# -*- coding: utf-8 -*-

'''
    Finite state machine library.
'''

class anything_else_cls:
    '''
        This is a surrogate symbol which you can use in your finite state machines
        to represent "any symbol not in the official alphabet". For example, if your
        state machine's alphabet is {"a", "b", "c", "d", fsm.anything_else}, then
        you can pass "e" in as a symbol and it will be converted to
        fsm.anything_else, then follow the appropriate transition.
    '''
    def __str__(self):
        return "anything_else"
    def __repr__(self):
        return "anything_else"

# We use a class instance because that gives us control over how the special
# value gets serialised. Otherwise this would just be `object()`.
anything_else = anything_else_cls()

def key(symbol):
    '''Ensure `fsm.anything_else` always sorts last'''
    return (symbol is anything_else, symbol)

class OblivionError(Exception):
    '''
        This exception is thrown while `crawl()`ing an FSM if we transition to the
        oblivion state. For example while crawling two FSMs in parallel we may
        transition to the oblivion state of both FSMs at once. This warrants an
        out-of-bound signal which will reduce the complexity of the new FSM's map.
    '''
    pass

class fsm:
    '''
        A Finite State Machine or FSM has an alphabet and a set of states. At any
        given moment, the FSM is in one state. When passed a symbol from the
        alphabet, the FSM jumps to another state (or possibly the same state).
        A map (Python dictionary) indicates where to jump.
        One state is nominated as a starting state. Zero or more states are
        nominated as final states. If, after consuming a string of symbols,
        the FSM is in a final state, then it is said to "accept" the string.
        This class also has some pretty powerful methods which allow FSMs to
        be concatenated, alternated between, multiplied, looped (Kleene star
        closure), intersected, and simplified.
        The majority of these methods are available using operator overloads.
    '''
    def __setattr__(self, name, value):
        '''Immutability prevents some potential problems.'''
        raise Exception("This object is immutable.")

    def __init__(self, alphabet, states, initial, finals, map):
        '''
            `alphabet` is an iterable of symbols the FSM can be fed.
            `states` is the set of states for the FSM
            `initial` is the initial state
            `finals` is the set of accepting states
            `map` may be sparse (i.e. it may omit transitions). In the case of omitted
            transitions, a non-final "oblivion" state is simulated.
        '''

        # Validation. Thanks to immutability, this only needs to be carried out once.
        if not initial in states:
            raise Exception("Initial state " + repr(initial) + " must be one of " + repr(states))
        if not finals.issubset(states):
            raise Exception("Final states " + repr(finals) + " must be a subset of " + repr(states))
        for state in map.keys():
            for symbol in map[state]:
                if not map[state][symbol] in states:
                    raise Exception("Transition for state " + repr(state) + " and symbol " + repr(symbol) + " leads to " + repr(map[state][symbol]) + ", which is not a state")

        # Initialise the hard way due to immutability.
        self.__dict__["alphabet"] = set(alphabet)
        self.__dict__["states"  ] = set(states)
        self.__dict__["initial" ] = initial
        self.__dict__["finals"  ] = set(finals)
        self.__dict__["map"     ] = map

    def accepts(self, input):
        '''
            Test whether the present FSM accepts the supplied string (iterable of
            symbols). Equivalently, consider `self` as a possibly-infinite set of
            strings and test whether `string` is a member of it.
            This is actually mainly used for unit testing purposes.
            If `fsm.anything_else` is in your alphabet, then any symbol not in your
            alphabet will be converted to `fsm.anything_else`.
        '''
        state = self.initial
        for symbol in input:
            if anything_else in self.alphabet and not symbol in self.alphabet:
                symbol = anything_else

            # Missing transition = transition to dead state
            if not (state in self.map and symbol in self.map[state]):
                return False

            state = self.map[state][symbol]
        return state in self.finals

    def __contains__(self, string):
        '''
            This lets you use the syntax `"a" in fsm1` to see whether the string "a"
            is in the set of strings accepted by `fsm1`.
        '''
        return self.accepts(string)

    def reduce(self):
        '''
            A result by Brzozowski (1963) shows that a minimal finite state machine
            equivalent to the original can be obtained by reversing the original
            twice.
        '''
        return reversed(reversed(self))

    def __repr__(self):
        string = "fsm("
        string += "alphabet = " + repr(self.alphabet)
        string += ", states = " + repr(self.states)
        string += ", initial = " + repr(self.initial)
        string += ", finals = " + repr(self.finals)
        string += ", map = " + repr(self.map)
        string += ")"
        return string

    def __str__(self):
        rows = []

