235 lines
6.9 KiB
Python
235 lines
6.9 KiB
Python
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""" Generic Unification algorithm for expression trees with lists of children
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This implementation is a direct translation of
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Artificial Intelligence: A Modern Approach by Stuart Russel and Peter Norvig
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Second edition, section 9.2, page 276
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It is modified in the following ways:
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1. We allow associative and commutative Compound expressions. This results in
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combinatorial blowup.
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2. We explore the tree lazily.
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3. We provide generic interfaces to symbolic algebra libraries in Python.
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A more traditional version can be found here
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http://aima.cs.berkeley.edu/python/logic.html
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"""
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from sympy.utilities.iterables import kbins
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class Compound:
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""" A little class to represent an interior node in the tree
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This is analogous to SymPy.Basic for non-Atoms
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"""
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def __init__(self, op, args):
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self.op = op
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self.args = args
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def __eq__(self, other):
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return (type(self) is type(other) and self.op == other.op and
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self.args == other.args)
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def __hash__(self):
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return hash((type(self), self.op, self.args))
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def __str__(self):
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return "%s[%s]" % (str(self.op), ', '.join(map(str, self.args)))
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class Variable:
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""" A Wild token """
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def __init__(self, arg):
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self.arg = arg
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def __eq__(self, other):
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return type(self) is type(other) and self.arg == other.arg
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def __hash__(self):
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return hash((type(self), self.arg))
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def __str__(self):
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return "Variable(%s)" % str(self.arg)
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class CondVariable:
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""" A wild token that matches conditionally.
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arg - a wild token.
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valid - an additional constraining function on a match.
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"""
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def __init__(self, arg, valid):
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self.arg = arg
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self.valid = valid
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def __eq__(self, other):
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return (type(self) is type(other) and
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self.arg == other.arg and
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self.valid == other.valid)
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def __hash__(self):
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return hash((type(self), self.arg, self.valid))
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def __str__(self):
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return "CondVariable(%s)" % str(self.arg)
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def unify(x, y, s=None, **fns):
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""" Unify two expressions.
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Parameters
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==========
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x, y - expression trees containing leaves, Compounds and Variables.
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s - a mapping of variables to subtrees.
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Returns
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=======
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lazy sequence of mappings {Variable: subtree}
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Examples
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========
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>>> from sympy.unify.core import unify, Compound, Variable
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>>> expr = Compound("Add", ("x", "y"))
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>>> pattern = Compound("Add", ("x", Variable("a")))
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>>> next(unify(expr, pattern, {}))
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{Variable(a): 'y'}
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"""
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s = s or {}
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if x == y:
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yield s
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elif isinstance(x, (Variable, CondVariable)):
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yield from unify_var(x, y, s, **fns)
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elif isinstance(y, (Variable, CondVariable)):
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yield from unify_var(y, x, s, **fns)
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elif isinstance(x, Compound) and isinstance(y, Compound):
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is_commutative = fns.get('is_commutative', lambda x: False)
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is_associative = fns.get('is_associative', lambda x: False)
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for sop in unify(x.op, y.op, s, **fns):
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if is_associative(x) and is_associative(y):
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a, b = (x, y) if len(x.args) < len(y.args) else (y, x)
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if is_commutative(x) and is_commutative(y):
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combs = allcombinations(a.args, b.args, 'commutative')
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else:
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combs = allcombinations(a.args, b.args, 'associative')
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for aaargs, bbargs in combs:
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aa = [unpack(Compound(a.op, arg)) for arg in aaargs]
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bb = [unpack(Compound(b.op, arg)) for arg in bbargs]
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yield from unify(aa, bb, sop, **fns)
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elif len(x.args) == len(y.args):
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yield from unify(x.args, y.args, sop, **fns)
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elif is_args(x) and is_args(y) and len(x) == len(y):
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if len(x) == 0:
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yield s
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else:
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for shead in unify(x[0], y[0], s, **fns):
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yield from unify(x[1:], y[1:], shead, **fns)
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def unify_var(var, x, s, **fns):
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if var in s:
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yield from unify(s[var], x, s, **fns)
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elif occur_check(var, x):
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pass
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elif isinstance(var, CondVariable) and var.valid(x):
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yield assoc(s, var, x)
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elif isinstance(var, Variable):
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yield assoc(s, var, x)
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def occur_check(var, x):
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""" var occurs in subtree owned by x? """
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if var == x:
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return True
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elif isinstance(x, Compound):
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return occur_check(var, x.args)
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elif is_args(x):
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if any(occur_check(var, xi) for xi in x): return True
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return False
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def assoc(d, key, val):
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""" Return copy of d with key associated to val """
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d = d.copy()
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d[key] = val
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return d
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def is_args(x):
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""" Is x a traditional iterable? """
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return type(x) in (tuple, list, set)
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def unpack(x):
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if isinstance(x, Compound) and len(x.args) == 1:
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return x.args[0]
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else:
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return x
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def allcombinations(A, B, ordered):
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"""
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Restructure A and B to have the same number of elements.
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Parameters
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==========
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ordered must be either 'commutative' or 'associative'.
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A and B can be rearranged so that the larger of the two lists is
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reorganized into smaller sublists.
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Examples
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========
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>>> from sympy.unify.core import allcombinations
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>>> for x in allcombinations((1, 2, 3), (5, 6), 'associative'): print(x)
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(((1,), (2, 3)), ((5,), (6,)))
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(((1, 2), (3,)), ((5,), (6,)))
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>>> for x in allcombinations((1, 2, 3), (5, 6), 'commutative'): print(x)
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(((1,), (2, 3)), ((5,), (6,)))
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(((1, 2), (3,)), ((5,), (6,)))
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(((1,), (3, 2)), ((5,), (6,)))
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(((1, 3), (2,)), ((5,), (6,)))
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(((2,), (1, 3)), ((5,), (6,)))
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(((2, 1), (3,)), ((5,), (6,)))
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(((2,), (3, 1)), ((5,), (6,)))
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(((2, 3), (1,)), ((5,), (6,)))
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(((3,), (1, 2)), ((5,), (6,)))
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(((3, 1), (2,)), ((5,), (6,)))
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(((3,), (2, 1)), ((5,), (6,)))
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(((3, 2), (1,)), ((5,), (6,)))
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"""
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if ordered == "commutative":
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ordered = 11
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if ordered == "associative":
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ordered = None
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sm, bg = (A, B) if len(A) < len(B) else (B, A)
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for part in kbins(list(range(len(bg))), len(sm), ordered=ordered):
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if bg == B:
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yield tuple((a,) for a in A), partition(B, part)
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else:
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yield partition(A, part), tuple((b,) for b in B)
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def partition(it, part):
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""" Partition a tuple/list into pieces defined by indices.
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Examples
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========
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>>> from sympy.unify.core import partition
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>>> partition((10, 20, 30, 40), [[0, 1, 2], [3]])
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((10, 20, 30), (40,))
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"""
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return type(it)([index(it, ind) for ind in part])
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def index(it, ind):
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""" Fancy indexing into an indexable iterable (tuple, list).
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Examples
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========
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>>> from sympy.unify.core import index
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>>> index([10, 20, 30], (1, 2, 0))
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[20, 30, 10]
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"""
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return type(it)([it[i] for i in ind])
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