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-====== Lab 02 - Turing Machines =======
+====== Lab 02 - Introduction to Turing Machines =======
+**Key concepts**
+  - How is a Turing Machine (TM) defined?
+  - What is a **configuration**?
+  - How is the execution of a TM defined?
+==== 1. Intro ====
+A **Turing Machine** consists of:
+    * an **alphabet** $math[\Sigma]
+    * a set of **states** $math[K]
+    * an **initial** state $math[q_0]
+    * a **transition function** $math[\delta : K \times \Sigma \rightarrow K \times \Sigma \times \{L,H,R\}]
+    * a set of **final** states $math[F \subseteq K]
+Which of the following components of an **assembly language** would best correspond to the above? $math[K,\Sigma, \delta, q_0, F]
+  * the processor
+  * the memory
+  * registers
+  * assembly instructions
+==== 1. A few basic Turing Machines ====
+**1.1** What does the following TM do?
+$math[M=(K,\Sigma,q_0,\delta,F)] where $math[K=\{q_0,q_1,q_2\}], $math[F=\{q_2\}], $math[\Sigma=\{0,1,\#\}] and $math[\delta] is defined as below:
+^              ^ 0                ^ 1                ^ #                 ^
+| $math[q_0]   | $math[(q_0,1,R)] | $math[(q_0,0,R)] | $math[(q_1,0,L)]  |
+| $math[q_1]   | $math[(q_1,0,L)] | $math[(q_1,1,L)] | $math[(q_2,\#,R)]  |
+**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 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.
+**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 ====
+How would the following algorithm be represented as a Turing Machine:
+<code>
+Algorithm(vector V, integer M) {
+   integer s = 0
+   for-each x in V
+      s += x
+   if (s > M)
+   then  return 1
+   else  return 0
+}
+</code>
+Helpful questions:
+  * how should the tape be organised?
+  * when should the machine accept?
+  * how would ''foreach x in V'' be implemented?
+  * how would ''s += x'' 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 ====
+  * 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 **reverses** a given binary string (always accepts)