Fuzzy Modalities Fuzzifying Modal Algebra Jules Desharnais, Bernhard M¨ oller Universit´ e Laval Qu´ ebec, University of Augsburg RAMiCS 2014 J. Desharnais/B. M¨ oller RAMiCS 2014
Why? Why? basic idea: bring together the concepts of fuzzy semirings (say, fuzzy relations or matrices) modal semirings (domain/codomain and box/diamond) domain and codomain “measure” enabledness in transition systems this motivated investigating modal fuzzy semirings not a new kind of algebraic “meta-system” for various fuzzy logics rather apply and re-use an existing well-established algebraic system for the particular case of fuzzy systems J. Desharnais/B. M¨ oller – 2 – RAMiCS 2014
Basics of Fuzziness Basics of Fuzziness fuzzy relations: mappings from pairs of elements into the interval [0 , 1] values can be interpreted as transition probabilities or as capacities and in various other ways now take up the above idea of measuring enrich fuzzy semirings with domain/codomain operators apply the corresponding modal operators in the description and derivation of systems or algorithms in that realm seems to be a novel approach J. Desharnais/B. M¨ oller – 3 – RAMiCS 2014
Basics of Fuzziness adapt classical relational operators: ( R ⊔ S )( x, y ) = max ( R ( x, y ) , S ( x, y )) ( R ⊓ S )( x, y ) = min ( R ( x, y ) , S ( x, y )) ( R ; S )( x, y ) = sup min ( R ( x, z ) , S ( z, y )) z with these operations, fuzzy relations form an idempotent semiring J. Desharnais/B. M¨ oller – 4 – RAMiCS 2014
Basics of Fuzziness weak notion of complementation R ( x, y ) = df 1 − R ( x, y ) main problem in transferring domain to fuzzy semirings: the original axiomatisation of domain used a Boolean subring of the overall semiring as the target set of the domain operator generally not present in the fuzzy case however, using weak negation and the concepts of t-norm and t-conorm (see below for the details), a substitute for Boolean algebra can be defined J. Desharnais/B. M¨ oller – 5 – RAMiCS 2014
Overview Overview new axiomatisation of two variants of domain and codomain in the more general setting of idempotent left semirings avoiding complementation hence applicable to fuzzy relations for idempotent semirings such an axiomatisation has been given by Desharnais/Struth (2011) here: more general case of idempotent left semirings in which left distributivity of multiplication over addition and right annihilation of zero are not required J. Desharnais/B. M¨ oller – 6 – RAMiCS 2014
Overview also, we weaken the domain axioms by requiring only isotony rather than distributivity over addition surprisingly, still a wealth of properties known from the semiring case persists in the more general setting however, it is no longer true that complemented subidentities are domain elements not really disturbing, though, because the fuzzy world has its own view of complementation anyway we show how the domain operators extend to matrices some applications of these are sketched J. Desharnais/B. M¨ oller – 7 – RAMiCS 2014
Algebraic Basis Algebraic Basis an idempotent left (or lazy) semiring, briefly an IL-semiring, is a quintuple ( S, + , 0 , · , 1) with the following properties: ( S, + , 0) is a commutative monoid and + is idempotent ( S, · , 1) is a monoid · is right-distributive over + and left-strict: ( a + b ) · c = a · c + b · c 0 · a = 0 · is right-isotone w.r.t. the subsumption order a ≤ b ⇔ df a + b = b , which can be axiomatised as a · b ≤ a · ( b + c ) J. Desharnais/B. M¨ oller – 8 – RAMiCS 2014
Algebraic Basis an idempotent right semiring is defined symmetrically an I-semiring is a structure which is both an idempotent left and right semiring; hence its multiplication is both left and right distributive over its addition and its 0 is a left and right annihilator J. Desharnais/B. M¨ oller – 9 – RAMiCS 2014
Predomain Predomain general semiring elements abstractly model transition systems over states predicates on states can be modelled by sub-identities, i.e., by elements ≤ 1 multiplication p · a of a transition element a by a sub-identity p means restriction of a to the starting states characterised by p former approaches used tests, i.e., sub-identities with a complement relative to 1 , and hence involved the Boolean operation of negation as mentioned in the introduction, we want to avoid that and hence give the following new axiomatisation of a (pre)domain operation, whose range will replace the set of tests J. Desharnais/B. M¨ oller – 10 – RAMiCS 2014
