Algorithmic Correspondence for Hybrid Logic with Binder
aa r X i v : . [ c s . L O ] F e b Algorithmic Correspondence for Hybrid Logicwith Binder
Zhiguang Zhao
School of Mathematics and Statistics, Taishan University, Tai’an, 271000, [email protected]
Abstract
In the present paper, we develop the algorithmic correspondence the-ory for hybrid logic with binder H (@ , ↓ ). We define the class of Sahlqvistinequalities for H (@ , ↓ ), each inequality of which is shown to have a first-order frame correspondent effectively computable by an algorithm ALBA ↓ . Keywords : correspondence theory, hybrid logic with binder, ALBAalgorithm
Hybrid Logic
Hybrid logics [2] refer to a number of extensions of modal logicwhere it is possible to refer to states by the so-called nominals which are trueat exactly one world. In addition, there are different connectives that can beadded to the hybrid language to further extend the expressive power. Two suchexamples are the satisfaction operator @ i which allows one to jump to the worlddenoted by the nominal i , and the binder ↓ x which binds the current worldand can refer to the world later in the formula. Correspondence Theory
Correspondence theory concerns the relation be-tween modal formulas and first-order formulas. We say that a modal formulaand a first-order formula correspond to each other if they are valid on the sameclass of Kripke frames. Early results concerning correspondence theory areSahlqvist’s [17] and van Benthem’s [20], who gave a syntactic characterizationof certain modal formulas (later called
Sahlqvist formulas ) which have first-ordercorrespondents and they are canonical, which implies that any normal modallogic axiomatized with Sahlqvist formulas is strongly complete with respect toits Kripke frames. The Sahlqvist-van Benthem algorithm [17, 20] was given totransform a Sahlqvist formula into its first-order correspondent.1 orrespondence Theory for Hybrid Logic
In the literature, there aremany existing works on the correspondence theory for hybrid logic [1, 3, 6, 11,12, 13, 14, 15, 18, 19]. In particular, ten Cate et al. [19] showed that any hybridlogic obtained by adding modal Sahlqvist formulas to the basic hybrid logic His strongly complete. Gargov and Goranko showed that any extension of H withpure axioms (formulas containing no propositional variables but only possiblynominals) is strongly complete. In [19] it was shown that these two resultscannot be combined in general, since there is a modal Sahlqvist formula anda pure formula which together give a Kripke-incomplete logic when added toH. Conradie and Robinson [11] investigated to what extent these two resultscan be combined. In the end of [11], it was mentioned that a further directionwould be to extend results concerning extending correspondence theory to moreexpressive hybrid languages e.g. hybrid logic with binder, which is the focus ofthe present paper.
Unified correspondence
The present paper belongs to the theory of unifiedcorrespondence [8, 4]. One major part of this theory is the algorithm
ALBA (Ack-ermann Lemma Based Algorithm), which computes the first-order correspon-dents of input formulas/inequalities and is guaranteed to succeed on Sahlqvistinequalities.
Structure of the paper
In the present paper, we will use the algorithmicmethodology to provide a correspondence theory treatment of hybrid logic withbinder. Section 2 presents preliminaries on hybrid logic with binder, includingsyntax and semantics. Section 3 provides preliminaries on algorithmic corre-spondence theory. Section 4 defines Sahlqvist inequalities. Section 5 gives theAckermann Lemma Based Algorithm (
ALBA ↓ ) for hybrid logic with binder. Sec-tion 6 gives the soundness proof of the algorithm. Section 7 shows that ALBA ↓ succeeds on Sahlqvist inequalities. Section 8 gives conclusions. In the present section we collect the preliminaries on hybrid logic with binder.For more details, see [2, Chapter 14].
Definition 2.1.
Given countably infinite sets
Prop of propositional variables,
Nom of nominals,
Svar of state variables, which are pairwise disjoint, the hybridlanguage H (@ , ↓ ) is defined as follows: ϕ ::= ⊥ | ⊤ | p | i | x | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | ✷ ϕ | ✸ ϕ | @ i ϕ | @ x ϕ |↓ x.ϕ, where p ∈ Prop , i ∈ Nom , x ∈ Svar . We also use
Prop ( α ) to denote thepropositional variables occuring in α . We use the notation ϕ ( ~p ) to indicate that2he propositional variables occur in ϕ are all in ~p . We call a formula pure if itdoes not contain propositional variables. In the present article we will consideronly the hybrid language with one unary modality.Throughout the article, we will also make substantial use of the followingexpressions (see [21] for more details): Definition 2.2. • An inequality is of the form ϕ ≤ ψ , where ϕ and ψ are formulas. • A quasi-inequality is of the form ϕ ≤ ψ & . . . & ϕ n ≤ ψ n ⇒ ϕ ≤ ψ . • A Mega-inequality is defined inductively as follows:
Mega ::=
Ineq | Mega & Mega | ∀ x ( Mega )where
Ineq is an inequality, & is the meta-conjunction and ∀ x is a uni-versal state quantifier. • A universally quantified inequality is defined as ∀ x . . . ∀ x n ( ϕ ≤ ψ ). • A quasi-universally quantified inequality is defined as UQIneq & . . . & UQIneq n ⇒ UQIneq where
