sci.logic: Re: Program implementing Post Tag System as applied to a Turing Machine
Re: Program implementing Post Tag System as applied to a Turing Machine
Source: http://sci.tech−archive.net/Archive/sci.logic/2004−09/0771.html
From: Alex Vinokur (alexvn_at_big−foot.com) Date: 09/09/04
Date: Thu, 9 Sep 2004 10:43:55 +0300
"Eray Ozkural exa"
wrote in message news:fa69ae35.0409080832.7ee8ae89@posting.google.com... > "Alex Vinokur" wrote in message news:<2q62saFqmuv9U1@uni−berlin.de>... > > Is there any C/C++−program which implements Post Tag System as applied to a Turing Machine? >> > > In other words, let T be some (one−tape) Turing Machine M = where > > * Q = {q0, q1, ..., qk, qfin} is a set of states; > > * A = {i1, ..., in, a1, ..., am} is a set of the tape alphabet; > > * I = {i1, ..., in} is a set of the input alphabet; > > * q0 is the initial state; > > * qfin is the halting state; > > * b is the blank symbol; > > * s : QxA −> QxAx{L, R, S} is the transition table, where L is left shift, R is right shift, S is no shift. >> > > Further, let w be some input word of M. >> > > Is there any C/C++−program which gets as input > > * set of states Q, > > * tape alphabet A, > > * blank symbol b, > > * transition table s, > > * input word w > > and produces as output > > * Post Tag System string. > > A TM to Tag System compiler? Interesting idea. If you take it in the > Platonist sense, there is at least one such program since TMs are > equivalent to Post Tag Systems :) You could use the representation in > Minsky's 1961 and Cocke and Minsky's 1964 papers I suppose as > mentioned on mathworld: > http://mathworld.wolfram.com/TagSystem.html > > I tried some google queries and found such a program by André Betz : > > http://forum.wolframscience.com/showthread.php?s=&threadid=239 Re: Program implementing Post Tag System as applied to a Turing Machine 1
sci.logic: Re: Program implementing Post Tag System as applied to a Turing Machine > > −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− > TM2TS.java converts from Turing Machine Definition with 2 Symbols to > Tag System Definition > −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−− > > He says that it is described in his book and home page: > http://www.andrebetz.de Thanks. > > You can re−implement this code in C++ and make it run faster than the > Java version, I'm sure. > > It looks like there is at least one other person who is interested in > the practical implementation of these theoretical devices as much as > yourself. I wonder what are you working on, could you explain briefly? > [snip] Of course. My question has to do with invocations of Universal Turing Machines. 1. Problem definition. ====================== Actually each Universal Turing Machine (UTM) consists of three components: * TM−to−UTM translator (Encoder), * UTM itself (i.e., Turing Machine with Universal Table Transition), * UTM−to−TM translator (Decoder). Let T be some Turing Machine (with T−alphabet, T−states, T−table−transition), let U be some Universal Turing Machine (with U−alphabet, U−states, U−table−transition). Let T2U be TM−to−UTM translator, U2T be UTM−to−TM translator of machine U. A Universal Turing Machine invocation (for some input) is performed in three stages. Input −−−−− a is the T−alphabet, s is the T−states, t is the T−table−transition, x is a word in T−alphabet. Step−1 (Encoding) −−−−−−−−−−−−−−−−− Re: Program implementing Post Tag System as applied to a Turing Machine 2
sci.logic: Re: Program implementing Post Tag System as applied to a Turing Machine y = T2U (a, s, t, x), y is a word in U−alphabet; Step−2 (Computing) −−−−−−−−−−−−−−−−−− z = U(y), z is a word in U−alphabet; Step−3 (Decoding) −−−−−−−−−−−−−−−−− w = U2T (a, s, t, z), w is a word in T−alphabet. Checking −−−−−−−− Must be: w == T(x). 2. Turing Machine Simulator =========================== I have developed C++ Simulator of a Turing Machine: * http://semillon.wpi.edu/~aofa/AofA/msg00020.html * http://sourceforge.net/project/showfiles.php?group_id=79519&package_id=80949 * http://alexvn.freeservers.com/s1/turing.html. That program simulates a Nondeterministic/Deterministic Multitape Turing Machine. Detailed log file is generated. Resources used (input size, output size, TM−space, TM−time) are computed as well. 3. Universal Turing Machine Simulator (not using Post Tag System) ================================================================= I have developed C++ Simulator of a Universal Turing Machine as well: * http://semillon.wpi.edu/~aofa/AofA/msg00024.html * http://alexvn.freeservers.com/s1/utm.html * http://sourceforge.net/project/showfiles.php?group_id=79519&package_id=80937 That UTM can simulate the behavior of a Multitape Turing Machine. The package consists of two executable files : * t2u − TM−to−UTM translator (Encoder), t2u generates several output files, one of them is used as input of the utm. * utm − UTM Simulator itself with builtin comprehensive UTM−to−TM translator (Decoder); Note. That Decoder decodes not only halting configuration of the UTM but also its intermediate configurations corresponding configurations of simulated TM. To invoke that UTM (actually its Transition Table) we don't need to use special UTM Simulator. We could use Simulator of regular Turing Machine (see previous paragraph) with Universal Transition Table. But I was interested in getting intermediate configurations of simulated TM; Re: Program implementing Post Tag System as applied to a Turing Machine 3
sci.logic: Re: Program implementing Post Tag System as applied to a Turing Machine so, I needed comprehensive run−time Decoder. To do that I wrote separate UTM Simulator which includes such a decoder. Of course, there are quite a few differences between UTM Simulator and Simulator of regular TM. Encoder and Decoder for that UTM don't use Post Tag System.
4. Invocation of Arbitrary Universal Turing Machine (probably) using Post Tag System ==================================================================================== I would like to use my Turing Machine Simulator to invoke various UTMs: Minsky's, Rogozhin's, Wolfram's UTMs (see http://mathworld.wolfram.com/UniversalTuringMachine.html) and possibly other UTMs. How to do that? We can see (for instance, from previous paragraph about UTM which doesn't use Post Tag System) the creating Encoder and Decoder for UTM is quite complex procedure. I supposed (I am not sure that I am right) that translators TM−to−UTM (Encoder) and UTM−to−TM (Decoder) could be based on Post Tag System for _ANY_ Universal Turing Machine. That might ensure comparatively uniform approach to invocation of different UTMs. So, to invoke some UTM we should create Post Tag System based Encoder and Decoder. For that reason, I tried to start with searching Post Tag System translators.
−− Alex Vinokur http://mathforum.org/library/view/10978.html http://sourceforge.net/users/alexvn
Re: Program implementing Post Tag System as applied to a Turing Machine
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