A compact proof of decidability for regular expression equivalence

Transcription

1 A compct proof of decidbility for regulr expression equivlence ITP 2012 Princeton, USA Andre Asperti Deprtment of Computer Science University of Bologn 25/08/2011

2 Abstrct We introduce the notion of pointed regulr expression nd use it to get 1 compct formliztion of the reltion between regulr expressions nd deterministic finite utomt 2 formlly verified, efficient lgorithm for testing regulr expression equivlence.

3 Content 1 Mny different techniques for building DFAs 2 Pointed Regulr Expressions 3 Forml definition nd semntics 4 ɛ-closure nd moves 5 Discussion nd Conclusions

4 Content 1 Mny different techniques for building DFAs 2 Pointed Regulr Expressions 3 Forml definition nd semntics 4 ɛ-closure nd moves 5 Discussion nd Conclusions

5 Content 1 Mny different techniques for building DFAs 2 Pointed Regulr Expressions 3 Forml definition nd semntics 4 ɛ-closure nd moves 5 Discussion nd Conclusions

6 Content 1 Mny different techniques for building DFAs 2 Pointed Regulr Expressions 3 Forml definition nd semntics 4 ɛ-closure nd moves 5 Discussion nd Conclusions

7 Content 1 Mny different techniques for building DFAs 2 Pointed Regulr Expressions 3 Forml definition nd semntics 4 ɛ-closure nd moves 5 Discussion nd Conclusions

8 Mny different techniques for building DFAs Thompson s lgorithm ε ε ε ε ε ε ε

9 Mny different techniques for building DFAs Brzozowski s derivtives (e) () = ɛ (b) = (e 1 + e 2 ) = (e 1 ) + (e 2 ) { (e 1 )e2 + (e 2 ) if nullble e 1 (e 1 e 2 ) = (e 1 )e2 otherwise (e ) = (e)e

10 Mny different techniques for building DFAs McNughton nd Ymd s lgorithm t 1,2 * 1,2 f 1,2 f ,2,3 1,2 f b ,2,3 f f $ 1,2, f 3 4 b b 5 b $ followpos 1,2,3 1,2,3 4 5

11 Pointed Regulr Expressions Pointed regulr expressions Intuition: mrk the positions inside the regulr expression which hve been reched fter reding some prefix of the input string. These pointed expression re the sttes of the DFA.

12 Pointed Regulr Expressions Exmple: (+b)*b Initil position: ( + b )* b

13 Pointed Regulr Expressions Exmple: (+b)*b Moves w.r.t. nd b: b ( + b )* b ( + b )* b

14 Pointed Regulr Expressions Exmple: (+b)*b b ( + b )* b b ( + b )* b b ( + b )* b

15 Pointed Regulr Expressions Exmple: (c+bc)* c ( c b c) * c c b 00 ( c + b c) 0 00 ( c + b c) * * b b ( c + b c) * b c

16 Pointed Regulr Expressions Exmple:(+ɛ)(b*+b)b 4 2 ( + ε)( b * + b) b 1 ( + ε)( b * + b) b ( ε * + )( b + b) b b ( + ε)( b * + b) b 3 b ( + ε)( * b + b) b 5 b b 6 ( + ε)( b * + b) b 7 b ( + ε)( b * + b) b b 9 ( + ε)( b * + b) b b 8 ( + ε)( b * + b) b b b

17 Pointed Regulr Expressions Exmple:(ɛ + + )() ( ε + + ) ( )* ( ε + + ) ( )* ( ε + + ) ( )*

18 Forml definition nd semntics Forml definition Pointed item:, ɛ,,, i 1 i 2, i 1 + i 2, i Pointed rergulr expression (pre): i, b : Bool b is true if there is point t the end of the expression.

