✏️ Editing: KanExtension.py Read Only

class LeftKanExtension:

# Implementation of algorithms described by Brown and Heyworth (ref.251)
    # and Heyworth (ref.253).

def __init__(self, mod, A, B, R, X, F):
        # External subsystem dependencies
        #    mod.KnuthBendix
        #    mod.FiniteAutomaton
        #    mod.SolveFSA
        #    mod.Cat
        #       mod.Cat.Function
        #       mod.Cat.Functor
        #       mod.Cat.check_graph
        #       mod.Cat.check_rules

self.mod = mod
        self.Cat = mod.Cat

self.Cat.check_graph(A)
        self.Cat.check_graph(B)
        self.Cat.check_rules(R, B)

self.A = A
        self.B = B
        self.R = [(tuple(g), tuple(h)) for (g, h) in R]
        self.X = X
        self.F = F

self.general_procedure()

def general_procedure(self):
        self.initialize_tables()
        self.make_confluent_system()
        self.make_automaton()  # self.make_catalogue()
        self.make_natural_transformation()

def initialize_tables(self):
        self.obj_to_str_table = {}
        self.str_to_obj_table = {}
        self.make_initial_rules()

def make_initial_rules(self):

# Algorithm 6.1 in (251)

Re = []

def add_rule(a, b):
            aw = self.make_word(a)
            bw = self.make_word(b)
            if aw != bw:
                Re.append((aw, bw))

for a in self.A.arrows:
            srca = self.A.source(a)
            tgta = self.A.target(a)
            XA = self.X.fo(srca)
            Xa = self.X.fa(a)
            Fa = tuple(self.F.fa(a))
            Fsrca = self.F.fo(srca)
            Ftgta = self.F.fo(tgta)
            if Fa:
                t = Fsrca
                for b in Fa:
                    srcb = self.B.source(b)
                    if srcb != t:
                        raise ValueError(
                            'Arrow [%s] with source %s does not compose with target %s' % (b, srcb, t))
                    t = self.B.target(b)
                if t != Ftgta:
                    raise ValueError(
                        'Arrow %s with target %s does not compose with %s' % (Fa, t, Ftgta))
            else:
                if Fsrca != Ftgta:
                    raise ValueError(
                        'Source %s does not match target %s' % (Fsrca, Ftgta))
            for x in XA:
                add_rule(((srca, x),) + Fa, ((tgta, Xa(x)),))

Rk = [(self.make_word(x), self.make_word(y)) for (x, y) in self.R]

self.Re = Re
        self.Rk = Rk
        self.Rinit = Re + Rk

def make_confluent_system(self):
        self.rs = self.mod.KnuthBendix(self.Rinit, delim='.')
        self.Rconf = self.rs.reductions

def make_automaton(self):
        # Make nondeterministic finite automaton

def target(e):
            if len(e) == 1 and isinstance(e[0], tuple):
                return self.F.fo(e[0][0])
            else:
                return self.B.target(e[-1])

XA = []
        for A in self.A.objects:
            for x in self.X.fo(A):
                XA.append(((A, x),))

follows = dict([(B, []) for B in self.B.objects])
        for b, (srcb, tgtb) in list(self.B.arrows.items()):
            follows[srcb].append((b, tgtb))

IR = dict([(self.make_term(u), self.make_term(v))
                   for u, v in self.Rconf])

pplR = {}
        for l, r in self.Rconf:
            t = self.make_term(l)
            for i in range(1, len(t)):
                pplR[t[:i]] = 1

s0 = ('s0',)

fsa = self.mod.FiniteAutomaton(s0)

for xi in XA:
            if xi not in IR:
                fsa.add_transition(s0, xi[0], xi)

for xi in XA:
            for b, tgtb in follows[target(xi)]:
                bterm = (b,)
                xib = xi + bterm
                if xib in pplR:
                    fsa.add_transition(xi, b, xib, tgtb)
                elif (bterm in pplR and xib not in IR):
                    fsa.add_transition(xi, b, bterm, tgtb)
                elif xib not in IR:
                    fsa.add_transition(xi, b, tgtb)

