Time Modalities over Many-valued Logics Time Modalities over Many-valued Logics Achille Frigeri Dipartimento di Matematica “Francesco Brioschi” Politecnico di Milano joint work with Nicholas Fiorentini, Liliana Pasquale, and Paola Spoletini September 19, 2012 Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 1 / 25
Time Modalities over Many-valued Logics Introduction Fuzzy logic Fuzzy Logic is a logical system which is an extension of multivalued logic and is intended to serve, as a logic of approximate reasoning Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 2 / 25
Time Modalities over Many-valued Logics Introduction Fuzzy Logic vs. Probability Fuzzy logic It deals with not measurable events The definition of the considered events is vague Ex.: Tomorrow will be cold Probability It deals with observable events whose occurrence is uncertain Ex.: Tomorrow the temperature will be 10 ◦ C at 12:00 Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 3 / 25
Time Modalities over Many-valued Logics Fuzzyfication From crisp to fuzzy connectives The semantics of existing fuzzy temporal operators is based on the idea of replacing classical connectives or propositions with their fuzzy counterparts. Fuzzy LTL (FLTL) [Lamine, Kabanza]: LTL in which Boolean operators are interpreted as in Zadeh interpretation Do not allow to represent additional temporal properties, such as almost always, soon. Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 4 / 25
Time Modalities over Many-valued Logics Fuzzyfication From fuzzy connectives to fuzzy modalities Introduction of proper fuzzy temporal operators to represent short/long time distance in which a specific property must be satisfied Lukasiewicz TL (FLTL) [Thiele, Kalenka]: LTL with short/medium/long term operators No specific fuzzy semantics for temporal modalities: depend on the interpretation given to a (sub-)argument, which is an untimed fuzzy formula. Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 5 / 25
Time Modalities over Many-valued Logics Fuzzyfication FTL: Fuzzy Time modalities in LTL We want to add temporal modalities such as “often”, “soon”, etc. This kind of modalities may be useful when we need to specify situations when a formula is slightly satisfied, since an event happens a little bit later than expected, when a property is always satisfied except for a small set of time instants, or a property is maintained for a time interval which is slightly smaller than the one. The underlying logic is a t -norm based logic. Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 6 / 25
Time Modalities over Many-valued Logics Fuzzyfication t -norm/conorm, implication & negation boundary value commutativity associativity monotonicity ⊖ 0 = 1 α ≤ β ⇒ ⊖ α ≥ ⊖ β negation - - ⊖ 1 = 0 α ⊗ 0 = 0 β ≥ γ ⇒ α ⊗ β ≥ α ⊗ γ t-norm yes yes α ⊗ 1 = α α ⊗ β ≤ α α ⊕ 0 = α β ≥ γ ⇒ α ⊕ β ≥ α ⊕ γ t-conorm yes yes α ⊕ 1 = 1 α ⊕ β ≥ α 1 � β = β α ≤ β ⇒ α � γ ≥ β � γ implication 0 � β = α � 1 = 1 no no β ≤ γ ⇒ α � β ≤ α � γ α � 0 = ⊖ α α � β ≥ max {⊖ α, β } Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 7 / 25
Time Modalities over Many-valued Logics Fuzzyfication Zadeh logic & t -norm based logics Zadeh G¨ odel-Dummett Łukasiewicz Product � 1 , α = 0 � 1 , α = 0 ⊖ α 1 − α 1 − α 0 , α > 0 0 , α > 0 α ⊗ β min { α, β } min { α, β } max { α + β − 1 , 0 } α · β α ⊕ β max { α, β } max { α, β } min { α + β, 1 } α + β − α · β � 1 , α ≤ β � 1 , α ≤ β α � β max { 1 − α, β } min { 1 − α + β, 1 } β, α > β β/α, α > β Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 8 / 25
