Introduction (Un)decidability on modal MTL logics Undecidability of some modal MTL logics (formerly product logics) Amanda Vidal Institute of Computer Science, Czech Academy of Sciences September 6, 2016 1 / 19
Introduction (Un)decidability on modal MTL logics Contents 1. Introduction 2. (Un)decidability on modal MTL logics Reducing to PCP The Global modal logic case The Local modal logic case 2 / 19
Introduction (Un)decidability on modal MTL logics Contents 1. Introduction 2. (Un)decidability on modal MTL logics Reducing to PCP The Global modal logic case The Local modal logic case 3 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability 4 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability ◮ many-valued cases: ? no (usual) FMP or (known) R.E axiomatization...for instance 4 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability ◮ many-valued cases: ? no (usual) FMP or (known) R.E axiomatization...for instance ◮ Validity in the expansion of Gödel logic with modal operators does not enjoy FMP with the usual Kripke semantics, but it is decidable [Metcalfe et.al.] 4 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability ◮ many-valued cases: ? no (usual) FMP or (known) R.E axiomatization...for instance ◮ Validity in the expansion of Gödel logic with modal operators does not enjoy FMP with the usual Kripke semantics, but it is decidable [Metcalfe et.al.] ◮ Similar concerning validity and >0-sat in FDL (multi-modal variation) over Product logic [Cerami et. al] 4 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability ◮ many-valued cases: ? no (usual) FMP or (known) R.E axiomatization...for instance ◮ Validity in the expansion of Gödel logic with modal operators does not enjoy FMP with the usual Kripke semantics, but it is decidable [Metcalfe et.al.] ◮ Similar concerning validity and >0-sat in FDL (multi-modal variation) over Product logic [Cerami et. al] ◮ The previous case with involutive negation or allowing GCI (some globally valid formulas) is undecidable [Baader et.al] 4 / 19
Introduction (Un)decidability on modal MTL logics Introduction ◮ Many normal (classical) modal logics: finite model property + finite axiomatizability ⇒ decidability ◮ many-valued cases: ? no (usual) FMP or (known) R.E axiomatization...for instance ◮ Validity in the expansion of Gödel logic with modal operators does not enjoy FMP with the usual Kripke semantics, but it is decidable [Metcalfe et.al.] ◮ Similar concerning validity and >0-sat in FDL (multi-modal variation) over Product logic [Cerami et. al] ◮ The previous case with involutive negation or allowing GCI (some globally valid formulas) is undecidable [Baader et.al] ◮ ... 4 / 19
Introduction (Un)decidability on modal MTL logics MTL Kripke-models A = � A , ⊙ , ⇒ , min , 1 , 0 , � a complete MTL algebra (conm. integral bounded prelinear residuated lattices = algebras in the variety generated by all left-continuous t-noms). Language: & , ∧ , → , 0 plus two unary (modal) symbols ( ✷ , ✸ ) 5 / 19
Introduction (Un)decidability on modal MTL logics MTL Kripke-models A = � A , ⊙ , ⇒ , min , 1 , 0 , � a complete MTL algebra (conm. integral bounded prelinear residuated lattices = algebras in the variety generated by all left-continuous t-noms). Language: & , ∧ , → , 0 plus two unary (modal) symbols ( ✷ , ✸ ) Definition A (crisp) A Kripke model M is a tripla � W , R , e � where: ◮ R ⊆ W × W ( Rus stands for � u , s � ∈ R ) ◮ e : W × Var → A uniquelly extended by: ◮ e ( u , ϕ & ψ ) = e ( u , ϕ ) ⊙ e ( u , ψ ) ; e ( u , ϕ → ψ ) = e ( u , ϕ ) ⇒ e ( u , ψ ) ; e ( u , ϕ ∧ ψ ) = min { e ( u , ϕ ) , e ( u , ψ ) } ; e ( e , 0 ) = 0 ◮ e ( u , ✷ ϕ ) = inf { e ( s , ϕ ) : Rus } ◮ e ( u , ✸ ϕ ) = sup { e ( s , ϕ ) : Rus } 5 / 19
Introduction (Un)decidability on modal MTL logics Modal MTL logics C a class of complete MTL-algebras. ◮ (Global deduction) : Γ � C ϕ iff [ ∀ u ∈ W e ( u , [ Γ ]) ⊆ { 1 } ] implies [ ∀ u ∈ W e ( u , ϕ ) = 1 ] for all A Kripke models M with A ∈ C . Γ � f C ϕ for denoting the same relation over finite (i.e., finite W) Kripke models. 6 / 19
