Finite Playground

To verify is human; to prove, divine.


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Three group-unrelated representation theorems

A representation theorem in general means a canonical way of expressing a class of objects using another class of objects, usually more fundamental or easier to grasp.  Beside the well-known group representation theory, which concerns about viewing (finite or Lie) groups as matrices, here are three lesser known (to me!) representation theorems.

Birkhoff’s Representation Theorem.  Every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.

Riesz–Markov–Kakutani Representation Theorem.  Any positive linear functional on the space of compactly supported continuous functions on a locally compact Hausdorff space can be viewed as integration against a measure.

Kapovich-Millson Universality Theorem.  Any compact smooth manifold is diffeomorphic to a component of the configuration space of some planar linkage.

 

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Random stuff, 2016-01-08

Doing some random readings, also trying to revive the blog :(


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Weakly (?) digest


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Hindman’s Theorem

While reading the posts on ultrafilters by Terry Tao [1,2], I find the following result interesting:

Hindman’s theorem (a special case).  Suppose we color the natural numbers using k colors.  Then there exists a color c and an infinite set S, all colored with c, such that every finite sum of over S has color c as well.

Also, this theorem is claimed to be rather unpleasant to prove if we insist on not using ultrafilters.  


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Compilation of basic results related to regular language

* Results related to Pumping Lemma

Theorem.  A language L accepted by a DFA A of n states is

  1. nonempty if and only if A accepts a sentence of length less lean n.
  2. infinite if and only if A accepts a sentence of length \ell with n \le \ell < 2n.

The union of the following languages, L_1 \cup L_2, is non-regular, but nevertheless it cannot be proven using pumping lemma.

L_1 = \{u_1 v_1 w v_2 u_2 : u_1,u_2,v_1,v_2,w \in [4]^*, and either v_1=w, v_1=w, or v_2=v_2\}

L_2 = \{w : w\in [4]^*, and precisely 1/7 of symbols in w are 3’s \}

* Regular languages are precisely constant-space languages

Theorem.  A regular language can be recognized by a two-way read-only Turing machine.

Proof sketch.  Record the configuration of each time we first visit the i-th cell.  Define a finite automaton to simulate the same behavior.

Theorem.  \textsc{Space}(o(\log\log n)) is the same as \textsc{Space}(1), both are the class of regular languages.  That is, having an algorithm that uses o(\log\log n)-space is equal to not using any space.

Proof sketch.  If a Turing machine uses s(n) space, there are 2^{O(s(n))} possible configurations and 2^{2^{O(s(n))}} possible crossing sequences.  If s(n) = o(\log\log n) then there are two crossing sequences that are the same, and we can further shorten a shortest string accepted by the machine by the following lemma:

Lemma.  Let M be a machine and x be the input.  Suppose two crossing sequence C_i and C_j are equal; then by removing the substring of x from index i+1 to j-i (assume i < j), we get another string that is accepted by the machine M.

This is a contradiction.

The following language is non-regular but can be decided in O(\log\log n) space:

L = \{\#(0)_2\#(1)_2\# \ldots \#(2^k-1)_2\# : k \ge 0\},

where (i)_2 is the binary representation of i.

* Generating function of regular language

For a given language L \subseteq \Sigma^*,

S_L(z) = \sum_{n\ge 0} |L \cap \Sigma^n| z^n

is the generating function of L.  Using analytic combinatorics we can prove the following useful facts.

Theorem.  For a regular language L, the generating function S_L(z) is rational.

Theorem.  Let A=(Q,\Sigma,\delta,q_0,F) be a finite automaton of a language L.  Then the generating function of L is the following rational function determined under matrix form

S_L(z) = [q_0]^T(I-z[\delta])^{-1}[F],

where [\delta](i,j) = | \{ number of different labels between q_i and q_j \}|; [q_0] and [F] as characteristic vectors.

* Minimizing finite automata

Brzozowski’s algorithm use only power set construction \textsc{Power} and edge reversal \textsc{Rev}.  One can observe that reversing edges of a DFA gives an NFA of the reverse language.  Then the power set construction gives a minimum DFA.  This algorithm takes exponential time in the worst case.

Hopcroft’s algorithm is the fastest algorithm known; runs in O(n \log n) time.  Try to partition the states using the Myhill-Nerode equivalence relation.

* Relation with AC^0

Theorem.  \mathsf{REG} and \mathsf{AC}^0 are incomparable.

Proof.  Parity is not in \mathsf{AC}^0.  Palindrome/addition/\{ww : w \in \Sigma^*\} are not regular.