From Linear Temporal Logic to Rewrite Propositions Towards a New Model-Checking Approach P.-C. Héam, Vincent Hugot, O. Kouchnarenko {pheam,vhugot,okouchna}@femto-st.fr University of Franche-Comté DGA & INRIA/CASSIS & FEMTO-ST (DISC) June 24, 2012
Introduction: A Model-Checking Proposal Preliminaries & Problem Statement The Proposed Approach Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 2/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 3/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Model-Checking Process Proposal R. Courbis, P.-C. Héam, O. Kouchnarenko in CIAA 2009, [1] “The system R satisfies the property”. . . R , Π | = � ( X ⇒ • Y ) R is a Term Rewriting System (TRS) X, Y ⊆ R are sets of rules Π ⊆ T ( A ) is the initial language Example: X = “ask PIN code” = { ask } Y = “authenticate or cancel” = { auth 1 , auth 2 , can } 4/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Model-Checking Process Proposal R. Courbis, P.-C. Héam, O. Kouchnarenko in CIAA 2009, [1] “The system R satisfies the property”. . . R , Π | = � ( X ⇒ • Y ) . . . is equivalent to the Rewrite Proposition (RP). . . = ∅ ∧ X ( R ∗ ( Π )) ⊆ Y − 1 ( T ( A )) � X ( R ∗ ( Π )) � [ R \ Y ] . . . semi-decided by TAGED-based procedure IsEmpty ( OneStep ( R \ Y, Approx ( A , R )) , X ) and Subset ( OneStep ( X, Approx ( A , R )) , Backward ( Y )) , where Lang( A ) = Π , Lang( Approx ( A , R )) ⊇ R ∗ (Lang( A )) is given in [2, 3], and assuming Y is left-linear. 5/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 6/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Our Goals. . . . . . make it work! 1 Generalise translation into Rewrite Propositions (RP) From three specific formulæ [1] to a fragment of LTL 2 Generalise translation from RP to TAGED semi-algos At least for a fragment of possible RP Relatively easy. . . 3 Combine them into a full (semi-)verification chain The present work deals with the first step only 7/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 8/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal General Idea: Example (1 of 3) Preliminaries & Problem Statement What We Want: Generalisation The Proposed Approach Intuition: No Syntactic Translation Intuition: No Syntactic Translation R. Courbis, P.-C. Héam, O. Kouchnarenko in CIAA 2009, [1] 1 R , Π | = � ( X ⇒ • Y ) X ( R ∗ ( Π )) = ∅ ∧ X ( R ∗ ( Π )) ⊆ Y − 1 ( T ( A )) � � [ R \ Y ] 2 R , Π | = ¬ Y ∧ � ( • Y ⇒ X ) [ R \ X ] ( R ∗ ( Π )) � � Y ( Π ) = ∅ ∧ Y = ∅ 3 R , Π | = � ( X ⇒ ◦ � ¬ Y ) � �� R ∗ � X ( R ∗ ( Π )) Y = ∅ 9/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal Maximal Rewrite Words Preliminaries & Problem Statement Temporal Logic & Semantics The Proposed Approach Rewrite Propositions & Statement Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 10/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal Maximal Rewrite Words Preliminaries & Problem Statement Temporal Logic & Semantics The Proposed Approach Rewrite Propositions & Statement Maximal Rewrite Words 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 ∗ u n v n t n X 11/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal Maximal Rewrite Words Preliminaries & Problem Statement Temporal Logic & Semantics The Proposed Approach Rewrite Propositions & Statement Maximal Rewrite Words Coding the Behaviour of the System Finite or Infinite Words on R : � ⊳ ⊳ � � = N ∪ { + ∞ } = � 1, n � → R W N n ∈ N Maximal Rewrite Words of R , Originating in Π : R � Π � is the set of words w ∈ W such that ∃ u 0 ∈ Π : ∃ u 1 , . . . , u # w ∈ T ( A ) : ∀ k ∈ dom w, w ( k ) − − − → u k ∧ # w ∈ N ⇒ R ( { u # w } ) = ∅ u k − 1 Notations: ⊳ Length # w ∈ N of a word w : # w = Card ( dom w ) . 12/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal Maximal Rewrite Words Preliminaries & Problem Statement Temporal Logic & Semantics The Proposed Approach Rewrite Propositions & Statement Introduction: A Model-Checking Proposal 1 General Idea: Example (1 of 3) What We Want: Generalisation Intuition: No Syntactic Translation Preliminaries & Problem Statement 2 Maximal Rewrite Words Temporal Logic & Semantics Rewrite Propositions & Statement The Proposed Approach 3 Weak and Strong Semantics Signatures for Implication Translation Rules 13/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
Introduction: A Model-Checking Proposal Maximal Rewrite Words Preliminaries & Problem Statement Temporal Logic & Semantics The Proposed Approach Rewrite Propositions & Statement Formula ϕ ∈ LTL: ≈ Finite-LTL [4] ϕ := X | ¬ ϕ | ϕ ∧ ϕ | • m ϕ | ◦ m ϕ | ϕ U ϕ X ∈ ℘ ( R ) ⊤ | ⊥ | ϕ ∨ ϕ | ϕ ⇒ ϕ | ♦ ϕ | � ϕ m ∈ N . Semantics of LTL: ( w, i ) | = X iff i ∈ dom w and w ( i ) ∈ X ( w, i ) | iff ( w, i ) | = ¬ ϕ / ϕ = ( w, i ) | = ( ϕ ∧ ψ ) iff ( w, i ) | = ϕ and ( w, i ) | = ψ = • m ϕ ( w, i ) | iff i + m ∈ dom w and ( w, i + m ) | = ϕ = ◦ m ϕ ( w, i ) | iff i + m / ∈ dom w or ( w, i + m ) | = ϕ � ∃ j ∈ dom w : j � i ∧ ( w, j ) | = ψ ( w, i ) | = ϕ U ψ iff ∧ ∀ k ∈ � i, j − 1 � , ( w, k ) | = ϕ For any w ∈ W , i ∈ N 1 , m ∈ N and X ∈ ℘ ( R ) . 14/35 IJCAR’12 Vincent HUGOT LTL → Rewrite Propositions
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