Introduction (Un)decidability on modal MTL logics Modal MTL logics C a class of complete MTL-algebras. ◮ (Global deduction) : Γ � C ϕ iff [ ∀ u ∈ W e ( u , [ Γ ]) ⊆ { 1 } ] implies [ ∀ u ∈ W e ( u , ϕ ) = 1 ] for all A Kripke models M with A ∈ C . Γ � f C ϕ for denoting the same relation over finite (i.e., finite W) Kripke models. ◮ (Local deduction) : Γ ⊢ 4 C ϕ iff ∀ u ∈ W [ e ( u , [ Γ ]) ⊆ { 1 } implies e ( u , ϕ ) = 1 ] for all transitive A Kripke models M with A ∈ C . Γ ⊢ f 4 C ϕ for denoting the same relation over finite transitive Kripke models 6 / 19
Introduction (Un)decidability on modal MTL logics Contents 1. Introduction 2. (Un)decidability on modal MTL logics Reducing to PCP The Global modal logic case The Local modal logic case 7 / 19
Introduction (Un)decidability on modal MTL logics Undecidability results For n < ω , a MTL-algebra is n -contractive iff it validates the equation x n → x n + 1 = 1 A class of MTL-algebras is non contractive iff, for all n , it contains some non n -contractive algebra. Theorem Let C be a non contractive class of complete MTL-algebras. For arbitrary Γ ∪ { ϕ } the following are undecidable: 8 / 19
Introduction (Un)decidability on modal MTL logics Undecidability results For n < ω , a MTL-algebra is n -contractive iff it validates the equation x n → x n + 1 = 1 A class of MTL-algebras is non contractive iff, for all n , it contains some non n -contractive algebra. Theorem Let C be a non contractive class of complete MTL-algebras. For arbitrary Γ ∪ { ϕ } the following are undecidable: 1. Γ � C ϕ 2. Γ � f C ϕ (global deduction) 8 / 19
Introduction (Un)decidability on modal MTL logics Undecidability results For n < ω , a MTL-algebra is n -contractive iff it validates the equation x n → x n + 1 = 1 A class of MTL-algebras is non contractive iff, for all n , it contains some non n -contractive algebra. Theorem Let C be a non contractive class of complete MTL-algebras. For arbitrary Γ ∪ { ϕ } the following are undecidable: 1. Γ � C ϕ 2. Γ � f C ϕ (global deduction) 3. Γ ⊢ 4 C ϕ 4. Γ ⊢ f 4 C ϕ (local deduction in transitive frames) 8 / 19
Introduction (Un)decidability on modal MTL logics Post Correspondence Problem An instance of the PCP is a list of pairs � v 1 , w 1 � . . . � v n , w n � where v i , w i are numbers in base s ≥ 2. 9 / 19
Introduction (Un)decidability on modal MTL logics Post Correspondence Problem An instance of the PCP is a list of pairs � v 1 , w 1 � . . . � v n , w n � where v i , w i are numbers in base s ≥ 2. It is undecidable whether there exist i 1 , . . . , i k such that v i 1 · · · v i k = w i 1 · · · w i k 9 / 19
Introduction (Un)decidability on modal MTL logics Post Correspondence Problem An instance of the PCP is a list of pairs � v 1 , w 1 � . . . � v n , w n � where v i , w i are numbers in base s ≥ 2. It is undecidable whether there exist i 1 , . . . , i k such that v i 1 · · · v i k = w i 1 · · · w i k ⇒ ab = a · s � b � + b , where � b � is ◮ a , b numbers in base s = the length of b (in base s ). 9 / 19
Introduction (Un)decidability on modal MTL logics Post Correspondence Problem An instance of the PCP is a list of pairs � v 1 , w 1 � . . . � v n , w n � where v i , w i are numbers in base s ≥ 2. It is undecidable whether there exist i 1 , . . . , i k such that v i 1 · · · v i k = w i 1 · · · w i k ⇒ ab = a · s � b � + b , where � b � is ◮ a , b numbers in base s = the length of b (in base s ). ◮ we can exploit the conjunction operation to express concatenation (using powers over some y ”non-contractive”) 9 / 19
Introduction (Un)decidability on modal MTL logics The global modal logic case Given a PCP instance P there is a finite set of formulas Γ g ( P ) ∪ { ϕ g } such that P is SAT ⇐ ⇒ Γ g ( P ) � � C ϕ g ⇒ Γ g ( P ) � f Moreover Γ g ( P ) � C ϕ g ⇐ C ϕ g . 10 / 19
Introduction (Un)decidability on modal MTL logics The global modal logic case Given a PCP instance P there is a finite set of formulas Γ g ( P ) ∪ { ϕ g } such that P is SAT ⇐ ⇒ Γ g ( P ) � � C ϕ g ⇒ Γ g ( P ) � f Moreover Γ g ( P ) � C ϕ g ⇐ C ϕ g . ◮ Proving = ⇒ will not be hard (constructing a model using the solution of P ). 10 / 19
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