arXiv subjects

M. Utkan Gezer

Publications and source records attributed to M. Utkan Gezer.

5 recordsLinked to original sources

arXiv2026

Real-Time, Constant-Space, Constant-Randomness Verifiers

We study the class of languages that have membership proofs which can be verified by real-time finite-state machines using only a constant number of random bits, regardless of the size of their inputs. Since any further restriction on the verifiers would preclude the verification of nonregular languages, this is the tightest computational budget which allows the checking of externally provided proofs to have meaningful use. We provide a full characterization of this class of languages in terms of a restricted version of the one-way nondeterministic multihead finite automaton model. For any $k>0$, there exist languages that cannot be recognized by any $k$-head one-way nondeterministic finite automaton, but that are nonetheless real-time verifiable in this sense. The set of nonpalindromes, which cannot be recognized by any one-way multihead deterministic finite automaton, is also demonstrated to be verifiable within these restrictions.

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arXiv2025

$\rm P$ has polynomial-time finite-state verifiers

Interactive proof systems whose verifiers are constant-space machines have interesting features that do not have counterparts in the better studied case where the verifiers operate under reasonably large space bounds. The language verification power of finite-state verifiers is known to be sensitive to the difference between private and public randomization. These machines also lack the capability of imposing worst-case superlinear bounds on their own runtime, and long interactions with untrustable provers can involve the risk of being fooled to loop forever. We analyze such verifiers under different bounds on the numbers of private and public random bits that they are allowed to use. This separate accounting for the private and public coin budgets as resource functions of the input length provides interesting characterizations of the collections of the associated languages. When the randomness bound is constant, the verifiable class is $\rm NL$ for private-coin machines, but equals just the regular languages when one uses public coins. Increasing the public coin budget while keeping the number of private coins constant augments the power: We show that the set of languages that are verifiable by such machines in expected polynomial time (with an arbitrarily small positive probability of looping) equals the complexity class $\rm P$. This hints that allowing a minuscule probability of looping may add significant power to polynomial-time finite-state automata, since it is still not known whether those machines can verify all of $\rm P$ when required to halt with probability 1, even with no bound on their private coin usage. We also show that logarithmic-space machines which hide a constant number of their coins are limited to verifying the languages in $\rm P$.

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arXiv2024

Unconditional proofs of quantumness between small-space machines

A proof of quantumness is a protocol through which a classical machine can test whether a purportedly quantum device, with comparable time and memory resources, is performing a computation that is impossible for classical computers. Existing approaches to provide proofs of quantumness depend on unproven assumptions about some task being impossible for machines of a particular model under certain resource restrictions. We study a setup where both devices have space bounds $\mathit{o}(\log \log n)$. Under such memory budgets, it has been unconditionally proven that probabilistic Turing machines are unable to solve certain computational problems. We formulate a new class of problems, and show that these problems are polynomial-time solvable for quantum machines, impossible for classical machines, and have the property that their solutions can be "proved" by a small-space quantum machine to a classical machine with the same space bound. These problems form the basis of our newly defined protocol, where the polynomial-time verifier's verdict about the tested machine's quantumness is not conditional on an unproven weakness assumption.

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arXiv2021

Constant-Space, Constant-Randomness Verifiers with Arbitrarily Small Error

We study the capabilities of probabilistic finite-state machines that act as verifiers for certificates of language membership for input strings, in the regime where the verifiers are restricted to toss some fixed nonzero number of coins regardless of the input size. Say and Yakaryılmaz showed that the class of languages that could be verified by these machines within an error bound strictly less than $1/2$ is precisely NL, but their construction yields verifiers with error bounds that are very close to $1/2$ for most languages in that class when the definition of "error" is strengthened to include looping forever without giving a response. We characterize a subset of NL for which verification with arbitrarily low error is possible by these extremely weak machines. It turns out that, for any $\varepsilon>0$, one can construct a constant-coin, constant-space verifier operating within error $\varepsilon$ for every language that is recognizable by a linear-time multi-head nondeterministic finite automaton (2nfa($k$)). We discuss why it is difficult to generalize this method to all of NL, and give a reasonably tight way to relate the power of linear-time 2nfa($k$)'s to simultaneous time-space complexity classes defined in terms of Turing machines.

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arXiv2019

Windable Heads & Recognizing NL with Constant Randomness

Every language in NL has a $k$-head two-way nondeterministic finite automaton (2nfa($k$)) recognizing it. It is known how to build a constant-space verifier algorithm from a 2nfa($k$) for the same language with constant-randomness, but with error probability $\tfrac{k^2-1}{2k^2}$ that can not be reduced further by repetition. We have defined the unpleasant characteristic of the heads that causes the high error as the property of being "windable". With a tweak on the previous verification algorithm, the error is improved to $\tfrac{k_{\textrm{W}}^2-1}{2k_{\textrm{W}}^2}$, where $k_{\textrm{W}} \le k$ is the number of windable heads. Using this new algorithm, a subset of languages in NL that have a 2nfa($k$) recognizer with $k_{\textrm{W}} \le 1$ can be verified with arbitrarily reducible error using constant space and randomness.

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