One possible answer to this is that there is no way that the system can halt according to the definition I've given, so it cannot be universal. Is there at least one mapping function that makes this system universal? The mapping function is limited in that it never changes the active tape element by more than one element along the tape, it doesn't change any tape elements but the previously active element, and non-previously-active elements don't affect the output of the function with respect to the colour other elements, which element is active, and which state the system is in." There is a fixed rule that is a function that maps the present situation of the system (the colour of each tape element, which tape element is active, and what state the system is in) to the 'next' situation of the system, and the system continues applying this rule forever, iterating constantly from one situation to the next. "The system has a tape with elements each of which can contain a finite number of possible 'colours' one of those elements is 'active' at any given moment, and the system can be in one of a finite number of states. The real problem here seems to be that the definition of universality has somehow got confused with the definition of things like 'Turing machine' and 'linear bounded automaton' to some extent.įor example, consider the following definition of a system: (I'm replying to a proposed counterexample to a definition of universality I use in my proof a specific explanation as to how the counterexample is based on inconsistent assumptions is given as part of this message, but it's mostly general discussion in an attempt to pin down how the problem arose in the first place and how the definition of 'universality' (and 'Turing machine', for that matter) isn't very clear at all.)
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