Predomain a prepredomain IL-semiring is a structure ( S, � ) , where S is an IL-semiring and the prepredomain operator � : S → S satisfies � a ≤ 1 ✭ sub-id ✮ � 0 ≤ 0 ✭ strict ✮ a ≤ � a · a ✭ dom1 ✮ ✭ dom1 ✮ strengthens to the equality a = � by isotony of · , Ax. a · a this means that restriction to all starting states is no actual restriction J. Desharnais/B. M¨ oller – 11 – RAMiCS 2014
Predomain � is called a predomain operator if additionally a ≤ � � ( a + b ) ✭ isot ✮ � ( � b · a ) ≤ � b ✭ dom2 ✮ ✭ isot ✮ states that � is isotone Ax. Ax. ✭ dom2 ✮ means that after restriction the remaining starting states satisfy the restricting predicate J. Desharnais/B. M¨ oller – 12 – RAMiCS 2014
Predomain a predomain operator � is called a domain operator if additionally it satisfies the locality axiom � b ) ≤ � ( a · � ( a · b ) ✭ loc ✮ Ax. ✭ loc ✮ again strengthens to an equality it means that the domain of a · b is not determined by the inner structure or the final states of b ; information about � b in interaction with a suffices Mace4 shows that these axioms are independent by � S we denote the image of S under � , by p, q, . . . elements of � S J. Desharnais/B. M¨ oller – 13 – RAMiCS 2014
(Pre)Domain Calculus (Pre)Domain Calculus as in the classical case, Axs. ✭ dom1 ✮ and ✭ dom2 ✮ combine into � a ≤ p ⇔ a ≤ p · a ✭ llp ✮ this characterises � a as the least left-preserver of a in � S and often allows a calculation of � a using indirect equality unlike in the classical case, predomain is not characterised uniquely by the axioms however, if two domain operators have the same range, by the above remark they coincide J. Desharnais/B. M¨ oller – 14 – RAMiCS 2014
(Pre)Domain Calculus Theorem a selection of predomain laws in an IL-semiring: � p = p (stability) predomain is fully strict, i.e., � a = 0 ⇔ a = 0 S forms an upper semilattice with p ⊔ q = � � ( p + q ) predomain preserves arbitrary existing suprema � ( a + b ) = � a ⊔ � b we have the absorption laws p · ( p ⊔ q ) = p and p ⊔ ( p · q ) = p hence ( � S, · , ⊔ ) is a lattice ( a · b ) ≤ � � ( a · � b ) � ( a · b ) ≤ � a predomain satisfies the partial import/export law � ( p · a ) ≤ p · � a p · q = � ( p · q ) ; hence � S is closed under · J. Desharnais/B. M¨ oller – 15 – RAMiCS 2014
(Pre)Domain Calculus Lemma additional properties of a domain operator in an IL-semiring: ( a · b ) = � ✭ loc ✮ strengthens to the equality � ( a · � b ) domain satisfies the full import/export law � ( p · a ) = p · � a in an I-semiring, the lattice ( � S, · , ⊔ ) is distributive J. Desharnais/B. M¨ oller – 16 – RAMiCS 2014
Fuzzy Domain Operators Fuzzy Domain Operators we now apply this to the fuzzy setting first we generalise the notion of t-norms and pseudo-complementation to general IL-semirings, in particular to semirings that do not just consist of the interval [0 , 1] (as, say, a subset of the real numbers) and where that interval is not necessarily linearly ordered J. Desharnais/B. M¨ oller – 17 – RAMiCS 2014
Fuzzy Domain Operators consider an IL-semiring S with the interval [0 , 1] = df { x | x ≤ 1 } a t-norm is a binary operator � : [0 , 1] × [0 , 1] → [0 , 1] that is isotone in both arguments, associative and commutative and has 1 as unit the definition implies p � q ≤ p, q in a predomain IL-semiring the operator · restricted to [0 , 1] is a t-norm a weak complement operator in an IL-semiring is a function ¬ : [0 , 1] → [0 , 1] that is an order-antiisomorphism, i.e., is bijective and satisfies p ≤ q ⇔ ¬ q ≤ ¬ p , such that additionally ¬¬ p = p this implies ¬ 0 = 1 and ¬ 1 = 0 J. Desharnais/B. M¨ oller – 18 – RAMiCS 2014
Fuzzy Domain Operators moreover, if the IL-semiring has a t-norm � the associated t-conorm � is defined as the analogue of the De Morgan dual of the t-norm: p � q = df ¬ ( ¬ p � ¬ q ) Lemma assume a predomain IL-semiring with weak negation p ≤ p � q if p � q is the infimum of p and q then p ⊔ q = p � q J. Desharnais/B. M¨ oller – 19 – RAMiCS 2014
Fuzzy Domain Operators next, we deal with a special t-norm and its associated t-conorm Lemma consider the sub-interval I = df [0 , 1] of the real numbers with x � y = df min ( x, y ) and x � y = df max ( x, y ) then ( I, � , 0 , � , 1) is an I-semiring and the identity function is a domain operator on I since this domain operator is quite boring, in a later section we will turn to matrices over I , where the behaviour becomes non-trivial J. Desharnais/B. M¨ oller – 20 – RAMiCS 2014
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