Algorithmic Correspondence and Completeness in Modal Logic. III. Extensions of the Algorithm SQEMA with Substitutions.

Dimiter Vakarelov
Sofia University "St. Kliment Ohridski", Sofia, Bulgaria, faculty of mathematics and informatics
ABSTRACT In earlier papers we have introduced an algorithm, SQEMA, for computing firstorder equivalents and proving canonicity of modal formulae. However, SQEMA is not complete with respect to the so called complex Sahlqvist formulae. In this paper we, first, introduce the class of complex inductive formulae, which extends both the class of complex Sahlqvist formulae and the class of polyadic inductive formulae, and second, extend SQEMA to SQEMAsub by allowing suitable substitutions in the process of transformation. We prove the correctness of SQEMAsub with respect to local equivalence of the input and output formulae and dpersistence of formulae on which the algorithm succeeds, and show that SQEMAsub is complete with respect to the class of complex inductive formulae.

Article: Algorithmic correspondence and completeness in modal logic. V. Recursive extensions of SQEMA
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ABSTRACT: The previously introduced algorithm SQEMA computes firstorder frame equivalents for modal formulae and also proves their canonicity. Here we extend SQEMA with an additional rule based on a recursive version of Ackermann's lemma, which enables the algorithm to compute local frame equivalents of modal formulae in the extension of firstorder logic with monadic least fixedpoints FOμ. This computation operates by transforming input formulae into locally frame equivalent ones in the pure fragment of the hybrid μcalculus. In particular, we prove that the recursive extension of SQEMA succeeds on the class of ‘recursive formulae’. We also show that a certain version of this algorithm guarantees the canonicity of the formulae on which it succeeds.Journal of Applied Logic 12/2010; · 0.40 Impact Factor 
Conference Paper: Algorithmic Definability and Completeness in Modal Logic.
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ABSTRACT: One of the nice features of modal languages is that sometimes they can talk about abstract properties of the corresponding semantic structures. For instance the truth of the modal formula \square pÞ p\square p\Rightarrow p in the Kripke frame (W,R) is equivalent to the reflexivity of the relation R. Using a terminology from modal logic [13], we say that the condition of reflexivity – ( ∀ x)(xRx), is a firstorder equivalent of the modal formula \square pÞ p\square p\Rightarrow p, or, that the formula \square pÞ p\square p\Rightarrow p is firstorder definable by the condition ( ∀ x)(xRx). More over, adding the formula \square pÞ p\square p\Rightarrow p to the axioms of the minimal modal logic K we obtain a complete logic with respect to the class of reflexive frames and the completeness proof can be done by the well known in modal logic canonical method (such formulas are called canonical). Let us note that definability and completeness are some of the good features in the applications of modal logic, and hence it is important to have algorithmic methods for establishing such properties. In our talk we will describe several algorithmic approaches to this problem.Foundations of Information and Knowledge Systems, 6th International Symposium, FoIKS 2010, Sofia, Bulgaria, February 1519, 2010. Proceedings; 01/2010
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Fundamenta Informaticae XX (2009) 1–371
IOS Press
Algorithmic correspondence and completeness in modal logic.
III. Extensions of the algorithm SQEMA with substitutions
Willem Conradie
Department of Mathematics,
University of Johannesburg
Johannesburg, South Africa
wconradie@uj.ac.za
Valentin Goranko
Department of Informatics and Mathematical Modelling
Technical University of Denmark
vfgo@imm.dtu.dk
Dimiter Vakarelov
Faculty of Mathematics and Computer Science,
Sofia University
Sofia, Bulgaria
dvak@fmi.unisofia.bg
Abstract. In earlier papers we have introduced an algorithm, SQEMA, for computing firstorder
equivalents and proving canonicity of modal formulae. However, SQEMA is not complete with
respect to the so called complex Sahlqvist formulae. In this paper we, first, introduce the class
of complex inductive formulae, which extends both the class of complex Sahlqvist formulae and
the class of polyadic inductive formulae, and second, extend SQEMA to SQEMAsubby allowing
suitable substitutions in the process of transformation. We prove the correctness of SQEMAsubwith
respect to local equivalence of the input and output formulae and dpersistence of formulae on which
the algorithm succeeds, and show that SQEMAsubis complete with respect to the class of complex
inductive formulae.
Keywords: SQEMA, correspondence, dpersistence, complex Sahlqvist formulae.
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2W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Introduction
This paper is in the field of algorithmic correspondence and completeness theory in modal logic. The first
general result in this field was the celebrated Sahlqvist’s theorem [22]. It introduces a large class of modal
formulae (subsequently called Sahlqvist formulae) which are firstorder definable and canonical. More
over, the proof of Sahlqvist’s definability theorem, also obtained independently by van Benthem [34],
provides an effective procedure, viz. the method of minimal valuations, for computing the firstorder
equivalents of the formulae in that class. For a long time the class of Sahlqvist formulae was considered
as the optimal syntactically defined class with these two properties. In [9] the class of Sahlqvist formulae
was extended to cover polyadic modal languages, but without extending the original Sahlqvist class on
monadic languages. Recently, several new effective extensions or analogs of the Sahlqvist class have
been obtained:
• the class of inductive formulae [15, 17, 5] for arbitrary polyadic modal languages,
• the class of inductive hybrid formulae, [16] (see also [25], [4]),
• the class of complex Sahlqvist formulae [26] (for the ordinary modal language).
