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Feb 28, 2023 •142 likes •1.97k views

Tree Automata, Approximations, and Constraints for Verification Ph.D. thesis defence for Vincent Hugot , Supervised by O. Kouchnarenko and P.-C. Ham {pheam,vhugot,okouchna}@femto-st.fr University of Franche-Comt DGA & Inria/CASSIS &

  1. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking LTL on Rewrite Sequences 1 Statement of the Central Problem Our Approach: An Overview TAGE With a Bounded Number of Constraints 2 Global Equality Constraints Overview of the Results Other Works and Some Perspectives 3 Results on SAT & Tree-Walking Automata Perspectives and Questions Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 8 / 24

  2. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 R , Π | = ¬ X : “The first transition, if it occurs, is not by X ” π 1 ≡ X ( Π ) = ∅ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  3. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : “The first transition, if it occurs, is not by X ” π 1 ≡ X ( Π ) = ∅ 2 X : “There is a first transition, and it is by X ” π 2 ≡ [ R \ X ]( Π ) = ∅ ? Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  4. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : “The first transition, if it occurs, is not by X ” π 1 ≡ X ( Π ) = ∅ 2 X : “There is a first transition, and it is by X ” π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  5. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : “The first transition, if it occurs, is not by X ” π 1 ≡ X ( Π ) = ∅ 2 X : “There is a first transition, and it is by X ” π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : “No transition that occurs is by X ” π 3 ≡ X ( R ∗ ( Π )) = ∅ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  6. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : “No transition that occurs is by X ” π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : “All transitions that occur are by X ” Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  7. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : “No transition that occurs is by X ” π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : “All transitions that occur are by X ” π 4 ≡ π 2 [ R ∗ ( Π ) /Π ] ≡ [ R \ X ]( R ∗ ( Π )) = ∅ ∧ R ∗ ( Π ) ⊆ X − 1 ( T ) ? Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  8. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : “No transition that occurs is by X ” π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : “All transitions that occur are by X ” π 4 ≡ π 2 [ R ∗ ( Π ) /Π ] ≡ [ R \ X ]( R ∗ ( Π )) = ∅ ∧ R ∗ ( Π ) ⊆ X − 1 ( T ) ? ω -language! Too strong Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  9. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : “No transition that occurs is by X ” π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : “All transitions that occur are by X ” π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  10. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 4 � X : 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . ϕ : π = “ R , Π | = ϕ is translated by π ” “for all executions, ϕ is satisfied” ∀ x.P ( x ) ∧ ∀ x.Q ( x ) ⇐ ⇒ ∀ x. ( P ( x ) ∧ Q ( x )) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  11. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 4 � X : 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . ∀ x.P ( x ) ∧ ∀ x.Q ( x ) ⇐ ⇒ ∀ x. ( P ( x ) ∧ Q ( x )) 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = ∀ x.P ( x ) ∨ ∀ x.Q ( x ) = ⇒ ∀ x. ( P ( x ) ∨ Q ( x )) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  12. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : ∧ Π ⊆ X − 1 ( T ) π 2 ≡ [ R \ X ]( Π ) = ∅ 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 4 � X : 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . ∀ x.P ( x ) ∧ ∀ x.Q ( x ) ⇐ ⇒ ∀ x. ( P ( x ) ∧ Q ( x )) 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = ∀ x.P ( x ) ∨ ∀ x.Q ( x ) = ⇒ ∀ x. ( P ( x ) ∨ Q ( x )) 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = ∀ x. ¬ P ( x ) � = ¬∀ x.P ( x ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  13. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  14. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  15. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  16. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  17. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  18. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] � ( X ⇒ • Y ) : π 0 ≡ π 7 [ R ∗ ( Π ) /Π ] Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  19. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] � ( X ⇒ • Y ) : π 0 ≡ π 7 [ R ∗ ( Π ) /Π ] What about • Y ⇒ X ? Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  20. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation 1 ¬ X : π 1 ≡ X ( Π ) = ∅ 2 X : π 2 ≡ [ R \ X ]( Π ) = ∅ ∧ Π ⊆ X − 1 ( T ) 3 � ¬ X : π 3 ≡ X ( R ∗ ( Π )) = ∅ ≡ π 1 [ R ∗ ( Π ) /Π ] 4 � X : π 4 ≡ [ R \ X ]( R ∗ ( Π )) = ∅ 5 Conjunction: if ϕ : π 5 and ψ : π ′ 5 then ϕ ∧ ψ : π 5 ∧ π ′ 5 . 6 Disjunction: π 6 ∨ π ′ ⇒ R , Π | = ϕ ∨ ψ 6 = 7 Negation: R , Π | / ϕ � = R , Π | = ¬ ϕ : “NNF” required = 8 Implication: X ⇒ • Y : π 7 ≡ [ R \ Y ]( X ( Π )) = ∅ ∧ X ( Π ) ⊆ Y − 1 ( T ) X : π 2 , Y : π ′ 2 ≡ π 2 [ Y/X ] , π 7 ≡ π ′ 2 [ X ( Π ) /Π ] � ( X ⇒ • Y ) : π 0 ≡ π 7 [ R ∗ ( Π ) /Π ] What about • Y ⇒ X ? Other techniques (signatures,. . . ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 9 / 24

