Minimal information needed for determine some function
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
asked Dec 23 at 9:45
ll ed
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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
asked Dec 23 at 9:45
ll ed
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
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
– Avi Tal
Dec 23 at 14:28
add a comment |
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
asked Dec 23 at 9:45
ll ed
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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
asked Dec 23 at 9:45
ll ed
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 23 at 9:45
ll ed
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 23 at 9:45
ll ed
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Dec 23 at 9:45
ll ed
261
asked Dec 23 at 9:45
ll ed
261
261
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
ll ed is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
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
– Avi Tal
Dec 23 at 14:28
add a comment |
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
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
– Avi Tal
Dec 23 at 14:28
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
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
– Avi Tal
Dec 23 at 14:28
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
– Avi Tal
Dec 23 at 14:28
add a comment |
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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.
answered Dec 23 at 14:05
Aryeh
5,51211838
add a comment |
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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.
answered Dec 23 at 14:05
Aryeh
5,51211838
add a comment |
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.
answered Dec 23 at 14:05
Aryeh
5,51211838
add a comment |
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.
answered Dec 23 at 14:05
Aryeh
5,51211838
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.
answered Dec 23 at 14:05
Aryeh
5,51211838
edited Dec 23 at 16:51
answered Dec 23 at 14:05
Aryeh
5,51211838
answered Dec 23 at 14:05
Aryeh
5,51211838
answered Dec 23 at 14:05
Aryeh
5,51211838
5,51211838
add a comment |
add a comment |
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If we use circuit complexity as a measure of information then Shannon showed that most boolean n-ary functions have exponential circuit complexity.
– Mohammad Al-Turkistany
Dec 23 at 14:05
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
– Avi Tal
Dec 23 at 14:28