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2. Lexical Analysis Prof. O. Nierstrasz Thanks to Jens Palsberg and Tony Hosking for their kind permission to reuse and adapt the CS132 and CS502 lecture notes. http://www.cs.ucla.edu/~palsberg/ http://www.cs.purdue.edu/homes/hosking/ Roadmap > > > > Regular languages Finite automata recognizers From regular expressions to deterministic finite automata Limits of regular languages See, Modern compiler implementation in Java (Second edition), chapter 2. 2 Roadmap > > > > Regular languages Finite automata recognizers From regular expressions to deterministic finite automata Limits of regular languages 3 Scanner • map characters to tokens x = x + y <id,x> = <id,x> + <id,y> • character string value for a token is a lexeme • eliminates white space (tabs, blanks, comments etc.) • a key issues is speed use specialized recognizer 4 Specifying patterns A scanner must recognize various parts of the language’s syntax Some parts are easy: White space <ws> ::= <ws> ’ ’ | <ws> ’\t’ | ’’ | ’\t’ Keywords and operators specified as literal patterns: do, end Comments opening and closing delimiters: /* … */ 5 Specifying patterns Other parts are much harder: Identifiers alphabetic followed by k alphanumerics (_, $, &, …)) Numbers integers: 0 or digit from 1-9 followed by digits from 0-9 decimals: integer ’.’ digits from 0-9 reals: (integer or decimal) ’E’ (+ or —) digits from 0-9 complex: ’(’ real ’,’ real ’)’ We need an expressive notation to specify these patterns! 6 Operations on languages A language is a set of strings Operation Definition Union L M = { s s L or s M } Concatenation LM = { st s L and t M } Kleene closure L* = I=0, Li Positive closure L+ = I=1, Li 7 Regular expressions describe regular languages Regular expressions over an alphabet : 1. is a RE denoting the set {} 2. If a , then a is a RE denoting {a} 3. If r and s are REs denoting L(r) and L(s), then: > > > > > (r) is a RE denoting L(r) (r)(s) is a RE denoting L(r) L(s) (r)(s) is a RE denoting L(r)L(s) (r)* is a RE denoting L(r)* If we adopt a precedence for operators, the extra parentheses can go away. We assume closure, then concatenation, then alternation as the order of precedence. 8 Examples identifier letter (a b c … z A B C … Z ) digit (0123456789) id letter ( letter digit )* numbers integer (+— ) (0(123… 9) digit * ) decimal integer . ( digit )* real ( integer decimal ) E (+ —) digit * complex ’(‘ real ’,’ real ’)’ We can use REs to build scanners automatically. 9 Algebraic properties of REs rs = sr is commutative r(st) = (rs)t is associative r (st) = (rs)t concatenation is associative r(st) = rsrt (st)r = srtr r = r r = r concatenation distributes over r * = (r)* is contained in * r ** = r* * is idempotent is the identity for concatenation 10 Examples Let = {a,b} > ab denotes {a,b} > (ab) (ab) denotes {aa,ab,ba,bb} > a* denotes {,a,aa,aaa,…} > (ab)* denotes the set of all strings of a’s and b’s (including ), i.e., (ab)* = (a*b*)* > aa*b denotes {a,b,ab,aab,aaab,aaaab,…} 11 Roadmap > > > > Regular languages Finite automata recognizers From regular expressions to deterministic finite automata Limits of regular languages 12 Recognizers From a regular expression we can construct a deterministic finite automaton (DFA) letter (a b c … z A B C … Z ) digit (0123456789) id letter ( letter digit )* 13 Code for the recognizer 14 Tables for the recognizer Two tables control the recognizer char_class next_state char a-z A-Z 0-9 other value letter letter digit other letter digit other 0 1 3 3 1 1 1 2 2 — — — 3 — — — To change languages, we can just change tables 15 Automatic construction > Scanner generators automatically construct code from regular expression-like descriptions — construct a DFA — use state minimization techniques — emit code for the scanner (table driven or direct code ) > A key issue in automation is an interface to the parser > lex is a scanner generator supplied with UNIX — emits C code for scanner — provides macro deﬁnitions for each token (used in the parser) 16 Grammars for regular languages Provable fact: — For any RE r, there exists a grammar g such that L(r) = L(g) The grammars that generate regular sets are called regular grammars Deﬁnition: In a regular grammar, all productions have one of two forms: 1. A aA 2. A a where A is any non-terminal and a is any terminal symbol These are also called type 3 grammars (Chomsky) 17 More regular languages Example: the set of strings containing an even number of zeros and an even number of ones The RE is (0011)*((0110)(0011)*(0110)(0011)*)* 18 More regular expressions What about