  21. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Translation Rules, by Examples A dozen rules, e.g. conjunction: � Π � σ � ϕ ∧ ψ � � � Π � σ � ϕ � ∧ � Π � σ � ψ � always (simplest case): � Π � ε � � ϕ � � � R ∗ ( Π ) � ⋆ ε � ϕ � positive literal: � � Π � σ � X � ( σ \ X ) ⊳ � h ( σ \ X ) = ε h ( σ \ X )− 1 � h ( σ \ X ) � � Π k σ \ X ⊆ R − 1 ( T ) = ∅ Π ∧ σ \ X k ∈ ∇ σ,k = 0 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 10 / 24

  22. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References LTL → Rewrite Proposition Derivation Tree Derivation tree: automatic translation and proof � � Π � ε � � ( X ⇒ • Y ) � � � R ∗ ( Π ) � ⋆ ε � X ⇒ • Y � � � R ∗ ( Π ) � � X � R | N 1 � � • Y � � � R ∗ ( Π ) � � X � R | N 1 � � ◦ Y � ∧ X ( R ∗ ( Π )) ⊆ R − 1 ( T ) . � � X ( R ∗ ( Π )) � ⋆ ε � Y � [ R \ Y ]( X ( R ∗ ( Π ))) = ∅ Optional global optimisation phase: R − 1 ( T ) → Y − 1 ( T ) . Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 11 / 24

  23. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Translatable Fragment Exactly rewrite-translatable fragment: X ∈ ℘ ( R ) , m ∈ N ϕ : = ⊤ | ⊥ | X | ¬ X | ϕ ∧ ϕ | ψ ⇒ ϕ | • ϕ | ◦ ϕ | � ϕ ψ : = ⊤ | ⊥ | X | ¬ X | ψ ∨ ψ | ψ ∧ ψ | • ψ | ◦ ψ | Φ Φ : = at least ε -stabilisable � ϕ Practical pre-experimental evaluation: good partial support of [Dwyer et al., 1999] patterns. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 12 / 24

  24. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References LTL on Rewrite Sequences Perspectives (Translation Into Rewrite Proposition) R Π ϕ δ 1 , . . . , δ n π To Rew. Prop. To Approx. R , Π | = ϕ ? [Héam et al., 2012a] Int. Conf. IJCAR’12, Manchester Extensions : Past-Time and Existential LTL Dealing with eventuality by studying “exhaustion”: e.g. ♦ ¬ { f ( x ) → x } holds with bounded f -height & no intro Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 13 / 24

  25. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References LTL on Rewrite Sequences Perspectives (Approximated Decision Procedures) R Π ϕ δ 1 , . . . , δ n π To Rew. Prop. To Approx. = ϕ ? R , Π | Coping with more non-linearity – e.g. protocols, rewrite steps e.g. f ( x, x ) → g ( x ) , f ( x ) → g ( x, x ) ,. . . Tractable algorithmic toolbox for TAGE Last points ⇒ closer study of TAGE complexity Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 14 / 24

  26. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking LTL on Rewrite Sequences 1 Statement of the Central Problem Our Approach: An Overview TAGE With a Bounded Number of Constraints 2 Global Equality Constraints Overview of the Results Other Works and Some Perspectives 3 Results on SAT & Tree-Walking Automata Perspectives and Questions Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 14 / 24

  27. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] TAGE, TA = , Positive TAGED, A = � A , Q, F, ∆, ≅ � : � A , Q, F, ∆ � vanilla tree automaton ta ( A ) ≅ equality constraints , ≅ ⊆ Q 2 Constraint p ≅ q : run ρ of A on t : run of ta ( A ) on t satisfying ≅ : ∀ α, β ∈ P ( t ) ; ρ ( α ) ≅ ρ ( β ) ⇒ t | α = t | β accepting run : accepting for ta ( A ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 15 / 24

  28. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v = f f f f a a a a a a a Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 16 / 24

  29. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u, ρ u = f q f and v = f f ^ f ^ f a q q a q a q a q a q a a Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 16 / 24

  30. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v, ρ v = f q f f f f ^ a ^ q q a a a a a q a q Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 16 / 24

  31. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking LTL on Rewrite Sequences 1 Statement of the Central Problem Our Approach: An Overview TAGE With a Bounded Number of Constraints 2 Global Equality Constraints Overview of the Results Other Works and Some Perspectives 3 Results on SAT & Tree-Walking Automata Perspectives and Questions Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 16 / 24