19 Forml definition nd semntics Semntics Intuition: Union of ll lnguges strting t the given points. The crrier i of n item i is its underlying r.e. = ɛ = {ɛ} = = {} i 1 + i 2 = i 1 i 2 i 1 i 2 = i 1 i 2 i 2 i = i ( i ) i, F = i i, T = i {ɛ}

20 Forml definition nd semntics An importnt remrk For ny i, ɛ i hence ɛ i, b b = T

21 ɛ-closure nd moves ɛ-closure The (i) opertion propgtes point inside n item i. Remrk ( ) goes from items to pres. ( ) =, F where (ɛ) = ɛ, T i 1, b 1 i 2, b 2 = i 1 + i 2, b 1 b 2 () =, F nd ( ) =, F e 1 i 2 = (i 1 +i 2 ) = (i 1 ) (i 2 ) let i 1, b 1 = e 1 in (i 1 i 2 ) = (i 1 ) i 2 if b 1 then let i 2, b 2 = (i 2 ) in i 1 i 2, b 2 (i ) = (fst( (i))), T else i 1 i 2, F

22 ɛ-closure nd moves lifted constructions Similrly to, we cn lift conctention nd str from items to pres: e 1 e 2 = let i 2, b 2 = e 2 in let i, b = e 1 i 2 in i, b b 2 e = let i, b = e in if b then (fst( (i))), T else i, F

23 ɛ-closure nd moves Moves Lifted constructions permit to define moves in very elegnt wy: move(, ) = emptyset, F move(ɛ, ) = epsilon, F move(c, ) = c, F move( c, ) = c, == c move(i 1 + i 2, ) = move(i 1, ) move(i 2, ) move(i 1 i 2, ) = move(i 1, ) move(i 2, ) move(i, ) = move(i, )

24 ɛ-closure nd moves Min Results for ll nd w :: w i w move(i, ) hence w i ɛ move (i, w) = i, b b = T

25 Discussion nd Conclusions Relted works (theory) The reference pper for pointed regulr expressions is the following report: Asperti, Tssi nd Scerdoti Coen. Regulr Expressions, u point. eprint rxiv: , A similr notion hs been independently introduced in Fischer, Huch nd Wilke. A ply on regulr expressions: functionl perl. ICFP 2010, Bltimore, Mrylnd.

26 Discussion nd Conclusions Relted works (formliztion) system pproch reference COQ Thompson s Bribnt nd Pous lgorithm An efficient coq tctic for deciding kleene lgebrs ITP 2010, LNCS 6172 COQ prtil Almeid, Moreir, Pereir nd de Sous derivtives Prtil Derivtive Automt Formlized in Coq IAA 2010, LNCS 6482 Isbelle Brzozowski s Kruss nd Nipkow derivtives Regulr Expression Equivlence nd Reltion Algebr JAR 2012 Isbelle prtil Wu, Zhng, nd Urbn derivtives A formlistion of the myhill-nerode theorem bsed on regulr expressions. ITP 2011, LNCS 6898 SSReflect Brzozowski s Coqund nd Siles derivtives A decision procedure for regulr expression equivlence in type theory. CPP 2011, LNCS 7086

27 Discussion nd Conclusions Discussion All our proofs hve been formlized nd checked in Mtit Due to their lgebric nture, working with pointed expressions t forml level is rel plesure. Proofs hve strong equtionl flvor, re short nd elegnt. A self contined snpshot of the Mtit librry up to the correctness proof of the bisimilrity test tkes bout 3400 lines; the prt concerning regulr lnguges tkes less thn 1200 lines.

28 Discussion nd Conclusions Performnce A couple of exmples: version of Bezout s identity n c. x, y.n = x + yb expressed s the following regulr expression problem A(, b, c) = (0 c )0 + (0 + 0 b ) (0 + 0 b ) Antimirov s problem, consists in proving the following equlity: B(n) = (ɛ n 1 )( n )

29 Discussion nd Conclusions Performnce We compre our technique (pres) with tht of Coqund&Siles (C&S); execution times hve been computed on mchine with Pentium M Processor GHz nd 1GB of RAM. problem nswer pres C&S problem nswer pres C&S A(3, 5, 8) yes B(6) yes A(4, 5, 11) no B(8) yes A(4, 5, 12) yes B(10) yes A(5, 6, 19) no B(12) yes A(5, 6, 20) yes B(14) yes A(5, 7, 23) no B(16) yes A(5, 7, 24) yes B(18) yes

