Log depth circuits for division and related problems
SIAM Journal on Computing
Journal of the ACM (JACM)
Time—space tradeoffs for satisfiability
Journal of Computer and System Sciences - Eleventh annual conference on computational learning theory&slash;Twelfth Annual IEEE conference on computational complexity
Time-space tradeoffs for branching programs
Journal of Computer and System Sciences
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
Time-space trade-off lower bounds for randomized computation of decision problems
Journal of the ACM (JACM)
Determinism versus nondeterminism for linear time RAMs with memory restrictions
Journal of Computer and System Sciences - STOC 1999
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
On determinism versus non-determinism and related problems
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
On determinism versus non-determinism and related problems
SFCS '83 Proceedings of the 24th Annual Symposium on Foundations of Computer Science
Separating the polynomial-time hierarchy by oracles
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Oblivious RAMs without cryptogrpahic assumptions
Proceedings of the forty-second ACM symposium on Theory of computing
Lower bounds for RAMs and quantifier elimination
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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For each natural number d we consider a finite structure md whose universe is the set of all 0,1-sequence of length n=2d, each representing a natural number in the set [0,1,...,2n-1] in binary form. The operations included in the structure are the four constants 0,1,2n-1,n, multiplication and addition modulo 2n, the unary function min[2x, 2n-1], the binary functions ⌊ x/y⌋ (with ⌊ x/0 ⌋ =0), max(x,y), min(x,y), and the boolean vector operations ∧,∨,- defined on 0,1 sequences of length n by performing the operations on all components simultaneously. These are essentially the arithmetic operations that can be performed on a RAM, with wordlength n, by a single instruction. We show that there exists a term (that is, an algebraic expression) F(x,y) built up from the mentioned operations, with the only free variables x,y, such that for all terms G(y), which is also built up from the mentioned operations, the following holds. For infinitely many positive integers d, there exists an a ∈ md such that the following two statements are not equivalent: (i) md |= ∃ x, F(x,a), (ii) md models G(a)=0. In other words, the question whether an existential statement, depending on the parameter a ∈ md is true or not, cannot be decided by evaluating an algebraic expression at a. Another way of formulating the theorem, in a slightly stronger form, is, that over the structures md, quantifier elimination is not possible in the following sense. Let calm be a first-order language with equality, containing function symbols for all of the mentioned arithmetic operations. Then there exists an existential first-order formula φ(y) of calm, containing a single existential quantifier and the only free variable y, such that for each propositional formula P(y) of calm, we have that for infinitely many positive integers d, φ(y) and P(y) are not equivalent on md, that is, md |= - ∀ y, φ(y) P(y). We also show that the theorem, in both forms, remains true if the binary operation min [xy, 2n-1] is added to the structure md. A general theorem is proved as well, which describes sufficient conditions for a set of operations on a sequence of structures kd, d=1,2,... which guarantees that the analogues of the mentioned theorems holds for the structures kd too.