# On the difference between one and many

## Abstract

We examine the following question: ‘Given a problem, is it more difficult to tell how many solutions the problem has than just deciding whether it has a solution?’. We show, that in specific cases, the question can be put into a mathematically meaningful form, namely when we can translate ‘number of solutions’ as ‘number of distinct accepting computations of a nondeterministic Turing machine’ (perhaps with appropriate weights). In this context, as we show, these questions are equivalent to problems about probabilistic machines (in the sense of Gill (9)).

In the first part of the paper we examine time-bounded computations, and justify our claim that this formalization is really the ma thematical form of the question above by exhibiting a unifying model (the treshold machine) which has a special subcases the nondeterministic and the probabilistic machines. We show that natural complete problems exist and prove some elementary properties of the model.

In the second part we examine tape-bounded machines. We show that probabilistic tape-bounded machines may be simulated by deterministic Turing machines with only a polynomial increase in the amount of tape needed. This settles an open problem of Gill's (9).

This is a very powerful and perhaps unexpected result: it is the best known situation in which we are able to show that powerful ‘extras’ like nondeterminism, get us only a polynomial improvement. The result is similar in content to Savitch's celebrated simulation of non-deterministic machines (20). The proof is completely unrelated to Savitch's (his construction does not work in the probabilistic case) and is quite involved, using some powerful recent results in complexity theory (10) (18) (4).

## Keywords

Polynomial Time Span Tree Turing Machine Random Access Machine Probabilistic Machine## Bibliography

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