Deterministic Frequency Pushdown Automata

A set L is (m,n)-computable iff there is a mechanism which on input of n different words produces n conjectures whether these words are in L, respectively, such that at least m of these conjectures are right. Prior studies dealt with (m,n)computable sets in the contexts of recursion theory, complexity theory and the theory of finite automata. The present work aims to do this with respect to computations by deterministic pushdown automata (using one common stack while processing all input words in parallel). We prove the existence of a deterministic context-free language L which is recognised by an (1, 1)-DPDA but fails to be recognised by any (m,n)-DPDA, where n ≥ 2 and m ≥ n/2+1. This answers a question posed by Eli Shamir at LATA 2013. Furthermore, it is shown that there is a language L such that, for all m,n with m ≤ n/2, L can be recognised by an (m,n)-DPDA but, for all m,n with 1 ≤ m ≤ n, L cannot be recognised by (m,n)-DFA.

[1]  Gerd Wechsung,et al.  Time Bounded Frequency Computations , 1997, Inf. Comput..

[2]  Robert McNaughton,et al.  The Theory of Automata, a Survey , 1961, Adv. Comput..

[3]  John Case,et al.  Learning Recursive Functions from Approximations , 1997, J. Comput. Syst. Sci..

[4]  Frank Stephan,et al.  Recursion Theoretic Properties of Frequency Computation and Bounded Queries , 1995, Inf. Comput..

[5]  Efim B. Kinber,et al.  Frequency Computation and Bounded Queries , 1996, Theor. Comput. Sci..

[6]  Volker Diekert,et al.  Regular frequency computations , 2005, Theor. Comput. Sci..

[7]  Rusins Freivalds Complexity of Probabilistic Versus Deterministic Automata , 1991, Baltic Computer Science.

[8]  A. F. Adams,et al.  The Survey , 2021, Dyslexia in Higher Education.

[9]  Rusins Freivalds,et al.  Frequency Prediction of Functions , 2011, MEMICS.

[10]  STACS 2002 , 2002, Lecture Notes in Computer Science.

[11]  Martin Kummer A Proof of Beigel's Cardinality Conjecture , 1992, J. Symb. Log..

[12]  Gerd Wechsung,et al.  Time bounded frequency computations , 1997, Proceedings of Computational Complexity. Twelfth Annual IEEE Conference.

[13]  Carl H. Smith,et al.  On learning multiple concepts in parallel , 1993, COLT '93.

[14]  Rusins Freivalds,et al.  On the Size Complexity of Deterministic Frequency Automata , 2013, LATA.

[15]  Valentina S. Harizanov,et al.  Frequency Computations and the Cardinality Theorem , 1992, J. Symb. Log..

[16]  Till Tantau,et al.  Comparing Verboseness for Finite Automata and Turing Machines , 2002, Theory of Computing Systems.

[17]  David S. Tartakoff,et al.  Review: A. M. Barzdin, On a Class of Turing Machines (Minsky Machines) , 1967 .

[18]  Rusins Freivalds,et al.  Why Sometimes Probabilistic Algorithms Can Be More Effective , 1996, MFCS.