  32. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References TA = versus TA = k Restriction on the kind of constraints: Rigid Automata (RTA) Same expressive power as TA = Less compact representations Linear emptiness / finiteness tests, vs. ExpTime -complete Applications: [Jacquemard et al., 2009, Vacher, 2010] What of the number of constraints? TA = k A = � Σ, Q, F, ∆, ≅ � : TA = A � Σ, Q, F, ∆, ≅ � ≅ such that Card ( ≅ ) � k Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 17 / 24

  33. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Summary of Results [Héam et al., 2012c] Int. Conf. CIAA’12, Porto Strict hierarchy of powers: L ( TA = k ) ⊂ L ( TA = k + 1 ) Emptiness linear for TA = 1 , ExpTime-complete TA = 2 Finiteness polynomial for TA = 1 , ExpTime-complete for TA = 2 NP-complete membership becomes polynomial if k fixed. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  34. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Summary of Results [Héam et al., 2012c] Int. Conf. CIAA’12, Porto Strict hierarchy of powers: L ( TA = k ) ⊂ L ( TA = k + 1 ) σ u i ∈ T ( A i ) ∀ i u k u k σ u k − 1 u k − 1 σ u 1 u 1 ⊥ , Emptiness linear for TA = 1 , ExpTime-complete TA = 2 Finiteness polynomial for TA = 1 , ExpTime-complete for TA = 2 NP-complete membership becomes polynomial if k fixed. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  35. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Summary of Results [Héam et al., 2012c] Int. Conf. CIAA’12, Porto Strict hierarchy of powers: L ( TA = k ) ⊂ L ( TA = k + 1 ) Emptiness linear for TA = 1 , ExpTime-complete TA = 2 . . . u n σ σ σ σ u n − 1 σ u 1 u 2 u 3 . . . σ σ σ σ u u u u u Finiteness polynomial for TA = 1 , ExpTime-complete for TA = 2 NP-complete membership becomes polynomial if k fixed. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  36. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Summary of Results [Héam et al., 2012c] Int. Conf. CIAA’12, Porto Strict hierarchy of powers: L ( TA = k ) ⊂ L ( TA = k + 1 ) Emptiness linear for TA = 1 , ExpTime-complete TA = 2 Finiteness polynomial for TA = 1 , ExpTime-complete for TA = 2 Reduction of emptiness to finiteness. NP-complete membership becomes polynomial if k fixed. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  37. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Summary of Results NP-complete membership becomes polynomial if k fixed. − h ∈ H t ← P − → G P S t • • • G 1 = { p, q } C G 1 C G 1 G 2 = { r, s, t } C G 3 • • • • C G 2 G 3 = { q x } . . . • • • • • • • • C G n G n Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  38. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking LTL on Rewrite Sequences 1 Statement of the Central Problem Our Approach: An Overview TAGE With a Bounded Number of Constraints 2 Global Equality Constraints Overview of the Results Other Works and Some Perspectives 3 Results on SAT & Tree-Walking Automata Perspectives and Questions Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 18 / 24

  39. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References TAGE SAT & Tree-Walking Overloops [Héam et al., 2010] Int. Workshop CSTVA’10, Paris [Héam et al., 2011] Int. Conf. CIAA’11, Blois [Héam et al., 2012b] Int. Journal Theo. Comp. Sci. SAT Encoding for TAGE membership & optimisations. Formal treatment of tree-walking loops for transformation into bottom-up TA; revealed missing factor in space Σ × T × 2 Q 2 . Introduced tree-walking overloops : restores T × 2 Q 2 , smaller automata in practice in extensive random tests. Shown overloops upper-bound is | T | · 2 | Q | log 2 ( | Q | + 1 ) in the deterministic case. Note that exponential is unavoidable. Polynomial overloops-based approximation to TWA emptiness, vs. ExpTime -c. Very precise in random tests. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 19 / 24

  40. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Polynomial Approximation for Emptiness Random tests 1 Ad-hoc scheme: ≈ 20 000 TWA, 2 � | Q | � 20 , | ∆ | ≈ 3 × | Q | , 75 % of empty languages, only two Unknown instead of Empty . 2 Uniform scheme [Héam et al., 2009], REGAL back-end for FSA generation [Bassino et al., 2007]. 2 000 deterministic and complete TWA uniformly generated for each 2 � | Q | � 25 . Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 20 / 24

  41. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Polynomial Approximation for Emptiness Random tests 55 Approximation – 2000 samples Approximation – 200 samples 50 Exact – 200 samples 45 40 35 30 ∅ % 25 20 15 10 5 0 2 4 6 8 10 12 14 16 18 20 22 24 | Q | Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 20 / 24