        # top row
        row = ["", "name", "final?"]
        row.extend(str(symbol) for symbol in sorted(self.alphabet, key=key))
        rows.append(row)

        # other rows
        for state in self.states:
            row = []
            if state == self.initial:
                row.append("*")
            else:
                row.append("")
            row.append(str(state))
            if state in self.finals:
                row.append("True")
            else:
                row.append("False")
            for symbol in sorted(self.alphabet, key=key):
                if state in self.map and symbol in self.map[state]:
                    row.append(str(self.map[state][symbol]))
                else:
                    row.append("")
            rows.append(row)

        # column widths
        colwidths = []
        for x in range(len(rows[0])):
            colwidths.append(max(len(str(rows[y][x])) for y in range(len(rows))) + 1)

        # apply padding
        for y in range(len(rows)):
            for x in range(len(rows[y])):
                rows[y][x] = rows[y][x].ljust(colwidths[x])

        # horizontal line
        rows.insert(1, ["-" * colwidth for colwidth in colwidths])

        return "".join("".join(row) + "\n" for row in rows)

    def concatenate(*fsms):
        '''
            Concatenate arbitrarily many finite state machines together.
        '''
        alphabet = set().union(*[fsm.alphabet for fsm in fsms])

        def connect_all(i, substate):
            '''
                Take a state in the numbered FSM and return a set containing it, plus
                (if it's final) the first state from the next FSM, plus (if that's
                final) the first state from the next but one FSM, plus...
            '''
            result = {(i, substate)}
            while i < len(fsms) - 1 and substate in fsms[i].finals:
                i += 1
                substate = fsms[i].initial
                result.add((i, substate))
            return result

        # Use a superset containing states from all FSMs at once.
        # We start at the start of the first FSM. If this state is final in the
        # first FSM, then we are also at the start of the second FSM. And so on.
        initial = set()
        if len(fsms) > 0:
            initial.update(connect_all(0, fsms[0].initial))
        initial = frozenset(initial)

        def final(state):
            '''If you're in a final state of the final FSM, it's final'''
            for (i, substate) in state:
                if i == len(fsms) - 1 and substate in fsms[i].finals:
                    return True
            return False

        def follow(current, symbol):
            '''
                Follow the collection of states through all FSMs at once, jumping to the
                next FSM if we reach the end of the current one
                TODO: improve all follow() implementations to allow for dead metastates?
            '''
            next = set()
            for (i, substate) in current:
                fsm = fsms[i]
                if substate in fsm.map:
                    if symbol in fsm.map[substate]:
                        next.update(connect_all(i, fsm.map[substate][symbol]))
                    elif  anything_else in fsm.map[substate] and symbol not in fsm.alphabet:
                        next.update(connect_all(i, fsm.map[substate][anything_else]))
            if len(next) == 0:
                raise OblivionError
            return frozenset(next)

        return crawl(alphabet, initial, final, follow).reduce()

    def __add__(self, other):
        '''
            Concatenate two finite state machines together.
            For example, if self accepts "0*" and other accepts "1+(0|1)",
            will return a finite state machine accepting "0*1+(0|1)".
            Accomplished by effectively following non-deterministically.
            Call using "fsm3 = fsm1 + fsm2"
        '''
        return self.concatenate(other)

    def star(self):
        '''
            If the present FSM accepts X, returns an FSM accepting X* (i.e. 0 or
            more Xes). This is NOT as simple as naively connecting the final states
            back to the initial state: see (b*ab)* for example.
        '''
        alphabet = self.alphabet

        initial = {self.initial}

        def follow(state, symbol):
            next = set()
            for substate in state:
                if substate in self.map and symbol in self.map[substate]:
                    next.add(self.map[substate][symbol])

                # If one of our substates is final, then we can also consider
                # transitions from the initial state of the original FSM.
                if substate in self.finals \
                and self.initial in self.map \
                and symbol in self.map[self.initial]:
                    next.add(self.map[self.initial][symbol])

            if len(next) == 0:
                raise OblivionError

            return frozenset(next)

        def final(state):
            return any(substate in self.finals for substate in state)

        return crawl(alphabet, initial, final, follow) | epsilon(alphabet)

    def times(self, multiplier):
        '''
            Given an FSM and a multiplier, return the multiplied FSM.
        '''
        if multiplier < 0:
            raise Exception("Can't multiply an FSM by " + repr(multiplier))

        alphabet = self.alphabet

        # metastate is a set of iterations+states
        initial = {(self.initial, 0)}

        def final(state):
            '''If the initial state is final then multiplying doesn't alter that'''
            for (substate, iteration) in state:
                if substate == self.initial \
                and (self.initial in self.finals or iteration == multiplier):
                    return True
            return False

