Speech Recognition CSCI-GA Fall Homework Assignment #1: Automata Operations. Instructor: Eugene Weinstein. Due Date: October 17th

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1 Speech Recognition CSCI-GA Fall 2013 Homework Assignment #1: Automata Operations Instructor: Eugene Weinstein Due Date: October 17th Note: It is advised, but not required, to use the OpenFST library to complete this assignment. In order to receive full credit, you must give the full sequence of shell commands and give the code of any script/program used to complete each problem. Note that not every single automata operation required to complete this assignment has been covered in class, and you may need to do some research on your own in the relevant documentation. A. Regular expressions and automata. Give epsilon free, deterministic, and minimal finite state acceptors accepting only the strings matching the following regular expressions: a (a ba)* ba* ( a b) *(baab)* ( a b) * baab

2 Producing deterministic and minimal versions of the acceptor (for first regular expression): $ cat > syms.txt a 1 b 2 $ cat > 1a1.txt 0 1 a 1 1 a 1 2 b 2 1 a 1 3 b 3 4 a 4 4 a 4 fstcompile acceptor isymbols=syms.txt 1a1.txt > 1a1.fst cat 1a1.fst fstdeterminize fstminimize > 1a1 opt.fst fstdraw isymbols=syms.txt acceptor portrait 1a1 opt.fst dot Tpng o 1a1 opt.png B. Weighted acceptors and transducers 1. Give a weighted finite state acceptor in the tropical semiring producing strings of length four over the alphabet Σ = { a, b} where a is three times as likely to be produced than b. l n(0.25) = ; l n(0.75) = Give a weighted finite state transducer T which can be used to count the number of and b symbols accepted by a (possibly weighted) finite state acceptor A. State the semiring over which your method gives the desired result, and give the precise sequence of automata operations required (hint: the composition A T will come in handy here). a The counting transducer T is over the log semiring. Note that T (x) = 1 x {a, b } *. Let C = Π o (A T ), where Π o is the projection to the output operation. Then e C(x) is the count of occurrences of the string x in A.

3 3. Randomly generate 100 strings according to the constraints of part 1: the alphabet is Σ = { a, b}, the string is of length four and a is more likely than b. You may do this using the OpenFST tools, or using a script of your own. # In order to generate the strings, we will sample from the acceptor of part 1 above. $ cat fstcompile acceptor isymbols=syms.txt 1b.fst 0 1 a b a b a b a b *ctrl d* # Generate 4 letter strings and put them all into strings.txt, one per line $ for i in {1..100}; do fstrandgen weighted select=log_prob 1b.fst fstprint isymbols=syms.txt awk '{print $3}' tr '\n' ' '; echo; sleep 2; done > strings.txt 4. Apply the transducer T you created in part 2 to the strings from part 3. Does the count of a and b symbols match the distribution of symbols given in part1? # Make a FST archive (far) of the generated strings: $ farcompilestrings symbols=syms.txt strings.txt > strings.far $ farextract strings.far $ echo fstcompile > union.fst # Empty transducer # Union together all string transducers produced $ for file in `ls strings.txt *`; do fstunion union.fst $file > foo.fst; mv foo.fst union.fst ; done # Convert to log semiring and sort on output label for composition $ fstprint union.fst fstcompile arc_type=log fstarcsort sort_type=olabel > union log.fst $ cat fstcompile arc_type=log isymbols=syms.txt osymbols=syms.txt count.fst 0 0 a <eps> 0 0 b <eps> 0 1 a a 0 1 b b 1 1 a <eps> 1 1 b <eps> 1 *ctrl d* $ fstcompose fsts/union log.fst count.fst fstproject project_output fstrmepsilon fstdeterminize fstminimize fstprint acceptor isymbols=syms.txt # output looks like: 0 1 a b The final answer is obtained as e , e The counts add up to 400, and

4 match our expected distribution. C. Camel casing with automata. 1. Download the list of 100 most common English words from 2. From this list, build a camel casing transducer T, i.e., that which, when composed with acceptor A accepting lowercase strings consisting of words in the list, can be used to produce a string acceptor A c, where the strings are camel cased. For example, if the input acceptor accepts the strings gowithme and dosomework, the output acceptor should accept the strings GoWithMe and DoSomeWork. Do not assume that the words are space separated. $ head words.txt the be to of and a in that have I # See appendix below for code listing of words.py $ cat words.txt./words.py fstcompile isymbols=syms.txt osymbols=syms.txt fstclosure > camelcase.fst The desired camelcased acceptor is found as A c = Π o (A T ) where Π o is projection on the output labels. 3. Repeat the process for just the first five words in the list to construct transducer T. Show this transducer.

5 4. Demonstrate that your transducer T works by showing the input acceptor A and the resulting transducer A T for both of the examples given in sub question 2 above. Be sure to also verify that no outputs are produced if the input does not consist solely of words in the list of 100 words. $ cat fstcompile isymbols=syms.txt acceptor > dosomework.fst 0 1 d 1 2 o 2 3 s 3 4 o 4 5 m 5 6 e 6 7 w 7 8 o 8 9 r 9 10 k 10 *ctrl d* $ fstcompose dosomework.fst camelcase.fst fstrmepsilon > dosomework camel.fst $ cat dosomework camel.fst fstdraw isymbols=syms.txt osymbols=syms.txt portrait dot Tpng odosomework camel.png 5. How many states and transitions does your transducer T have? Is it epsilon free and/or deterministic? If not, what ideas do you have for determinizing it (you do not have to implement them)? $ fstinfo camelcase.fst # of states 340 # of arcs 439

6 input deterministic output deterministic input/output epsilons input epsilons output epsilons n n y y y No, the transducer is not deterministic, and it has epsilons. The issue with determinizing such a transducer is that it s not functional, i.e., that is it does not map each input sequence to a unique output sequence. It s possible to determinize such a transducer by introducing special symbols at the end of each output string corresponding to more than one input path. With such symbols, the transducer becomes functional, and can then be determinized. The special symbols can then be replaced with epsilons. See the discussion about p subsequential transducers in Mehryar Mohri. Finite State Transducers in Language and Speech Processing. Computational Linguistics, 23:2, Appendix: words.py #!/usr/bin/python import sys state_num = 1 last_state = 0 for line in sys.stdin: for i, c in enumerate(line.rstrip()): input_sym = c.lower() if i == 0: last_state = 0 output_sym = c.upper() else: output_sym = input_sym print "%d %d %s %s" % ( last_state, state_num, input_sym, output_sym) last_state = state_num state_num += 1 print "%d" % last_state