Topic #16 – Models of Computation

Finite State Machines/Automata (FSM/FSA)

  • Simple models of computation.

  • We can define a very very very very simple computer in terms of
    • Input

    • Output

    • States

  • And then we can geek out over how much work this little computer can do
    • Like, what types of problems this computer can solve.

  • FSM’s are VERY simple computers.

  • But, just because they’re simple, doesn’t mean we can’t use them

  • In fact, these little guys are powerful enough to solve A LOT problems we deal with every day.
    • Vending Machines

    • Traffic Lights

    • Elevators

    • Locks/Safe

    • Regular Expressions

  • Let’s say we want a lock that has the combination 7, 7, 3.

  • Here’s a FSM for that lock

_images/FSM_lock.png
  • Or, we can go the other way and say, given this machine, what input can it accept.

_images/FSM_regex.png

Activity

What input strings can this machine accept? What I mean is, what strings will get this machine to it’s final state.

  • Think about it this way. Think of how simple the computational system is that is powerful enough to unlock a lock

  • Computer Scientists like to think about what else can a computer this powerful do?

Pushdown Automata (PDA) and Context-Free Grammars

  • FSM’s are cool and all, but they can only solve certain types of problems. If we want to solve more complex problems (for lack of a better term), we need a more powerful machine.

  • PDA’s are more powerful computational models

  • They can solve strictly more problems than FSMs.
    • Anything a FSM can do, a PDA can do

    • There are a lot of things a PDA can do, that a FSM cannot

  • PDA’s accept what we call Context-Free Grammars.

  • Here’s an example of doing the same thing as the 2nd FSM.

    S –> aA

    S –> bB

    A –> aA

    A –> b

    B –> bB

    B –> a

  • Here, the upper case letters are like special symbols that mean you can replace them in a string with whatever is on the right hand side.

  • The lower case letters are just letters.

  • Start with S
    • String: ‘S’

  • We have two options, let’s go with the first
    • String: ‘aA’

  • We now have an A, so let’s go with the first options
    • String: ‘aaA’

  • Let’s do it again
    • String: ‘aaaA’

  • Let’s do it 4 more times
    • String: ‘aaaaaaaA’

  • Now let’s go with the second option
    • String: ‘aaaaaaab’

  • No more upper case letters, so we’re done.
    • String: ‘aaaaaaab’

  • Here’s another one

    S –> aSa

    S –> bSb

    S –> epsilon/ ‘’

    (epsilon means empty string)

Activity

What strings can this system create? Try to generate a few and see if you can see the big picture on what it’s doing.

  • Another one

    S –> SS

    S –> (S)

    S –> ()

    S –> epsilon/ ‘’

Activity

What strings can this system create? Try to generate a few and see if you can see the big picture on what it’s doing.

Context-Sensitive Grammars

  • Context matters now

    S –> abc

    S –> aAbc

    Ab –> bA

    Ac –> Bbcc

    bB -> Bb

    aB –> aa

    aB –> aaA

Turing Machines

_images/TuringMachine.jpg
  • Turing Machines are even more powerful models of computation

  • Basically, the computers we use today are kinda’ like these
    • They’re not built like these, but they are as powerful

    • Can solve the same problems.

  • Also, we often say that our brains are at least as powerful as a Turing Machine.
    • At least?

For next class