  42. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Size Comparison: Loops vs. Overloops One Example & Uniform Generation Scheme For X : loops � B l � = 1986 ; overloops � B o � = 95 ; deterministic minimal � B m � = 56 ; smallest known non-deterministic � B s � = 34 . Loops 60 times worse than manual optimal; overloops 3 times . � = 1617 , � B ′ � � B ′ � o � = 78 . Orthogonal to post-processing cleanup: l � � B ′ � � B l � � B l � � ≈ � B o � l � � B o � ≈ 20.9 o � ≈ 20.7 o � ≈ 1.2 . and and � B ′ � � B ′ � � B ′ l Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 21 / 24

  43. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Size Comparison: Loops vs. Overloops One Example & Uniform Generation Scheme 18 � B l � � B l � / � B o � 16 � B ′ l � (clean) � B ′ l � / � B ′ o � � B o � � B l � / � B ′ l � � B ′ � B o � / � B ′ 14 3 o � (clean) o � 12 10 5 10 Ratio 2 � B � � 8 6 1 4 2 70 2 3 4 5 6 7 2 3 4 5 6 | Q A | | Q A | Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 21 / 24

  44. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking LTL on Rewrite Sequences 1 Statement of the Central Problem Our Approach: An Overview TAGE With a Bounded Number of Constraints 2 Global Equality Constraints Overview of the Results Other Works and Some Perspectives 3 Results on SAT & Tree-Walking Automata Perspectives and Questions Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 21 / 24

  45. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Conclusion / Summary Generalisation of the translation R Π ϕ δ 1 , . . . , δ n π To Rew. Prop. To Approx. R , Π | = ϕ ? Study of complexity of bounded global constraints Improved loops-based methods for tree-walking automata Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 22 / 24

  46. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Perspectives Full TAGE may not be required for X ( Π ) ; flat constraints ensure polynomial emptiness decision; are they enough? Implemented algorithmic toolbox for these automata. Rewrite propositions go beyond LTL (e.g. ∃ -LTL). What is their full expressive power ? Intermix state and transition -based properties. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 23 / 24

  47. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Questions ? Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 24 / 24

  48. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Supported Fragment, In Practice Partially Supported Patterns From [Dwyer et al., 1999] Scope Pattern Global Before After Between Until Support Absence 41 5 12 18 9 48% Universality 110 1 5 2 1 96% Existence 12 1 4 8 1 0% Bound Exist. 0 0 0 1 0 0% Response 241 1 3 0 0 99% Precedence 25 0 1 0 0 96% Resp. Chain 8 0 0 0 0 0% Prec. Chain 1 0 0 0 0 0% Support 95% 0% 32% 0% 0% 83% Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 25 / 24

  49. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Formal Tools for Verification Reliable Software Software failure is undesirable . . . Ariane 5, Therac-25, Mariner I, Phobos I, XA/21 USA & Canada Northeast 2003 blackout, MIM-104 Patriot anti-missile, Mars Climate Orbiter, Mars Polar Lander, Mars Global Surveyor space probes,. . . . . . hence the need for formal verification methods. E.G. With Hoare logic , correctness is a mathematical theorem. Precondition, code, post-condition: { ⊤ } x : = y { x = y } . Manual proofs require mathematical ingenuity. Automation ? Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 26 / 24

  50. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Model-Checking Rewrite Sequences Coding the Behaviour of the System: � ( X ⇒ • Y ) X r ′ i ∈ Y r i ∈ X u i v i t i R ∗ r ′ j ∈ Y r j ∈ X u j v j t j R ∗ X t 0 ∈ Π X r ′ k ∈ Y r k ∈ X R ∗ . . . . . . . . . r ′ n ∈ Y r n ∈ X R ∗ t n u n v n X Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 27 / 24

  51. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Maximal Rewrite Words Coding the Behaviour of the System X r ′ i ∈ Y r i ∈ X t i u i v i R ∗ r ′ j ∈ Y r j ∈ X t j u j v j R ∗ X t 0 ∈ Π X r ′ k ∈ Y r k ∈ X R ∗ . . . . . . . . . r ′ r n ∈ X n ∈ Y R ∗ u n v n t n X Executions may or may not terminate : finite and infinite words. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 28 / 24

  52. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Maximal Rewrite Words Coding the Behaviour of the System Finite or infinite words on R : � � � 1, n � → R � N = N ∪ { + ∞ } W = n ∈ N Notation: length # w ∈ N : # w = Card ( dom w ) . Maximal rewrite words of R , originating in Π : � Π � is the set of words w ∈ W such that ∃ u 0 ∈ Π : ∃ u 1 , . . . , u # w ∈ T : ∀ k ∈ dom w, w ( k ) → u k ∧ # w ∈ N ⇒ R ( { u # w } ) = ∅ u k − 1 − − − Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 28 / 24