UQIneq , UQIneq i are universally quantified inequalities.We will find it easy to work with inequalities ϕ ≤ ψ in place of implicativeformulas ϕ → ψ in Section 4. Definition 2.3. A Kripke frame is a pair F = ( W, R ) where W is the domainof F , the accessibility relation R is a binary relation on W . A pointed Kripkeframe is a pair ( F , w ) where w ∈ W . A Kripke model is a pair M = ( F , V )where V : Prop ∪ Nom → P ( W ) is a valuation on F such that for all nominals i ∈ Nom , V ( i ) is a singleton subset of W .An assignment g on M = ( W, R, V ) is a map g : Svar → W . Given anassignment g , x ∈ Svar , w ∈ W , we can define g xw , the x -variant of g such that g xw is the same as g except that g xw ( x ) = w .Now the satisfaction relation can be defined as follows: given any Kripkemodel M = ( W, R, V ), any assignment g on M , any w ∈ W ,3 , g, w (cid:13) ⊥ : never; M , g, w (cid:13) ⊤ : always; M , g, w (cid:13) p iff w ∈ V ( p ); M , g, w (cid:13) i iff { w } = V ( i ); M , g, w (cid:13) x iff g ( x ) = w ; M , g, w (cid:13) ¬ ϕ iff M , g, w ϕ ; M , g, w (cid:13) ϕ ∧ ψ iff M , g, w (cid:13) ϕ and M , g, w (cid:13) ψ ; M , g, w (cid:13) ϕ ∨ ψ iff M , g, w (cid:13) ϕ or M , g, w (cid:13) ψ ; M , g, w (cid:13) ϕ → ψ iff M , g, w ϕ or M , g, w (cid:13) ψ ; M , g, w (cid:13) ✷ ϕ iff ∀ v ( Rwv ⇒ M , g, v (cid:13) ϕ ); M , g, w (cid:13) ✸ ϕ iff ∃ v ( Rwv and M , g, v (cid:13) ϕ ); M , g, w (cid:13) @ i ϕ iff M , g, V ( i ) (cid:13) ϕ ; M , g, w (cid:13) @ x ϕ iff M , g, g ( x ) (cid:13) ϕ ; M , g, w (cid:13) ↓ x.ϕ iff M , g xw , w (cid:13) ϕ .For any formula ϕ , we let J ϕ K M ,g = { w ∈ W | M , g, w (cid:13) ϕ } denote the truth set of ϕ in ( M , g ). The formula ϕ is globally true on ( M , g ) (notation: M , g (cid:13) ϕ )if M , g, w (cid:13) ϕ for every w ∈ W . We say that ϕ is valid on a Kripke frame F (notation: F (cid:13) ϕ ) if ϕ is globally true on ( F , V, g ) for every valuation V andevery assignment g .For the semantics of inequalities, quasi-inequalities, mega-inequalities, uni-versally quantified inequalities, quasi-universally quantified inequalities, theyare given as follows: Definition 2.4. • An inequality is interpreted as follows:(
W, R, V ) , g (cid:13) ϕ ≤ ψ iff(for all w ∈ W, if ( W, R, V ) , g, w (cid:13) ϕ, then ( W, R, V ) , g, w (cid:13) ψ ) . • A quasi-inequality is interpreted as follows:(
W, R, V ) , g (cid:13) ϕ ≤ ψ & . . . & ϕ n ≤ ψ n ⇒ ϕ ≤ ψ iff( W, R, V ) , g (cid:13) ϕ ≤ ψ holds whenever ( W, R, V ) , g (cid:13) ϕ i ≤ ψ i for all 1 ≤ i ≤ n. • A Mega-inequality is interpreted as follows: – ( W, R, V ) , g (cid:13) Ineq iff the inequality holds as defined in the definitionabove; – ( W, R, V ) , g (cid:13) Mega & Mega iff ( W, R, V ) , g (cid:13) Mega and ( W, R, V ) , g (cid:13) Mega ; – ( W, R, V ) , g (cid:13) ∀ x ( Mega ) iff (
W, R, V ) , g xw (cid:13) Mega for all w ∈ W ;4 A universally quantified inequality is interpreted as follows:(
W, R, V ) , g (cid:13) ∀ x . . . ∀ x n ( ϕ ≤ ψ ) iff for all w , . . . , w n ∈ W , ( W, R, V ) , g x ,...,x n w ,...,w n (cid:13) ϕ ≤ ψ ; • A quasi-universally quantified inequality is interpreted as follows:(
W, R, V ) , g (cid:13) UQIneq & . . . & UQIneq n ⇒ UQIneq iff(
W, R, V ) , g (cid:13) UQIneq holds whenever (
W, R, V ) , g (cid:13) UQIneq i for all 1 ≤ i ≤ n. The definitions of validity are similar to formulas. It is easy to see that(
W, R, V ) , g (cid:13) ϕ ≤ ψ iff ( W, R, V ) , g (cid:13) ϕ → ψ . In this section, we give necessary preliminaries on the correspondence algorithm
ALBA ↓ for hybrid logic with binder in the style of [5, 7, 21]. The algorithm ALBA ↓ transforms the input hybrid formula ϕ ( ~p ) ≤ ψ ( ~p ) into an equivalentset of pure quasi-(universally quantified) inequalities which does not containoccurrences of propositional variables, and therefore can be translated into thefirst-order correspondence language via the standard translation of the expandedlanguage of hybrid logic with binder (see page 6).The ingredients for the algorithmic correspondence proof to go through canbe listed as follows: • An expanded hybrid modal language as the syntax of the algorithm, aswell as its interpretations in the relational semantics; • An algorithm
ALBA ↓ which transforms a given hybrid inequality ϕ ( ~p ) ≤ ψ ( ~p ) into equivalent pure quasi-(universally quantified) inequalities Pure ( ϕ ( ~p ) ≤ ψ ( ~p )); • A soundness proof of the algorithm; • A syntactically identified class of inequalities on which the algorithm issuccessful; • A first-order correspondence language and first-order translation whichtransforms pure quasi-(universally quantified) inequalities into their equiv-alent first-order correspondents.In the remainder of the paper, we will define an expanded hybrid modal lan-guage which the algorithm will manipulate (Section 3.1), define the first-ordercorrespondence language of the expanded hybrid modal language and the stan-dard translation (Section 3.2). We give the definition of Sahlqvist inequalities(Section 4), define a modified version of the algorithm
ALBA ↓ (Section 5), andshow its soundness (Section 6) and success on Sahlqvist inequalities (Section 7).5 .1 The expanded hybrid modal language In the present subsection, we give the definition of the expanded hybrid modallanguage , which will be used in the execution of the algorithm: ϕ ::= ⊥ | ⊤ | p | i | x | ¬ ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | ✷ ϕ | ✸ ϕ | @ i ϕ | @ x ϕ |↓ x.ϕ | (cid:4) ϕ | ♦ ϕ | A ϕ | E ϕ | ∀ xϕ | ∃ xϕ For (cid:4) and ♦ , they are interpreted as the box and diamond modality onthe inverse relation R − . A and E are global box and diamond modalitiesrespectively, ∀ xϕ indicates that for all x -variant g xv of g , ( W, R, V ) , g xv , w (cid:13) ϕ ,and ∃ xϕ is the corresponding existential statement.For the semantics of the expanded hybrid modal language, the additionalsemantic clauses can be given as follows: M , g, w (cid:13) (cid:4) ϕ iff for all v s.t. ( v, w ) ∈ R , M , g, v (cid:13) ϕ M , g, w (cid:13) ♦ ϕ iff there exists a v s.t. ( v, w ) ∈ R and M , g, v (cid:13) ϕ M , g, w (cid:13) A ϕ iff for all v ∈ W , M , g, v (cid:13) ϕ M , g, w (cid:13) E ϕ iff there exists a v ∈ W s.t. M , g, v (cid:13) ϕ M , g, w (cid:13) ∀ xϕ iff for all v ∈ W , M , g xv , w (cid:13) ϕ M , g, w (cid:13) ∃ xϕ iff there exists a v ∈ W s.t. M , g xv , w (cid:13) ϕ . In the first-order correspondence language, we have a binary relation symbol R corresponding to the binary relation, a set of constant symbols i correspondingto each nominal i , a set of unary predicate symbols P corresponding to eachpropositional variable p . The state variables x correspond to individual variables x in the first-order language. Definition 3.1.
