Minimal information needed for determine some function












5












$begingroup$


From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of information" is sufficient to describe a continuous function completely.

Is there a similar notion for computable functions (or Turing machines)? i.e.:

Let $T : {0,1}^* rightarrow {0,1}$ be a Turing machine and let ${0,1}^{n_0}$ be the set of all binary string of lengh $n_0$. Describing $T$ on ${0,1}^{n_0}$ trivially requires $2^{n_0}$ values. Can we impose conditions on $T$, which will reduce $2^{n_0}$ (like continuity in the continuous domain)? Can we hope for non-trivial such conditions?






cc.complexity-theory computability turing-machines boolean-functions it.information-theory







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$endgroup$












  • $begingroup$
    If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
    $endgroup$
    – Mohammad Al-Turkistany
    Dec 23 '18 at 14:05










  • $begingroup$
    There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
    $endgroup$
    – Avi Tal
    Dec 23 '18 at 14:28
















5












$begingroup$


From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of information" is sufficient to describe a continuous function completely.

Is there a similar notion for computable functions (or Turing machines)? i.e.:

Let $T : {0,1}^* rightarrow {0,1}$ be a Turing machine and let ${0,1}^{n_0}$ be the set of all binary string of lengh $n_0$. Describing $T$ on ${0,1}^{n_0}$ trivially requires $2^{n_0}$ values. Can we impose conditions on $T$, which will reduce $2^{n_0}$ (like continuity in the continuous domain)? Can we hope for non-trivial such conditions?






cc.complexity-theory computability turing-machines boolean-functions it.information-theory







share|cite|improve this question









$endgroup$












  • $begingroup$
    If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
    $endgroup$
    – Mohammad Al-Turkistany
    Dec 23 '18 at 14:05










  • $begingroup$
    There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
    $endgroup$
    – Avi Tal
    Dec 23 '18 at 14:28














5












5








5





$begingroup$


From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of information" is sufficient to describe a continuous function completely.

Is there a similar notion for computable functions (or Turing machines)? i.e.:

Let $T : {0,1}^* rightarrow {0,1}$ be a Turing machine and let ${0,1}^{n_0}$ be the set of all binary string of lengh $n_0$. Describing $T$ on ${0,1}^{n_0}$ trivially requires $2^{n_0}$ values. Can we impose conditions on $T$, which will reduce $2^{n_0}$ (like continuity in the continuous domain)? Can we hope for non-trivial such conditions?






cc.complexity-theory computability turing-machines boolean-functions it.information-theory







share|cite|improve this question









$endgroup$




From calculus, we know that if someone has a continuous function $f$, it is enough to know $f$'s values on the rationals in order to know $f$ on the entire line. In some sense, a "countable amount of information" is sufficient to describe a continuous function completely.

Is there a similar notion for computable functions (or Turing machines)? i.e.:

Let $T : {0,1}^* rightarrow {0,1}$ be a Turing machine and let ${0,1}^{n_0}$ be the set of all binary string of lengh $n_0$. Describing $T$ on ${0,1}^{n_0}$ trivially requires $2^{n_0}$ values. Can we impose conditions on $T$, which will reduce $2^{n_0}$ (like continuity in the continuous domain)? Can we hope for non-trivial such conditions?






cc.complexity-theory computability turing-machines boolean-functions it.information-theory




cc.complexity-theory computability turing-machines boolean-functions it.information-theory






share|cite|improve this question













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asked Dec 23 '18 at 9:45









ll edll ed

261




261












  • $begingroup$
    If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
    $endgroup$
    – Mohammad Al-Turkistany
    Dec 23 '18 at 14:05










  • $begingroup$
    There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
    $endgroup$
    – Avi Tal
    Dec 23 '18 at 14:28


















  • $begingroup$
    If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
    $endgroup$
    – Mohammad Al-Turkistany
    Dec 23 '18 at 14:05










  • $begingroup$
    There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
    $endgroup$
    – Avi Tal
    Dec 23 '18 at 14:28
















$begingroup$
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
$endgroup$
– Mohammad Al-Turkistany
Dec 23 '18 at 14:05




$begingroup$
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
$endgroup$
– Mohammad Al-Turkistany
Dec 23 '18 at 14:05












$begingroup$
There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
$endgroup$
– Avi Tal
Dec 23 '18 at 14:28




$begingroup$
There is a constant c such that every n-ary boolean function has circuit complexity at most 2^n/(c*n). Theorem 5, p.3 in jeffe.cs.illinois.edu/teaching/497/13-circuits.pdf
$endgroup$
– Avi Tal
Dec 23 '18 at 14:28










1 Answer
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$begingroup$

A natural condition is to bound the number of states of $T$ --- say, at most $n$. The set
$mathcal{C}_n$
of all TM's on at most $n$ states is finite --- in fact, of cardinality $2^{O(n)}$. It is well-known that finite classes have a finite teaching dimension,
http://www.cs.columbia.edu/~djhsu/coms6998-f17/student-work/scribe-notes/teaching_dimension.pdf
of size $d_n=O(|mathcal{C}_n|)$ --- meaning that for each $Minmathcal{C}_n$, some set of at most $d_n$ labeled strings uniquely identifies $M$. Taking the union of these "teaching sets" over all $Minmathcal{C}_n$ gives you a finite set $SsubsetSigma^*$, which belongs to $Sigma^{le n_0}$ for some $n_0$. Now you required that all of the examples be of the same length, but other than that, this satisfies your criteria.