  53. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Syntax and Semantics for LTL Close to Finite-LTL [Manna and Pnueli, 1995] ϕ : = X | ¬ ϕ | ϕ ∧ ϕ | • m ϕ | ◦ m ϕ | ϕ U ϕ X ∈ ℘ ( R ) ⊤ | ⊥ | ϕ ∨ ϕ | ϕ ⇒ ϕ | ♦ ϕ | � ϕ m ∈ N . ⇔ ( w, i ) | = X i ∈ dom w and w ( i ) ∈ X ( w, i ) | = ¬ ϕ ⇔ ( w, i ) | / ϕ = ⇔ ( w, i ) | = ( ϕ ∧ ψ ) ( w, i ) | = ϕ and ( w, i ) | = ψ = • m ϕ ( w, i ) | ⇔ i + m ∈ dom w and ( w, i + m ) | = ϕ = ◦ m ϕ ( w, i ) | ⇔ i + m / ∈ dom w or ( w, i + m ) | = ϕ � ∃ j ∈ dom w : j � i ∧ ( w, j ) | = ψ ( w, i ) | = ϕ U ψ ⇔ ∧ ∀ k ∈ � i, j − 1 � , ( w, k ) | = ϕ For any w ∈ W , i ∈ N 1 , m ∈ N and X ∈ ℘ ( R ) . Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 29 / 24

  54. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Syntax and Semantics for LTL Close to Finite-LTL [Manna and Pnueli, 1995] ϕ : = X | ¬ ϕ | ϕ ∧ ϕ | • m ϕ | ◦ m ϕ | ϕ U ϕ X ∈ ℘ ( R ) ⊤ | ⊥ | ϕ ∨ ϕ | ϕ ⇒ ϕ | ♦ ϕ | � ϕ m ∈ N . ( w, i ) | = X ⇔ i ∈ dom w and w ( i ) ∈ X ⇔ ( w, i ) | = ¬ ϕ ( w, i ) | / ϕ = ( w, i ) | = ( ϕ ∧ ψ ) ⇔ ( w, i ) | = ϕ and ( w, i ) | = ψ = • m ϕ ⇔ ( w, i ) | i + m ∈ dom w and ( w, i + m ) | = ϕ = ◦ m ϕ ( w, i ) | ⇔ i + m / ∈ dom w or ( w, i + m ) | = ϕ ⇔ ∀ j ∈ dom w, j � i ⇒ ( w, j ) | ( w, i ) | = � ϕ = ϕ For any w ∈ W , i ∈ N 1 , m ∈ N and X ∈ ℘ ( R ) . Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 29 / 24

  55. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Syntax and Semantics for LTL Close to Finite-LTL [Manna and Pnueli, 1995] ( w, i ) | ⇔ i ∈ dom w and w ( i ) ∈ X = X ( w, i ) | = ¬ ϕ ⇔ ( w, i ) | / ϕ = ⇔ = ϕ and ( w, i ) | ( w, i ) | = ( ϕ ∧ ψ ) ( w, i ) | = ψ = • m ϕ ( w, i ) | ⇔ i + m ∈ dom w and ( w, i + m ) | = ϕ = ◦ m ϕ ⇔ ( w, i ) | i + m / ∈ dom w or ( w, i + m ) | = ϕ ( w, i ) | = � ϕ ⇔ ∀ j ∈ dom w, j � i ⇒ ( w, j ) | = ϕ For any w ∈ W , i ∈ N 1 , m ∈ N and X ∈ ℘ ( R ) . Satisfaction: = ϕ ⇐ ⇒ ( w, 1 ) | w | = ϕ R , Π | = ϕ ⇐ ⇒ ∀ w ∈ � Π � , w | = ϕ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 29 / 24

  56. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Rewrite Propositions Problem Statement: First Translation Step Rewrite proposition π on R , from Π ; has a trivial truth value γ : = ℓ = ∅ | ℓ ⊆ ℓ π : = γ | γ ∧ γ | γ ∨ γ ℓ : = Π | T | X ( ℓ ) | X − 1 ( ℓ ) | X ∗ ( ℓ ) X ∈ ℘ ( R ) Problem statement : translations into RP Input: R , ϕ ∈ LTL, Π ⊆ T Output: RP π such that: R , Π | = ϕ ⇐ ⇒ π (exact translation) R , Π | = ϕ ⇐ = π (under-approximated translation) R , Π | = ϕ = ⇒ π ( over-approximated translation ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 30 / 24

  57. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Intuitions for the Translation Boundaries of the Translatable Fragment R ∗ ( Π ) hides traces : ♦ X probably untranslatable. So are { ♦ , U , W , R , . . . } . Formulæ in sanitised form : negation on literals. Not exactly NNF. ( A ∨ B ) ⇒ C ( A ⇒ C ) ∧ ( B ⇒ C ) ( ¬ A ∧ ¬ B ) ∨ C Preprocessing to fit translatable “antecedent/consequent” form. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 31 / 24

