Zero-Error Affine, Unitary, and Probabilistic OBDDs

Ibrahimov, Rishat; Khadiev, Kamil; Prūsis, Krišjānis; Vihrovs, Jevgēnijs; Yakaryılmaz, Abuzer

doi:10.48550/arxiv.1703.07184

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…This measure is an analog of a number of states for finite automaton and OBDDs can be seen as nonuniform finite automata (see for example [4]). As for many other computation models, it is possible to consider quantum OBDDs, and during the last decade they have been studied vividly [1,5,43,48,47,2,24,23,20,19,21,31,30].…”

Section: Introductionmentioning

confidence: 99%

Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

Khadiev¹,

Khadieva²,

Knop³

2022

Preprint

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In this paper, we study quantum Ordered Binary Decision Diagrams(OBDD) model; it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique ("reordering") for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function REQ such that the deterministic OBDD complexity of it is at least 2 Ω(n/ log n) , and the quantum OBDD complexity of it is at most O(n 2 / log n). It is the biggest known gap for explicit functions not representable by OBDDs of a linear width. Another function(shifted equality function) allows us to obtain a gap 2 Ω(n) vs O(n 2 ). Moreover, we prove the bounded error quantum and probabilistic OBDD width hierarchies for complexity classes of Boolean functions. Additionally, using "reordering" method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (k-OBDD) of polynomial width, for k = o(n/ log 3 n). We prove a similar hierarchy for bounded error probabilistic k-OBDDs of polynomial, superpolynomial and subexponential width.

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Section: Introductionmentioning

confidence: 99%

Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

Khadiev¹,

Khadieva²,

Knop³

2022

Preprint

Self Cite

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“…AfAs was formally defined in [7], and it was shown that they are more powerful than PFAs and quantum finite automata (QFAs) in bounded-error and unbounded-error settings, but their nondeterministic version is equivalent to nondeterministic QFAs. Since then, AfAs and their different generalizations (e.g., OBDDs and using counters) have been investigated in a series of work [28,14,22,17,29,15,16].…”

Section: Introductionmentioning

confidence: 99%

Affine automata verifiers

Khadieva¹,

Yakaryılmaz²

2021

Preprint

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We initiate the study of the verification power of AfAs as part of Arthur-Merlin (AM) proof systems. We show that every unary language is verified by a real-valued AfA verifier. Then, we focus on the verifiers restricted to have only integer-valued or rational-valued transitions. We observe that rational-valued verifiers can be simulated by integer-valued verifiers, and, their protocols can be simulated in nondeterministic polynomial time. We show that this bound tight by presenting an AfA verifier for NP-complete problem SUBSETSUM. We also show that AfAs can verify certain non-affine and non-stochastic unary languages.

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“…The computational power of AfAs and their generalizations have been examined and compared with their probabilistic and quantum counterparts in a series of papers [23,5,9,18,12,24,10,11,13,14]. In the unbounded-error and boundederror language recognition modes, AfAs are more powerful than PFAs and QFAs, where the latter models recognize all and only stochastic and regular languages, respectively [20,19,16,27,17].…”

Section: Introductionmentioning

confidence: 99%

Improved constructions for succinct affine automata

Yakaryılmaz¹

2021

Preprint

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Affine finite automata (AfA) can be more succinct than probabilistic and quantum finite automata when recognizing some regular languages with bounded-error. In this paper, we improve previously known constructions given for the succinctness of AfAs in three ways. First, we replace some of fixed error bounds with arbitrarily small error bounds. Second, we present new constructions by using less states than the previous constructions. Third, we show that any language recognized by a nondeterministic finite automaton (NFA) is also recognized by boundederror AfAs having one more state, and so, AfAs inherit all succinct results by NFAs. As a special case, we also show that any language recognized by a NFA is recognized by AfAs with zero error if the number of accepting path(s) for each member is exactly the same number. Keywords: succinctness • state complexity • affine automata • quantum automata • probabilistic automata • linear systems • bounded error • onesided error • zero error 3 We refer the reader to [8] for certain discussions about interference with historical remarks.

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Zero-Error Affine, Unitary, and Probabilistic OBDDs

Cited by 3 publications

References 22 publications

Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

Exponential Separation between Quantum and Classical Ordered Binary Decision Diagrams, Reordering Method and Hierarchies

Affine automata verifiers

Improved constructions for succinct affine automata

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