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Universal Turing Machine CSE2303 Formal Methods I Lecture 16

Overview Tables for Turing Machines Encoding Decoding Definition of a Universal Turing Machine Algorithm for a Universal Turing Machine

Assumptions Input Alphabet a b Tape Alphabet a b # Start State –Numbered 1. Halt State –Numbered 2

Example START 1 34 HALT 2 (a, a, R) (b, b, R) ( , , R)

TM for a n b n a n HALT 2START ( , , L) (a, a, R) (b, a, R) (a, , L) (a, a, L) (b, b, L) (a, a, L) (b, b, R) (a, a, R) (a,#, R) (#,#, R) ( , , R)

Table

Conditions to Check Check whether there is a row with a 1 in the From column. Check that there is no row with a 2 in the From column. Check there are no two rows with the same numbers in the From and the same letter in the Read column.

Coding Integer n –Code as: a n b

Coding the Table

Encoding of the TM abaaabaabbbaaabaaabaaaabaaabaaaabababbaa aabaaaabababbaaaabaaaaabaaaaaaaaaabaaaa aababaabaaaaaabaaaaaabaaaabaaaaaabaaaaa aabbabaaaaaaaaabaaaaaaaabaabaaaaaaaaaab aaaaaaaaabaabaaaaaaaaaaabaaaaaaaaabaaaa aaaaaaaaaabaaaaaaaaabababaaaaaaaaaababb bbbbabaabbabab

Code Word Language (CWL) CWL is the regular language (a + ba + b (a + b) 5 )* Words which encode a TM belong to CWL. Note: Not all words in CWL encode a TM

Decode abaaabaaaababaaabababbaaabaaaabababbaaaabaabbabab

Algorithm 1. Count the initial clump of a ’s. 2. Count the next clump of a ’s. 3. Read the next two letters. 4. Read the next two letters. 5. Read the next letter. 6. Repeat until there are no more letters.

Definition An Universal Turing Machine (UTM) is A Turing Machine Can run any TM on any input data.

Example Turing Machine abaaabaaaabababababbaaabaabaaaab Data baa HALT 2START 1 3 (a, a, R) (b, b, R)

Input for UTM #abaaabaaaabababababbaaabaabaaaab$baa  Start of Tape Turing Machine (encoded) Mark to indicate start of Data Data

Algorithm for an UTM Initialize Tape Remember first letter of data Find first instruction Is the next state HALT? Update Tape Remember current letter of data Find next instruction YES NO HALT Find & remember what to write & where to move Update Tape Find & remember what to write & where to move

Revision Know how to encode a Turing Machine. Know how to decode Turing Machine representation. Know what is a Universal Turing Machine, and what it does. Preparation Read –Chapter 25 in the Text Book.