  58. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures Implication: Girdling the Future Idea: ϕ ⇒ ψ ? ϕ as an assumption , i.e. a model of ϕ : ξ ( ϕ ) �� � � � 1, n � ∪ { ω } � Σ = → ℘ ( R ) × ℘ ( N ) . n ∈ N Notations: σ ∈ Σ compactly as σ = � f | S � = � ∂σ | ∇ σ � , or in extenso as � f ( 1 ) , f ( 2 ) , . . . , f ( # σ ) � f ( ω ) | S � . X ∧ ◦ 1 Y ∧ ◦ 2 � Z � � Example: ξ = � X, Y � Z | N 1 � Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 32 / 24

  59. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures Implication: Girdling the Future �� � � � Σ = � 1, n � ∪ { ω } → ℘ ( R ) × ℘ ( N ) . n ∈ N Notations: σ ∈ Σ compactly as σ = � f | S � = � ∂σ | ∇ σ � , or in extenso as � f ( 1 ) , f ( 2 ) , . . . , f ( # σ ) � f ( ω ) | S � . X ∧ ◦ 1 Y ∧ ◦ 2 � Z � � Example: ξ = � X, Y � Z | N 1 � Constrained Words: � Π � σ � = { w ∈ � Π � | # w ∈ ∇ σ ∧ ∀ k ∈ dom w, w ( k ) ∈ σ [ k ] } ∀ Π ⊆ T , ϕ ∈ A -LTL , � Π � ξ ( ϕ ) � = { w ∈ � Π � | w | = ϕ } Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 32 / 24

  60. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures: the Transformation ξ ( · ) Modelling the Antecedent to Girdle the Future ξ ( ⊥ ) = �� ∅ | ∅ � ξ ( ⊤ ) = �� R | N � = ε ξ ( X ) = � X � R | N 1 � ξ ( ¬ X ) = � R \ X � R | N � ξ ( • m ϕ ) = ξ ( ϕ ) ◮ m ξ ( ◦ m ϕ ) = ξ ( ϕ ) ⊲ m ∞ � � � ξ ( ϕ ∧ ψ ) = ξ ( ϕ ) � ξ ( ψ ) ξ ( � ϕ ) = ξ ( ϕ ) ⊲ m m = 0 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 33 / 24

  61. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures: the Transformation ξ ( · ) Modelling the Antecedent to Girdle the Future ξ ( ⊥ ) = �� ∅ | ∅ � ξ ( ⊤ ) = �� R | N � = ε ξ ( X ) = � X � R | N 1 � ξ ( ¬ X ) = � R \ X � R | N � ξ ( • m ϕ ) = ξ ( ϕ ) ◮ m ξ ( ◦ m ϕ ) = ξ ( ϕ ) ⊲ m ∞ � � � ξ ( ϕ ∧ ψ ) = ξ ( ϕ ) � ξ ( ψ ) ξ ( � ϕ ) = ξ ( ϕ ) ⊲ m m = 0 σ ◮ m = Strong Shift Right = � R 1 , . . . , R m , ∂σ ( 1 ) , . . . , ∂σ ( # σ ) � ∂σ ( ω ) | ( ∇ σ \ { 0 } ) + m � σ ⊲ m = Weak Shift Right = � R 1 , . . . , R m , ∂σ ( 1 ) , . . . , ∂σ ( # σ ) � ∂σ ( ω ) | � 0, m � ∪ ( ∇ σ + m ) � Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 33 / 24

  62. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures: the Transformation ξ ( · ) Modelling the Antecedent to Girdle the Future ξ ( ⊥ ) = �� ∅ | ∅ � ξ ( ⊤ ) = �� R | N � = ε ξ ( X ) = � X � R | N 1 � ξ ( ¬ X ) = � R \ X � R | N � ξ ( • m ϕ ) = ξ ( ϕ ) ◮ m ξ ( ◦ m ϕ ) = ξ ( ϕ ) ⊲ m ∞ � � � ξ ( ϕ ∧ ψ ) = ξ ( ϕ ) � ξ ( ψ ) ξ ( � ϕ ) = ξ ( ϕ ) ⊲ m m = 0 Product Property: � Π � σ � σ ′ � = � Π � σ � ∩ � Π � σ ′ � ρ = � X ′ � Z ′ | N 3 � Example: σ = � X, Y � Z | N 2 � σ � ρ = � X ∩ X ′ , Y ∩ Z ′ � Z ∩ Z ′ | N 3 � Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 33 / 24

  63. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Signatures: the Transformation ξ ( · ) Modelling the Antecedent to Girdle the Future ξ ( ⊥ ) = �� ∅ | ∅ � ξ ( ⊤ ) = �� R | N � = ε ξ ( X ) = � X � R | N 1 � ξ ( ¬ X ) = � R \ X � R | N � ξ ( • m ϕ ) = ξ ( ϕ ) ◮ m ξ ( ◦ m ϕ ) = ξ ( ϕ ) ⊲ m ∞ � � � ξ ( ϕ ∧ ψ ) = ξ ( ϕ ) � ξ ( ψ ) ξ ( � ϕ ) = ξ ( ϕ ) ⊲ m m = 0 ∞ ∞ ∞ � � � ◦ m ϕ � ϕ ⇔ � Π � σ n � = � Π � σ n � m = 0 n = 0 n = 0 ∞ ∞ � � � � � � σ ◮ n and σ ⊲ n converge ∀ σ ∈ Σ n = 0 n = 0 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 33 / 24

