Bala Ravikumar

Minimal NFA problems are hard

Finite automata (FA’s) are of fundamental importance in theory and in applications. The following basic minimization problem is studied: Given a DFA (deterministic FA), find a minimum equivalent no...

Relating the type of ambiguity of finite automata to the succinctness of their representation

We consider the problem of how the size of a nondeterministic finite automaton (nfa) representing a regular language depends on the degree of ambiguity of the nfa. We obtain results for the unary and bounded inputs, and partial results for the unrestricted inputs. One of the main results of this paper shows that for unrestricted inputs, deterministic, unambiguous and nondeterministic machines form a hierarchy with respect to the number of states, solving an open problem of Stearns and Hunt. We also propose a new approach to the study of the succinctness of representation through regularity preserving closure properties and obtain some results in this direction.

A note on the space complexity of some decision problems for finite automata

In this note, we establish the space complexity of decision problems (such as membership, nonemptiness and equivalence) for some finite automata. Our study includes 2-way infinite automata with a pebble.

On sparseness, ambiguity and other decision problems for acceptors and transducers

We consider some decision problems on sparseness, degrees of ambiguity and multiple valuedness concerning finite-state and pushdown acceptors and transducers. A language L is sparse if there is a polynomial P such that the number of strings of length n in L is atmost P(n). A recognizer (transducer) is of polynomial ambiguity (valued) if there exists a polynomial P such that the number of derivations (outputs) for any input of length n is at most P(n). We relate these problems and show that they are decidable for finite-state devices. For cfls, only the sparseness problem is decidable. We also study some properties of structure generating function defined as f L (z)=Σa n z n , where a n is the number of strings of length n in a language L. Our results are useful in proving the non-regularity/non-context-freeness of some languages.

A simplified NP-complete MAXSAT problem

It is shown that the MAX2SAT problem is NP-complete even if every variable appears in at most three clauses. However, if every variable appears in at most two clauses, it is shown that it (and even the general MAXSAT problem) can be solved in linear time. When every variable appears in at most three clauses, we give an exact algorithm for MAXSAT that takes at most O(3n2n) steps where n is the number of variables.

Coping with known patterns of lies in a search game

Abstract In this paper we study a two-person search game in which player A thinks of a real number z ϵ[0, 1), and player B asks Q questions of the yes/no answer type to confine z to a subset R Q of as small a size as possible. The responses of player A to the queries of B need not all be true, but the nature of lies characterized by a set X of binary strings of length Q . We show that the smallest possible worst-case size of R Q is 2 - Q | X |, and there exist algorithms for player B to achieve this. We also carry out an average-case analysis of the problem.

On selecting the largest element in spite of erroneous information

In this paper, we study the problem of finding the largest of a set of n distinct integers using comparison queries which receive “yes” or “no” answers, but some of which may be erroneous. If at most e queries can receive erroneous answers, we prove that (e+1)n−1 comparisons are necessary and sufficient to find the largest. If there is further restriction that errors are confined to “no” answers and that all “yes” answers are guaranteed to be correct, then 2n+2e−4 comparisons are sufficient. This contrasts with earlier results relating to errors in binary search procedures where both versions of the problem have the same complexity.

THE STRUCTURE AND COMPLEXITY OF MINIMAL NFA’S OVER A UNARY ALPHABET

Many difficult open problems in theoretical computer science center around non-determinism. We study the fundamental problem of converting a given deterministic finite automaton (DFA) to a minimal nondeterministic finite automaton (NFA). Despite extensive work on finite automata, this fundamental problem has remained open. Recently, we studied this problem and showed that this (and related) problems are computationally hard.11 Herewe study the restriction of this problem to the case when the input DFA is over a one-letter alphabet. Even in this restricted case the problem is computationally hard even though our evidence of hardness is different from (and is weaker than) the standard ones such as NP-hardness. Let A→B denote the problem of converting a finite automaton of type A to a minimal finite automaton of type B. Our main result is that DFA→NFA, when the input is a unary cyclic DFA (a DFA whose graph is a simple cycle), is in NP but not in P unless NP⊆DTIME (nO(log n)). Our work was also motivated by the problem of finding structurally simple “normal forms” of NFA’s over a unary alphabet. We present some normal forms for minimal NFA’s over a unary alphabet and present an application to lower bounds on the size complexity of an NFA. In fact, the normal form result is used in a nontrivial manner to show the NP membership result stated above.

Some subclasses of context-free languages in NC 1

Abstract We obtain a characterization of strings in D2 (the one-sided Dyck language over two letters) and use this characterization to show that D2 is in NC1 (under ALOGTIME reductions). Our technique can easily be extended to show that Dk (the one-sided Dyck language over k letters) is in NC1 for any k. We also show that structured context-free languages (cfls), bracketed cfls and deterministic linear cfls are in NC1.

Some observations concerning alternating Turing machines using small space

Abstract We make some observations concerning alternating Turing machines operating in small space. For example, we show that alternating Turing machines using o(log n) space are more powerful than nondeterministic Turing machines using the same space-bound. In fact, we show that there is a language over a unary alphabet that can be accepted by an on-line alternating Turing machine in log n space, but not by any off-line nondeterministic Turing machine in o(log n) space. We also investigate the weak vs. strong space bounds and on-line vs. off-line machines at these low tape bounds.