• classes of formulae having equivalents in the firstorder logic with least fix points [8, 13, 17, 20,
31, 32, 35, 36].
Because of the undecidability of the class of firstorder definable modal formulae [3], the hierarchy
of effective extensions of the Sahlqvist class concerning firstorder definability is infinite and all further
syntactic extensions are bound to be increasingly more complicated. In [6, 7] another approach has been
proposed: instead of syntactic extensions of the Sahlqvist class, an algorithm, SQEMA (SecondOrder
Quantifier Elimination for Modal formulae using Ackermann’s lemma), was developed to compute first
order equivalents of modal formulae with unary modalities, further extended in [7] to polyadic and hybrid
modal languages. It has been proved in [6, 7] that SQEMA is correct with respect to local equivalence
of the input and output formulae, and that the formulae for which it succeeds are locally dpersistent
(respectively, locally dipersistent for the case of languages with nominals and converse modalities), and
hence canonical in the respective senses.
With respect to firstorder correspondence, our approach was preceded and influenced by two earlier
developed algorithms for the elimination of secondorder quantifiers over predicate variables, viz. SCAN
[12, 11] and DLS [10, 21, 23, 19]. Each of them, applied to the negation of the standard translation of
a modal formula into monadic secondorder logic, attempts to eliminate all occurring existentially quan
tified predicate variables and thus to compute a firstorder correspondent. To that aim, SCAN employs
a modification of the resolution method, while DLS is based on a result by Ackermann [1] (see also
the references above, as well as [6, 7]), allowing explicit elimination, up to logical equivalence, of an
existentially quantified secondorder predicate variable.
Let us note that both SCAN and DLS use Skolemization of the input and, after the quantifier elimina
tion procedure, a procedure attempting reverse Skolemization (deSkolemization, or unSkolemization)
is applied. That procedure is not always successful, which may lead to (sometimes unnecessary) failure
of the main algorithm. To avoid the necessity for deSkolemization, SQEMA does not use the stan
dard translation into the firstorder logic but works directly on modal formulae and includes only a very
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions3
restricted form of Skolemization, viz. only Skolem constants, introduced as nominals (an algorithm
working directly with modal formulae was also considered in [24]). Thus, SQEMA attempts to eventu
ally transform modal formulae into pure formulae in an appropriate hybrid modal language, from which
the local firstorder equivalent is extracted. In order to eliminate the propositional variables, SQEMA
uses a modal version of Ackermann’s lemma, formulated in terms of propositional modal logic, while
the original lemma formulated by Ackermann and used in DLS is in terms of secondorder logic. Further
information about SCAN, DLS, and SQEMA can be found in the recent book on secondorder quantifier
elimination [13].
An implementation of a variant of SQEMA for monadic languages extended with nominals and
universal modality has been realized by Dimiter Georgiev (see [14]) as a master project, and works
online at http://fmi.unisofia.bg/fmi/logic/sqema.
The starting point of the present paper is the fact that none of the versions of SQEMA mentioned
above is complete for the class of so called complex Sahlqvist formulae [26, 27]. This is an interesting
phenomenon, because all complex Sahlqvist formulae can be effectively translated to Sahlqvist formulae
for which all current versions of SQEMA succeed. The translation of complex Sahlqvist formulae into
Sahlqvist formulae was constructed in [26] by means of quite complex reversible Boolean substitutions
(preserving local firstorder equivalents and dpersistence), effectively computed from the input complex
formula. In the present paper we have extended SQEMA with a mechanism for applying such substi
tutions, which enables the new extension, denoted by SQEMAsub, to succeed on all complex inductive
formulae – a natural polyadic extension of the class of complex Sahlqvist formulae. We prove that all
formulae for which SQEMAsubsucceeds are firstorder definable and canonical, thus implying that this
is the currently largest effective extension of the Sahlqvist class of firstorder definable and canonical
modal formulae.