The standard translation of the expanded hybrid modal lan-guage is defined as follows: • ST x ( ⊥ ) := x = x ; • ST x ( ⊤ ) := x = x ; • ST x ( p ) := P x ; • ST x ( i ) := x = i ; • ST x ( y ) := x = y ; • ST x ( ¬ ϕ ) := ¬ ST x ( ϕ ); • ST x ( ϕ ∧ ψ ) := ST x ( ϕ ) ∧ ST x ( ψ ); • ST x ( ϕ ∨ ψ ) := ST x ( ϕ ) ∨ ST x ( ψ ); 6 ST x ( ϕ → ψ ) := ST x ( ϕ ) → ST x ( ψ ); • ST x ( ✷ ϕ ) := ∀ y ( Rxy → ST y ( ϕ )) ( y does not occur in ϕ ); • ST x ( ✸ ϕ ) := ∃ y ( Rxy ∧ ST y ( ϕ )) ( y does not occur in ϕ ); • ST x (@ i ϕ ) := ∃ y ( y = i ∧ ST y ( ϕ )) ( y does not occur in ϕ ); • ST x (@ z ϕ ) := ∃ y ( y = z ∧ ST y ( ϕ )) ( y does not occur in ϕ ); • ST x ( ↓ y.ϕ ) := ∃ y ( y = x ∧ ST x ( ϕ )); • ST x ( (cid:4) ϕ ) := ∀ y ( Ryx → ST y ( ϕ )) ( y does not occur in ϕ ); • ST x ( ♦ ϕ ) := ∃ y ( Ryx ∧ ST y ( ϕ )) ( y does not occur in ϕ ); • ST x ( A ϕ ) := ∀ yST y ( ϕ ) ( y does not occur in ϕ ); • ST x ( E ϕ ) := ∃ yST y ( ϕ ) ( y does not occur in ϕ ); • ST x ( ∀ yϕ ) := ∀ yST x ( ϕ ); • ST x ( ∃ yϕ ) := ∃ yST x ( ϕ ).It is easy to see that this translation is correct: Proposition 3.2.
For any Kripke model M , any assignment g on M , any w ∈ W and any expanded hybrid modal formula ϕ , M , g, w (cid:13) ϕ iff M , g xw (cid:15) ST x ( ϕ ) , where x is a fresh variable not occurring in ϕ .For inequalities, quasi-inequalities, mega-inequalities, universally quantifiedinequalities and quasi-universally quantified inequalities, the standard transla-tion is given in a global way: Definition 3.3. • ST ( ϕ ≤ ψ ) := ∀ x ( ST x ( ϕ ) → ST x ( ψ )) ( x does not occurin ϕ and ψ ); • ST ( ϕ ≤ ψ & . . . & ϕ n ≤ ψ n ⇒ ϕ ≤ ψ ) := ST ( ϕ ≤ ψ ) ∧ . . . ∧ ST ( ϕ n ≤ ψ n ) → ST ( ϕ ≤ ψ ); • ST ( Mega & Mega ) = ST ( Mega ) ∧ ST ( Mega ); • ST ( ∀ x ( Mega )) := ∀ xST ( Mega ); • ST ( ∀ x . . . ∀ x n Ineq ) := ∀ x . . . ∀ x n ST ( Ineq ); • ST ( UQIneq & . . . & UQIneq n ⇒ UQIneq ) := ST ( UQIneq ) ∧ . . . ∧ ST ( UQIneq n ) → ST ( UQIneq ). 7 roposition 3.4.