The bottom line is that for fixed $n$, the resulting $n_0$ will be a constant.






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    $begingroup$

    A natural condition is to bound the number of states of $T$ --- say, at most $n$. The set
    $mathcal{C}_n$
    of all TM's on at most $n$ states is finite --- in fact, of cardinality $2^{O(n)}$. It is well-known that finite classes have a finite teaching dimension,
    http://www.cs.columbia.edu/~djhsu/coms6998-f17/student-work/scribe-notes/teaching_dimension.pdf
    of size $d_n=O(|mathcal{C}_n|)$ --- meaning that for each $Minmathcal{C}_n$, some set of at most $d_n$ labeled strings uniquely identifies $M$. Taking the union of these "teaching sets" over all $Minmathcal{C}_n$ gives you a finite set $SsubsetSigma^*$, which belongs to $Sigma^{le n_0}$ for some $n_0$. Now you required that all of the examples be of the same length, but other than that, this satisfies your criteria.



    The bottom line is that for fixed $n$, the resulting $n_0$ will be a constant.






    share|cite|improve this answer











    $endgroup$


















      2












      $begingroup$

      A natural condition is to bound the number of states of $T$ --- say, at most $n$. The set
      $mathcal{C}_n$
      of all TM's on at most $n$ states is finite --- in fact, of cardinality $2^{O(n)}$. It is well-known that finite classes have a finite teaching dimension,
      http://www.cs.columbia.edu/~djhsu/coms6998-f17/student-work/scribe-notes/teaching_dimension.pdf
      of size $d_n=O(|mathcal{C}_n|)$ --- meaning that for each $Minmathcal{C}_n$, some set of at most $d_n$ labeled strings uniquely identifies $M$. Taking the union of these "teaching sets" over all $Minmathcal{C}_n$ gives you a finite set $SsubsetSigma^*$, which belongs to $Sigma^{le n_0}$ for some $n_0$. Now you required that all of the examples be of the same length, but other than that, this satisfies your criteria.



      The bottom line is that for fixed $n$, the resulting $n_0$ will be a constant.






      share|cite|improve this answer











      $endgroup$
















        2












        2








        2





        $begingroup$

        A natural condition is to bound the number of states of $T$ --- say, at most $n$. The set
        $mathcal{C}_n$
        of all TM's on at most $n$ states is finite --- in fact, of cardinality $2^{O(n)}$. It is well-known that finite classes have a finite teaching dimension,
        http://www.cs.columbia.edu/~djhsu/coms6998-f17/student-work/scribe-notes/teaching_dimension.pdf
        of size $d_n=O(|mathcal{C}_n|)$ --- meaning that for each $Minmathcal{C}_n$, some set of at most $d_n$ labeled strings uniquely identifies $M$. Taking the union of these "teaching sets" over all $Minmathcal{C}_n$ gives you a finite set $SsubsetSigma^*$, which belongs to $Sigma^{le n_0}$ for some $n_0$. Now you required that all of the examples be of the same length, but other than that, this satisfies your criteria.



        The bottom line is that for fixed $n$, the resulting $n_0$ will be a constant.






        share|cite|improve this answer











        $endgroup$



        A natural condition is to bound the number of states of $T$ --- say, at most $n$. The set
        $mathcal{C}_n$
        of all TM's on at most $n$ states is finite --- in fact, of cardinality $2^{O(n)}$. It is well-known that finite classes have a finite teaching dimension,
        http://www.cs.columbia.edu/~djhsu/coms6998-f17/student-work/scribe-notes/teaching_dimension.pdf
        of size $d_n=O(|mathcal{C}_n|)$ --- meaning that for each $Minmathcal{C}_n$, some set of at most $d_n$ labeled strings uniquely identifies $M$. Taking the union of these "teaching sets" over all $Minmathcal{C}_n$ gives you a finite set $SsubsetSigma^*$, which belongs to $Sigma^{le n_0}$ for some $n_0$. Now you required that all of the examples be of the same length, but other than that, this satisfies your criteria.



        The bottom line is that for fixed $n$, the resulting $n_0$ will be a constant.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 23 '18 at 16:51

























        answered Dec 23 '18 at 14:05









        AryehAryeh

        5,53211838




        5,53211838






























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