MODEL INDEPENDENCE 4 3 MODEL INDEPENDENCE Variations of

MODEL INDEPENDENCE

4. 3 MODEL INDEPENDENCE Variations of RM proposed: cost of an operation is proportional to the length of its operands. => bounded RMs (size of numbers stored in registers is bounded by a constant). - fixed # of CPU registers can also be used as index registers for memory addressing. Variations of TM - random access TM (access an arbitrary stored quantity in ct. time) - multitape TM (several tapes, heads, R/W etc. ) - off-line TM (mainly used in connection with space complexity, 3 tapes: read, write, scratch) still the time complexity is polynomial

4. 3 MODEL INDEPENDENCE CONT’D

4. 3 MODEL INDEPENDENCE CONT’D

4. 4 TURING MACHINES ENUMERATORS AS ACCEPTORS & TM- general computing engines. D) TM- can be viewed as language acceptors (like FA) - need to adopt additional connections. - the string to be tested is on the tape. - the TM – accepts the string if it stops on symbol Yes. - rejects the string if it stops on symbol No. - TM may not always stop as a language acceptor. - consider a machine that can list all strings in the language but cannot always decide membership.

4. 4 TURING MACHINES ENUMERATORS CONT’D Þ TM AS ACCEPTORS & is able to answer Yes – when all strings in language are listed and stops as soon as the desired string appears in the list. No – it might fail to stop when machine is fed a string that is not in the language. FAILURE TO STOP IS NOT A CONDITION THAT CAN BE DETECTED. (Since we never know if the machine might not stop in just one more step)

4. 4 TURING MACHINES ENUMERATORS CONT’D AS ACCEPTORS & D) => an enumerating TM is one that lists all strings in the language. (this listing is generally in an arbitrary order). NTM vs DTM similar to NFA vs DFA. S) If there is any way for a TM to accept its input, it will do so. The NFA, whenever faced with a choice, automatically (and at no cost) choses the correct next step. (in practice, that comes at a considerable cost-since much information concerning consequences of taking various alternatives need to be considered).

4. 4 TURING MACHINES ENUMERATORS CONT’D AS ACCEPTORS & Þ TM viewed as a purely DTM that is a “prolific breeder”. AND { � This determinism requires counting – since the machine cannot answer no until all its descendants have answered “no”. } OR { � This machine can answer Yes as soon as it found one Asymmetry: AND vs OR. } Theorem 4. 2: Any language accepted by a NTM can be accepted by a DTM

4. 4 TURING MACHINES ENUMERATORS CONT’D AS ACCEPTORS & Theorem 4. 2: Any language accepted by a NTM can be accepted by a DTM. Proof: a simulation of a NTM. - the simulation must halt whenever the TM halts (including when NTM has accepting path in a tree of computations that includes Nondeterministic paths) - breadth-first search of the tree of possible computations of the NTM.