The paper is organized as follows. Section 1 contains an informal introduction to polyadic modal
logic, modal algebras over Kripke frames and the Ackermann lemma formulated in terms of modal al
gebras. It also provides an example of the latter lemma’s application which illustrates the intuition upon
which SQEMA is based. The section also contains the formal definition of SQEMA and the formulation
of its basic metaproperties which will be used later on in the paper. In Section 2 we introduce the notion
of reversible substitution and define two large classes of polyadic modal formulae: the class of complex
recursive formulae which extends the class of regular formulae introduced in [17], and the class of com
plex polyadic inductive formulae, extending the class of polyadic inductive formulae [15, 17]. We also
give an example of an inductive complex modal formula for which SQEMA does not succeed. Section
3 is devoted to the study of a special class of so called complex substitutions, on which SQEMAsubis
based. Section 4 is devoted to an effective translation Θ of the class of complex inductive formulae
into the class of inductive formulae by means of special type of reversible Boolean substitutions. This
implies a generalization of the Sahlqvist Theorem both on its definability and canonicity part to the class
of inductive complex modal formulae. Section 5 is preparatory for the definition of SQEMAsub. Here
we introduce the notion of complex normal form and a special translation Σ which is the main tool in
SQEMAsub. Section 6 is devoted to the definition of SQEMAsub. We first discuss SQEMAsubinfor
mally, motivating its internal structure, which contains as a subprogram the former algorithm SQEMA
and a new block SUB performing some transformations based on reversible Boolean substitutions. We
illustrate the algorithm with some examples for which it succeeds but for which SQEMA does not suc
ceed. We prove correctness and canonicity of SQEMAsuband its completeness with respect to the class
of complex inductive formulae. We conclude in section 7 where we also mention some open problems
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4W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
and future research agenda.
1. Background on polyadic modalities and the algorithm SQEMA
The version of the algorithm SQEMA introduced in [7] is designed to work on polyadic modal formulae.
Since the aim of the present paper is to introduce an extension of this algorithm, we invite the reader to
consult [7] as well as [2, 15, 17] for all formal definitions and motivating examples concerning polyadic
modal languages and inductive modal formulae. In this section we give an informal introduction to
polyadic modal logic, fix some notation, and provide some intuitions underlying the algorithm SQEMA
and the main results of the paper.
1.1.Polyadic modal logics
Standard polymodal propositional modal languages contain only unary modalities. With each class Σ
of relational structures containing only binary relations we may associate such a language, L(Σ), with
the modalities interpreted in Σ using the corresponding relations in the structures. One way to extend
this parallelism to arbitrary relational structures is to use modal operators with arbitrary arity, called
polyadic. Extending some notations from dynamic logic, we present standard polyadic modalities in
the form [α](A1,...,An) (generalizing the box modality [α]A) and ?α?(A1,...,An) (generalizing the
diamond modality ?α?A ). Here α is called a modal term of arity n (notation ρ(α) = n, where ρ is an
arity function) and in the semantics of [α](A1,...,An) and ?α?(A1,...,An) this term is associated to
a certain n+1ary relation Rα(w,w1,...,wn). Using the standard notation for the satisfiability relation
in modal logic (see for instance [2]) we express the semantics of polyadic modalities as follows:
(M,w) ? ?α?(A1,...,An) iff there exist w1,...,wnsuch that Rα(w,w1,...,wn) and
(M,wi) ? Ai, for each 1 ≤ i ≤ n,
(M,w) ? [α](A1,...,An) if, for all w1,...,wnsuch that Rα(w,w1,...,wn), it is the case that
(M,wi) ? Ai, for some 1 ≤ i ≤ n.
Obviously, if n = 1 then this semantics coincides with the standard Kripke semantics for the unary
modalities. The above semantics shows that the modality [α](A1,...,An) is dual to the modality
?α?(A1,...,An) and the following equivalences are valid which obviously generalize the corresponding
equivalences for the unary case:
[α](A1,...,An) ↔ ¬?α?(¬A1,...,¬An) and
?α?(A1,...,An) ↔ ¬[α](¬A1,...,¬An),
Note that the case n = 0 is also included and in this case the two modal operators ?α? and [α] have
no arguments and are treated as constants and the corresponding relation Rαis an unary relation, i.e., a
subset of the universe of the model M. The semantics of these constants is the following:
(M,w) ? ?α? iff w ∈ Rα,
(M,w) ? [α] iff w ?∈ Rα.
Let us denote by ιnthe modal term which in any model has the following interpretation as n+1ary
identity: Rιn(w,w1,...,wn) iff w = w1= ... = wn. Then the modality ?ιn?(A1...,An) is semanti
cally equivalent with the conjunction A1∧ ... ∧ An, and the modality [ιn](A1...,An) is semantically
equivalents with the disjunction A1∨ ... ∨ An. This fact allows one to treat classical conjunctions and
disjunctions as polyadic modalities which considerably simplifies the theory of polyadic modal logic (see
[15, 17]).
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions5
As in dynamic logic, modal terms can be composed subject to some obvious arity constraints. We
shall illustrate this construction with an example. Let α be a modal term of arity 2 (ρ(α) = 2), let β,γ
be modal terms of arbitrary arity, say ρ(β) = 2 and ρ(γ) = 3 and let Rα,Rβ,Rγbe the corresponding
relations in some model. Then we may define a new relation S by the following natural definition:
S(x,x1,x2,x3,x4,x5)iffthereexisty1,y2suchthatRα(x,y1,y2), Rβ(y1,x1,x2)andRγ(y2,x3,x4,x5).
With this construction in mind, it is natural to consider the relation S corresponding to the composed
modal term α(β,γ), called the composition of α,β and γ in this order. The following equivalence is true
for this composition:
[α(β,γ)](A1,A2,A3,A4,A5) ↔ [α]([β](A1,A2),[γ](A3,A4,A5).
