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Theory of Computation What types of things are computable? How can we demonstrate what things are computable? Foundations • Cantor’s Set Theory - contradictions • Multiple sizes to infinity. • There exists at least one set bigger than the universal set. • Hilbert’s rigor • Find an algorithm that will generate proofs for all true statements • Gödel’s Incompleteness Theorem • No such algoritm exists. • Even worse, there exist true statements for which no proof can ever be found. • In any mathematical system there will either be true statements that cannot be proven or false statements that can. Computability • The question then becomes, can we at least find an algorithm that can find any proof that does exist? – Church & Kleene and Post found no. – Turing developed the “universal algorithm machine”, now called the Turing maching. This machine also had tasks it could not perform. Languages and grammars • A language is a set of strings. • A grammar is the set of rules that define what strings are valid members of a language. Rule structure • Rules consist of three types of symbols: • Terminals are symbols that cannot be further expanded. • Nonterminals are symbols that are not part of the language’s alphabet, but can be expanded into larger substrings • , which represents the empty string. • Assume the alphabet is {a, b}. An example grammar might be: S aSb S ba where a and b are terminals and S is a nonterminal. • Note that rules can be defined recursively.z Formal grammars • A grammar is thus the set of generation rules that define the valid strings of the language. • Any string that can be generated by the grammar is a valid string in the language, and any string in the language that is valid can be generated by the grammar • The type of grammar used by a language is determined by the restrictiveness of the rules. Regular grammars • Regular grammars are ones where the the rules have a nonterminal on the left and on the right either: • The empty string • A single terminal • A single terminal and a single nonterminal » Left regular grammars are ones where the nonterminal is to the left of the terminal; in right regular grammars, the opposite is true. Example regular grammars S aB S aB B bB B bB B Ba What language does What language does this define? this define? ab* ab*a State machines • Designed to allow for the modeling of transitions from one state to another. • Consist of states representing the current state of the machine and transitions between those states. • Entry Action - the action to perform when the state is entered • Exit Action - the action to perform when the state is exited • Input Action - an action to perform in a particular state with a a particular input (usually involves following a transition) • Have a start state and an acceptor state • Transitions define how the machine changes from one state to another. They usually define the condition(s) under which the machine changes states • Transition Action - the action to perform when following a Finite Automata • Particular type of state machine that has no actions at all. Only states and transitions. Limits of Regular languages and finite automata • What types of languages can’t FA’s accept? In other words, what limits are there on the complexity of regular languages? • FA’s lack memory, so that you can’t have one part of a regular language dependent on another part. Context-free grammars In CFGs, the rules all take the form: Nw where N is a single non-terminal and w is some finite string of terminals and non-terminals, in any order. Whereas in regular languages, nonterminals were restricted as to where they could appear in the rules, now they can appear anywhere. Hence the term context-free. S aSb S ab CFG example S aB SU S bA SV Aa U TaU A aS U TaT A bAA V TbV Bb V TbT B bS T aTbT B aBB T bTaT T Pushdown Automata • Pushdown automata extend FA’s in one very important way. We are now given a stack on which we can store information. This works like a standard LIFO stack, where information gets pushed onto the top and popped off the top. • This means that we can now choose transitions based not just on the input, but also based on what’s on the top of the stack. • We also now have transition actions available to us. We can either push a specific element to the top of the stack, or Limits on PDAs and CFGs • Adding memory is nice, but there are still significant limits on what a PDA can accomplish. • Can a PDA be constructed that can do arithmetic? • How, or why not? The Turing Machine • Four components: – A tape of infinite length, divided into cells, which can contain one character of data – A head which can read and write in the current cell, and which can move the tape one cell at a time left or right. – A program, which is the set of rules that tells us what to do. Our transitions between states will now specify the letter we should read from the tape, the letter we should write to the tape, and the direction we should move the tape.