30 Discussion nd Conclusions Bibliogrphy Alfred V. Aho, Monic S. Lm, Rvi Sethi, nd Jeffrey D. Ullmn. Compilers: Principles, Techniques, nd Tools. Person Eduction Inc., José Bcelr Almeid, Nelm Moreir, Dvid Pereir, nd Simão Melo de Sous. Prtil derivtive utomt formlized in coq. In Implementtion nd Appliction of Automt - 15th Interntionl Conference, CIAA 2010, Winnipeg, MB, Cnd, LNCS 6482, pges Springer, Vlentin Antimirov. Prtil derivtives of regulr expressions nd finite utomton constructions. Theoreticl Computer Science, 155: , Andre Asperti nd Jeremy Avigd. Zen nd the rt of formliztion. Mthemticl Structures in Computer Science, 21(4): , Andre Asperti, Wilmer Ricciotti, Cludio Scerdoti Coen, nd Enrico Tssi. The Mtit interctive theorem prover. In Proceedings of the 23rd Interntionl Conference on Automted Deduction (CADE-2011), Wroclw, Polnd, volume 6803 of LNCS, 2011.

31 Discussion nd Conclusions Bibliogrphy Andre Asperti, Wilmer Ricciotti, Cludio Scerdoti Coen, nd Enrico Tssi. Hints in unifiction. In TPHOLs 2009, volume 5674 of LNCS, pges Springer-Verlg, Andre Asperti, Enrico Tssi, nd Cludio Scerdoti Coen. Regulr expressions, u point. eprint rxiv: , Gérrd Berry nd Rvi Sethi. From regulr expressions to deterministic utomt. Theor. Comput. Sci., 48(3): , Thoms Bribnt nd Dmien Pous. An efficient coq tctic for deciding kleene lgebrs. In Proceedings of Interctive Theorem Proving, ITP 2010, Edinburgh, UK, volume 6172 of LNCS, pges Springer, Anne Brüggemnn-Klein. Regulr expressions into finite utomt. Theor. Comput. Sci., 120(2): , 1993.

32 Discussion nd Conclusions Bibliogrphy Chi-Hsing Chng nd Robert Pige. From regulr expressions to df s using compressed nf s. In Combintoril Pttern Mtching, Third Annul Symposium, CPM 92, Tucson, Arizon, USA, April 1992, Proceedings, LNCS 644, pges Springer, Thierry Coqund nd Vincent Siles. A decision procedure for regulr expression equivlence in type theory. In Proceedings of Certified Progrms nd Proofs, CPP 2011, Kenting, Tiwn, volume 7086 of Lecture Notes in Computer Science, pges Springer, Sebstin Fischer, Frnk Huch, nd Thoms Wilke. A ply on regulr expressions: functionl perl. In Proceeding of the 15th ACM SIGPLAN interntionl conference on Functionl progrmming, ICFP 2010, Bltimore, Mrylnd., pges ACM, Georges Gonthier nd Assi Mhboubi. An introduction to smll scle reflection in coq. Journl of Formlized Resoning, 3(2):95 152, Gérrd P. Huet. Residul theory in lmbd-clculus: A forml development. J. Funct. Progrm., 4(3): , 1994.

33 Discussion nd Conclusions Bibliogrphy Lucin Ilie nd Sheng Yu. Follow utomt. Inf. Comput., 186(1): , Alexnder Kruss nd Tobis Nipkow. Proof perl: Regulr expression equivlence nd reltion lgebr. Journl of Automted Resoning, published on line, R. McNughton nd H. Ymd. Regulr expressions nd stte grphs for utomt. Ieee Trnsctions On Electronic Computers, 9(1):39 47, Scott Owens, John H. Reppy, nd Aron Turon. Regulr-expression derivtives re-exmined. J. Funct. Progrm., 19(2): , Ken Thompson. Regulr expression serch lgorithm. Communictions of ACM, 11: , Chunhn Wu, Xingyun Zhng, nd Christin Urbn. A formlistion of the myhill-nerode theorem bsed on regulr expressions. ITP 2011, Berg en Dl, The Netherlnds, LNCS 6898, pges Springer, 2011.