asr_lab5_solutions.ipynb

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    "# Lab 5  - WFST operations\n",
    "\n",
    "So far we've used WFSTs mainly as a usual structure for encoding and traversing HMMs.  In this lab we'll move away from HMM acoustic modelling and look at how WFST operations can be used to avoid the need for specialised algorithms in speech and language processing.  It is intended to give you insight into how these operations are used to construct HMMs encapsulating langauge model, pronunciation and acoustic modelling assumptions &ndash; the so-called \"HCLG\" WFST.\n",
    "\n",
    "This lab will focus on the lexicon transducer, $L$, and grammar transducer, $G$.\n",
    "\n",
    "We'll use some of the following operations, defined by Openfst:\n",
    "* `fst.determinize(f)` creates determinized version of `f`\n",
    "* `fst.compose(f1,f2)` composes FSTs `f1` and `f2`\n",
    "* `fst.shortestpath(f)` returns the shortest path (in terms of weight) through `f` from the start to a final state\n",
    "* `f.minimize()` creates minimized version of `f`\n",
    "* `f.project(project_output=False)` for every arc in `f`, copies the input label to the output label (or vice versa, if `project_output=True`).\n",
    "* `f.rmepsilon()` removes epsilon transitions &ndash; those arcs where both input and output labels are empty\n",
    "\n",
    "For efficiency, the compostion of `f1` and `f2` requires either the output arcs of `f1` or input arcs of `f2` to be sorted prior to `compose()` being called.  You can do this by calling `f1.arcsort(sort_type='olabel')` or `f2.arcsort(sort_type='ilabel')`.\n",
    "\n",
    "The functions above assume that `openfst_python` has been imported as `fst`.  Note that the first three functions above return a new WFST; the others modify the WFST *in place*, meaning that the original WFST is modified directly.\n",
    "\n",
    "For convenience, we've provided a python module `helper_functions` that provides the `parse_lexicon()` and `generate_symbol_tables()` from the [Lab 1 solutions](https://github.com/Ore-an/asr_lab1/blob/master/asr_lab1_solutions.ipynb).  And here is a function to generate an $L$ transducer:"
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   "source": [
    "import openfst_python as fst\n",
    "from helper_functions import parse_lexicon, generate_symbol_tables\n",
    "\n",
    "lex = parse_lexicon('lexicon.txt')\n",
    "word_table, phone_table, state_table = generate_symbol_tables(lex)  # we won't use state_table in this lab\n",
    "\n",
    "def generate_L_wfst(lex):\n",
    "    \"\"\" Express the lexicon in WFST form\n",
    "    \n",
    "    Args:\n",
    "        lexicon (dict): lexicon to use, created from the parse_lexicon() function\n",
    "    \n",
    "    Returns:\n",
    "        the constructed lexicon WFST\n",
    "    \n",
    "    \"\"\"\n",
    "    L = fst.Fst()\n",
    "    \n",
    "    # create a single start state\n",
    "    start_state = L.add_state()\n",
    "    L.set_start(start_state)\n",
    "    \n",
    "    for (word, pron) in lex.items():\n",
    "        \n",
    "        current_state = start_state\n",
    "        for (i,phone) in enumerate(pron):\n",
    "            next_state = L.add_state()\n",
    "            \n",
    "            if i == len(pron)-1:\n",
    "                # add word output symbol on the final arc\n",
    "                L.add_arc(current_state, fst.Arc(phone_table.find(phone), \\\n",
    "                                                 word_table.find(word), None, next_state))\n",
    "            else:\n",
    "                L.add_arc(current_state, fst.Arc(phone_table.find(phone),0, None, next_state))\n",
    "            \n",
    "            current_state = next_state\n",
    "                          \n",
    "        L.set_final(current_state)\n",
    "        \n",
    "    L.set_input_symbols(phone_table)\n",
    "    L.set_output_symbols(word_table)                      \n",
    "    \n",
    "    return L\n",
    "\n",
    "L = generate_L_wfst(lex)\n",
    "L.arcsort()"
   ]
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   "metadata": {},
   "source": [
    "For the exercises, here are two functions to generate linear WFSTs for an arbitary sequence of phones or words.  (Yes, they are really just variants of the same function!)"