The above considerations show that we may have different modal languages depending on the set τ
of modal terms with their predefined arity, called a modal similarity type. A modal similarity type τ
and a set Θ of propositional variables together uniquely determine (by a simultaneous induction) the set
of all (composed) terms MTτand the set of all formulae. This language is denoted by Lτ(Θ). If the
particular set of proposition letters Θ over which the language is built is not important, we will omit it
and simply write Lτ. We will always assume that modal languages contain the identity modal terms ιn.
Similarity types are important in the formal definition of the semantics of a given modal language.
Namely, given a type τ, we consider the class of τframes. These are relational structures of the form
F = (W,{Rα})α∈τ, with Rαa (ρ(α) + 1)ary relation for each α ∈ τ.
As in dynamic logic with inverse operations α−1on modal terms (also called converse operations),
we may consider a generalization of this operation in polyadic modal logic. For the binary case we have
the following condition between Rαand Rα−1: Rα−1(x,y) iff R−1
For the polyadic case, if ρ(α) = n, then we have n inverses α−i, i = 1,...,n, with the following
semantics: Rα−i(x,y1,...,yi,...,yn) iff R−i
defined as Rα(yi,y1...,x,...,yn)), i.e., the first and (i + 1)st arguments are interchanged.
The following equivalence is always true for the inverse modalities, which generalize the unary
case in the obvious way. Let M be any model in which Rαand Rα−i are interpreted. Then M ?
B ∨ [α](A1,...,Ai,...,An) iff M ? [α−i](A1,...,B,...,An) ∨ Ai, i = 1,...,n.
The extension of the language Lτ with inverse operations is denoted by Lτ(r)and is called com
pletely reversive extension of Lτ([7]).
We will consider also hybrid modal languages containing nominals, – special variables, true in
exactly one point. The hybrid extensions of Lτand Lτ(r)will be denoted by Ln
formulae which do not contain any propositional variables but only (possibly) nominals are called pure
formulae. Hybrid languages are often extended with the universal modality corresponding to a special
term U such that in the semantics RU= W × W, i.e., the largest relation in the frame F. A formula A
in a hybrid language is called a pure formula if it does not contain propositional variables.
A well known natural translation of a formula A (in any of the above mentioned modal languages)
into a firstorder formula ST(A,x) with only one free variable x can be defined, and we invite the reader
to consult [2] or [7] for its definition. Let us note that the standard translation of a pure formula is a
firstorder condition, a fact which will be used in the final stage of algorithm SQEMA and its extension
SQEMAsubfor obtaining the desired local firstorder equivalent of the input formula.
Lastly some terminology relating to the various notions of equivalence for formulae. Two Ln
formulas ϕ and ψ are semantically equivalent (denoted ϕ ≡ ψ) if they are true at exactly the same
states in all τmodels; locally frame equivalent if they are valid at exactly the same states in all τ
α(x,y)(=defRα(y,x)).
α(x,y1...,yi,...,yn) where R−i
α(x,y1...,yi,...,yn) is
τand Ln
τ(r)([7]). Hybrid
τ(r)
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6W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
frames; locally equivalent if they are valid at exactly the same states in all τgeneral frames.
1.2. Modal algebras
For a given similarity type τ, let F = (W,{Rα}α∈τ) be a τframe. For any (n + 1)place rela
tion Rαon W, not necessarily corresponding to a given modal τterm, we define two nary opera
tions over subsets A1,...,Anof W as follows: ?α?(A1,...,An) =def {x ∈ W : (∃y1...yn ∈
W)(Rα(x,y1,...,yn) and yi∈ Aifor all 1 ≤ i ≤ n} and [α](A1...,An) =def¬?α?(¬A1,...¬An).
Let B(F) be the Boolean algebra of all subsets of W augmented with the operations ?α?, α ∈ τ. We
will use the standard notion for the logical operations of negation ¬, conjunction ∧, and disjunction ∨,
to denote the corresponding Boolean operations of complement, meet and join, and 0 = ∅,1 = W will
be respectively the zero and the unit of the algebra.
The algebra obtained in this way will be called modal algebra over F. Modal algebras may have
richer signatures than the signature of the corresponding modal language. Namely, even if the modal
language does not contain inverses, we allow the application of the operations [α−i] and ?α−i? to subsets
of W, having in mind the natural assumption that Rα−i = R−i
Modal algebras can be used to simplify some semantical definitions, when one regards modal τ
formulae as polynomials over a modal algebra B(F) with propositional variables and nominals ranging
over subsets and singleton subsets of W, respectively, and modal operations [α] and ?α? interpreted as
the operations corresponding to the relation Rα. In this way modal formulae will denote subsets of W.
Now validity of a formula A in F is equivalent to the fact that the equation A = 1 is identically true in
B(F), i.e., A = 1 for all possible values of the variables and nominals occurring in A. Local validity at
a point x ∈ W is equivalent to the fact that x ∈ A is true identically.
