[ a brief summary ]

1. Logical Aspects

The overwhelming majority of logical systems presupposes that we can extract whatever large volume of information in whatever small amount of time. This is an implicit presupposition of almost all systems of inference. Really, logical inference can be however long and logical rules of inference can be however complex. Moreover, a set of axioms can be defined in a such way, that nobody knows how much time we need to establish that a formula is an axiom. For real time systems, we have limited amount of time to finish an inference.

We accept three following (obvious enough) statements:

• Every Real System Works in Real Time
• Any Extraction of Information Takes Time
• Speed of Information Extraction is Limited

Every logical inference demands the extraction of information. For instance, we have to decide if there exists a rule that is applicable to the present formulae, if a formula is an axiom, or if a formula has been deduced earlier, etc. Under our presuppositions, it means that every logical inference takes time. So if a system has a time limit for the inference (is time-dependent), not every inference can be realized. Such limitations increase the importance of calculus selection.

Different calculuses that define that the same logics have different "streams" of deducible formulae. In this sense, they define the different logics.

Using the 1-st Calculus in the n seconds, the "top" part of formulae will be deduced.

Using the 2-st Calculus in the n seconds, the "bottom" part of formulae will be deduced.

With regard to this issue, Prof. Peter Kugler mentioned the importance of using the parallel inference procedures to address the problem of perceptional measurement.

Logic with Limited Resources is not a temporal logic. Temporal Logic deals with the "internal" time. Temporal Logic formalizes reasoning using the time-dependent statements. Logic with Limited Resources deals with "external" time, or, in other words, metatime. Metatime is the time in which the actually pragmatics of logical entailment lives. The initial ideas on such kinds of logics can be found in the papers by A.S.Esenin-Volpin (mid 60's). The importance of investigating the real time system logics has been pointed out by Prof. D. A. Pospelov (Columbus, 1995)

2. Unpredictability of Events

We will assume the following:

• An event seems to be unpredictable if it occurs earlier than we can extract all the information which is necessary for predicting it.

For example, even if we have a object, which is well defined by a system of linear differential equations, the state of the object seems to be unpredictable, if this state occurs earlier, than we can find a solution for the system. For instance, the movement of classical particles can be described by huge system of equations ...

From the presuppositions 1-4, it follows:

Corollary. Relative to real time system, there exist unpredictable events.

Due to H. Poincare, there are two major causes of randomness of events:

• the complexity of the system (as in the case with Brawn movement)
• the small causes entail huge changes ( we can see it with the cone which is staying on the vertex)

The first case have been discussed above. The second case concerns the preciseness of measurement. There are Parameters which We Can Not Measure with Required Precision.

Any measurement is a kind of extraction of information. Higher precision means extraction of more information. Therefore,

• In limited time, we can reach only the limited precision.

If a parameter changes quickly enough, we can reach only the limited precision before this parameter changes. If we cannot measure the parameters with required preciseness, some states of the system will be unpredictable (random) for us.

Corollary. There are unmeasurable parameters for the systems working in real time.