Contents 1. Introduction 2. (Un)decidability on modal MTL logics - PowerPoint PPT Presentation

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