# Parallel Time and Quantifier Prefixes

##### References

1,542 citations

### "Parallel Time and Quantifier Prefix..." refers background or methods in this paper

...These machine models allow one to define the classes PR and PARR of subsets S ⊆ R∞ decidable, respectively, in polynomial and parallel polynomial time (see [2] for details) as well as their exponential versions EXPR and PAREXPR....

[...]

...Indeed, while the alternation of k blocks of quantifiers leads to the classes Σ R and Π R of the polynomial hierarchy PHR over the reals [2] which is included in PARR [1], the unrestricted alternation of polynomially many quantifiers yields a class PATR (from polynomial alternating time) which strictly includes PARR [6]....

[...]

...We consider sequential machines over R as originally defined in [3] (see also [2])....

[...]

1,340 citations

### "Parallel Time and Quantifier Prefix..." refers background in this paper

...The following result can be seen as a real version of Savitch’s Theorem [9]....

[...]

...In [9] Savitch proved that NPSPACE = PSPACE....

[...]

1,081 citations

383 citations

### "Parallel Time and Quantifier Prefix..." refers background or methods in this paper

...On the other hand, the first term in the upper bound for the parallel time of the algorithm in [1] vanishes (since t = 1) and its second term becomes ( ω ∏...

[...]

...Indeed, while the alternation of k blocks of quantifiers leads to the classes Σ R and Π R of the polynomial hierarchy PHR over the reals [2] which is included in PARR [1], the unrestricted alternation of polynomially many quantifiers yields a class PATR (from polynomial alternating time) which strictly includes PARR [6]....

[...]

...14 with bounds of similar order would require a thorough reworking of the algorithms of [1] allowing for a recursive computation of nested lists of sign conditions for real and Boolean quantifier elimination altogether, with tighter size and computation time bounds when dealing with Boolean quantifiers....

[...]

...(iii) One of the main results in [1] shows that quantified sentences over R can be decided in parallel time ( ω ∏...

[...]

351 citations

### "Parallel Time and Quantifier Prefix..." refers background in this paper

...[7] J. H. Davenport and J. Heintz....

[...]

...The following lemma has its origin in a paper by Davenport and Heintz [7]....

[...]