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L10-4up - Programmability from silicon to bits 0110100110...

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L10 - Models of Computation 1 6.004 – Fall 2010 10/12/10 Programmability from silicon to bits 0110100110 1110010010 0011100011 the Big Ideas of Computer Science modified 10/10/10 11:41 Quiz 2: FRIDAY! L10 - Models of Computation 2 6.004 – Fall 2010 10/12/10 6.004 Roadmap Sequential logic: FSMs CPU Architecture: interpreter for coded programs Programmability: Models Interpretation; Programs; Languages; Translation Beta implementation Pipelined Beta Software conventions Memory architectures Fets & voltages Logic gates Combinational logic circuits L10 - Models of Computation 3 6.004 – Fall 2010 10/12/10 FSMs as Programmable Machines ROM-based FSM sketch: Given i, s, and o, we need a ROM organized as: 2 i+s words x (o+s) bits i s 0...01 0...00 0...00 10110 011 o 2 i + s s N+1 o s N i inputs outputs 2 (o+s)2 i+s (some may be equivalent) An FSM’s behavior is completely determined by its ROM contents. So how many possible i-input, o-output, FSMs with s-state bits exist? L10 - Models of Computation 4 6.004 – Fall 2010 10/12/10 Big Idea #1: FSM Enumeration GOAL: List all possible FSMs in some canonical order. • INFINITE list, but • Every FSM has an entry and an associated index. 0...01 0...00 0...00 10110 011 s N+1 o s N i inputs outputs 2 8 FSMs 2 64 Every possible FSM can be associated with a number. We can discuss the i th FSM What if s=2, i=o=1??
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L10 - Models of Computation 5 6.004 – Fall 2010 10/12/10 Some Perennial Favorites... FSM 837 modulo 3 counter FSM 1077 4-bit counter FSM 1537 lock for 6.004 Lab FSM 89143 Steve’s digital watch FSM 22698469884 Intel Pentium CPU – rev 1 FSM 784362783 Intel Pentium CPU – rev 2 FSM 72698436563783 Intel Pentium II CPU Reality: The integer indexes of actual FSMs are much bigger than the examples above. They must include enough information to constitute a complete description of each device’s unique structure. L10 - Models of Computation 6 6.004 – Fall 2010 10/12/10 Models of Computation The roots of computer science stem from the study of many alternative mathematical “models” of computation, and study of the classes of computations they could represent. An elusive goal was to find an “ultimate” model, capable of representing all practical computations... We’ve got FSMs ... what else do we need? switches gates combinational logic memories FSMs Are FSMs the ultimate digital computing device? L10 - Models of Computation 7 6.004 – Fall 2010 10/12/10 FSM Limitations Despite their usefulness and flexibility, there exist common problems that cannot be computed by FSMs. For instance: Paren Checker “(()())” OK Paren Checker “(())())” Nix Well-formed Parentheses Checker: Given any string of coded left & right parens, outputs 1 if it is balanced, else 0. Simple, easy to describe. Is this device equivalent to one of our enumerated FSMs??? PROBLEM: Requires ARBITRARILY many states, depending on input. Must "COUNT" unmatched LEFT parens. An FSM can only keep track of a finite number of unmatched parens: for every FSM, we can find a string it can’t check.
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