Fall 2003Costas Busch - RPI1 Turing Machines (TMs) Linear Bounded Automata (LBAs)

Fall 2003Costas Busch - RPI2 Input string Working space in tape Turing Machine (TM) Infinite Tape Finite State Control Unit

Fall 2003Costas Busch - RPI3 Left-end marker Input string Right-end marker Working space in tape All computation is done between end markers Linear Bounded Automaton (LBA) Finite State Control Unit

Fall 2003Costas Busch - RPI4 We define LBA’s as NonDeterministic Open Problem: NonDeterministic LBA’s have same power with Deterministic LBA’s ?

Fall 2003Costas Busch - RPI5 Example languages accepted by LBAs: LBA’s have more power than NPDA’s LBA’s have also less power than Turing Machines

Fall 2003Costas Busch - RPI6 Linear Bounded Automata (LBAs) are the same as Turing Machines with one difference: The input string tape space is the only tape space allowed to use

Fall 2003Costas Busch - RPI7 The Chomsky Hierarchy

Fall 2003Costas Busch - RPI8 Unrestricted Grammars: Productions String of variables and terminals String of variables and terminals

Fall 2003Costas Busch - RPI9 Example unrestricted grammar:

Fall 2003Costas Busch - RPI10 A language is recursively enumerable (r.e.) if and only if is generated by an unrestricted grammar Theorem: S is r.e. if there exists an algorithm A that enumerates the members of S (A need not necessarily terminate for non- members of S) S is recursive if there exists a decision algorithm that determines if x is a member of S

Fall 2003Costas Busch - RPI11 Context-Sensitive Grammars: and: Productions String of variables and terminals String of variables and terminals

Fall 2003Costas Busch - RPI12 The language is context-sensitive:

Fall 2003Costas Busch - RPI13 A language is context sensitive if and only if is accepted by a Linear-Bounded Automaton Theorem:

Fall 2003Costas Busch - RPI14 Non-recursively enumerable Recursively-enumerable Recursive Context-sensitive Context-free Regular The Chomsky Hierarchy