Most people, in their everyday reasoning, use perhaps poorly remembered
facts and intuitions.
This causes us to use phrases such as ``I believe ...,''
``I think ...,'' ``I suppose ....''
For computers to exhibit truly human behavior, the intuitive
nature of the human condition must somehow be reflected, even if
computers have a perfect memory.
Computers are rigorous by nature, so we must find a rigorous
way to appear intuitive.
In this talk, we suggest such a technique called
computing with confidence.
This technique is applied in the three different domains.
Firstly, we apply confidence techniques to ordinary computations.
What emerges is a model that is similar to, but mathematically
distinct from, limit computability.
The result is a more powerful model of effective computability.
Characterizations in terms of the arithmatic hierarchy are given.
Secondly, when confidence is applied to the traditional Gold model of learning,
a stronger model emerges where all the recursive functions are
learnable by a single strategy.
The interesting questions revolve around the learning of minimal programs.
Finally, the notion of confidence is applied to first order logic.
A sound and complete model of first order logic results.
The proofs, however, become infinite in length.
Constructive ordinals are used to measure the length of proofs.
Upper bounds for the lengths of proofs are given for all statments
based on their position in the arithmatic hierarchy.
(This talk is based on joint work with
Janis Barzdins and Rusins Freivalds (University of Latvia, Riga.)