the RE (ab)*abb ? State s0 has multiple transitions on a! It is a non-deterministic finite automaton 19 Finite automata A non-deterministic ﬁnite automaton (NFA) consists of: 1. a set of states S = { s0 , … , sn } 2. a set of input symbols (the alphabet) 3. a transition function move mapping state-symbol pairs to sets of states 4. a distinguished start state s0 5. a set of distinguished accepting or final states F A Deterministic Finite Automaton (DFA) is a special case of an NFA: 1. no state has a -transition, and 2. for each state s and input symbol a, there is at most one edge labelled a leaving s. A DFA accepts x iff there exists a unique path through the transition graph from the s0 to an accepting state such that the labels along the edges spell x. 20 DFAs and NFAs are equivalent 1. DFAs are clearly a subset of NFAs 2. Any NFA can be converted into a DFA, by simulating sets of simultaneous states: — each DFA state corresponds to a set of NFA states — possible exponential blowup 21 Roadmap > > > > Regular languages Finite automata recognizers From regular expressions to deterministic finite automata Limits of regular languages 22 NFA to DFA using the subset construction: example 1 23 Constructing a DFA from a regular expression > RE NFA — Build NFA for each term; connect with moves > NFA DFA — Simulate the NFA using the subset construction > DFA minimized DFA — Merge equivalent states > DFA RE — Construct Rkij = Rk-1ik (Rk-1kk)* Rk-1kj Rk-1ij 24 RE to NFA 25 RE to NFA example: (ab)*abb (ab) (ab)* abb 26 NFA to DFA: the subset construction add state P = -closure(s0) Input: NFA N unmarked to SD Output: DFA D with states SD and while unmarked state P in SD transitions TD such that L(D) = mark P L(N) for each input symbol a Method: Let s be a state in N and P be a set of states. Use the U = -closure(move(P,a)) following operations: if U SD > -closure(s) — set of states of N then add U unmarked to SD reachable from s by transitions alone TD[T,a] = U > -closure(P) — set of states of N end for reachable from some s in P by end while transitions alone -closure(s0) is the start state of D > move(T,a) — set of states of N to which there is a transition on input a A state of D is accepting if it from some s in P contains an accepting state of N 27 NFA to DFA using subset construction: example 2 A = {0,1,2,4,7} B = {1,2,3,4,6,7,8} C = {1,2,4,5,6,7} D = {1,2,4,5,6,7,9} E = {1,2,4,5,6,7,10} A B C D E a B B B B B b C D C E C 28 Roadmap > > > > Regular languages Finite automata recognizers From regular expressions to deterministic finite automata Limits of regular languages 29 Limits of regular languages Not all languages are regular! One cannot construct DFAs to recognize these languages: L = { pkqk } L = { wcwr w * } In general, DFAs cannot count! However, one can construct DFAs for: • Alternating 0’s and 1’s: ( 1)(01)*( 0) • Sets of pairs of 0’s and 1’s (01 10)+ 30 So, what is hard? Certain language features can cause problems: > Reserved words — PL/I had no reserved words — if then then then = else; else else = then > Significant blanks — FORTRAN and Algol68 ignore blanks — do 10 i = 1,25 — do 10 i = 1.25 > String constants — Special characters in strings — Newline, tab, quote, comment delimiter > Finite limits — Some languages limit identifier lengths — Add state to count length — FORTRAN 66 — 6 characters(!) 31 How bad can it get? Compiler needs context to distinguish variables from control constructs! 32 What you should know! What are the key responsibilities of a scanner? What is a formal language? What are operators over languages? What is a regular language? Why are regular languages interesting for defining scanners? What is the difference between a deterministic and a non-deterministic finite automaton? How can you generate a DFA recognizer from a regular expression? Why aren’t regular languages expressive enough for parsing? 33 Can you answer these questions? Why do compilers separate scanning from parsing? Why doesn’t NFA DFA translation normally result in an exponential increase in the number of states? Why is it necessary to minimize states after translation a NFA to a DFA? How would you program a scanner for a language like FORTRAN? 34 License > http://creativecommons.org/licenses/by-sa/2.5/ Attribution-ShareAlike 2.5 You are free: • to copy, distribute, display, and perform the work • to make derivative works • to make commercial use of the work Under the following conditions: Attribution. You must attribute the work in the manner specified by the author or licensor. Share Alike. 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