  64. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Rewrite Proposition → Procedure automatic kind inference and generation rules Kind inference: expressiveness required & assumptions α : TA ⊢ X ( α ) : TA = ⊳ α : TA , X : reg-pres ⊢ X ( α ) : TA ⊢ X − 1 ( T ) : TA = ⊳ X : left-lin ⊢ X − 1 ( T ) : TA α : TA = ⊢ ♮ α : TA , ♮ α : + α : TA ⊢ ♮ α : TA + + Procedure Generation: from languages to automata Γ � X − 1 ( T ) ⇒ Γ, � X : left-lin � � X − 1 ( T ) Γ � [ ℓ ֌ ∆, α ] � ∆ ⊢ ∗ α : TA � X ( ℓ ) ⇒ Γ, ∆, � X : reg-pres � � X ( α ) Γ � [ ℓ ֌ ∆, α ] � ∆ ⊢ ∗ α : TA = � X ( ℓ ) ⇒ Γ, ∆, � X : reg-pres � � X ( ♮ α ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 34 / 24

  65. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Supported Fragment, In Practice Partially Supported Patterns From [Dwyer et al., 1999] Scope Pattern Global Before After Between Until Support Absence 41 5 12 18 9 48% Universality 110 1 5 2 1 96% Existence 12 1 4 8 1 0% Bound Exist. 0 0 0 1 0 0% Response 241 1 3 0 0 99% Precedence 25 0 1 0 0 96% Resp. Chain 8 0 0 0 0 0% Prec. Chain 1 0 0 0 0 0% Support 95% 0% 32% 0% 0% 83% Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 35 / 24

  66. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata [Comon et al., 2008] Introduced in the fifties; regular tree languages : model-checking: programs, protocols,. . . automated theorem-proving XML schema and (esp. variants) query languages . . . and so much more Doesn’t deal with comparisons and non-linearity : { f ( u, u ) | u ∈ T ( Σ ) } e.g. password verification { f ( u, v ) | u, v ∈ T ( Σ ) , u � = v } e.g. primary keys R ( ℓ ) , ℓ regular, R a TRS e.g. { g ( x ) → f ( x, x ) } ( T ( A )) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 36 / 24

  67. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata Bottom-Up, Non-Deterministic, Finite Tree Automaton A = � A , Q, F, ∆ � : finite ranked alphabet A Q finite set of states F final states, F ⊆ Q ∆ finite set of transitions Transition r ∈ ∆ : σ ( q 1 , . . . , q n ) → q σ ∈ A n q 1 , . . . , q n , q ∈ Q Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 37 / 24

  68. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata Bottom-Up, Non-Deterministic, Finite A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } , Q = { q 0 , q 1 } , F = { q 1 } , ∆ = � ⊤ → q 1 , � � ⊥ → q 0 , ¬ ( q b ) → q ¬ b � b, b ′ ∈ { 0, 1 } � � ∧ ( q b , q b ′ ) → q b ∧ b ′ , ∨ ( q b , q b ′ ) → q b ∨ b ′ t = ∧ ¬ ∨ ∧ ⊥ ¬ ⊥ ⊤ ⊥ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 38 / 24

  69. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata Bottom-Up, Non-Deterministic, Finite → ∗ → ∗ → ∗ ∧ ∧ ∧ ∆ ∆ ∆ ¬ ∨ ¬ ∨ ¬ ∨ ∧ ⊥ ¬ ∧ q 0 ¬ q 0 q 0 q 1 ⊥ ⊤ ⊥ q 0 q 1 q 0 → ∆ q 1 ∧ q 1 q 1 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 38 / 24

  70. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata Runs and Languages The reduction t → ∗ ∆ q 1 is captured by the run : q 1 decorated: ε ∧ q 1 q 1 q 1 1 ¬ q 1 2 ∨ q 1 q 0 q 0 q 1 11 ∧ q 0 21 ⊥ q 0 22 ¬ q 1 q 0 q 1 q 0 111 ⊥ q 0 112 ⊤ q 1 221 ⊥ q 0 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 39 / 24

  71. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] TAGE, TA = , Positive TAGED, A = � A , Q, F, ∆, ≅ � : � A , Q, F, ∆ � vanilla tree automaton ta ( A ) ≅ equality constraints , ≅ ⊆ Q 2 Constraint p ≅ q : run ρ of A on t : run of ta ( A ) on t satisfying ≅ : ∀ α, β ∈ P ( t ) ; ρ ( α ) ≅ ρ ( β ) ⇒ t | α = t | β accepting run : accepting for ta ( A ) Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 40 / 24