        def follow(current, symbol):
            next = []
            for (substate, iteration) in current:
                if iteration < multiplier \
                and substate in self.map \
                and symbol in self.map[substate]:
                    next.append((self.map[substate][symbol], iteration))
                    # final of self? merge with initial on next iteration
                    if self.map[substate][symbol] in self.finals:
                        next.append((self.initial, iteration + 1))
            if len(next) == 0:
                raise OblivionError
            return frozenset(next)

        return crawl(alphabet, initial, final, follow).reduce()

    def __mul__(self, multiplier):
        '''
            Given an FSM and a multiplier, return the multiplied FSM.
        '''
        return self.times(multiplier)

    def union(*fsms):
        '''
            Treat `fsms` as a collection of arbitrary FSMs and return the union FSM.
            Can be used as `fsm1.union(fsm2, ...)` or `fsm.union(fsm1, ...)`. `fsms`
            may be empty.
        '''
        return parallel(fsms, any)

    def __or__(self, other):
        '''
            Alternation.
            Return a finite state machine which accepts any sequence of symbols
            that is accepted by either self or other. Note that the set of strings
            recognised by the two FSMs undergoes a set union.
            Call using "fsm3 = fsm1 | fsm2"
        '''
        return self.union(other)

    def intersection(*fsms):
        '''
            Intersection.
            Take FSMs and AND them together. That is, return an FSM which
            accepts any sequence of symbols that is accepted by both of the original
            FSMs. Note that the set of strings recognised by the two FSMs undergoes
            a set intersection operation.
            Call using "fsm3 = fsm1 & fsm2"
        '''
        return parallel(fsms, all)

    def __and__(self, other):
        '''
            Treat the FSMs as sets of strings and return the intersection of those
            sets in the form of a new FSM. `fsm1.intersection(fsm2, ...)` or
            `fsm.intersection(fsm1, ...)` are acceptable.
        '''
        return self.intersection(other)

    def symmetric_difference(*fsms):
        '''
            Treat `fsms` as a collection of sets of strings and compute the symmetric
            difference of them all. The python set method only allows two sets to be
            operated on at once, but we go the extra mile since it's not too hard.
        '''
        return parallel(fsms, lambda accepts: (accepts.count(True) % 2) == 1)

    def __xor__(self, other):
        '''
            Symmetric difference. Returns an FSM which recognises only the strings
            recognised by `self` or `other` but not both.
        '''
        return self.symmetric_difference(other)

    def everythingbut(self):
        '''
            Return a finite state machine which will accept any string NOT
            accepted by self, and will not accept any string accepted by self.
            This is more complicated if there are missing transitions, because the
            missing "dead" state must now be reified.
        '''
        alphabet = self.alphabet

        initial = {0 : self.initial}

        def follow(current, symbol):
            next = {}
            if 0 in current and current[0] in self.map and symbol in self.map[current[0]]:
                next[0] = self.map[current[0]][symbol]
            return next

        # state is final unless the original was
        def final(state):
            return not (0 in state and state[0] in self.finals)

        return crawl(alphabet, initial, final, follow).reduce()

    def reversed(self):
        '''
            Return a new FSM such that for every string that self accepts (e.g.
            "beer", the new FSM accepts the reversed string ("reeb").
        '''
        alphabet = self.alphabet

        # Start from a composite "state-set" consisting of all final states.
        # If there are no final states, this set is empty and we'll find that
        # no other states get generated.
        initial = frozenset(self.finals)

        # Find every possible way to reach the current state-set
        # using this symbol.
        def follow(current, symbol):
            next = frozenset([
                prev
                for prev in self.map
                for state in current
                if symbol in self.map[prev] and self.map[prev][symbol] == state
            ])
            if len(next) == 0:
                raise OblivionError
            return next

        # A state-set is final if the initial state is in it.
        def final(state):
            return self.initial in state

        # Man, crawl() is the best!
        return crawl(alphabet, initial, final, follow)
        # Do not reduce() the result, since reduce() calls us in turn

    def __reversed__(self):
        '''
            Return a new FSM such that for every string that self accepts (e.g.
            "beer", the new FSM accepts the reversed string ("reeb").
        '''
        return self.reversed()

    def islive(self, state):
        '''A state is "live" if a final state can be reached from it.'''
        reachable = [state]
        i = 0
        while i < len(reachable):
            current = reachable[i]
            if current in self.finals:
                return True
            if current in self.map:
                for symbol in self.map[current]:
                    next = self.map[current][symbol]
                    if next not in reachable:
                        reachable.append(next)
            i += 1
        return False

