Hyper-minimisation of weighted finite automata

Transcription

1 Hyper-minimistion of weighted finite utomt Dniel Quernheim nd ndres Mletti Institute for Nturl Lnguge Processing, University of Stuttgrt ugust 18, Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

2 Outline Unweighted cse Weighted cse onclusion. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

3 Outline Unweighted cse Weighted cse onclusion. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

4 Minimistion Prolem given DF, return equivlent DF such tht ll equivlent DF re lrger Theorem (Hopcroft 1971) DF minimistion cn e done in time O(n log n) n: numer of sttes. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

5 Minimistion Prolem given DF, return equivlent DF miniml Theorem (Hopcroft 1971) DF minimistion cn e done in time O(n log n) n: numer of sttes. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

6 Hyper-minimistion Definition DF, B lmost equivlent if L() nd L(B) hve finite difference Prolem [Bdr et l. 2009] given DF, return lmost equivlent DF such tht ll lmost equivlent DF re lrger Theorem (Holzer, 2009, Gwrychowsky, Jeż 2009) DF hyper-minimistion cn e done in time O(n log n). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

7 Hyper-minimistion Definition DF, B lmost equivlent if L() nd L(B) hve finite difference Prolem [Bdr et l. 2009] given DF, return lmost equivlent DF hyper-miniml Theorem (Holzer, 2009, Gwrychowsky, Jeż 2009) DF hyper-minimistion cn e done in time O(n log n). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

8 lmost equivlence Sttes re lmost equivlent if their right lnguges differ finitely q = {w Σ δ(q, w) F} E G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

9 Kernel nd premle sttes Definition premle stte: finite left lnguge kernel stte: infinite left lnguge q = {w Σ δ(q 0, w) = q} E G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

10 Structurl chrcteristion Theorem (Bdr et l. 2009) DF is hyper-miniml iff miniml no premle stte is lmost equivlent to nother stte. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

11 Structurl chrcteristion Theorem (Bdr et l. 2009) DF is hyper-miniml iff miniml no premle stte is lmost equivlent to nother stte Exmple (Not hyper-miniml) E G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

12 Hyper-minimistion premle lmost equivlent B D F E Tle: Merges E G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

13 Hyper-minimistion premle lmost equivlent B D F E Tle: Merges E G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

14 Motivtion ppliction In nturl lnguge processing utomt often deterministic for efficient evlution DF tend to e very ig good reduction potentil Often pproximtive dt lossy compression ok. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

15 Motivtion ppliction In nturl lnguge processing utomt often deterministic for efficient evlution DF tend to e very ig good reduction potentil Often pproximtive dt lossy compression ok ut DF re weighted onclusion Let s do hyper-minimistion for weighted DF!. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

16 Outline Unweighted cse Weighted cse onclusion. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

17 Weight structure Generl pproch works for semifields (semirings with mult. inverses). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

18 Weight structure Generl pproch works for semifields (semirings with mult. inverses) Presenttion here we use the field of rel numers Exmple (WDF) E 2 weight of is = 16 weight of is 0 G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

19 Weighted merge E 2 G I Weighted merge of F into E with fctor 2. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

20 Weighted merge E G I 2 Weighted merge of F into E with fctor 2. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

21 lmost equivlence Definition Two WDF, B re lmost equivlent if (w) B(w) for only finitely mny w Σ Definition Two sttes p, q re lmost equivlent if there is k R \ {0} such tht p (w) k q (w) for only finitely mny w Σ [Borchrdt: The Myhill-Nerode theorem for recognizle tree series. DLT 2003]. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

22 Exmple E 2 G I ɛ G I G nd I lmost equivlent. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

23 Exmple 0 B D F 4 H E G I G nd I lmost equivlent E nd F lmost equivlent (with fctor 2) E F Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

24 Exmple 0 B D F 4 H E G I G nd I lmost equivlent E nd F lmost equivlent (with fctor 2) nd B not lmost equivlent! B Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

25 Finding lmost equivlent sttes Definition co-kernel stte: infinite right lnguge co-premle stte: finite right lnguge E 2 G I. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

26 Signture stndrdistion Definition signture of q: ( δ(q, σ), c(q, σ) ) σ Σ Stndrdistion select miniml σ 0 Σ such tht δ(q, σ 0 ) is co-kernel djust trnsition weights: c (q, σ) = { c(q,σ) c(q,σ 0 ) if δ(q, σ) is co-kernel 1 otherwise. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

27 Signture stndrdistion E 2 G I Stndrdised signture of F F I leds to co-premle. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