For any Kripke model M , any assignment g on M , and in-equality Ineq , quasi-inequality
Quasi , mega-inequality
Mega , universally quanti-fied inequality
UQIneq , quasi-universally quantified inequality
QUQIneq , M , g (cid:13) Ineq iff M , g (cid:15) ST ( Ineq ); M , g (cid:13) Quasi iff M , g (cid:15) ST ( Quasi ); M , g (cid:13) Mega iff M , g (cid:15) ST ( Mega ); M , g (cid:13) UQIneq iff M , g (cid:15) ST ( UQIneq ); M , g (cid:13) QUQIneq iff M , g (cid:15) ST ( QUQIneq ) . In the present section, since we will use the algorithm
ALBA ↓ which is based onthe classsification of nodes in the signed generation trees of hybrid modal for-mulas, we will use the unified correspondence style definition (cf. [9, 10, 16, 21])to define Sahlqvist inequalities. We will collect all the necessary preliminarieson Sahlqvist inequalities. Definition 4.1 (Order-type of propositional variables) . (cf. [7, page 346]) Foran n -tuple ( p , . . . , p n ) of propositional variables, an order-type ε of ( p , . . . , p n )is an element in { , ∂ } n . We say that p i has order-type 1 if ε i = 1, and denote ε ( p i ) = 1 or ε ( i ) = 1; we say that p i has order-type ∂ if ε i = ∂ , and denote ε ( p i ) = ∂ or ε ( i ) = ∂ . Definition 4.2 (Signed generation tree) . (cf. [10, Definition 4]) The positive (resp. negative ) generation tree of any given formula ϕ is defined by first labellingthe root of the generation tree of ϕ with + (resp. − ) and then labelling thechildren nodes as follows: • Assign the same sign to the children nodes of any node labelled with ∨ , ∧ , ✷ , ✸ , ↓ x , (cid:4) , ♦ , A , E , ∀ x, ∃ x ; • Assign the opposite sign to the child node of any node labelled with ¬ ; • Assign the opposite sign to the first child node and the same sign to thesecond child node of any node labelled with → ; • Assign the same sign to the second child node labelled with @ (notice thatwe do not label the first child node with nominal or state variable).Nodes in signed generation trees are positive (resp. negative ) if they are signed+ (resp. − ). 8uter Inner+ ∨ ∧ ✸ ¬ ↓ x @ − ∧ ∨ ✷ ¬ ↓ x @ → + ∧ ✷ ¬ ↓ x @ − ∨ ✸ ¬ ↓ x @Table 1: Outer and Inner nodes.Signed generation trees will be used in the inequalities ϕ ≤ ψ , where thepositive generation tree + ϕ and the negative generation tree − ψ will be consid-ered. We will also say that an inequality ϕ ≤ ψ is uniform in a variable p i if alloccurrences of p i in + ϕ and − ψ have the same sign, and that ϕ ≤ ψ is ε - uniform in an array ~p if ϕ ≤ ψ is uniform in p i , occurring with the sign indicated by ε (i.e., p i has the sign + if ε ( p i ) = 1, and has the sign − if ε ( p i ) = ∂ ), for eachpropositional variable p i in ~p .For any given formula ϕ ( p , . . . p n ), any order-type ε over n , and any 1 ≤ i ≤ n , an ε -critical node in a signed generation tree of ϕ is a leaf node + p i when ε i = 1 or − p i when ε i = ∂ . An ε - critical branch in a signed generation tree isa branch from an ε -critical node. The ε -critical occurrences are intended to bethose which the algorithm ALBA ↓ will solve for. We say that + ϕ (resp. − ϕ ) agrees with ε , and write ε (+ ϕ ) (resp. ε ( − ϕ )), if every leaf node in the signedgeneration tree of + ϕ (resp. − ϕ ) is ε -critical.We will also use the notation + ψ ≺ ∗ ϕ (resp. − ψ ≺ ∗ ϕ ) to indicate that anoccurence of a subformula ψ inherits the positive (resp. negative) sign from thesigned generation tree ∗ ϕ , where ∗ ∈ { + , −} . We will write ε ( γ ) ≺ ∗ ϕ (resp. ε ∂ ( γ ) ≺ ∗ ϕ ) to indicate that the signed generation subtree γ , with the signinherited from ∗ ϕ , agrees with ε (resp. with ε ∂ ). We say that a propositionalvariable p is positive (resp. negative ) in ϕ if + p ≺ + ϕ (resp. − p ≺ + ϕ ). Definition 4.3. (cf. [10, Definition 5]) Nodes in signed generation trees arecalled outer nodes and inner nodes , according to Table 1.A branch in a signed generation tree is called a excellent branch if it is theconcatenation of two paths P and P , one of which might be of length 0, suchthat P is a path from the leaf consisting (apart from variable nodes) of innernodes only, and P consists (apart from variable nodes) of outer nodes only. Definition 4.4 (Sahlqvist inequalities) . (cf. [10, Definition 6]) For any order-type ε , the signed generation tree ∗ ϕ of a formula ϕ ( p , . . . p n ) is ε -Sahlqvist iffor all 1 ≤ i ≤ n , every ε -critical branch with leaf p i is excellent. An inequality ϕ ≤ ψ is ε -Sahlqvist if the signed generation trees + ϕ and − ψ are ε -Sahlqvist.An inequality ϕ ≤ ψ is Sahlqvist if it is ε -Sahlqvist) for some ε . ALBA ↓ In the present section, we define the correspondence algorithm