Let us note that the above treatment of modal formulae as algebraic expressions in modal algebras
has some additional features, which will beused subsequently in the algorithm SQEMA and its extension
SQEMAsub. Namely, some modal expressions over modal algebras code, in some sense, local and global
firstorder conditions of the frame F. Let us explain this with some examples.
x ∈ {y} means x = y.
Let Rαbe a binary relation in W. Then:
x ∈ ?α?{y} means Rα(x,y)
x ∈ ?α−1?{x} means Rα(x,x)  local reflexivity of Rαat x.
x ∈ [α][α]?α−1?{x} means (∀y,z)(Rα(x,y) ∧ Rα(y,z) → Rα(x,z))  the local transitivity of Rα
at x.
We will also use the following algebraic facts:
A → B = 1 iff A ⊆ B,
A ⊆ B iff ¬A ∨ B = 1,
x ∈ A iff {x} ⊆ A iff ¬{x} ∨ A = 1,
x ?∈ A iff {x} ∈ ¬A iff ¬{x} ∨ ¬A = 1,
A ∨ [α](B1,...,Bi,...,Bn) = 1 iff [α−i](B1,...,A,...,Bn) ∨ Bi= 1,
x ∈ [α](B1,...,Bi,...,Bn) iff [α−i](B1...,¬{x},...,Bn) ∨ Bi= 1.
(α−i)−i= α.
α.
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions7
1.3.Ackermann’s lemma and SQEMA, informally
The main transformation rule of the algorithm SQEMA ([6, 7]) is the so called AckermannRule, the
details of which will be recalled in the next section. The AckermannRule is based on the Ackermann
lemma introduced by Ackermann in [1] for elimination of secondorder quantifiers. We owe to A. Szałas
[23] the idea to apply the Ackermann lemma (in its original formulation) in modal definability theory. In
[6, 7] we used a modal version of the Ackermann lemma, whereas here we will give an algebraic version
of this lemma. Similar algebraic treatment of Ackermann’s lemma and its generalizations can be found
also in [28] – [32]. The algebraic reformulation of this lemma, in standard mathematical language, can
be seen as the statement of a kind of necessary and sufficient condition for a special system of equations
in modal algebras to have a solution. This makes the lemma more readily understandable and illustrates
the intuition behind SQEMA. Notwithstanding the extreme simplicity of its proof, this lemma is most
fruitfully applicable.
Lemma 1.1. Modal Ackermann lemma: an algebraic form. Let B(F) be a modal algebra over a
given τframe F = (W,R). Let A and B(q) be modal formulae over B(W) such that A does not contain
the variable p and B(q) be a formula having only positive occurrences of the variable q. Consider the
following system of equations with respect to p:
?????
Then (∗) has a solution for p in B(F) iff B(A) = 1.
Proof:
(⇒) Suppose that (∗) has a solution for p. Then A ∨ p = 1, which is equivalent to ¬p ⊆ A. Since B(q)
has only positive occurrences of q, it is upward monotone with respect to q. Hence B(¬p) ⊆ B(A) and,
since B(¬p) = 1, B(A) = 1.
(⇐) If B(A) = 1, then p = ¬A is a solution of (∗).
Now we will show how to apply Lemma 1.1 to obtain local firstorder equivalents of modal formulae.
As an example, consider the formula [α]p → [α][α]p. Let F = (W,{Rα}α∈τ) be a τframe and x ∈ W.
The local condition at x for this formula is (see the previous section) (∀p ⊆ W)(x ∈ ([α]p → [α][α]p)).
We will perform the following sequence of equivalent transformations of this condition and at the
end we will obtain the desired firstorder local equivalent.
(1) (∀p ⊆ W)(x ∈ ([α]p → [α][α]p)) iff
(2) ¬¬(∀p ⊆ W)(x ∈ [α]p → x ∈ [α][α]p) iff
(3) ¬(∃p ⊆ W)(x ∈ [α]p and x ?∈ [α][α]p) iff
(4) ¬(∃p ⊆ W)(¬{x} ∨ [α]p = 1 and x ∈ ¬[α][α]p) iff
(5) ¬(∃p ⊆ W)([α−1]¬{x} ∨ p = 1 and ¬{x} ∨ ¬[α][α]p = 1) iff
(6) ¬(∃p ⊆ W)([α−1]¬{x} ∨ p = 1 and ¬{x} ∨ ¬[α][α]¬¬p = 1) iff (by Lemma 1.1, with
B(q) = ¬{x} ∨ ¬[α][α]¬q))
(7) ¬(¬{x} ∨ ¬[α][α]¬[α−1]¬{x} = 1) iff
(8) ¬(x ?∈ [α][α]?α−1?{x}) iff
(9) x ∈ [α][α]?α−1?{x} iff
(∗)
A ∨ p
B(¬p)
=
=
1
1.
? ?
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8W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
(10) (∀y,z)(Rα(x,y) ∧ Rα(y,z) → Rα(x,z)) — the local transitivity of Rαat x.
