On the power of finite automata with both nondeterministic and probabilistic states

Anne Condon, Lisa Hellerstein, Samuel Pottle, Avi Wigderson

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Abstract

    We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where the players are limited to polynomial time and constant space. Dwork and Stockmeyer asked whether the above class of npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the `characteristic matrix' of L, namely the binary matrix with a row and column for each string of length ≤n, where entry [x,y] = 1 if and only if the string xyqqL. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model, and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices.

    Original languageEnglish (US)
    Title of host publicationConference Proceedings of the Annual ACM Symposium on Theory of Computing
    PublisherPubl by ACM
    Pages676-685
    Number of pages10
    ISBN (Print)0897916638
    StatePublished - 1994
    EventProceedings of the 26th Annual ACM Symposium on the Theory of Computing - Montreal, Que, Can
    Duration: May 23 1994May 25 1994

    Other

    OtherProceedings of the 26th Annual ACM Symposium on the Theory of Computing
    CityMontreal, Que, Can
    Period5/23/945/25/94

    Fingerprint

    Finite automata
    Polynomials
    Formal languages
    Tile
    Tapes

    ASJC Scopus subject areas

    • Software

    Cite this

    Condon, A., Hellerstein, L., Pottle, S., & Wigderson, A. (1994). On the power of finite automata with both nondeterministic and probabilistic states. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing (pp. 676-685). Publ by ACM.

    On the power of finite automata with both nondeterministic and probabilistic states. / Condon, Anne; Hellerstein, Lisa; Pottle, Samuel; Wigderson, Avi.

    Conference Proceedings of the Annual ACM Symposium on Theory of Computing. Publ by ACM, 1994. p. 676-685.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Condon, A, Hellerstein, L, Pottle, S & Wigderson, A 1994, On the power of finite automata with both nondeterministic and probabilistic states. in Conference Proceedings of the Annual ACM Symposium on Theory of Computing. Publ by ACM, pp. 676-685, Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, Montreal, Que, Can, 5/23/94.
    Condon A, Hellerstein L, Pottle S, Wigderson A. On the power of finite automata with both nondeterministic and probabilistic states. In Conference Proceedings of the Annual ACM Symposium on Theory of Computing. Publ by ACM. 1994. p. 676-685
    Condon, Anne ; Hellerstein, Lisa ; Pottle, Samuel ; Wigderson, Avi. / On the power of finite automata with both nondeterministic and probabilistic states. Conference Proceedings of the Annual ACM Symposium on Theory of Computing. Publ by ACM, 1994. pp. 676-685
    @inproceedings{4f18dff6cdff41e08e5acf472c53da34,
    title = "On the power of finite automata with both nondeterministic and probabilistic states",
    abstract = "We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where the players are limited to polynomial time and constant space. Dwork and Stockmeyer asked whether the above class of npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the `characteristic matrix' of L, namely the binary matrix with a row and column for each string of length ≤n, where entry [x,y] = 1 if and only if the string xyqqL. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model, and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices.",
    author = "Anne Condon and Lisa Hellerstein and Samuel Pottle and Avi Wigderson",
    year = "1994",
    language = "English (US)",
    isbn = "0897916638",
    pages = "676--685",
    booktitle = "Conference Proceedings of the Annual ACM Symposium on Theory of Computing",
    publisher = "Publ by ACM",

    }

    TY - GEN

    T1 - On the power of finite automata with both nondeterministic and probabilistic states

    AU - Condon, Anne

    AU - Hellerstein, Lisa

    AU - Pottle, Samuel

    AU - Wigderson, Avi

    PY - 1994

    Y1 - 1994

    N2 - We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where the players are limited to polynomial time and constant space. Dwork and Stockmeyer asked whether the above class of npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the `characteristic matrix' of L, namely the binary matrix with a row and column for each string of length ≤n, where entry [x,y] = 1 if and only if the string xyqqL. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model, and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices.

    AB - We study finite automata with both nondeterministic and random states (npfa's). We restrict our attention to those npfa's that accept their languages with a small probability of error and run in polynomial expected time. Equivalently, we study Arthur-Merlin games where the players are limited to polynomial time and constant space. Dwork and Stockmeyer asked whether the above class of npfa's accept only the regular languages (this was known if the automaton has only randomness or only nondeterminism). We show that the answer is yes in the case of npfa's with a 1-way input head. We also show that if L is a nonregular language, then either L or L̄ is not accepted by any npfa with a 2-way input head. Toward this end, we define a new measure of the complexity of a language L, called its 1-tiling complexity. For each n, this is the number of tiles needed to cover the 1's in the `characteristic matrix' of L, namely the binary matrix with a row and column for each string of length ≤n, where entry [x,y] = 1 if and only if the string xyqqL. We show that a language has constant 1-tiling complexity if and only if it is regular, from which the result on 1-way input follows. Our main result regarding the general 2-way input tape follows by contrasting two bounds: an upper bound of polylog(n) on the 1-tiling complexity of every language computed by our model, and a lower bound stating that the 1-tiling complexity of a nonregular language or its complement exceeds a function in 2Ω(√log n) infinitely often. The last lower bound follows by proving that the characteristic matrix of every nonregular language has rank n for infinitely many n. This is our main technical result, and its proof uses techniques of Frobenius and Iohvidov developed for Hankel matrices.

    UR - http://www.scopus.com/inward/record.url?scp=0027929414&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=0027929414&partnerID=8YFLogxK

    M3 - Conference contribution

    AN - SCOPUS:0027929414

    SN - 0897916638

    SP - 676

    EP - 685

    BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

    PB - Publ by ACM

    ER -