for Bi in self.B.objects:
            for b, tgtb in follows[Bi]:
                bterm = (b,)
                if bterm in pplR:
                    fsa.add_transition(Bi, b, bterm, tgtb)
                elif bterm not in IR:
                    fsa.add_transition(Bi, b, tgtb)

for u in pplR:
            if u in XA:
                continue
            for b, tgtb in follows[target(u)]:
                bterm = (b,)
                ub = u + bterm
                if ub in pplR:
                    fsa.add_transition(u, b, ub, tgtb)
                elif self.irreducible(ub):  # ub not in IR:
                    fsa.add_transition(u, b, tgtb)

def get_RS(Bi):
            finals = {}
            finals[Bi] = 1
            for xi in XA:
                if self.F.fo(xi[0][0]) == Bi:
                    finals[xi] = 1
            for u in pplR:
                if target(u) == Bi:
                    finals[u] = 1

for c in fsa.get_composites():
                for s in c:
                    if s not in finals:
                        break
                else:
                    finals[c] = 1

dfa = fsa.get_minimized_dfa(finals)
            regexp = self.mod.SolveFSA(dfa)
            return RegularSet(regexp)

KB = self.Cat.Function(get_RS, self.B.objects, None)

Kb = self.Cat.Function(
            lambda a: KanAction(self.B, KB, a, target,
                                self.irreducible, self.reduce),
            self.B.arrows,
            KanAction,
        )

self.KB = KB
        self.Kb = Kb
        self.K = self.Cat.Functor(KB, Kb)

def make_catalogue(self):
        # Catalogue the elements of the sets pointed to by extension functor K,
        # according to algorithm described in 7.1 in (251).

# Precondition:

# Tables initialized and a confluent system created.
        # The system is assumed to be finite, otherwise we won't terminate.

# Postcondition:

# Functor self.K represented as:
        #
        # self.K.tabo = self.KB = dict mapping,
        #               source: {each B in self.B.objects}
        #               target: sets represented as lists
        # self.K.taba = self.Kb = dict, mapping
        #               source: {each a in self.B.arrows}
        #               target: tabulated function, mapping
        #                       source: KB[source of a]
        #                       target: KB[target of a]

def target(e):
            if len(e) == 1:
                return self.F.fo(e[0][0])
            else:
                return self.B.target(e[-1])

def add_element(e):
            if self.irreducible(e):
                block.append(e)
                KB[target(e)].append(e)
            else:
                pass

KB = dict([(B, []) for B in self.B.objects])
        block = []

for A in self.A.objects:
            for x in self.X.fo(A):
                add_element(((A, x),))

while block:
            oblock = block
            block = []
            for e in oblock:
                tgt = target(e)
                for a in self.B.arrows:
                    if self.B.source(a) == tgt:
                        add_element(e + (a,))

Kb = {}

for a in self.B.arrows:
            src = KB[self.B.source(a)]
            tgt = KB[self.B.target(a)]
            tab = dict([(s, self.reduce(s + (a,))) for s in src])
            Kb[a] = self.Cat.Function(tab, src, tgt)

KB = self.Cat.Function(KB, self.B.objects, list(KB.values()))
        Kb = self.Cat.Function(Kb, self.B.arrows, list(Kb.values()))

self.KB = KB
        self.Kb = Kb
        self.K = self.Cat.Functor(KB, Kb)

def make_natural_transformation(self):