Note that from (2) to (6) we produce only transformations after the negation sign (the first negation
step) which is needed in order to turn the universal sentence (1) into an existential form and then to
prepare the equations for an application of the modal Ackermann lemma (in step (6)). In (8) we apply
the second negation step and obtain the needed local condition in a “coded” modal form, which in (10)
is “decoded” in its firstorder format.
The formal versions of SQEMA performs all these steps following strictly defined syntactic formal
transformation rules over some systems of “equations” which are analogs of the algebraic equations of
the above informal example.
1.4.The algorithm SQEMA
This subsection recalls the highlevel description of the algorithm SQEMA and its transformation rules,
and also some of its metaproperties.
1.4.1.Description of SQEMA.
Here we present briefly the basic algorithm SQEMA for reader’s convenience; for more detail see [6, 7].
First, some terminology — an expression of the form ϕ ∨ ψ with ϕ,ψ ∈ Ln
equation. A finite set of SQEMAequations is called a SQEMAsystem. For a system Sys, we let
Form(Sys) be the conjunction of all equations in Sys. Given a formula ϕ ∈ Lτ as input, SQEMA
processes it in three phases, with the goal to reduce ϕ first to a suitably equivalent pure, and then first
order formula.
Phase 1 (preprocessing) — The negation of ϕ is converted into negation normal form, and 3 and ∧
are distributed over ∨ as much as possible, by applying the equivalences 3(ψ ∨ γ) ≡ 3ψ ∨ 3γ and
δ ∧ (ψ ∨ γ) ≡ (δ ∧ ψ) ∨ (δ ∧ γ). For each disjunct of the resulting formula?ϕ?
of evaluation in a model, and not allowed to occur in the input formula ϕ. These are the initial systems
in the execution.
Phase 2 (elimination) — The algorithm now proceeds separately on each initial system, Sysi, by
applying to it the transformation rules listed below in section 1.4.2 (table 1). The aim is to eliminate from
the system all occurring propositional variables. If this is possible for each system, we proceed to phase
3, else the algorithm report failure and terminates. The rules in table 1 are to be read as rewrite rules,
i.e., they replace equations in systems with new equations or, in the case of the Ackermannrule, systems
with new systems. Note that each actual elimination of a variable is achieved through an application of
the Ackermannrule while the other rules are used to solve the system for the variable to be eliminated,
i.e., to bring the system into the right form for the application of this rule.
Phase 3 (translation) — This phase is reached only if all systems have been reduced to pure systems,
i.e., systems Sysiwith Form(Sysi) a pure formula. Let Sys1,...,Sysnbe these systems. Recall
ing that ϕ was the input to the algorithm, we will write pure(ϕ) for the formula (Form(Sys1) ∨ ··· ∨
Form(Sysn)). The algorithm now computes and returns, as local frame correspondent for the input
formula ϕ, the formula ∀y∃x0ST(¬pure(ϕ),x0) where y is the tuple of all occurring variables corre
sponding to nominals, but with yi(corresponding to the designated current state nominal i) left free,
since a local correspondent is being computed.
τ(r)is called a SQEMA
ia system Sysiis
formed consisting of the single equation ¬i ∨ ϕ?
i, where i is a reserved nominal used to denote the state
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions9
1.4.2.The transformation rules of SQEMA
Table 1 lists the transformation rules used by SQEMA. We have added the ∨rule in order to simplify
the Ackermann rule from [7] by enabling all equations of the type A∨p to be put together into one. Note
that, for monadic modalities, the 2 and 3rules simplify as follows:
(Monadic 2rule)
A ∨ [α]B
[α−1]A ∨ B
(Monadic inverse 2rule)
A ∨ [α−1]B
[α]A ∨ B
(Monadic 3rule)
¬j ∨ ?α?A
¬j ∨ ?α?k, ¬k ∨ A
where α is any unary modal term, and k is a fresh nominal not occurring in the premise. The algo
rithm can be strengthened further by adding more transformation rules facilitating some propositional
reasoning, as is done in [6, 7].
1.4.3. Some metaproperties of SQEMA
A. Correctness
A formula on which SQEMA succeeds will be called a SQEMAformula.
Theorem 1.1. (Correctness of SQEMA, [7])
Every SQEMAformula is locally framecorrespondent to the firstorder formula returned.
B. Canonicity
For a definition of descriptive frames see e.g., [7].
A formula ϕ is locally dpersistent, if, for every pointed descriptive frame (F,w) for the respective
language, it is the case that (F?,w) ? ϕ whenever (F,w) ? ϕ; ϕ is dpersistent if F?? ϕ whenever
F ? ϕ. Clearly, local dpersistence implies dpersistence.
Theorem 1.2. (Dpersistence,[7])
1. Every SQEMAformula in Lτis locally persistent with respect to the class of all descriptive τ
frames.
2. Every SQEMAformula in Lr(τ)is locally persistent with respect to the class of all reversive
descriptive τframes.
Corollary 1.1. (Canonicity of SQEMA, [7])
All formulae on which SQEMA succeeds are canonical.
