Deterministic One-Way Turing Machines with Sublinear Space

Kutrib, Martin; Provillard, Julien; Vaszil, György; Wendlandt, Matthias

doi:10.3233/fi-2015-1147

Cited by 3 publications

(8 citation statements)

References 13 publications

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“…The complexity class NSPACE(f (n)) is the set of decision problems that can be solved by a nondeterministic Turing machine M , using space f (n), where n is the length of the input [21]. The two-way Turing machines may move their head on the input tape into two-way (left and right directions) while the one-way Turing machines are not allowed to move the head on the input tape to the left [18]. The complexity class 1-NSPACE(f (n)) is the set of decision problems that can be solved by a nondeterministic one-way Turing machine M , using space f (n), where n is the length of the input [21].…”

Section: Background Theorymentioning

confidence: 99%

The Complexity of Mathematics

Vega¹

2020

Preprint

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The strong Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two primes. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). However, if this happens, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. In addition, if this happens, then the Twin prime conjecture is true. Moreover, if this happens, then the Beal's conjecture is true. Furthermore, if this happens, then the Riemann hypothesis is true. Since the weak Goldbach's conjecture was proven, then this will certainly happen.

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Section: Background Theorymentioning

confidence: 99%

The Complexity of Mathematics

Vega¹

2020

Preprint

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“…From the early days of automata and complexity theory, two different models of Turing machines are considered, the offline and online machines [18]. Each model has a read-only input tape and some work tapes [18].…”

Section: Background Theorymentioning

confidence: 99%

The Complexity of Number Theory

Vega¹

2020

Preprint

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The Goldbach's conjecture has been described as the most difficult problem in the history of Mathematics. This conjecture states that every even integer greater than 2 can be written as the sum of two primes. This is known as the strong Goldbach's conjecture. The conjecture that all odd numbers greater than 7 are the sum of three odd primes is known today as the weak Goldbach conjecture. A principal complexity class is NSPACE(S(n)) for some S(n). We show if the weak Goldbach's conjecture is true, then the problem PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n). This proof is based on the assumption that if some language belongs to NSPACE(S(n)), then the unary version of that language belongs to NSPACE(S(log n)) and vice versa. However, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples. Since Harald Helfgott proved that the weak Goldbach's conjecture is true, then the strong Goldbach's conjecture is true or this has an infinite number of counterexamples, where the case of infinite number of counterexamples statistically seems to be unlikely. In addition, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Twin prime conjecture is true. Moreover, if PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Beal's conjecture is true. Since the Beal's conjecture is a generalization of Fermat's Last Theorem, then this is also a simple and short proof for that Theorem. In mathematics, the Riemann hypothesis is consider to be the most important unsolved problem in pure mathematics. If PRIMES is not in NSPACE(S(n)) for all S(n) = o(log n), then the Riemann hypothesis is true.

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Section: Consequencesmentioning

confidence: 99%

“…From the early days of automata and complexity theory, two different models of Turing machines are considered, the offline and online machines [14]. Each model has a read-only input tape and some work tapes [14]. The offline machines may read their input two-way while the online machines are not allowed to move the input head to the left [14].…”

Section: Consequencesmentioning

confidence: 99%

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Logarithmic Space Verifiers on NP-complete

Vega¹

2019

Preprint

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P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? A precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have f ailed. NP is the complexity class of languages defined b y p olynomial t ime v erifiers M su ch th at wh en th e in put is an el ement of the language with its certificate, then M outputs a string which belongs to a single language in P. Another major complexity classes are L and NL. The certificate-based definition of NL is based on logarithmic space Turing machine with an additional special read-once input tape: This is called a logarithmic space verifier. NL is the complexity class of languages defined by logarithmic space verifiers M s uch t hat when t he i nput i s a n e lement o f t he l anguage with i ts c ertificate, th en M outputs 1. To attack the P versus NP problem, the NP-completeness is a useful concept. We demonstrate there is an NP-complete language defined by a logarithmic space verifier M such that when the input is an element of the language with its certificate, then M outputs a s tring which belongs to a single language in L. In this way, we obtain if L is not equal to NL, then P = NP. In addition, we show that L is not equal to NL. Hence, we prove the complexity class P is equal to NP.

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Deterministic One-Way Turing Machines with Sublinear Space

Cited by 3 publications

References 13 publications

The Complexity of Mathematics

The Complexity of Mathematics

The Complexity of Number Theory

Logarithmic Space Verifiers on NP-complete

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