# Precondition:
        # initial tables should be initialized
        # self.K.fo should exist

# Postcondition:
        #
        # self.nat[A] for A in self.A.objects

get_nat_memo = {}

def get_nat(A):
            if A in get_nat_memo:
                return get_nat_memo[A]

src = self.X.fo(A)
            tgt = self.K.fo(self.F.fo(A))
            tab = dict([(x, self.reduce(((A, x),))) for x in src])
            get_nat_memo[A] = self.Cat.Function(tab, src, tgt)
            return get_nat_memo[A]

self.nat = self.Cat.Function(get_nat, self.A.objects, None)

def make_word(self, x):
        ots = self.obj_to_str
        return '.'.join([ots(e) for e in x if e != ''])

def obj_to_str(self, x):
        otn = self.obj_to_str_table
        try:
            return otn[x]
        except KeyError:
            assert not (isinstance(x, tuple) and len(x) > 2)
            n = str(len(otn))
            #n = '%d:%s'%(len(otn), x)
            #n = str(x)
            otn[x] = n
            self.str_to_obj_table[n] = x
            return n

def str_to_obj(self, x):
        return self.str_to_obj_table[x]

def irreducible(self, x):
        tx = self.make_word(x)
        return tx == self.rs.reduce(tx)

def reduce(self, x):
        w = self.rs.reduce(self.make_word(x))
        return self.make_term(w)

def make_term(self, word):
        sto = self.str_to_obj_table
        return tuple([sto[s] for s in word.split('.') if s])

class KanAction:
    def __init__(self, B, KB, a, targetof, irreducible, reduce):
        srca = B.source(a)
        tgta = B.target(a)
        self.src = KB(srca)
        self.tgt = KB(tgta)
        self.a = a
        self.srca = srca
        self.targetof = targetof
        self.irreducible = irreducible
        self.reduce = reduce

def __call__(self, s):
        if self.targetof(s) != self.srca:
            raise TypeError('''\
Target of %r (= %r) does not match source of %r (= %r)''' % (
                s, self.targetof(s), self.a, self.srca))
        if not self.irreducible(s):
            raise TypeError('''\
Argument %r is reducible to %r; and is thus not in the source set K.fo(%r)''' % (
                s, self.reduce(s), self.srca))
        return self.reduce(s + (self.a,))

class RegularSet:
    # Wraps a regular expression;
    # provides a set protocol for the underlying set of sequences:
    #  o If the RE specifies a finite language, iteration over its strings
    #  [ o set inclusion ]

is_simplified = 0

def __init__(self, re):
        self.re = re

def __iter__(self):
        return iter(self.uniform)

def __getitem__(self, x):
        return self.uniform[x]

def __len__(self):
        return len(self.uniform)

def get_xs_covered(self, coverage):
        N = coverage
        X = self.re.limited(coverage)
        xs = X.sequni()
        return [tuple(x) for x in xs]

def get_uniform(self):
        self.simplify()
        return self.re.sequni()

uniform = property(fget=get_uniform)

def simplify(self):
        if not self.is_simplified:
            self.re = self.re.simplified()
            self.is_simplified = 1

class ObjectTester:
    def __init__(self, category_tester, object, code):
        self.category_tester = category_tester
        self.functor = category_tester.functor
        self.object = object
        self.code = code

def get_all_arrows(self):
        return self.category_tester.arrows[self.object]

def get_intermediate_test_code(self):
        return self.code

def get_python_test_source_code(self):
        cmap = {
            'aseq': 'assert e[%r] == e[%r]',
            'evalfa': 'e[%r] = fa[%r](e[%r])',
            'asfo': 'assert fo[%r](e[%r])'
        }

return '\n'.join([cmap[c[0]] % c[1:] for c in self.code])

def execode(self, arg):
        code = self.get_python_test_source_code()

e = {'arg': arg}
        d = {'fa': self.functor.fa,
             'fo': self.functor.fo,
             'e': e,
             }
        exec(code, d)
        return e

def intercode(self, arg):
        e = {'arg': arg}
        fa = self.functor.fa
        fo = self.functor.fo
        for c in self.code:
            a = c[0]
            if a == 'evalfa':
                dst, ar, src = c[1:]
                e[dst] = fa[ar](e[src])
            elif a == 'asfo':
                ob, src = c[1:]
                if not fo[ob](e[src]):
                    raise ValueError('Predicate failed')
            elif a == 'aseq':
                na, nb = c[1:]
                if e[na] != e[nb]:
                    raise ValueError('e[%r] != e[%r]' % (na, nb))
            else:
                raise ValueError('Invalid code: %r' % (a,))

