3

I am struggling with the following two problems:

  1. Suppose that $M$ is a structure with finite universe and finite alphabet. Show that the set of formulas $\{\varphi$ $\mid$ for every $M$-assignment $\nu$ of the variables, $(M,\nu) \models \varphi\}$ is computable.

  2. Give an example of a finite language such that the set of formulas $\{\varphi$ $\mid$ for every finite structure $M$ which interprets that language and every $M$-assignment $\nu$ of the variables, $(M,\nu) \models \varphi\}$ is not computable.

Note that in the above problems, "$(M,\nu) \models \varphi$" is to be read as "$\varphi$ holds in $M$ when the variables of $\varphi$ are evaluated in $M$ according to $\nu$."

Now let $V:= \{\varphi \mid \varphi$ is a validity $\}$, where "A formula $\varphi$ is a validity" means "For all $(M,\nu)$, if $(M,\nu)$ interprets all the nonlogical symbols of $\varphi$, then $(M,\nu) \models \varphi$." I know that $V \geq_m H$ and therefore $V$ is not computable. The example sought by the second problem would seem to involve this theorem.

Unfortunately I can't see much further beyond this. Would anyone be so kind to share any hints or remarks that would help me begin to understand how to work towards a solution of these?

Thank you so much!

Rebecca Bonham
  • 359
  • 1
    How do you define computable? Surely you can come up with an algorithm that solves 1. – Andrés E. Caicedo Nov 17 '18 at 21:23
  • Hi, thanks for your response. The definition of a computable set is this: https://en.wikipedia.org/wiki/Recursive_set – Rebecca Bonham Nov 17 '18 at 21:34
  • Intuitively, I understand the idea of algorithmically deciding whether each $\varphi$ is belongs to the set defined in the first problem, but am having trouble translating that idea into a computable function, i.e. exhibiting an algorithm, if that makes sense. – Rebecca Bonham Nov 17 '18 at 21:37
  • 1
    For part 1, what level of detail do you need to go into to describe the algorithm? If you are given a finite model, a formula $\phi$, it is a finite task to check whether $\phi$ holds under every interpretation of its free variables in $M$. (because it is a finite task to enumerate all the interpretations and then, for each interpretation it is a finite task to check whether $\phi$ holds). – Rob Arthan Nov 17 '18 at 23:41
  • @RobArthan Thanks much for your answer. I was just finishing writing down that very fact, and my next move was to be coming here and commenting that yes, this is basically a writeup of my intuitive understanding, but I still feel I should be more detailed in constructing an actual algorithm. Any thoughts in how that might be done are welcome, and should greatly aid my understanding. – Rebecca Bonham Nov 17 '18 at 23:48
  • Meanwhile in problem 2, I've found that Church's theorem on the Entscheidungsproblem essentially seems to give a proof of this. Does that seem correct? Would anyone care to share how they would use this result in answering the problem as stated? Thanks again. – Rebecca Bonham Nov 17 '18 at 23:51
  • 1
    If pressed to give more details of something like this in my own work, I would write some mathematical style pseudo-code. I can't really comment on how much explicit detail is required in your case. – Rob Arthan Nov 17 '18 at 23:53
  • @RobArthan Yes, that is a good idea. I will try my hand at writing something along those lines. I think not a tremendous amount of detail is needed but I'd like at least to sketch an outline of some effective procedure. – Rebecca Bonham Nov 17 '18 at 23:57
  • I don't know if anyone reading this is interested but I've now found some very useful material, very helpful in showing how to actually put together an argument for this type of problem, at this address: http://kilby.stanford.edu/~rvg/154/handouts/fol.html – Rebecca Bonham Nov 18 '18 at 00:21
  • Having said that, problem 2 requires me to give a specific example of a finite language such that, etc. Would it suffice to offer "first order logic" as such an example? Seems too simple. Am I missing something obvious? – Rebecca Bonham Nov 18 '18 at 01:02
  • Can we study with each other @RebeccaBonham ? – Idonotknow Nov 18 '18 at 16:44
  • I think it's safe to say that whenever anyone wants to talk about this stuff, I'll be right here! Here (related) is what I'm trying to understand today. – Rebecca Bonham Nov 19 '18 at 00:44

0 Answers0