28 Signture stndrdistion E 2 G I Stndrdised signture of F F I leds to co-premle c (F, ) = 1. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

29 Signture stndrdistion E 2 G I Stndrdised signture of F F I leds to co-premle c (F, ) = 1 F 4 H leds to co-kernel. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

30 Signture stndrdistion E 2 G I Stndrdised signture of F F I leds to co-premle c (F, ) = 1 F 4 H leds to co-kernel c (F, ) = 4 4 = 1. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

31 Signture stndrdistion E 2 G I Stndrdised signture of F F I leds to co-premle c (F, ) = 1 F 4 H leds to co-kernel c (F, ) = 4 4 = 1 Stndrdised signture ( I, 1, H, 1 ). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

32 Finding lmost equivlent sttes E G I 2 Signture mp:, 1,, 1 Blocks: {, I} sig(i) = (, 1,, 1 ) in mp! dd I to lock of (nd merge). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

33 Finding lmost equivlent sttes E G I 2 Signture mp:, 1,, 1 H, 1,, 1 H Blocks: {, I} sig(h) = ( H, 1,, 1 ) dd to mp. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

34 Finding lmost equivlent sttes E G I 2 Signture mp:, 1,, 1 H, 1,, 1 H Blocks: {, I, G} sig(g) = (, 1,, 1 ) in mp! dd G to (nd merge). Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

35 Finding lmost equivlent sttes 0 B D F 4 H E G I sig(f) = (, 1, H, 1 ) dd to mp Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F Blocks: {, I, G}. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

36 Finding lmost equivlent sttes 0 B D F 4 H E G I sig(e) = (, 1, H, 1 ) in mp! dd E to F (nd merge with fctor 1 2 ) Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F Blocks: {, I, G} {F, E}. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

37 Finding lmost equivlent sttes 0 B D F 4 H 4 E G I sig(d) = (, 1, F, 1 ) dd to mp Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D Blocks: {, I, G} {F, E}. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

38 Finding lmost equivlent sttes 0 B D F 4 H 4 E G I sig() = (, 1, F, 1 ) in mp! dd to D (nd merge) Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

39 Finding lmost equivlent sttes 4 4 E G I sig(b) = ( D, 1,, 4 ) dd to mp Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D D, 1,, 4 B Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

40 Finding lmost equivlent sttes 4 4 E Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D D, 1,, 4 B D, 1,, 1 G sig() = ( D, 1,, 1 ) dd to mp I Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

41 Finding lmost equivlent sttes 4 4 E G I sig(0) = (, 1, B, 1 ) dd to mp Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D D, 1,, 4 B D, 1,, 1, 1, B, 1 0 Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

42 Finding lmost equivlent sttes 4 4 E G I Blocks represent lmost equivlence Scling fctors used in merges Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D D, 1,, 4 B D, 1,, 1, 1, B, 1 0 Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

43 Finding lmost equivlent sttes 4 4 E G I Blocks represent lmost equivlence Scling fctors used in merges ut we merged kernel sttes Signture mp:, 1,, 1 H, 1,, 1 H, 1, H, 1 F, 1, F, 1 D D, 1,, 4 B D, 1,, 1, 1, B, 1 0 Blocks: {, I, G} {F, E} {D, }. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

44 Weighted merges merge of F into E with fctor 2 merge of D into with fctor E G I 2. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

45 onclusion Solved hyper-minimistion for WDF over semifields Open Optimise numer of errors Incrementl construction Extensions to WNF, more weight structures, etc.. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

46 onclusion Solved hyper-minimistion for WDF over semifields Open Optimise numer of errors Incrementl construction Extensions to WNF, more weight structures, etc. Thnk you for your ttention!. Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21

47 References ndrew Bdr, Vilim Geffert, nd In Shipmn. Hyper-minimizing minimized deterministic finite stte utomt. RIRO Theor. Inf. ppl., 43(1), Json Eisner. Simpler nd more generl minimiztion for weighted finite-stte utomt. In Proc. HLT-NL, Pweł Gwrychowski nd rtur Jeż. Hyper-minimistion mde efficient. In Proc. MFS, volume 5734 of LNS, Mrkus Holzer nd ndres Mletti. n n log n lgorithm for hyper-minimizing (minimized) deterministic utomton. Theor. omput. Sci., 411(38 39), John E. Hopcroft. n n log n lgorithm for Minimizing the Sttes in Finite utomton. In The Theory of Mchines nd omputtions. cdemic Press, Mletti nd D. Quernheim Hyper-minimistion of weighted finite utomt ugust 18, / 21