ALBA ↓ for hybridlogic with binder, in the style of [5, 7, 21]. The algorithm goes in three steps.1. Preprocessing and first approximation :9n the generation tree of + ϕ and − ψ ,(a) Apply the distribution rules:i. Push down + ✸ , −¬ , + ∧ , + ↓ x, +@ i , +@ x , − → by distributingthem over nodes labelled with + ∨ which are outer nodes, andii. Push down − ✷ , + ¬ , −∨ , − ↓ x, − @ i , − @ x , − → by distributingthem over nodes labelled with −∧ which are outer nodes.(b) Apply the splitting rules: α ≤ β ∧ γα ≤ β α ≤ γ α ∨ β ≤ γα ≤ γ β ≤ γ (c) Apply the monotone and antitone variable-elimination rules: α ( p ) ≤ β ( p ) α ( ⊥ ) ≤ β ( ⊥ ) β ( p ) ≤ α ( p ) β ( ⊤ ) ≤ α ( ⊤ )for β ( p ) positive in p and α ( p ) negative in p .We denote by Preprocess ( ϕ ≤ ψ ) the finite set { ϕ i ≤ ψ i } i ∈ I of inequalitiesobtained after the exhaustive application of the previous rules. Thenwe apply the following first approximation rule to every inequality in Preprocess ( ϕ ≤ ψ ): ϕ i ≤ ψ i i ≤ ϕ i ψ i ≤ ¬ i Here, i and i are special fresh nominals. Now we get a set of inequalities { i ≤ ϕ i , ψ i ≤ ¬ i } i ∈ I .2. The reduction stage :In this stage, for each { i ≤ ϕ i , ψ i ≤ ¬ i } , we apply the following rules toprepare for eliminating all the proposition variables in { i ≤ ϕ i , ψ i ≤ ¬ i } :(a) Substage 1: Decomposing the outer part
In the current substage, the following rules are applied to decomposethe outer part of the Sahlqvist signed formula:i. Splitting rules: α ≤ β ∧ γα ≤ β α ≤ γ α ∨ β ≤ γα ≤ γ β ≤ γ The discussion below relies on the definition of signed generation tree in Section 4. Inwhat follows, we identify a formula with its signed generation tree. i ≤ ✸ α j ≤ α i ≤ ✸ j ✷ α ≤ ¬ i α ≤ ¬ j ✷ ¬ j ≤ ¬ ii ≤ @ j α j ≤ α @ j α ≤ ¬ i α ≤ ¬ ji ≤ @ x αx ≤ α @ x α ≤ ¬ i α ≤ ¬ x i ≤↓ x.α i ≤ α [ i /x ] ↓ x.α ≤ ¬ i α [ i /x ] ≤ ¬ i α → β ≤ ¬ ij ≤ α β ≤ ¬ k j → ¬ k ≤ ¬ i The nominals introduced by the approximation rules must notoccur in the system before applying the rule, and α [ i /x ] indicatesthat all occurrences of x in α are replaced by i .iii. Residuation rules: i ≤ ¬ αα ≤ ¬ i ¬ α ≤ ¬ ii ≤ α (b) Substage 2: Decomposing the inner part
In the current substage, the following rules are applied to decomposethe inner part of the Sahlqvist signed formula:i. Splitting rules: α ≤ β ∧ γα ≤ β α ≤ γ α ∨ β ≤ γα ≤ γ β ≤ γ ii. Residuation rules: α ≤ ¬ ββ ≤ ¬ α ¬ α ≤ β ¬ β ≤ α ✸ α ≤ βα ≤ (cid:4) β α ≤ ✷ β ♦ α ≤ βα ≤ @ j β E α ∧ j ≤ β @ j β ≤ αβ ≤ j → A α α ≤ @ x β E α ∧ x ≤ β @ x β ≤ αβ ≤ x → A αα ≤↓ x.β ∀ y ( A ( y → α ) ∧ y ≤ β [ y/x ]) ↓ x.β ≤ α ∀ y ( β [ y/x ] ≤ y → E ( y ∧ α ))The state variables introduced by the residuation rules must notoccur in the system before applying the rule.11ii. Second splitting rule: ∀ x ( Mega & Mega ) ∀ x ( Mega ) ∀ x ( Mega )Here Mega and Mega denote mega-inequalities.(c) Substage 3: Preparing for the Ackermann rules
In this substage, we prepare for eliminating the propositional vari-ables by the Ackermann rules, with the help of the following packingrules:Packing rules: ∀ x ( α ≤ β )( ∃ xα ) ≤ β ∀ x ( β ≤ α ) β ≤ ( ∀ xα )where β does not contain occurrences of x .(d) Substage 4: The Ackermann stage
In this substage, we compute the minimal/maximal valuation forpropositional variables and use the Ackermann rules to eliminate allthe propositional variables. These two rules are the core of
ALBA ,since their application eliminates proposition variables. In fact, allthe preceding steps are aimed at reaching a shape in which the rulescan be applied. Notice that an important feature of these rules isthat they are executed on the whole set of (universally quantified)inequalities, and not on a single inequality.The right-handed Ackermann rule:The system α ≤ p ... α n ≤ p ∀ ~x ( β ≤ γ )... ∀ ~x m ( β m ≤ γ m )is replaced by ∀ ~x ( β [( α ∨ . . . ∨ α n ) /p ] ≤ γ [( α ∨ . . . ∨ α n ) /p ])... ∀ ~x m ( β m [( α ∨ . . . ∨ α n ) /p ] ≤ γ m [( α ∨ . . . ∨ α n ) /p ])where:i. p, ~x , . . . , ~x m do not occur in α , . . . , α n ;ii. Each β i is positive, and each γ i negative in p , for 1 ≤ i ≤ m ;iii. Each α i is pure. 12he left-handed Ackermann rule:The system p ≤ α ... p ≤ α n ∀ ~x ( β ≤ γ )... ∀ ~x m ( β m ≤ γ m )is replaced by ∀ ~x ( β [( α ∨ . . . ∨ α n ) /p ] ≤ γ [( α ∨ . . . ∨ α n ) /p ])... ∀ ~x m ( β m [( α ∨ . . . ∨ α n ) /p ] ≤ γ m [( α ∨ . . . ∨ α n ) /p ])where:i. p, ~x , . . . , ~x m do not occur in α , . . . , α n ;ii. Each β i is negative, and each γ i positive in p , for 1 ≤ i ≤ m .iii. Each α i is pure.3. Output : If in the previous stage, for some { i ≤ ϕ i , ψ i ≤ ¬ i } , thealgorithm gets stuck, i.e. some proposition variables cannot be eliminatedby the application of the reduction rules, then the algorithm halts andoutput “failure”. Otherwise, each initial tuple { i ≤ ϕ i , ψ i ≤ ¬ i } ofinequalities after the first approximation has been reduced to a set ofpure (universally quantified) inequalities Reduce ( ϕ i ≤ ψ i ), and then theoutput is a set of quasi-(universally quantified) inequalities { & Reduce ( ϕ i ≤ ψ i ) ⇒ i ≤ ¬ i : ϕ i ≤ ψ i ∈ Preprocess ( ϕ ≤ ψ ) } . Then the algorithm usethe standard translation to transform the quasi-(universally quantified)inequalities into first-order formulas. Finally, use universal quantifiers toquantify all free individual variables x and individual constants i in thestandard translation. ALBA ↓ In the present section, we will prove the soundness of the algorithm
ALBA ↓ withrespect to Kripke frames. The basic proof structure is similar to [21]. Theorem 6.1 (Soundness) . If ALBA ↓ runs successfully on ϕ ≤ ψ and outputs FO ( ϕ ≤ ψ ), then for any Kripke frame F = ( W, R ), F (cid:13) ϕ ≤ ψ iff F | = FO ( ϕ ≤ ψ ) . Proof.
