2

From http://www.cse.ohio-state.edu/~gurari/theory-bk/theory-bk-twoli1.html#30007-23021r2.2.4:

Let M = <Q, Σ, Δ, δ, q0, F> be the deterministic finite-state transducer whose transition diagram is given in Figure 2.E.2.

Figure 2.E.2

For each of the following relations find a finite-state transducer that computes the relation.

a. { (x, y) | x is in L(M), and y is in Δ* }.
b. { (x, y) | x is in L(M), y is in Δ*, and (x, y) is not in R(M) }.

Yes, this is HW, but I have been struggling with these questions and could at least use pointers. If you want to create your own c. and/or d. examples just to show me HOW to do it rather than lead me to the answers for a. and b. then obviously I'm fine with that.

Thanks in advance!

Community
  • 1
  • 1
oogly
  • 115
  • 1
  • 8

1 Answers1

2

Since you don't indicate what progress you've made so far, I'm going to assume that you've made no progress at all, and will give overall guidance for how you can approach this sort of problem.

  • First of all, examine the transition diagram. Do you understand what all the notations mean? Note that the transducer is described as deterministic. Do you understand what that means? Convince yourself that the transducer depicted in the transition diagram is, in fact, deterministic. Trace through it; try to get a sense for what inputs are accepted by the transducer, and what outputs it gives.
  • Next, figure out what L(M), Δ, and R(M) are for this transducer, since the questions refer to them. Do you know what those notations mean?
  • Do you know what it means for a transducer to compute a certain relation? Do you understand the { (x, y) | ... } notation for describing the relation?
  • Can you modify the transition diagram to eliminate the ε/0 transition and merge it into adjacent transitions (which then might output multiple symbols at a single transition)? (This can help, IMHO, with creating other transducers that accept the same input language. More so with part b, in this case, than part a.)
  • Describe for yourself the transducers you need to create, in a way that's independent of the original transducer. Will these transducers be deterministic?
  • Create the transition diagrams for these transducers.
ruakh
  • 175,680
  • 26
  • 273
  • 307
  • Ive made some progress, but the book doesnt do a great job with explaining in detail all the things that I do not grasp. 1) I see that it is deterministic since it has only one state transition for each input at that state. 2)From what I know, L(M) is the set of all accepting inputs which seem to be 0 in this case, and DELTA is the outputs 0,1,E (not sure how to make the symbol) 3) Relations are where I begin to get lost, but that x is the input and y is the output given that x is in Accepted Inputs and y is in all outputs) After that, im unsure where to go. – oogly Mar 04 '12 at 04:58
  • Ah. One thing that you're missing is that ε (read "epsilon") is *not* an element of Δ; rather, it's a special notation for the empty string. The transition from q₁ to q₂ outputs 0 without swallowing any symbol from the input, and the transition from q₂ to q₀ swallows a 1 from the input without outputting any symbol. (This also means that your explanation for why it's deterministic is incomplete: since it can transition from q₁ to q₂ without any input, it wouldn't be deterministic if any symbol caused state transitions out of both q₁ and q₂.) – ruakh Mar 04 '12 at 05:06
  • L(M) is the set of all strings that are accepted. In this case, `0` is indeed accepted — but so is (for example) `000`, or `00100`. (Just because q₂ is the "accepted" state, that doesn't mean you're never allowed to leave it. It just means that a string is part of L(M) if and only if it ends up leaving you at q₂.) – ruakh Mar 04 '12 at 05:07
  • Okay, I just thought when it was Delta* it meant the alphabet of the output including the empty string and Delta+ meant without the empty string. And the transition from q1 to q2 only has one transition with epsilon which is why I believe it to be deterministic. Per input(or lack of input, there is exactly one transition for that "input"). I REALLY only need to complete letter a. for this assignment and was hoping to learn how to do b. as well. But I would really appreciate you walking me through this if you have the time now. I want to learn it, I just cant find proper instruction to learn it – oogly Mar 04 '12 at 05:12
  • Ahh.. Okay, i see. So there are basically an infinite number of x... That is one dimension of the problem that I was not seeing. The FULL string, I was thinking only of the current input character. – oogly Mar 04 '12 at 05:16
  • Δ* means "the language consisting of symbols from Δ, repeated zero or more times". In this case, since Δ is {0,1}, Δ* is {ε,0,1,00,01,10,11,000,001,010,011,100,101,110,111,1000,...}. (And it *is* deterministic, but note that it would *not* be deterministic if q₁ and q₂ both had state-transitions for input 0, since q₁'s empty-string state-transition to q₂ would mean that there would be no deterministic way to choose between those two state-transitions.) – ruakh Mar 04 '12 at 05:17
  • I'll be going to bed shortly, but will happily give you more help with this tomorrow. – ruakh Mar 04 '12 at 05:18
  • Okay, it is due at midnight tonight, but I will turn in what I have and learn it tomorrow if you are willing and hopefully get late credit. But either way, I NEED to learn it. So I appreciate any and all help you can give. – oogly Mar 04 '12 at 05:21
  • @user1144605: Ping me when you're back online, and I'll try to walk you through it. – ruakh Mar 04 '12 at 14:48
  • Hey, I couldnt get online yesterday, but if you are around today I would love to learn! – oogly Mar 06 '12 at 02:58
  • I sent you a gmail chat invitation... if you use that. – oogly Mar 06 '12 at 03:13