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Introduction to Turing Machines and Computability in Programming Languages

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Explore the fundamental concepts of computability and programming languages, focusing on Turing machines as abstract computational models. Understand the difference between interpreters and compilers, and delve into essential topics like semantics, regular expressions, finite state automata, and context-free grammars. Discuss the significance of Turing completeness, the implications of the Church-Turing thesis, and the limitations imposed by the halting problem. Through hands-on examples, including unary operations and the Busy Beaver game, gain insight into the power and limitations of computation.

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Introduction to Turing Machines and Computability in Programming Languages

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  1. CS 360Programming Languages Introduction Computability Turing Machines

  2. Background • Some programming language • Some assembly programming (CS 281) • Regular Expressions (CS 270) • Finite State Automata (CS 270, ECE 200) • CF Grammars (CS 270)

  3. Topics • Computability • What is a language? • Interpreter vs. Compiler • Semantics • Operationally

  4. What is a language? A programming language is a notational system for describing computation in a machine-readable and human-readable form. - Kenneth Louden • So, what do we use them for? • What does it mean to be computable?

  5. Turing Machine • Abstract machine • Consists of: • A tape • A set of symbols • A read/write head • A finite set of instructions • A state register q

  6. Turing Machine • One state is the start state • Instructions are pentuplets: qc, ac → qn, an, d • q is the state • a is the symbol read or written • d is the direction the head moves • The tape is infinite (one- or two-ended)

  7. Turing “programs” • See http://ironphoenix.org/tril/tm/ for a nice simulator from Suzanne Britton Hello World 1,_ 2,H,> 2, _ 3,E,> 3,_ 4,L,> 4,_ 5,L,> 5,_ 6,O,> 6,_ H,!,> Unary Subtraction 1,_ 1,_,> 1,1 1,1,> 1,- 1,-,> 1,= 2,_,< 2,1 3,=,< 2,- H,_,< 3,1 3,1,< 3,- 4,-,< 4,_ 4,_,< 4,1 1,_,> • Now you try: • Unary addition • Copy a string • Reverse a string

  8. Turing Complete • The Turing machine is universal • It can be used to implement a Turing Machine • Any language that can be used to simulate a Turing Machine is Turing complete • So, HTML might be a language, but it is not a programming language

  9. Church-Turing Thesis • Alonso Church was also exploring what was computable, using lambda calculus • The Turing Machine can simulate the logic of any computer algorithm • May be less efficient • May use more instructions/memory • But, no less powerful • Anything that is effectively calculable can be decided by a Turing Machine

  10. Halting Problem • There are problems that are not decidable • It is not possible to write a program that can always determine if a given program (along with some input) would halt • I could sketch a proof here, but I’d be plagarising and there are some nice discussions out there • See http://www.cgl.uwaterloo.ca/~csk/halt/ , from Prof. Kaplan at Waterloo

  11. Busy Beaver • A game • Given n states, write a Turing program that makes a maximum number of marks on the tape, but still halts • Number gets large quickly • Has been shown to be the fastest growing of all functions • Undecidable (for n > some n0) • If we could compute the busy beaver value, we could solve the halting problem

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