An in-between "implicit" and "explicit" complexity: Automata

Implicit Computational Complexity makes two aspects implicit, by manipulating programming languages rather than models of com-putation, and by internalizing the bounds rather than using external measure. We survey how automata theory contributed to complexity with a machine-dependant with implicit bounds model

Finite state verifiers with constant randomness

We give a new characterization of $\mathsf{NL}$ as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio

Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature

On the power of real-time turing machines under varying specifications

We investigate the relative computing power of Turing machines with differences in the number of work tapes, heads pro work tape, instruction repertoire etc. We concentrate on the k-tape, k-head and k-head jump models as well as the 2-way multihead finite automata with and without jumps. Differences in computing power between machines of unlike specifications emerge under the real-time restriction. In particular it is shown that k+1 heads are more powerful than k heads for re

Finite state verifiers with both private and public coins

We consider the effects of allowing a finite state verifier in an interactive proof system to use a bounded number of private coins, in addition to "public" coins whose outcomes are visible to the prover. Although swapping between private and public-coin machines does not change the class of verifiable languages when the verifiers are given reasonably large time and space bounds, this distinction has well known effects for the capabilities of constant space verifiers. We show that a constant private-coin "budget" (independent of the length of the input) increases the power of public-coin interactive proofs with finite state verifiers considerably, and provide a new characterization of the complexity class $\rm P$ as the set of languages that are verifiable by such machines with arbitrarily small error in expected polynomial time.Comment: 18 pages, of which 5 pages are appendix, accepted for presentation in the conference ICTCS 2023, and is to be published in its proceeding