  72. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v = f f f f a a a a a a a Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 41 / 24

  73. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u, ρ u = f q f and v = f f ^ f ^ f a q q a q a q a q a q a a Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 41 / 24

  74. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] q ≅ ^ A = { a/ 0 , f/ 2 } , Q = { q, ^ q, q f } , F = { q f } , ^ q , and ∆ = { f (^ q, ^ q ) → q f , f ( q, q ) → q, f ( q, q ) → ^ q, a → q, a → ^ q } u = f and v, ρ v = f q f f f f ^ a ^ q q a a a a a q a q Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 41 / 24

  75. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } ⊎ X , Q = { q 0 , q 1 } ⊎ { v x | x ∈ X } and F = { q 1 } , new rules ⊤ → v x , ⊥ → v x , x ( q 0 , v x ) → q 1 , x ( v x , q 1 ) → q 0 for each x ∈ X , v x ≅ v x . ( x ∧ y ) ∨ ¬ x ≡ ∨ ∧ ¬ x y x ⊥ ⊤ ⊥ ⊤ ⊥ ⊤ Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 42 / 24

  76. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Tree Automata With Global Equality Constraints [Filiot et al., 2008] A = { ∧ , ∨ / 2 , ¬ / 1 , ⊤ , ⊥ / 0 } ⊎ X , Q = { q 0 , q 1 } ⊎ { v x | x ∈ X } and F = { q 1 } , new rules ⊤ → v x , ⊥ → v x , x ( q 0 , v x ) → q 1 , x ( v x , q 1 ) → q 0 for each x ∈ X , v x ≅ v x . ( x ∧ y ) ∨ ¬ x ≡ ∨ q 1 ∧ q 0 ¬ q 1 x q 0 y q 1 x q 0 ⊥ v x ⊤ q 1 ⊥ q 0 ⊤ v y ⊥ v x ⊤ q 1 Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 42 / 24

  77. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References TA versus RTA versus TA = Closure, Complexity and Decidability RTA ( p ≅ p ) TA = TA ∪ PTime PTime PTime ∩ PTime ExpTime ExpTime ∅ ∅ ¬ ExpTime NP -c (a) t ∈ L ( A ) ? NP -c PTime L ( A ) = ∅ ? linear-time linear-time ExpTime -c |L ( A ) | ∈ N ? ExpTime -c PTime PTime L ( A ) = T ( Σ ) ? ExpTime -c undecidable undecidable L ( A ) ⊆ L ( B ) ? ExpTime -c undecidable undecidable L ( � i A i ) = ∅ ? ExpTime -c ExpTime -c ExpTime -c (a) SAT solver approach: [Héam et al., 2010]. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 43 / 24

  78. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References TA versus RTA versus TA = Closure, Complexity and Decidability RTA ( p ≅ p ) TA = TA ∪ PTime PTime PTime ∩ PTime ExpTime ExpTime ∅ ∅ ¬ ExpTime NP -c (a) t ∈ L ( A ) ? NP -c PTime L ( A ) = ∅ ? linear-time linear-time ExpTime-c |L ( A ) | ∈ N ? PTime ExpTime-c PTime L ( A ) = T ( Σ ) ? ExpTime -c undecidable undecidable L ( A ) ⊆ L ( B ) ? ExpTime -c undecidable undecidable L ( � i A i ) = ∅ ? ExpTime -c ExpTime -c ExpTime -c (a) SAT solver approach: [Héam et al., 2010]. Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 43 / 24

  79. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References TA = versus TA = k Restriction on the kind of constraints: Rigid Automata (RTA) Same expressive power as TA = Less compact representations Linear emptiness / finiteness tests, vs. ExpTime -complete Applications: [Jacquemard et al., 2009, Vacher, 2010] What of the number of constraints? TA = k A = � Σ, Q, F, ∆, ≅ � : TA = A � Σ, Q, F, ∆, ≅ � ≅ such that Card ( ≅ ) � k Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 44 / 24

  80. Preliminaries LTL Checking Bounded TAGE Other Works Appendices References Expressive Power The Separation Languages L = ( ℓ k ) k ∈ N [Hugot, 2013] k � A i ⊎ { σ/ 3 , ⊥ / 0 } A i = { a i , b i / 0 , f i , g i / 2 } i = 1 ℓ 0 = { ⊥ } ∀ k � 1, ℓ k = { σ ( u, u, t k − 1 ) | u ∈ T ( A k ) , t k − 1 ∈ ℓ k − 1 } u i ∈ T ( A i ) ∀ i σ u k u k σ u k − 1 u k − 1 σ u 1 u 1 ⊥ , Tree (Not Quite) Regular Model-Checking Vincent HUGOT Ph.D. Defence 45 / 24

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