   ]
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   "source": [
    "def generate_linear_phone_wfst(phone_list):\n",
    "    \n",
    "    P = fst.Fst()\n",
    "    \n",
    "    current_state = P.add_state()\n",
    "    P.set_start(current_state)\n",
    "    \n",
    "    for p in phone_list:\n",
    "        \n",
    "        next_state = P.add_state()\n",
    "        P.add_arc(current_state, fst.Arc(phone_table.find(p), phone_table.find(p), None, next_state))\n",
    "        current_state = next_state\n",
    "        \n",
    "    P.set_final(current_state)\n",
    "    P.set_input_symbols(phone_table)\n",
    "    P.set_output_symbols(phone_table)\n",
    "    return P\n",
    "    \n",
    "def generate_linear_word_wfst(word_list):\n",
    "    \n",
    "    W = fst.Fst()\n",
    "    \n",
    "    current_state = W.add_state()\n",
    "    W.set_start(current_state)\n",
    "    \n",
    "    for w in word_list:\n",
    "        \n",
    "        next_state = W.add_state()\n",
    "        W.add_arc(current_state, fst.Arc(word_table.find(w), word_table.find(w), None, next_state))\n",
    "        current_state = next_state\n",
    "        \n",
    "    W.set_final(current_state)\n",
    "    W.set_input_symbols(word_table)\n",
    "    W.set_output_symbols(word_table)\n",
    "    return W"
   ]
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   "source": [
    "## Exercises\n",
    "\n",
    "1. Suppose you are given a sequence of phones, in the form `['p','ih','k','t']`, and the $L$ transducer created above.  Write a function that returns the matching word from the lexicon for any given phone sequence, or else `None` if no matching word is found.   Write two functions:\n",
    "  1. That works for $L$ as provided by the code above\n",
    "  2. That works only on a determinized version of $L$ &ndash; and test it on the output of `fst.determinize(L)`\n",
    "  \n",
    " This should enable you to see why determinization is a very useful WFST operation!\n",
    "  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "\n",
    "def transduce_sequence_nondet(f, in_seq):\n",
    "    \"\"\"Return transduced sequence given input sequence and non determinized FST\n",
    "    \n",
    "        Args:\n",
    "            f (fst.Fst()): a non determinized FST\n",
    "            in_seq (list[str]): the sequence of strings to transduce\n",
    "            \n",
    "        Returns:\n",
    "            out_seq (list[str]): the sequence of transduced symbols\n",
    "            \"\"\"\n",
    "    \n",
    "    seq_len = len(in_seq)\n",
    "    in_seq.append('<EOS>') # adding a padding symbol at the end for possible final eps traversal\n",
    "    eps = f.input_symbols().find('<eps>')\n",
    "    queue = [(f.start(), 0, [])]  # the tuple is (state, index in input sequence, output)\n",
    "    \n",
    "    while queue:\n",
    "        curr_state, i, output = queue.pop(0) # pop first element in list\n",
    "        \n",
    "        if i <= seq_len:  # <= because we could traverse epsilons even when the input sequence ended\n",
    "            label = f.input_symbols().find(in_seq[i])  # transform label into index in table\n",
    "            for arc in f.arcs(curr_state):\n",
    "                if arc.ilabel == label:\n",
    "                    new_output = output + [arc.olabel]\n",
    "                    queue.append((arc.nextstate, i+1, new_output))\n",
    "                elif arc.ilabel == eps and arc.nextstate != curr_state:\n",
    "                    new_output = output + [arc.olabel]\n",
    "                    queue.append((arc.nextstate, i, new_output))  # we're not advancing in the input sequence because it's epsilon\n",
    "                        \n",
    "        if i == seq_len:\n",
    "            final_weight = float(f.final(curr_state))\n",
    "            if final_weight != math.inf: # if this is a final state\n",
    "                out_seq = [f.output_symbols().find(w) for w in output if w != eps]  # find the labels in the table, remove epsilons\n",
    "                return out_seq\n",
    "            \n",
    "        \n",
    "    print(\"Can't transduce the sequence with provided FST\")  # return exits the function, so this is printed only when the stack is empty and we didn't find a path "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['picked']\n"
     ]
    }
   ],
   "source": [
    "seq = ['p','ih','k','t']\n",
    "print(transduce_sequence_nondet(L, seq))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "def transduce_sequence_det(f, seq):\n",
    "    \"\"\"Return transduced sequence given input sequence and determinized FST\n",
    "    \n",
    "        Args:\n",
    "            f (fst.Fst()): a determinized FST\n",
    "            in_seq (list[str]): the sequence of strings to transduce\n",
    "            \n",
    "        Returns:\n",
    "            out_seq (list[str]): the sequence of transduced symbols\n",
    "            \"\"\"\n",
    "    \n",
    "    seq_len = len(seq)\n",
    "    eps = f.input_symbols().find('<eps>')\n",
    "    curr_state = f.start()\n",
    "    output = []\n",
    "    \n",
    "    for i in range(seq_len):\n",
    "        found = False\n",
    "        label = f.input_symbols().find(seq[i])\n",
    "        for arc in f.arcs(curr_state):\n",
    "            if arc.ilabel == label:\n",
    "                output += [arc.olabel]\n",
    "                curr_state = arc.nextstate\n",
    "                found = True\n",