The proof goes similarly to [7, Theorem 8.1]. Let ϕ i ≤ ψ i , 1 ≤ i ≤ n denote the inequalities produced by preprocessing ϕ ≤ ψ after Stage 1, and { i ≤ ϕ i , ψ i ≤ ¬ i } denote the inequalities after the first-approximation rule, Reduce ( ϕ i ≤ ψ i ) denote the set of pure (universally quantified) inequalitiesafter Stage 2, and FO ( ϕ ≤ ψ ) denote the standard translation of the quasi-(universally quantified) inequalities into first-order formulas, then we have thefollowing chain of equivalences: 13t suffices to show the equivalence from (1) to (5) given below: F (cid:13) ϕ ≤ ψ (1) F (cid:13) ϕ i ≤ ψ i , for all 1 ≤ i ≤ n (2) F (cid:13) ( i ≤ ϕ i & ψ i ≤ ¬ i ) ⇒ i ≤ ¬ i for all 1 ≤ i ≤ n (3) F (cid:13) Reduce ( ϕ i ≤ ψ i ) ⇒ i ≤ ¬ i for all 1 ≤ i ≤ n (4) F (cid:13) FO ( ϕ ≤ ψ ) (5) • The equivalence between (1) and (2) follows from Proposition 6.2; • the equivalence between (2) and (3) follows from Proposition 6.3; • the equivalence between (3) and (4) follows from Propositions 6.4, 6.7,6.11, 6.12; • the equivalence between (4) and (5) Proposition 3.4.In the remainder of this section, we prove the soundness of the rules in Stage1, 2 and 3. Proposition 6.2 (Soundness of the rules in Stage 1) . For the distributionrules, the splitting rules and the monotone and antitone variable-eliminationrules, they are sound in both directions in F . Proof.
For the soundness of the distribution rules, it follows from the fact thatthe corresponding distribution laws are valid in F . Here we only list the distri-bution laws for @, ↓ (the others can be found in [21, Section 4.4]): • @ i ( α ∧ β ) ↔ (@ i α ∧ @ i β ); • @ i ( α ∨ β ) ↔ (@ i α ∨ @ i β ); • @ x ( α ∧ β ) ↔ (@ x α ∧ @ x β ); • @ x ( α ∨ β ) ↔ (@ x α ∨ @ x β ); • ↓ x. ( α ∧ β ) ↔ ( ↓ x.α ∧ ↓ x.β ); • ↓ x. ( α ∨ β ) ↔ ( ↓ x.α ∨ ↓ x.β ).For the soundness of the splitting rules and the monotone and antitonevariable elimination rules, similar to the same rules in [21, Section 4.4]. Proposition 6.3. (2) and (3) are equivalent, i.e. the first-approximation ruleis sound in F . Proof.
Similar to the soundness of the same rule in [21, Section 4.4].14he next step is to show the soundness of Stage 2, for which it suffices toshow the soundness of each rule in each substage.
Proposition 6.4.
The splitting rules, the approximation rules for ✸ , ✷ , @ , ↓ , → ,the residuation rules for ¬ in Substage 1 are sound in F . Proof.
For the splitting rules, the approximation rules for ✸ , ✷ , → , the residua-tion rules for ¬ in Substage 1, their soundness proofs are similar to the soundnessof the same rules in [21, Section 4.4]. The soundness of the approximation rulesfor @ and ↓ are proved in Proposition 6.5 and 6.6. Proposition 6.5.
The approximation rules for @ in Substage 1 are sound in F . Proof.
Since for @ i and @ x , the proofs are essentially the same, we only proveit for @ i .For the left approximation rule for @ i , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M ,1. if M , g (cid:13) i ≤ @ j α , then M , g (cid:13) j ≤ α ;2. if M , g (cid:13) j ≤ α , then M , g (cid:13) j ≤ α .For item 1, if M , g (cid:13) i ≤ @ j α , then M , g, V ( i ) (cid:13) @ j α , therefore M , g, V ( j ) (cid:13) α , thus M , g (cid:13) j ≤ α .For item 2, if M , g (cid:13) j ≤ α , then M , g, V ( j ) (cid:13) α , so M , g (cid:13) @ j α , therefore M , g, V ( i ) (cid:13) @ j α , thus M , g (cid:13) i ≤ @ j α .The right approximation rule for @ i is similar. Proposition 6.6.
The approximation rules for ↓ in Substage 1 are sound in F . Proof.
We prove it for the left approximation rule, the right rule being similar.For the left approximation rule for ↓ x , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M , M , g (cid:13) i ≤↓ x.α iff M , g (cid:13) i ≤ α [ i /x ] . Indeed, M , g (cid:13) i ≤↓ x.α iff M , g, V ( i ) (cid:13) ↓ x.α iff M , g xV ( i ) , V ( i ) (cid:13) α iff M , g xV ( i ) , V ( i ) (cid:13) α [ i /x ]iff M , g, V ( i ) (cid:13) α [ i /x ]iff M , g (cid:13) i ≤ α [ i /x ]. Proposition 6.7.
The splitting rules, the residuation rules for ¬ , ✸ , ✷ , @ , ↓ ,the second splitting rule in Substage 2 are sound in F .15 roof. The soundness proofs of the splitting rules, the residuation rules for ¬ , ✸ , ✷ in Substage 2 are similar to the soundness proofs of the same rules in[21, Section 4.4]. The soundness of the residuation rules for @ and ↓ are provedin Proposition 6.8 and 6.9, and the soundness proof of the second splitting ruleis in Proposition 6.10. Proposition 6.8.
The residuation rules for @ are sound in F . Proof.