def test(self, arg):
        return self.intercode(arg)

class CategoryTester:
    def __init__(self, mod, functor, arrows, get_arrow_name=None):
        self.mod = mod
        self.cat = functor.src
        self.functor = functor
        self.arrows = arrows
        if get_arrow_name is not None:
            self.get_arrow_name = get_arrow_name

def get_arrow_name(self, a):
        return '.'.join(a)

def get_eval_arrows_code(self, object, argname):
        fa = self.functor.fa

name = argname
        memo = {(): name}
        memolist = [((), name)]

codes = []

def eval_arrow(a):
            if a in memo:
                return memo[a]
            a0 = a[:-1]
            a1 = a[-1]
            name = self.get_arrow_name(a)
            na0 = eval_arrow(a0)
            #codes.append('%s = fa[%r](%s)'%(name, a1, na0))
            codes.append(('evalfa', name, a1, na0))
            memo[a] = name
            memolist.append((a, name))
            return name

for ar in self.arrows[object]:
            eval_arrow(ar)

return codes, memolist

def get_object_tester(self, object):
        code = self.get_test_object_code(object)
        return ObjectTester(self, object, code)

def get_test_inclusion_code(self, object, ml):
        codes = []
        src = self.functor.fo.src
        for arrow, value in ml:
            ob = object
            if arrow:
                ob = self.cat.graph.target(arrow[-1])
            #codes.append('assert fo[%r](%s)'%(ob, value))
            if src is None or ob in src:
                codes.append(('asfo', ob, value))
        return codes

def get_test_object_code(self, object):
        argname = 'arg'
        evalcodes, memolist = self.get_eval_arrows_code(object, argname)

relcodes = self.get_test_relations_code(object, memolist)

incodes = self.get_test_inclusion_code(object, memolist)

return evalcodes+relcodes+incodes

def get_test_relations_code(self, object, memolist):
        codes = []
        cat = self.cat
        fa = self.functor.fa
        memo = dict(memolist)

def teval_arrow(ar):
            if ar in memo:
                return memo[ar]
            a0 = teval_arrow(ar[:-1])
            name = self.get_arrow_name(ar)
            #codes.append('%s = fa[%r](%s)'%(name, ar[-1], a0))
            codes.append(('evalfa', name, ar[-1], a0))
            memo[ar] = name
            return name

# Check that the equality relations really match up
        # for all arrows in old memolist, i.e. original unique arrows
        # which is arguably overkill sometimes?..
        for a, b in cat.relations:
            a = tuple(a)
            b = tuple(b)
            src = cat.graph.source(a[0])
            for (arr, val) in memolist:
                if arr:
                    tgt = cat.graph.target(arr[-1])
                else:
                    tgt = object
                if src == tgt:
                    ara = arr + a
                    arb = arr + b
                    if ara != arb:
                        va = teval_arrow(ara)
                        vb = teval_arrow(arb)
                        assert va != vb
                        #codes.append('assert %s == %s'%(va, vb))
                        codes.append(('aseq', va, vb))
        return codes

def test_object(self, object, value):
        tester = self.get_object_tester(object)
        tester.test(value)
        return tester

def test_object_fail(self, object, value):
        try:
            self.test_object(object, value)
        except Exception:
            pass
        else:
            raise Exception('Exception excepted')

class _GLUECLAMP_:
    # 'imports'

def _get_KnuthBendix(self): return self._parent.KnuthBendix.KnuthBendix
    def _get_FiniteAutomaton(self): return self._parent.FSA.FiniteAutomaton
    def _get_SolveFSA(self): return self._parent.RE.SolveFSA
    def _get_Cat(self): return self._parent.Cat

# Main exported interface is the lke method
    # which provides a context for the LeftKanExtension class.

def lke(self, A, B, R, X, F):
        return LeftKanExtension(self, A, B, R, X, F)

