Lower Bounds for RAMs and Quantifier Elimination

We are considering RAMs \(N_{n}\), with wordlength \(n=2^{d}\), whose arithmetic instructions are the arithmetic operations multiplication and addition modulo \(2^{n}\), the unary function \( \min\lbrace 2^{x}, 2^{n}-1\rbrace\), the binary functions \(\lfloor x/y\rfloor \) (with \(\lfloor x/0 \rfloor =0\)), \(\max(x,y)\), \(\min(x,y)\), and the boolean vector operations \(\wedge,\vee,\neg\) defined on \(0,1\) sequences of length \(n\). It also has the other RAM instructions. The size of the memory is restricted only by the address space, that is, it is \(2^{n}\) words. The RAMs has a finite instruction set, each instruction is encoded by a fixed natural number independently of \(n\). Therefore a program \(P\) can run on each machine \(N_{n}\), if \(n=2^{d}\) is sufficiently large. We show that there exists an \(\epsilon>0\) and a program \(P\), such that it satisfies the following two conditions. (i) For all sufficiently large \(n=2^{d}\), if \(P\) running on \(N_{n}\) gets an input consisting of two words \(a\) and \(b\), then, in constant time, it gives a \(0,1\) output \(P_{n}(a,b)\). (ii) Suppose that \(Q\) is a program such that for each sufficiently large \(n=2^{d}\), if \(Q\), running on \(N_{n}\), gets a word \(a\) of length \(n\) as an input, then it decides whether there exists a word \(b\) of length \(n\) such that \(P_{n}(a,b)=0\). Then, for infinitely many positive integers \(d\), there exists a word \(a\) of length \(n=2^{d}\), such that the running time of \(Q\) on \(N_{n}\) at input \(a\) is at least \(\epsilon (\log d)^{\frac{1}{2}} (\log \log d)^{-1}\).