    def empty(self):
        '''
            An FSM is empty if it recognises no strings. An FSM may be arbitrarily
            complicated and have arbitrarily many final states while still recognising
            no strings because those final states may all be inaccessible from the
            initial state. Equally, an FSM may be non-empty despite having an empty
            alphabet if the initial state is final.
        '''
        return not self.islive(self.initial)

    def strings(self):
        '''
            Generate strings (lists of symbols) that this FSM accepts. Since there may
            be infinitely many of these we use a generator instead of constructing a
            static list. Strings will be sorted in order of length and then lexically.
            This procedure uses arbitrary amounts of memory but is very fast. There
            may be more efficient ways to do this, that I haven't investigated yet.
            You can use this in list comprehensions.
        '''

        # Many FSMs have "dead states". Once you reach a dead state, you can no
        # longer reach a final state. Since many strings may end up here, it's
        # advantageous to constrain our search to live states only.
        livestates = set(state for state in self.states if self.islive(state))

        # We store a list of tuples. Each tuple consists of an input string and the
        # state that this input string leads to. This means we don't have to run the
        # state machine from the very beginning every time we want to check a new
        # string.
        strings = []

        # Initial entry (or possibly not, in which case this is a short one)
        cstate = self.initial
        cstring = []
        if cstate in livestates:
            if cstate in self.finals:
                yield cstring
            strings.append((cstring, cstate))

        # Fixed point calculation
        i = 0
        while i < len(strings):
            (cstring, cstate) = strings[i]
            if cstate in self.map:
                for symbol in sorted(self.map[cstate], key=key):
                    nstate = self.map[cstate][symbol]
                    nstring = cstring + [symbol]
                    if nstate in livestates:
                        if nstate in self.finals:
                            yield nstring
                        strings.append((nstring, nstate))
            i += 1

    def __iter__(self):
        '''
            This allows you to do `for string in fsm1` as a list comprehension!
        '''
        return self.strings()

    def equivalent(self, other):
        '''
            Two FSMs are considered equivalent if they recognise the same strings.
            Or, to put it another way, if their symmetric difference recognises no
            strings.
        '''
        return (self ^ other).empty()

    def __eq__(self, other):
        '''
            You can use `fsm1 == fsm2` to determine whether two FSMs recognise the
            same strings.
        '''
        return self.equivalent(other)

    def different(self, other):
        '''
            Two FSMs are considered different if they have a non-empty symmetric
            difference.
        '''
        return not (self ^ other).empty()

    def __ne__(self, other):
        '''
            Use `fsm1 != fsm2` to determine whether two FSMs recognise different
            strings.
        '''
        return self.different(other)

    def difference(*fsms):
        '''
            Difference. Returns an FSM which recognises only the strings
            recognised by the first FSM in the list, but none of the others.
        '''
        return parallel(fsms, lambda accepts: accepts[0] and not any(accepts[1:]))

    def __sub__(self, other):
        return self.difference(other)

    def cardinality(self):
        '''
            Consider the FSM as a set of strings and return the cardinality of that
            set, or raise an OverflowError if there are infinitely many
        '''
        num_strings = {}
        def get_num_strings(state):
            # Many FSMs have at least one oblivion state
            if self.islive(state):
                if state in num_strings:
                    if num_strings[state] is None: # "computing..."
                        # Recursion! There are infinitely many strings recognised
                        raise OverflowError(state)
                    return num_strings[state]
                num_strings[state] = None # i.e. "computing..."

                n = 0
                if state in self.finals:
                    n += 1
                if state in self.map:
                    for symbol in self.map[state]:
                        n += get_num_strings(self.map[state][symbol])
                num_strings[state] = n

            else:
                # Dead state
                num_strings[state] = 0

            return num_strings[state]

        return get_num_strings(self.initial)

    def __len__(self):
        '''
            Consider the FSM as a set of strings and return the cardinality of that
            set, or raise an OverflowError if there are infinitely many
        '''
        return self.cardinality()

    def isdisjoint(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if they are disjoint
        '''
        return (self & other).empty()

    def issubset(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a subset
            of `other`... `self` recognises no strings which `other` doesn't.
        '''
        return (self - other).empty()

    def __le__(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a subset
            of `other`... `self` recognises no strings which `other` doesn't.
        '''
        return self.issubset(other)

    def ispropersubset(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a proper
            subset of `other`.
        '''
        return self <= other and self != other

    def __lt__(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a strict
            subset of `other`.
        '''
        return self.ispropersubset(other)

    def issuperset(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a
            superset of `other`.
        '''
        return (other - self).empty()

    def __ge__(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a
            superset of `other`.
        '''
        return self.issuperset(other)