    "                break  # no need to keep going through other arcs, as it's determinized\n",
    "        if not found:\n",
    "            print(\"Can't transduce the sequence with provided FST\")\n",
    "                \n",
    "    final_weight = float(f.final(curr_state))\n",
    "    if final_weight != math.inf: # if this is a final state\n",
    "        out_seq = [f.output_symbols().find(w) for w in output if w != eps]  # find the labels in the table, remove epsilons\n",
    "        return out_seq\n",
    "    else:  \n",
    "        print(\"Can't transduce the sequence with provided FST\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "['picked']\n"
     ]
    }
   ],
   "source": [
    "seq = ['p','ih','k','t']\n",
    "Ldet = fst.determinize(L)\n",
    "print(transduce_sequence_det(Ldet, seq))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "2. WFST composition allows you to achieve the same result much more easily.  Create a linear WFST, $P$, corresponding to a string of phones, and compute $P \\circ L$.  Then use the projection and epsilon removal operations to display just the matching word."
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    "seq = ['p','ih','k','t']\n",
    "P = generate_linear_phone_wfst(seq)\n",
    "P.arcsort(sort_type='ilabel')\n",
    "comp = fst.compose(P, L)\n",
    "comp.project(project_output=True).rmepsilon()"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "3. Modify your lexicon WFST slightly to allow a list of phones to be \"decoded\" to a sequence of multiple words from the lexicon, using composition.  Try it with `['p','eh','k','ah','v','p','iy','t','er']`.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# this modified the Lexicon WFST directly - you could have done it more simply by adding\n",
    "# extra code to the generate_L_wfst() function above\n",
    "\n",
    "start_state = L.start()\n",
    "for state in L.states():\n",
    "    if float(L.final(state)) != math.inf:\n",
    "        L.add_arc(state, fst.Arc(0, 0, None, start_state))  # add arc to start\n",
    "Ldet = fst.determinize(L)"
   ]
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    "seq = ['p','eh','k','ah','v','p','iy','t','er']\n",
    "P = generate_linear_phone_wfst(seq)\n",
    "P.arcsort(sort_type='ilabel')\n",
    "comp = fst.compose(P, L)\n",
    "comp.project(project_output=True).rmepsilon()"
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   "cell_type": "markdown",
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   "source": [
    "4. Now solve the reverse problem: create a word-sequence WFST, $W$, and use composition to expand it into a sequence of phones."
   ]
  },
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   "cell_type": "code",
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    "seq = ['peck', 'of', 'peter']\n",
    "W = generate_linear_word_wfst(seq)\n",
    "W.arcsort(sort_type='olabel')\n",
    "comp = fst.compose(L, W)\n",
    "comp.project().rmepsilon()"
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    "5. Another advantage of WFST composition to solve these kind of problems are that it is easy to encode uncertainty in the input (a bit like in real ASR).  For example, consider this WFST, in which the multiple arcs denote alternative phone transcriptions from the acoustic model:"
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    "def create_alt_phone_wfst(phone_alternatives):\n",
    "    \n",
    "    P = fst.Fst()\n",
    "    \n",
    "    current_state = P.add_state()\n",
    "    P.set_start(current_state)\n",
    "    \n",
    "    for alt in phone_alternatives:\n",
    "        \n",
    "        next_state = P.add_state()\n",
    "        for p in alt:\n",
    "            if p=='*':\n",
    "                P.set_final(current_state)\n",
    "            else:\n",
    "                P.add_arc(current_state, fst.Arc(phone_table.find(p), phone_table.find(p), None, next_state))\n",
    "        current_state = next_state\n",
    "        \n",
    "    P.set_final(current_state)\n",
    "    P.set_input_symbols(phone_table)\n",
    "    P.set_output_symbols(phone_table)\n",
    "    return P    \n",
    "    \n",
    "altP = create_alt_phone_wfst([['p'],['ay'],['p'],['er'],['p'],['eh','ih'],['k'],['t','<eps>'],['ah','<eps>'],['l','v','*'],['d','*']])\n",
    "altP"
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    "  Again, perform composition with your $L$ from Question 3, and observe the result.  (Notice particularly what happens to the `<eps>` transitions during composition.  "
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      "text/plain": [
       "<vector Fst at 0x7f6bc407a3b0>"
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     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "altP.arcsort()\n",
    "comp = fst.compose(altP, L)\n",
    "comp"
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    "6. We could have added weights to the arcs of the WFST above to describe the probability of the phone alternatives given by the acoustic model &ndash; this would have enabled you to find the most likely sequence of words.  Without this information, let's instead use a $G$ WFST to find the most likely sequence.  Let's assume that a word sequence taken from the passage \"peter piper picked a peck of pickled peppers\" is most likely.  Design a $G$ WFST that accepts any sequence of words from the lexicon, but adds a cost of 1.0 to any word transition not in the passage.  Given $G$, use composition to recover the most likely word sequence from the uncertain $P$."