Since for @ i and @ x , the proofs are essentially the same, we only proveit for @ i .For the left residuation rule for @ i , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M , M , g (cid:13) α ≤ @ j β iff M , g (cid:13) E α ∧ j ≤ β. Indeed, M , g (cid:13) α ≤ @ j β iff M , g (cid:13) α ≤ A ( j → β )iff M , g (cid:13) E α ≤ j → β iff M , g (cid:13) E α ∧ j ≤ β .For the right residuation rule for @ i , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M , M , g (cid:13) @ j β ≤ α iff M , g (cid:13) β ≤ j → A α. Indeed, M , g (cid:13) @ j β ≤ α iff M , g (cid:13) ¬ α ≤ ¬ @ j β iff M , g (cid:13) ¬ α ≤ @ j ¬ β iff M , g (cid:13) E ¬ α ∧ j ≤ ¬ β iff M , g (cid:13) β ≤ j → A α . Proposition 6.9.
The residuation rules for ↓ are sound in F . Proof.
For the left residuation rule for ↓ , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M , M , g (cid:13) α ≤↓ x.β iff M , g (cid:13) ∀ y ( A ( y → α ) ∧ y ≤ β [ y/x ]) . Indeed, 16 , g (cid:13) α ≤↓ x.β iff for all w ∈ W , if M , g, w (cid:13) α , then M , g, w (cid:13) ↓ x.β iff for all w ∈ W , if M , g yw , w (cid:13) α , then M , g yw , w (cid:13) ↓ x.β iff for all w ∈ W , if M , g yw (cid:13) y ≤ α , then M , g y,xw,w , w (cid:13) β iff for all w ∈ W , if M , g yw (cid:13) y ≤ α , then M , g y,xw,w , w (cid:13) β [ y/x ]iff for all w ∈ W , if M , g yw (cid:13) A ( y → α ), then M , g yw , w (cid:13) β [ y/x ]iff for all w ∈ W , if M , g yw (cid:13) A ( y → α ), then M , g yw (cid:13) y ≤ β [ y/x ]iff for all w ∈ W , M , g yw (cid:13) A ( y → α ) ∧ y ≤ β [ y/x ] (see below)iff M , g (cid:13) ∀ y ( A ( y → α ) ∧ y ≤ β [ y/x ]).Now it suffices to show that line-3 and line-2 are equivalent.line-3 ⇒ line-2: Assume line-3 holds. Take arbitrary w, v ∈ W . If M , g yw , v (cid:13) A ( y → α ) ∧ y , then M , g yw , v (cid:13) A ( y → α ), so y → α is true everywhere in ( M , g yw ),therefore M , g yw (cid:13) A ( y → α ). By line-3, we have M , g yw (cid:13) y ≤ β [ y/x ]. Since M , g yw , v (cid:13) y , we have M , g yw , v (cid:13) β [ y/x ].line-2 ⇒ line-3: Assume line-2 holds. For any w ∈ W , assume that M , g yw (cid:13) A ( y → α ). Then M , g yw (cid:13) A ( y → α ) ↔ ⊤ , so from line-2 one get M , g yw (cid:13) ⊤ ∧ y ≤ β [ y/x ], i.e. M , g yw (cid:13) y ≤ β [ y/x ].For the right residuation rule for ↓ , it suffices to show that for any Kripkemodel M = ( W, R, V ), any assignment g on M , M , g (cid:13) ↓ x.β ≤ α iff M , g (cid:13) ∀ y ( β [ y/x ] ≤ y → E ( y ∧ α )) . Indeed, M , g (cid:13) ↓ x.β ≤ α iff M , g (cid:13) ¬ α ≤ ¬ ↓ x.β iff M , g (cid:13) ¬ α ≤↓ x. ¬ β iff M , g (cid:13) ∀ y ( A ( y → ¬ α ) ∧ y ≤ ¬ β [ y/x ])iff M , g (cid:13) ∀ y ( β [ y/x ] ≤ ¬ ( A ( y → ¬ α ) ∧ y ))iff M , g (cid:13) ∀ y ( β [ y/x ] ≤ y → E ( y ∧ α )). Proposition 6.10.
The second splitting rule in Substage 2 is sound in F . Proof.
It follows immediately from the meta-equivalence that ∀ x ( α ∧ β ) ↔ ∀ xα ∧∀ xβ . Proposition 6.11.
The packing rules in Substage 3 are sound in F . Proof.
We only prove the soundness of the first packing rule, the other is similar.For the left packing rule, to show its soundness, it suffices to show that forany Kripke model M = ( W, R, V ), any assignment g on M , M , g (cid:13) ∀ x ( α ≤ β ) iff M , g (cid:13) ( ∃ xα ) ≤ β, where β does not contain occurrences of x . Indeed,17 , g (cid:13) ∀ x ( α ≤ β )iff for all w ∈ W , M , g xw (cid:13) α ≤ β iff for all w, v ∈ W , if M , g xw , v (cid:13) α , then M , g xw , v (cid:13) β iff for all w, v ∈ W , if M , g xw , v (cid:13) α , then M , g, v (cid:13) β (since β does not contain occurrences of x )iff for all v ∈ W , if there exists a w ∈ W such that M , g xw , v (cid:13) α , then M , g, v (cid:13) β iff for all v ∈ W , if M , g, v (cid:13) ∃ xα , then M , g, v (cid:13) β iff M , g (cid:13) ( ∃ xα ) ≤ β . Proposition 6.12.
The Ackermann rules in Substage 4 are sound in F . Proof.
The proof is similar to the soundness of the Ackermann rules in [21,Section 4.4].
ALBA ↓ In the present section we show that
ALBA ↓ succeeds on all Sahlqvist inequalities.The proof structure is similar to [21, Section 4.5]. Theorem 7.1.
ALBA ↓ succeeds on all Sahlqvist inequalities. Definition 7.2 (Definite ε -Sahlqvist inequality, see Definition 31 in [21]) . Givenan order type ε , ∗ ∈ {− , + } , the signed generation tree ∗ ϕ of the term ϕ ( p , . . . , p n )is definite ε -Sahlqvist if there is no + ∨ , −∧ occurring in the outer part on an ε -critical branch. An inequality ϕ ≤ ψ is definite ε -Sahlqvist if the trees + ϕ and − ψ are both definite ε -Sahlqvist. Lemma 7.3.
