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aa:lab:02 [2020/10/07 15:48]
pdmatei 		aa:lab:02 [2020/10/22 10:25] (current)
pdmatei
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-====== Lab 02 - Turing Machines =======+====== Lab 02 - Introduction to Turing Machines =======
  
 **Key concepts** **Key concepts**
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-**1.2** Write a TM which **accepts** only if the **input** is a binary encoding of a **even** natural number.+**1.2** Write a TM which **enters the final state** only if the **input** is a binary encoding of an **even** natural number.
  
-**1.3** Write a TM which **accepts** only if the **input** is a binary encoding of a **even** natural number.+**1.3** Write a TM which verifies ​if a given word over alphabet $math[{A,​B}] contains ​the sequence ''​ABA''​.
  
-**1.4** ​ Write a TM which adds **5** to a number encoded in binary on the tape. The machine will always accept.+**1.4** Write a TM which adds **5** to a number encoded in binary on the tape. 
  
-**1.5** Write a machine ​which checks if a symbol ​is present on the tape. The machine should accept if this is so.+**1.5** Write a TM which checks if a binary number ''​x'' ​is strictly larger than ''​y''​. Hint: use a separator symbol between words.
  
 ==== 2. Algorithms and Turing Machines ==== ==== 2. Algorithms and Turing Machines ====
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    ​for-each x in V    ​for-each x in V
       s += x       s += x
-   if (x > 1000)+   if (s > M)
    ​then ​ return 1    ​then ​ return 1
    ​else ​ return 0    ​else ​ return 0
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   * how would ''​foreach x in V''​ be implemented?​   * how would ''​foreach x in V''​ be implemented?​
   * how would ''​s += x''​ be implemented?​   * how would ''​s += x''​ be implemented?​
-  * how would ''​if (x > 1000) then ... else ...''​ be implemented ?+  * how would ''​if (s > M) then ... else ...''​ be implemented ?
  
 +/*
  
 +**Answer:**
 +  * The input of the tape should contain each element of the vector ''​v'',​ encoded in binary, separated by a special character (e.g. ''​@''​). The last number in the sequence will be separated by another character (e.g. ''​!''​) from the value ''​M''​.
 +  * The machine should accept if the algorithm returns 1, that is, if the sum of elements of the array is greater than ''​M''​
 +  * Before executing the foreach, the TM should //​allocate//​ part of its tape for the sum ''​s''​ which is initially 0. The LHS of the tape could be used.
 +  * The foreach can be easily implemented by moving the cursor between ''​@''​ characters.
 +  * For a particular ''​x'',​ we can implement binary adding between ''​x''​ and the current ''​s'':​
 +    * M should go back-and-forth between the location of ''​s''​ and that of the current ''​x''​.
 +    * As each bit of ''​x''​ is processed, it should be erased (writing ''#''​) so that we can easily skip to the current ''​x''​.
 +  * After ''​!''​ is read on the tape, we know we have finished the ''​foreach''​. We can then implement a bit-wise comparison of the values ''​s''​ and ''​M'',​ which would now be the current value of the tape. The machine accepts if ''​s > M''​.
  
- +*/ 
- +==== More practice exercises ====
-Homework:+
   * Write a TM which verifies if a string has the **same number** of ones and zeroes. Give hints - live (what should the machine do?)   * Write a TM which verifies if a string has the **same number** of ones and zeroes. Give hints - live (what should the machine do?)
   * write a TM which **accepts** a given regular expression   * write a TM which **accepts** a given regular expression
   * write a TM which **reverses** a given binary string (always accepts)   * write a TM which **reverses** a given binary string (always accepts)