# Other functions - examples of applications of Kan extension
    # in alphabetic order

def arrows_map(self, cat, from_objects=0, coverage=1):
        if from_objects:
            cat = cat.get_dual()

A = self.Cat.Graph(cat.graph.objects, [])
        B = cat.graph
        R = cat.relations
        X = self.Cat.Functor(lambda x: [1], lambda x: lambda y: y)
        F = self.Cat.Functor(lambda x: x, lambda x: [])
        ke = self.lke(A, B, R, X, F)

memo = {}

def get_arrows(object):
            if object in memo:
                return memo[object]
            re = ke.K.fo[object].re.rempretup()
            if from_objects:
                re = re.reversed()
            if str(coverage).startswith('length'):
                maxlen = int(coverage[6:])
                ar = []
                xs = re.get_words_memo()
                for i in range(1, maxlen+1):
                    ar.extend([tuple(x) for x in xs.get_words_of_length(i)])
            else:
                re = re.limited(coverage)
                xs = re.sequni()
                ar = [tuple(x) for x in xs]
            memo[object] = ar
            return ar

return self.Cat.Function(
            get_arrows,
            src=ke.K.fo.src,
            tgt=None
        )

def category_tester(self, functor, arrows=None, coverage=1):
        if isinstance(functor, tuple):
            fo, fa, src = functor
            if fo is None:
                def fo(x): return lambda y: 1
            functor = self.Cat.Functor(fo, fa, src)
        if arrows is None:
            arrows = self.arrows_map(
                functor.src, from_objects=1, coverage=coverage)
        return CategoryTester(self, functor, arrows)

def coequalizer(self, S0, S1, f0, f1):
        # Given
        #
        # S0, S1 sets (objects that can be iterated over)
        # f0, f1 functions from S0 to S1
        #
        # Return a coequalizing function,
        # such that in the following diagram:

#
        #  S0 ===== S0
        #  |        |
        #  | f0     | f1
        #  |        |
        #  V        V
        #  S1 ===== S1 ==== coequalizing_function.src
        #  |
        #  | coequalizing_function
        #  |
        #  V
        #  coequalizing_function.tgt

# both paths from S0 to coequalizing_function.tgt will be equivalent,
        # and coequalizing_function.tgt is a colimit of all such sets.
        #
        # The coequalizing_function object is callable with
        # an argument from S1, and has the following attributes:
        #   .src                is identical to S1
        #   .tgt                is a set in iterable form
        #   .asdict()           returns a dict representing the mapping

objects = [0, 1]
        arrows = {'a0': (0, 1), 'a1': (0, 1)}
        A = self.Cat.Graph(objects, arrows)

Xo = self.Cat.Function({0: S0, 1: S1}, objects, [S0, S1])
        Xa = self.Cat.Function({'a0': f0, 'a1': f1}, arrows, [f0, f1])
        X = self.Cat.Functor(Xo, Xa)

colimit_object, colimit_functions = self.colimit(A, X)
        return colimit_functions[1]

def colimit(self, A, X):
        # According to 9.6 in (ref.251)

B = self.Cat.Graph([0], {})
        R = []
        F = self.Cat.Functor(lambda x: 0, lambda x: ())

lka = self.lke(A, B, R, X, F)

colimit_object = lka.KB[0]
        colimit_functions = lka.nat

# Reduce elements to a smaller (but isomorphic) form
        # I.E since elements are all of the form
        #       ((A, X),)
        # they can be reduced to the form
        #       (A, X)
        #

colimit_object = [x[0] for x in colimit_object]

colimit_functions = dict([
            (A, self.Cat.Function(
                dict([(a, k[0])
                      for (a, k) in list(cof.items())]),
                cof.src,
                colimit_object,
            )
            )
            for (A, cof) in list(colimit_functions.items())])

return colimit_object, colimit_functions

def test_arrows(self, functor, object, value):
        # Application of arrow listing to test sequencing
        # Discussed in Notes Mar 9 2005

tester = self.category_tester(functor)
        return tester.test_object(object, value)

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