str = str(words[idx-1]) # Get generator string (use idx-1 for 0-based list)
+                if exp == 1:
+                    l2_parts.append(gen_str)
                 else:
-                    l4.append([m_split[0], 1])
-            l5 = [[convert_back(w), e] for w, e in l4]
-
-        if display:
-            print(l1)
-            print(l5)
+                    # Handle potential need for parentheses around generator string if it contains spaces etc.
+                    # For permutations, str() usually produces cycle notation without spaces,
+                    # so parentheses might not be strictly needed, but safer? Let's omit for now.
+                    l2_parts.append(f"{gen_str}^{exp}")
+            l2 = "*".join(l2_parts)
+
+            if display:
+                print(l1)
+                print(syllable_list_indices) # Display the new index-based list
+
+            if as_list:
+                return syllable_list_indices
+            else:
+                return l1, l2
 
-        if as_list:
-            return l5
         else:
-            return l1, l2
+            raise ValueError(f"Unknown algorithm specified: '{algorithm}'. Choose 'syllable' or 'legacy'.")
 
 
 cdef class SymmetricGroupElement(PermutationGroupElement):