Let { ϕ i ≤ ψ i } i ∈ I = Preprocess ( ϕ ≤ ψ ) obtained by exhaustiveapplication of the rules in Stage 1 on an input ε -Sahlqvist inequality ϕ ≤ ψ .Then each ϕ i ≤ ψ i is a definite ε -Sahlqvist inequality. Proof.
Same as the proof of [21, Lemma 32].
Definition 7.4 (Inner ε -Sahlqvist signed generation tree, see Definition 33 in[21]) . Given an order type ε , ∗ ∈ {− , + } , the signed generation tree ∗ ϕ of theterm ϕ ( p , . . . , p n ) is inner ε -Sahlqvist if its outer part P on an ε -critical branchis always empty, i.e. its ε -critical branches have inner nodes only. Lemma 7.5.
Given inequalities i ≤ ϕ i and ψ i ≤ ¬ i obtained from Stage 1where + ϕ i and − ψ i are definite ε -Sahlqvist, by applying the rules in Substage1 of Stage 2 exhaustively, the inequalities that we get are in one of the followingforms:1. pure inequalities which does not have occurrences of propositional vari-ables;2. inequalities of the form i ≤ α or x ≤ α where + α is inner ε -Sahlqvist;18. inequalities of the form β ≤ ¬ i or β ≤ ¬ x where − β is inner ε -Sahlqvist. Proof.
Same as the proof of [21, Lemma 34].
Lemma 7.6.
Assume we have an inequality i ≤ α or β ≤ ¬ i where + α and − β are inner ε -Sahlqvist, by applying the rules in Substage 2 of Stage 2, we have(universally quantified) inequalities ( k can be 0 where a universally quantifiedinequality becomes an inequality) of the following form:1. ∀ x . . . ∀ x k ( α ≤ p ), where ε ( p ) = 1, α is pure;2. ∀ x . . . ∀ x k ( p ≤ β ), where ε ( p ) = ∂ , β is pure;3. ∀ x . . . ∀ x k ( α ≤ γ ), where α is pure and + γ is ε ∂ -uniform;4. ∀ x . . . ∀ x k ( γ ≤ β ), where β is pure and − γ is ε ∂ -uniform. Proof.
The proof is similar to [21, Lemma 35]. First of all, from the rulesof the Substage 2 of Stage 2, it is easy to see that from the given inequality,what we will obtain would be a set of mega-inequalities, and by applying thesecond splitting rule we would get universally quantified inequalities of the form ∀ x . . . ∀ x k ( γ ≤ δ ). Now it suffices to check the shape of γ and δ . (From nowon we call γ ≤ δ the head of the universally quantified inequality.)Notice that for each input inequality, it is of the form i ≤ α , x ≤ α or β ≤ ¬ i , β ≤ ¬ x , where + α and − β are inner ε -Sahlqvist. By applying thesplitting rules and the residuation rules in this substage, it is easy to checkthat the head of the (universally quantified) inequality will have one side of theinequality pure, and the other side still inner ε -Sahlqvist. By applying theserules exhaustively, one will either have p as the non-pure side (with this p ona critical branch), or have an inner ε -Sahlqvist signed generation tree with nocritical branch, i.e., ε ∂ -uniform. Lemma 7.7.
Assume we have (universally quantified) inequalities of the formas described in Lemma 7.6. Then we can get (universally quantified) inequalitiesof the following form:1. α ≤ p where ε ( p ) = 1, α is pure;2. p ≤ α where ε ( p ) = ∂ , α is pure;3. ∀ x . . . ∀ x k ( α ≤ γ ), where α is pure and + γ is ε ∂ -uniform;4. ∀ x . . . ∀ x k ( γ ≤ β ), where β is pure and − γ is ε ∂ -uniform. Proof.
For universally quantified inequalities of form 1 and 2 in Lemma 7.6,we can apply the packing rule since p does not contain occurrences of statevariables. For universally quantified inequalities of form 3 and 4 in Lemma 7.6,we do not need to apply any rules in this stage.19 emma 7.8. Assume we have (universally quantified) inequalities of the formas described in Lemma 7.7, the Ackermann lemmas are applicable and thereforeall propositional variables can be eliminated.
Proof.
Immediate observation from the requirements of the Ackermann lemmas.
Proof of Theorem 7.1.
The proof is similar to [21, Theorem 4.2]. Assume wehave an ε -Sahlqvist inequality ϕ ≤ ψ as input. By Lemma 7.3, we get a setof definite ε -Sahlqvist inequalities. Then by Lemma 7.5, we get inequalitiesas described in Lemma 7.5. By Lemma 7.6, we get the universally quantifiedinequalities as described. Therefore by Lemma 7.7, we can apply the packingrules to get inequalities and universally quantified inequalities as described inthe lemma. Finally by Lemma 7.8, the (universally quantified) inequalities arein the right shape to apply the Ackermann rules, and thus we can eliminate allthe propositional variables and the algorithm succeeds on the input. In the present paper, we investigates the correspondence theory for hybrid logicwith binder H (@ , ↓ ). We define the class of Sahlqvist H (@ , ↓ )-inequalities, andshow that each of these inequalities has a first-order frame correspondent by analgorithm ALBA ↓ .For future directions, we consider the canonicity theory for H (@ , ↓ ), i.e.which class of H (@ , ↓ )-formulas is preserved under taking canonical extensionsor MacNeille completions, as well as develop the correspondence theory andcanonicity theory for other very expressive hybrid languages, e.g. with ∃ , ⇓ , Σbinders.
Acknowledgement
The research of the author is supported by the ChineseNational Social Science Fund of China (Project No. 18CZX063), Taishan Uni-versity Starting Grant “Studies on Algebraic Sahlqvist Theory” and the Tais-han Young Scholars Program of the Government of Shandong Province, China(No.tsqn201909151).
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