C. Completeness
For the definition of polyadic inductive formula we refer to the paper [17] (see also [15, 16, 7]. Later
on in the paper these formulae will be discussed as special case of complex inductive formulae.
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10W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Table 1.
SQEMA Transformation Rules
Rules for connectives
C ∨ (A ∧ B)
C ∨ A,C ∨ B
(∧rule)
A ∨ C,B ∨ C
(A ∧ B) ∨ C
(∨rule)
C ∨ (A ∨ B)
(C ∨ A) ∨ B
(leftshift ∨rule)
(C ∨ A) ∨ B
C ∨ (A ∨ B)
(rightshift ∨rule)
A ∨ [γ](B1,...,Bn)
[γ−i](B1,...,Bi−1,A,Bi+1,...,Bn) ∨ Bi
(2rule)
A ∨ [γ−i](B1,...,Bn)
[γ](B1,...,Bi−1,A,Bi+1,...,Bn) ∨ Bi
(inverse 2rule)
¬j ∨ ?γ?(A1,...,An)
¬j ∨ ?γ?(k1,...,kn), ¬k1∨ A1,...,¬kn∨ An
(3rule∗)
∗where the kiare new nomi
nals not occurring in the sys
tem.
Polarity switching rule
Substitute ¬p for every occurrence of p in the system.
Ackermannrule
The system
?????????????????
A ∨ p
B1(p)
···
Bm(p)
C1
···
Ck
is replaced by
??????????????????
B1(A/¬p)
...
Bm(A/¬p)
C1
···
Ck
where:
1. p does not occur in A,C1,...,Ck;
2. Form(B1) ∧ ··· ∧ Form(Bm) is negative in p.
3. Bi(A/¬p) means that ¬p in Biis replaced by A.
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions11
Theorem 1.3. (Completeness of SQEMA w.r.t. inductive formulae, [7])
SQEMA succeeds on all conjunctions of polyadic inductive formulae.
Examples of how SQEMA works on different formulae, including polyadic inductive formulae, can
be found in [7]. Later we will present an example of inductive complex modal formula (to be defined in
the next section) on which SQEMA fails.
2.Inductive complex modal formulae and reversible substitutions
In this section we introduce two large classes of modal formulae: the recursive complex modal formu
lae (RCMformulae) and their subclass of inductive complex modal formulae (ICMformulae).
The class of the ICMformulae was introduced in [27] under the name ‘complex polyadic Sahlqvist
formulae’. It simultaneously extends both the class of inductive formulae and the class of complex
Sahlqvist formulae [26]. The adjective ‘complex’ comes from the fact that these formulae are built
over some special Boolean formulae, called ‘complex variables’ in [26]. All ICMformulae are first
order definable and canonical, but the current version of SQEMA does not succeed on all of them (see
examples 2.2 and 2.3, below). One of our main objectives in this paper is to extend SQEMA with
an additional module which performs special substitutions which enables it to succeeds on all ICM
formulae.
2.1.Substitutions
We adopt the standard definition of (uniform) substitution ([2]) as a mapping in the set of formulae
acting on them homomorphically. This means that a substitution S can be defined if we first specify it
on propositional variables and then extend it by induction for arbitrary formulae as follows: S(¬A) =
¬S(A), S(A ◦ B) = S(A) ◦ S(B) where ◦ is any binary Boolean connective, and S[α](A1,...,An) =
[α](S(A1),...,S(An))
We will usually denote substitutions by S,T. Sometimes we will be interested in substitutions acting
on a fixed set of propositional variables. In such a case we assume that they act on all other variables
identically. The following observation is immediate.
Fact 2.1. Local frame validity is preserved by uniform substitutions and by modus ponens.
If A,B(p) ∈ Ln
uniform substitution of A for all occurrences of p.
r(τ)we will write B(A/p), or simply B(A), for the formula obtained from B(p) by
Definition 2.1. Let p = ?p1,...,pn? and q = ?q1,...,qm? be two disjoint lists of different proposi
tional variables, and S a substitution which maps the variables in p to formulae built over q, and acts
identically on variables not in p. We say that S is a reversible substitution if there is a substitution T
that maps the variables in q to formulae built over p, acts on variables not in q identically, and is such
that T(S(pi)) ≡ pi, for i = 1,...,n (and, consequently, T(S(A)) ≡ A for any formula A containing
only propositional variables in p). We then say that p is the domain of S, q is a codomain of S, and T
reverses S.
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12W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Note that if a substitution T reverses a substitution S, then T need not be reversible itself, because of its
action on variables not in the range of S. The following lemma follows immediately from fact 2.1 and
the definition of a reversible substitution.
Lemma 2.1. Let S be a reversible substitution with a domain p and a codomain q. Then A is locally
equivalent to S(A) for every formula A.
Clearly, the requirement for S to be a reversible substitution is essential, e.g., consider S such that
S(p) = q ∨ ¬q and take A = p.
A simple example of reversible substitutions, as employed by SQEMA, is polarity change: S(p) :=
¬p. Nontrivial examples of reversible Boolean substitutions can be found in [26, 29]; more such exam
ples are provided further in the paper.
