Unambiguous automata inference by means of states-merging methods François Coste, Daniel Fredouille IRISA-INRIA, Campus de Beaulieu Rennes Cedex France

D. Fredouille and F. Coste, Unambiguous Automata Inference2 Definitions n Alphabet:  = {a,b} n Word:abbabbabbaaa  n Language:L n Automaton:  *  *  I- Automata inference L={a+b}*a{a+b}

D. Fredouille and F. Coste, Unambiguous Automata Inference3 Classes of automata (1/3) n Nondeterministic Automata (NFA) n Deterministic Automata (DFA) –one outgoing transition per input symbol I- Automata inference a a L={a+b}*a{a+b}

D. Fredouille and F. Coste, Unambiguous Automata Inference4 Classes of automata (2/3) n Unambiguous Automata (UFA) [SH85] –one acceptance per word I- Automata inference a b a b b a a b b b L={a+b}*a{a+b} n NFA  UFA  DFA

D. Fredouille and F. Coste, Unambiguous Automata Inference5 Automata inference Examples Counter-examples I- Automata inference S ={aa,abab} + S ={ba,abbb} - L={a+b}*a{a+b}

D. Fredouille and F. Coste, Unambiguous Automata Inference6 Why this study ? n State of the art: DFA inference n Our goal: introducing some amount of non-determinism n Why ? –NFA << DFA –inferring with less data –inferring “explicit” representations n Method: –extending classical DFA inference algorithm I- Automata inference

D. Fredouille and F. Coste, Unambiguous Automata Inference7 II - Study of the DFA inference framework

D. Fredouille and F. Coste, Unambiguous Automata Inference8 Search space for NFAs [DMV94] UA MCA II - The DFA search space

D. Fredouille and F. Coste, Unambiguous Automata Inference9 Counter-examples : compatibility UA S  L   - S  L =  - (compatible) (incompatible) MCA II - The DFA search space

D. Fredouille and F. Coste, Unambiguous Automata Inference10 The search space for DFA UA MCA Deterministic merging State merging II - The DFA search space

D. Fredouille and F. Coste, Unambiguous Automata Inference11  q1,q2  Q,  w   *: w  pref(q1)  w  pref(q2)  state-merging(q1,q2) II - The DFA search space Merging for determinisation procedure

D. Fredouille and F. Coste, Unambiguous Automata Inference12  q1,q2  Q,  w   *: w  pref(q1)  w  pref(q2)  state-merging(q1,q2) II - The DFA search space Merging for determinization procedure

D. Fredouille and F. Coste, Unambiguous Automata Inference13  q1,q2  Q,  w   *: w  pref(q1)  w  pref(q2)  state-merging(q1,q2) II - The DFA search space Merging for determinization procedure

D. Fredouille and F. Coste, Unambiguous Automata Inference14 Deterministic merging operator = state-merging + merging for determinization II - The DFA search space n Very commonly used [OG92, LPP98,...] n Demonstration of formal properties –Merging for determinization Enables to reach the “closest” DFA from the original NFA –Deterministic merging Enables to reach all derived DFA from a given DFA –... (see tech. rep.)

D. Fredouille and F. Coste, Unambiguous Automata Inference15 IV - From DFA to UFA inference or how to introduce some amount of non- determinism in inference

D. Fredouille and F. Coste, Unambiguous Automata Inference16 Inferring non-deterministic representations: the choice of UFA III - DFA to UFA inference n Why UFA ? –unity in the search space (like DFA) NFA UFA DFA UA MCA({aaaaa})

D. Fredouille and F. Coste, Unambiguous Automata Inference17 Merging for disambiguisation procedure III - DFA to UFA inference  q1,q2  Q,  w1,w2   *: w1  pref(q1)  w1  pref(q2)  w2  suff(q1)  w2  suff(q2)  state-merging(q1,q2)

D. Fredouille and F. Coste, Unambiguous Automata Inference18 Merging for disambiguisation procedure III - DFA to UFA inference  q1,q2  Q,  w1,w2   *: w1  pref(q1)  w1  pref(q2)  w2  suff(q1)  w2  suff(q2)  state-merging(q1,q2)

D. Fredouille and F. Coste, Unambiguous Automata Inference19 Unambiguous merging = state-merging + merging for disambiguisation III - DFA to UFA inference n Finer operator than merging for determinization n Demonstration of formal properties –Merging for disambiguisation Enables to reach the “closest” UFA from the original NFA –unambiguous merging Enables to reach all derived UFA from a given UFA –... (see tech. rep.)

D. Fredouille and F. Coste, Unambiguous Automata Inference20 IV - Comparative experiments - Inference algorithms - Benchmarks - Experimental results

D. Fredouille and F. Coste, Unambiguous Automata Inference21 Algorithms n UFA –Hill-climbing heuristic n DFA –EDSM heuristic [LPP98] n RFSA –DeLeTe II [DLT01] IV - Comparative experiments –Hill-climbing heuristic

D. Fredouille and F. Coste, Unambiguous Automata Inference22 Counter-example use for DFA and UFA inference n Compatibility [DMV94] –generalization of, stopped by n Functionality [AS95] –generalization of and, stopped by empty intersection IV - Comparative experiments S + S - S + S -

D. Fredouille and F. Coste, Unambiguous Automata Inference23 Benchmarks n [DLT01] –Generation: DFA, NFA, Regular Expression –4 sizes of training sample –30 languages generated for each generation mode and sample size n + UFA generator n Evaluation based on –average recognition level on test sets –matches between recognition level IV - Comparative experiments

D. Fredouille and F. Coste, Unambiguous Automata Inference24 Results n Best algorithms w.r.t. benchmarks –DFA bench:UFA inference with hill-climbing –UFA bench:DFA inference with hill-climbing  UFA inference with hill-climbing –NFA bench: RFSA inference –Reg. Expr.:RFSA inference  DFA inference with hill-climbing ? IV - Comparative experiments

D. Fredouille and F. Coste, Unambiguous Automata Inference25 Results n Heuristic: –Hill-climbing >> EDSM when inferring DFAs for NFA/Regular Expression/UFA bench. n Counter-examples: –Compatibility  Functionality IV - Comparative experiments

D. Fredouille and F. Coste, Unambiguous Automata Inference26 Results (matches) IV - Comparative experiments

D. Fredouille and F. Coste, Unambiguous Automata Inference27 Conclusion n UFA inference –Merging for disambiguisation –Heuristic n Comparison with EDSM & DeLeTe II Perspectives n Speeding up the algorithm n Application n Using properties of the DFA/UFA space

D. Fredouille and F. Coste, Unambiguous Automata Inference28 References n [AS95] Alquézar, Sanfeliu, “Incremental grammatical inference from positive and negative data using unbiased finite state automata”, SSPR’94 n [DMV94] Dupont and al. “What is the search space of the regular inference ?”, ICGI ’94 n [DLT00] Denis and al., “Learning regular languages using nondeterministic automata”, ICGI ’00 n [SH85] Stearns, Hunt, “On the equivalence and containment problems for unambiguous regular expressions, regular grammars and finite automata”, SIAM vol 14 n [tech. rep.] Coste, Fredouille “What is the search space for NFA, UFA and DFA inference ?”, IRISA