The use of NFC in transportation information system for visually impaired persons

Jerabek, Michal; Krčál, Jan

doi:10.2495/icte130041

Cited by 1 publication

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“…The fact that the identities of Mat d+1 (F) are also identities of Mat d (F) is easy to show. The fact that the chain above is strictly decreasing can be proved either by a elementary methods [12] or as a corollary of [1].…”

Section: Stratificationmentioning

confidence: 99%

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Witnessing matrix identities and proof complexity

Tzameret

2018

Int. J. Algebra Comput.

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Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of d × d matrices over a field F, is a non-commutative polynomial f (x 1 , . . . , x n ) over F such that f vanishes on every d × d matrix assignment to its variables.We focus on arithmetic proofs, which are proofs of polynomial identities operating with arithmetic circuits and whose axioms are the polynomial-ring axioms (these proofs serve as an algebraic analogue of the Extended Frege propositional proof system; and over GF (2) they constitute formally a sub-system of Extended Frege [9]). We introduce a decreasing in strength hierarchy of proof systems within arithmetic proofs, in which the dth level is a sound and complete proof system for proving d × d matrix identities (over a given field). For each level d > 2 in the hierarchy, we establish a proof-size lower bound in terms of the number of variables in the matrix identity proved: we show the existence of a family of matrix identities f n with n variables, such that any proof of f n = 0 requires Ω(n 2d ) number of lines.The lower bound argument uses fundamental results from the theory of algebras with polynomial identities together with a generalization of the arguments in [7]. Specifically, we establish an unconditional lower bound on the minimal number of generators needed to generate a matrix identity, where the generators are substitution instances of elements from any given finite basis of the matrix identities; a result that might be of independent interest.We then set out to study matrix identities as hard instances for (full) arithmetic proofs. We present two conjectures, one about non-commutative arithmetic circuit complexity and the other about proof complexity, under which up to exponentialsize lower bounds on arithmetic proofs (in terms of the arithmetic circuit size of the identities proved) hold. Finally, we discuss the applicability of our approach to strong propositional proof systems such as Extended Frege.

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Section: Stratificationmentioning

confidence: 99%

“…While [7] uses the commutator [x, y] to define the s-polynomials, we consider the higher order commutativity axiom S 2d instead. It is possible to show that S 2d has sufficient properties for the lower bound as the commutator [x, y] (see Lemmas 7,8,12).…”

Section: Proving the Main Lower Bound: Generative Complexity Lower Bomentioning

confidence: 99%

Witnessing matrix identities and proof complexity

Tzameret

2018

Int. J. Algebra Comput.

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The use of NFC in transportation information system for visually impaired persons

Cited by 1 publication

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Witnessing matrix identities and proof complexity

Witnessing matrix identities and proof complexity

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