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    "**Note on this solution**\n",
    "\n",
    "Peter and Andrea independently came up with solutions to this question.  It is well worth looking at both!"
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   "source": [
    "def generate_G_wfst_peter(wseq):\n",
    "    \n",
    "    G = fst.Fst()\n",
    "    \n",
    "    start_state = G.add_state()\n",
    "    G.set_start(start_state)\n",
    "    \n",
    "    prev_state = None\n",
    "    \n",
    "    for w in wseq.split():\n",
    "        \n",
    "        current_state = G.add_state()\n",
    "        \n",
    "        # add transition from the start with cost 1\n",
    "        G.add_arc(start_state, fst.Arc(word_table.find(w), word_table.find(w), 1.0, current_state))\n",
    "        \n",
    "        # arc from previous word with cost of zero\n",
    "        if prev_state:\n",
    "            G.add_arc(prev_state, fst.Arc(word_table.find(w), word_table.find(w), 0, current_state))\n",
    "        \n",
    "        # <eps> transition back to the start\n",
    "        G.add_arc(current_state, fst.Arc(0, 0, 0, start_state))\n",
    "    \n",
    "        prev_state = current_state\n",
    "    \n",
    "    G.set_final(start_state)\n",
    "        \n",
    "    G.set_input_symbols(word_table)\n",
    "    G.set_output_symbols(word_table)  \n",
    " \n",
    "    return G\n",
    "\n",
    "string = \"peter piper picked a peck of pickled peppers\"\n",
    "G = generate_G_wfst_peter(string)\n",
    "G\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
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    "def generate_G_wfst_andrea(wseq):\n",
    "    G = fst.Fst()\n",
    "    \n",
    "    word2state = {} # we will map each word in the lexicon to a state\n",
    "    state2word = {} # and vice-versa; note this is different from the symbol table mapping\n",
    "    \n",
    "    # create a single start state\n",
    "    start_state = G.add_state()\n",
    "    G.set_start(start_state)\n",
    "    \n",
    "    word2state['<s>'] = start_state  # we're using a start-of-sentence token to not overload epsilon\n",
    "    state2word[start_state] = '<eps>'  # epsilon is fine when we're looking up the state to put a label on the arc\n",
    "    \n",
    "    wseq = '<s> ' + wseq\n",
    "    \n",
    "    for word in lex.keys():\n",
    "        idx = G.add_state()\n",
    "        word2state[word] = idx \n",
    "        state2word[idx] = word\n",
    "        \n",
    "    bigrams = [w for w in zip(wseq.split(\" \")[:-1], wseq.split(\" \")[1:])] # zipping together the list with itself with offset 1\n",
    "    passage_state_trans = [(word2state[x], word2state[y]) for x,y in bigrams]  # state indexes for transitions existing in the passage, the ones we don't have to penalize\n",
    "    \n",
    "    for state1 in G.states():\n",
    "        if state1 != start_state:\n",
    "            G.set_final(state1)\n",
    "        for state2 in G.states():\n",
    "            if (state1, state2) in passage_state_trans:\n",
    "                weight = 0\n",
    "            else:\n",
    "                weight = 1.0\n",
    "            word = state2word[state2]\n",
    "            label = word_table.find(word)\n",
    "\n",
    "            G.add_arc(state1, fst.Arc(label, label, weight, state2))\n",
    "    G.set_input_symbols(word_table)\n",
    "    G.set_output_symbols(word_table)\n",
    "    return G\n",
    "\n",
    "string = \"peter piper picked a peck of pickled peppers\"\n",
    "G2 = generate_G_wfst_andrea(string)\n",
    "G2 = fst.determinize(G2)\n",
    "G2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
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    "# Composition with Peter's G wfst\n",
    "\n",
    "G.arcsort(sort_type='olabel')\n",
    "comp = fst.compose(uncertainP, G)\n",
    "comp.rmepsilon()\n",
    "\n",
    "# Note that by default when the weight is 0 it's not printed out in the graph\n",
    "# so the arcs that don't have a 1 are the most probable (log(0) > log(-1))"
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    "# Composition with Andrea's G wfst - notice how the results are the same\n",