    def ispropersuperset(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a proper
            superset of `other`.
        '''
        return self >= other and self != other

    def __gt__(self, other):
        '''
            Treat `self` and `other` as sets of strings and see if `self` is a
            strict superset of `other`.
        '''
        return self.ispropersuperset(other)

    def copy(self):
        '''
            For completeness only, since `set.copy()` also exists. FSM objects are
            immutable, so I can see only very odd reasons to need this.
        '''
        return fsm(
            alphabet = self.alphabet,
            states   = self.states,
            initial  = self.initial,
            finals   = self.finals,
            map      = self.map,
        )

    def derive(self, input):
        '''
            Compute the Brzozowski derivative of this FSM with respect to the input
            string of symbols. <https://en.wikipedia.org/wiki/Brzozowski_derivative>
            If any of the symbols are not members of the alphabet, that's a KeyError.
            If you fall into oblivion, then the derivative is an FSM accepting no
            strings.
        '''
        try:
            # Consume the input string.
            state = self.initial
            for symbol in input:
                if not symbol in self.alphabet:
                    if not anything_else in self.alphabet:
                        raise KeyError(symbol)
                    symbol = anything_else

                # Missing transition = transition to dead state
                if not (state in self.map and symbol in self.map[state]):
                    raise OblivionError

                state = self.map[state][symbol]

            # OK so now we have consumed that string, use the new location as the
            # starting point.
            return fsm(
                alphabet = self.alphabet,
                states   = self.states,
                initial  = state,
                finals   = self.finals,
                map      = self.map,
            )

        except OblivionError:
            # Fell out of the FSM. The derivative of this FSM is the empty FSM.
            return null(self.alphabet)

def null(alphabet):
    '''
        An FSM accepting nothing (not even the empty string). This is
        demonstrates that this is possible, and is also extremely useful
        in some situations
    '''
    return fsm(
        alphabet = alphabet,
        states   = {0},
        initial  = 0,
        finals   = set(),
        map      = {
            0: dict([(symbol, 0) for symbol in alphabet]),
        },
    )

def epsilon(alphabet):
    '''
        Return an FSM matching an empty string, "", only.
        This is very useful in many situations
    '''
    return fsm(
        alphabet = alphabet,
        states   = {0},
        initial  = 0,
        finals   = {0},
        map      = {},
    )

def parallel(fsms, test):
    '''
        Crawl several FSMs in parallel, mapping the states of a larger meta-FSM.
        To determine whether a state in the larger FSM is final, pass all of the
        finality statuses (e.g. [True, False, False] to `test`.
    '''
    alphabet = set().union(*[fsm.alphabet for fsm in fsms])

    initial = dict([(i, fsm.initial) for (i, fsm) in enumerate(fsms)])

    # dedicated function accepts a "superset" and returns the next "superset"
    # obtained by following this transition in the new FSM
    def follow(current, symbol):
        next = {}
        for i in range(len(fsms)):
            if symbol not in fsms[i].alphabet and anything_else in fsms[i].alphabet:
                actual_symbol = anything_else
            else:
                actual_symbol = symbol
            if i in current \
            and current[i] in fsms[i].map \
            and actual_symbol in fsms[i].map[current[i]]:
                next[i] = fsms[i].map[current[i]][actual_symbol]
        if len(next.keys()) == 0:
            raise OblivionError
        return next

    # Determine the "is final?" condition of each substate, then pass it to the
    # test to determine finality of the overall FSM.
    def final(state):
        accepts = [i in state and state[i] in fsm.finals for (i, fsm) in enumerate(fsms)]
        return test(accepts)

    return crawl(alphabet, initial, final, follow).reduce()

def crawl(alphabet, initial, final, follow):
    '''
        Given the above conditions and instructions, crawl a new unknown FSM,
        mapping its states, final states and transitions. Return the new FSM.
        This is a pretty powerful procedure which could potentially go on
        forever if you supply an evil version of follow().
    '''

    states = [initial]
    finals = set()
    map = {}

    # iterate over a growing list
    i = 0
    while i < len(states):
        state = states[i]

        # add to finals
        if final(state):
            finals.add(i)

        # compute map for this state
        map[i] = {}
        for symbol in sorted(alphabet, key=key):
            try:
                next = follow(state, symbol)

                try:
                    j = states.index(next)
                except ValueError:
                    j = len(states)
                    states.append(next)

            except OblivionError:
                # Reached an oblivion state. Don't list it.
                continue

            map[i][symbol] = j

        i += 1

    return fsm(
        alphabet = alphabet,
        states   = range(len(states)),
        initial  = 0,
        finals   = finals,
        map      = map,
    )
