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Minimize the number of states in a DFA Algorithm (3.6, page 142): Input: a DFA M

Minimize the number of states in a DFA Algorithm (3.6, page 142): Input: a DFA M output: a minimum state DFA M’ If some states in M ignore some inputs, add transitions to a “dead” state. Let P = {All accepting states, All nonaccepting states} Let P’ = {} Loop: for each group G in P do

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Minimize the number of states in a DFA Algorithm (3.6, page 142): Input: a DFA M

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  1. Minimize the number of states in a DFA • Algorithm (3.6, page 142): • Input: a DFA M • output: a minimum state DFA M’ • If some states in M ignore some inputs, add transitions to a “dead” state. • Let P = {All accepting states, All nonaccepting states} • Let P’ = {} • Loop: for each group G in P do Partition G into subgroups so that s and t (in G) belong to the same subgroup if and only if each input a moves s and t to the same state of the same P-groups put the new subgroups in P’ if (P != P’) {P = P’; goto loop} • Remove any dead states and unreachable states.

  2. Example: minimize the DFA for (ab|ba)a* • Example: minimize the DFA for Fig 3.29 (pages 121) • Questions: How can we implement Lex? %% BEGIN {return(BEGINNUMBER);} END {return(ENDNUMBER);} IF {return(IFNUMBER);} %%

  3. Lex internal: • construct an NFA to recognize the sum of all patterns • convert the NFA to a DFA (record all accepting states for each individual pattern). • Minimize the DFA (separate distinct accepting states for the initial pattern). • Simulate the DFA to termination (that is, no further transitions) • Find the last DFA state entered that holds an accepting NFA state (this picks the longest match). If no such state, then it is an invalid token.

  4. Chapter 4: Syntax analysis • Syntax analysis is done by the parser. • Detects and reports any syntax errors. • Produces a parse tree from which intermediate code can be generated. token Rest of front end Lexical analyzer Parse tree Int. code Source program parser Request for token Symbol table

  5. The syntax of a programming language is described by a context-free grammar (Backus-Naur Form (BNF)). • A grammar gives a precise syntactic specification of a language. • From some classes of grammars, tools exist that can automatically construct an efficient parser. These tools can also detect syntactic ambiguities and other problems automatically. • A compiler based on a grammatical description of a language is more easily maintained and updated.

  6. A grammar G = (N, T, P, S) • N is a finite set of non-terminal symbols • T is a finit set of terminal symbols • P is a finit subset of • An element is written as • S is a distinguished symbol in N and is called the start symbol. • Language defined by a grammar • We say “aAb derives awb in one step”, denoted as “aAb=>awb”, if A->w is a production and a and b are arbitrary strings of terminal or nonterminal symbols. • We say a1 derives am if a1=>a2=>…=>am, written as a1=>am • The languages L(G) defined by G are the set of strings of the terminals w such that S=>w. * *

  7. Example: A->aA A->bA A->a A->b

  8. Chomsky Hierarchy (classification of grammars) • A grammar is said to be • regular if it is • right-linear, where each production in P has the form, or . Here, A and B are non-terminals and w is a terminal • left-linear • context-free if each production in P is of the form , where and • context sensitive if each production in P is of the form where • unrestricted if each production in P is of the form where

  9. Context-free grammar is sufficient to describe most programming languages. • Example: a grammar for arithmetic expressions. <expr> -> <expr> <op> <expr> <expr> -> ( <expr> ) <expr> -> - <expr> <expr> -> id <op> -> + | - | * | / derive -(id) from the grammar: <expr> => -<expr> => - (<expr>) =>-(id) sentence: a strings of terminals that can be derived from S sentential form: a strings of terminals or none terminals that can be derived from S.

  10. derive id + id * id from the grammar: E=>E+E=>E+E*E=>E+E*id=>E+id*id=>id+id*id • leftmost/rightmost derivation -- each step replaces leftmost/rightmost non-terminal. E=>E+E=>id+E=>id+E*E=>id+id*E=>id+id*id • Parse tree: • A parse tree pictorially shows how the start symbol of a grammar derives a specific string in the language. Given a context-free grammar, a parse tree has the following properties: • The root is labeled by the start symbol • Each leaf is labeled by a token or the empty string • Each interior node is labeled by a nonterminal • If A is a non-terminal labeling some interior node and abcdefg..z are the labels of the children of that node from left to right, then A->abcdefg..z is a production of the grammar.

  11. The leaves of the parse tree read from left to right is called “yield” of the parse tree. It is equivalent to the string derived from the nonterminal at the root of the parse tree. • An ambiguous grammar is one that can generate two or more parse trees that yield the same string • E.G string -> string + string string->string - string string ->0|1|2|3|4|5|6|7|8|9 string=>string + string =>string - string + string => 9 -5 + 2 string=>string - string=>string - string + string =>9-5+2

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