    "# even thought the G wfsts are very different.\n",
    "\n",
    "G2.arcsort(sort_type='olabel')\n",
    "comp2 = fst.compose(uncertainP, G2)\n",
    "comp2.rmepsilon()"
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    "fst.shortestpath(comp)"
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  },
  {
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   "metadata": {},
   "source": [
    "## If you have more time"
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   "metadata": {},
   "source": [
    "Use WFST composition to implement a \"predictive text\"-style algorithm, that, given a partial phone sequence such as `['p']` or `['p','ih']`, returns a WFST giving all matching words.  You'll need to make some special modifications to $P$ or $L$, or both. On a determinized $L$ transducer this is a highly efficient way of solving this problem."
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   "execution_count": 36,
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     "execution_count": 36,
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   "source": [
    "# There are very many ways this problem can be solved. Our skeleton code add extra arcs with \n",
    "# special <rho> symbol. This symbol will represent unterminated sequences, and will be transduced\n",
    "#  to words to be output at every intermediate state.\n",
    "# The same <rho> symbol is added the end of the partial pronunciation.\n",
    "\n",
    "\n",
    "# In a real application, the lexicon would be determinised *before* the <rho> arcs are added\n",
    "\n",
    "def generate_predictive_L_wfst(lex):\n",
    "    \"\"\" express the lexicon in WFST form s.t. composition with partial sequence gives matching words\n",
    "    \n",
    "    Args:\n",
    "        lexicon (dict): lexicon to use, created from the parse_lexicon() function\n",
    "    \n",
    "    Returns:\n",
    "        the constructed WFST\n",
    "    \n",
    "    \"\"\"\n",
    "    Lpred = fst.Fst()\n",
    "    rho = phone_table.add_symbol('<rho>')\n",
    "    # create a single start state\n",
    "    start_state = Lpred.add_state()\n",
    "    Lpred.set_start(start_state)\n",
    "    \n",
    "    for (word, pron) in lex.items():\n",
    "        state_list = []\n",
    "        current_state = start_state\n",
    "        for (i,phone) in enumerate(pron):\n",
    "            next_state = Lpred.add_state()\n",
    "            state_list.append(next_state)\n",
    "            \n",
    "            if i == len(pron)-1:\n",
    "                # add word output symbol on the final arc\n",
    "                Lpred.add_arc(current_state, fst.Arc(phone_table.find(phone), \\\n",
    "                                                 word_table.find(word), None, next_state))\n",
    "            else:\n",
    "                Lpred.add_arc(current_state, fst.Arc(phone_table.find(phone), 0, None, next_state))\n",
    "            \n",
    "            current_state = next_state\n",
    "\n",
    "        Lpred.set_final(current_state)\n",
    "        for state in state_list:\n",
    "            if state != current_state:\n",
    "                Lpred.add_arc(state, fst.Arc(phone_table.find('<rho>'), word_table.find(word), None, current_state))\n",
    "            else:\n",
    "                Lpred.add_arc(state, fst.Arc(phone_table.find('<rho>'), 0, None, current_state)) # in the final state the word was already output, so no word on output\n",
    "        \n",
    "    Lpred.set_input_symbols(phone_table)\n",
    "    Lpred.set_output_symbols(word_table)                      \n",
    "    \n",
    "    return Lpred\n",
    "\n",
    "Lpred = generate_predictive_L_wfst(lex)\n",
    "Lpred.arcsort()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
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    "seq = ['p']\n",
    "seq += ['<rho>']\n",
    "Ppred = generate_linear_phone_wfst(seq)\n",
    "Ppred.arcsort(sort_type='ilabel')\n",
    "comp = fst.compose(Ppred, Lpred)\n",
    "comp.project(True).rmepsilon()"
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