Time Modalities over Many-valued Logics Fuzzyfication Syntax Syntax ϕ := p | ¬ ϕ | ϕ ∼ ϕ | O ϕ | ϕ T ϕ Unary modalities F , ( F t ) eventually G , ( G t ) , AG , ( AG t ) globally & almost globally (or often) X , S oon next & soon W t , L t within & lasts t instants Binary modalities U , ( U t ) , AU , ( AU t ) until & almost until Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 9 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Fuzzy satisfiability It is defined w.r.t. a linear structure ( S, s 0 , π, L ) An strictly decreasing avoiding function η : Z → [0 , 1] : η ( i ) = 1 , ∀ i ≤ 0 , and η ( n η ) = 0 for some n η ∈ N . ⊆ S ω × F × [0 , 1] , where Fuzzy satisfiability relation | = ( π | = ϕ ) = ν ∈ [0 , 1] means that the truth degree of ϕ along π is ν . Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 10 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Connectives t -norm substitutes ∧ t -conorm substitutes ∨ ( π i | = p ) = L ( s i )( p ) , ( π i | = ¬ ϕ ) = ⊖ ( π i | = ϕ ) , ( π i | = ϕ ∧ ψ ) = ( π i | = ϕ ) ⊗ ( π i | = ψ ) , ( π i | = ϕ ∨ ψ ) = ( π i | = ϕ ) ⊕ ( π i | = ψ ) , ( π i | = ϕ ⇒ ψ ) = ( π i | = ϕ ) � ( π i | = ψ ) , Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 11 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Next and Soon X has the same semantics of its corresponding LTL operator : ( π i | = X ϕ ) = ( π i +1 | = ϕ ) . S oon extends X by tolerating at most n η time instants of delay: i + n η ( π i | ( π j | � = S oon ϕ ) = = ϕ ) · η ( j − i − 1) . j = i +1 Remark: ( π i | = X ϕ ) ≤ ( π i | = S oon ϕ ) . Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 12 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Next and Soon: example n 0 1 2 3 4 η ( n ) 1 0.73 0.69 0.26 0 π 0 | = p 1 0.51 0.75 0.99 1 π 0 | = S oon p = 1 · 0 . 51 ⊕ 0 . 73 · 0 . 75 ⊕ 0 . 69 · 0 . 99 ⊕ 0 . 26 · 1 0 . 6831 (Z) = 1 (Ł) ∼ 0 . 928 ( Π ) Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 13 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Eventually F and F t maintain the semantics of LTL operator F : i + t ( π i | ( π j | � = F t ϕ ) = = ϕ ) , j = i ( π i | ( π j | t → + ∞ ( π i | � = F ϕ ) = = ϕ ) = lim = F t ϕ ) . j ≥ i Remark: F is well defined by monotonicity and if t ≤ t ′ : ( π i | = ϕ ) ≤ ( π i | = F t ϕ ) ≤ ( π i | = F t ′ ϕ ) ≤ ( π i | = F ϕ ) . Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 14 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Within W t is inherently bounded: i + t + n η − 1 ( π i | ( π j | � = W t ϕ ) = = ϕ ) · η ( j − t − i ) . j = i W t p means p is supposed to hold in at least one of the next t instant or, possibly, in the next t + n η . In the last case we apply a penalization for each instant after the t -th. Remark W 0 ϕ ≡ S oon ϕ W t ϕ ≡ F t ϕ ∨ X t +1 S oon ϕ ( π i | = W t ϕ ) ≥ ( π i | = F t ϕ ) lim t → + ∞ ( π i | = W t ϕ ) = ( π i | = F ϕ ) Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 15 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Always G and G t extend the semantics of G : ( π i | j = i ( π j | = G t ϕ ) = � i + t = ϕ ) , ( π i | j ≥ i ( π j | = ϕ ) = lim t → + ∞ ( π i | = G ϕ ) = � = G t ϕ ) . Remark: G is well defined by monotonicity and if t ≤ t ′ : ( π i | ≤ ( π i | = G t ϕ ) ≤ ( π i | = G ϕ ) = G t ′ ϕ ) ≤ ( π i | = G 1 ϕ ) = ( π i | = ϕ ∧ X ϕ ) ≤ ( π i | = G 0 ϕ ) = ( π i | = ϕ ) Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 16 / 25
Time Modalities over Many-valued Logics Fuzzyfication Semantics Almost always (Often) AG and AG t evaluate a property over a path π i , by avoiding at most n η evaluations of this property, and introducing a penalization for each avoided case. Let I t be the initial segment of N of length t + 1 and P k ( I t ) the set of subsets of I t of cardinality k : ( π i | h ∈ H ( π i + h | � = AG t ϕ ) = max j ∈ I t max H ∈P t − j ( I t ) = ϕ ) · η ( j ) ( π i | = AG ϕ ) = lim t → + ∞ ( π i | = AG t ϕ ) Remark: the sequence ( π i | = AG t ϕ ) t ∈ N is not monotonic. Still, the semantics of AG is well-defined. Achille Frigeri (Politecnico di Milano) Time Modalities over Many-valued Logics September 19, 2012 17 / 25
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