2.2. Substitutions producing inductive formulae: an informal discussion
Since the definitions of recursive and inductive complex modal formulae are complicated, we will begin
with some concrete motivating examples. For simplicity we will start with an example in the basic
monomodal language with the usual box and diamond modalities ? and 3. Consider the formula
A = 3?(p1∨ p2) ∧ 3?(p1∨ ¬p2) ∧ 3?(¬p1∨ p2) → ?3(p1∧ p2).
It is not a Sahlqvist formula, nor even an inductive one, and the standard Sahlqvistvan Benthem substi
tution method does not work on it. However, note that p1∧ p2≡ (p1∨ p2) ∧ (p1∨ ¬p2) ∧ (¬p1∨ p2),
hence A can be obtained, up to local equivalence, from the formula
A?= 3?q1∧ 3?q2∧ 3?q3→ ?3(q1∧ q2∧ q3).
by applying the substitution:
T(q1)
T(q2)
T(q3)
=
=
=
p1∨ p2,
p1∨ ¬p2,
¬p1∨ p2.
The formula A?is a Sahlqvist formula and it locally corresponds to the following ChurchRosserlike
firstorder property of a binary relation R:
xRy1∧ xRy2∧ xRy3∧ xRy4→ (∃z)(y1Rz ∧ y2Rz ∧ y3Rz ∧ y4Rz).
Thus, A is a local consequence from A?. Conversely, A?can be obtained, up to local equivalence, from
A by means of the following substitution:
(CR4)
S(p1)
S(p2)
=
=
q1∧ q2,
(q1∧ ¬q2) ∨ (q1∧ q3).
After simple Boolean transformations one can obtain the formula
A??= 3?q1∧ 3?(¬q1∨ q2) ∧ 3?(¬q1∨ ¬q2∨ q3) → ?3(q1∧ q2∧ q3).
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions13
Using the valid implications q2→ ¬q1∨ q2and q3→ ¬q1∨ ¬q2∨ q3, and the monotonicity of ? and
3, one can then easily obtain A?as a local consequence from A??.
Thus, the two formulae A and A?are locally equivalent. In particular, the formula A locally cor
responds to the firstorder formula (CR4), too1. Furthermore, note that the substitution T reverses the
substitution S: T(S(pi)) ≡ pi, for every variable piin the domain of S.
In order to see the general pattern of transformation between A and A?, let us denote D1= p1∨ p2,
D2= p1∨ ¬p2and D3= ¬p1∨ p2. Then A can be presented in the following way:
A = 3?D1∧ 3?D2∧ 3?D3→ ?3(D1∧ D2∧ D3).
Notationally, A and A?look quite similar, the only difference being that the elementary disjunctions
D1, D2and D3in A replace the variables q1,q2and q3in A?. In [26] such elementary disjunctions
are called complex variables, because they code in some way ordinary variables, and the respective
extension of the Sahlqvist class defined in [26] — complex Sahlqvist formulae. It is not, however, true
in general that complex formulae can be obtained from Sahlqvist formulae simply by replacing their
different variables by different elementary disjunctions as in the above example, because nonfirstorder
definable modal formulae can be obtained in such a way, too. Consider, for instance, the Sahlqvist
formula 3?q1→ 3?(q1∧ q2) ∨ 3?(q1∧ q3) and replace q1,q2,q3by D1,D2,D3, respectively. After
some Boolean simplifications we obtain the formula 3?(p1∨ p2) → 3?p1∨ 3?p2, which is not
firstorder definable [33].
The next example of complex formula is in a polyadic language, where α,β, and γ are modal terms
of suitable arities:
B = [α](¬[β](¬p1∨ p2),¬[β](p1∨ ¬p2),¬[β](p1∨ p2),?γ?((¬p1∨ p2),(p1↔ p2),(p1∧ p2))).
Modulo semantic equivalences the formula B can be rewritten, using the elementary disjunctions D1,
D2, D3, as:
B = [α](¬[β]D3,¬[β]D2,¬[β]D1,?γ?(D3,D3∧ D2,D3∧ D2∧ D1)).
That formula can be obtained by an obvious substitution from
B?= [α](¬[β]q1,¬[β]q2,¬[β]q3,?γ?(q1,q1∧ q2,q1∧ q2∧ q3)).
Here B is a complex formula and B?is an inductive formula with headed boxes [β]q1, [β]q2and [β]q3
and a negated headless box (positive part) ?γ?(q2,q2∧ q3,q2∧ q3∧ q1). Note that B and B?are again
related by means of a pair of reversible substitutions S?and T?, where:
T?(q1) = D3, T?(q2) = D2, T?(q3) = D1.
S?(p1) = ¬q1∨ (q2∧ q3), S?(p2) = q1∧ (¬q2∨ q3).
From the example above one can see that the complex variables D1,D2,D3appear in B only as the
heads of the headed boxes and in the positive part ?γ? in special ‘complex blocks’ based on a given order
1This was first established by using the algorithm SCAN, thanks to a suggestion of Andreas Herzig, before complex formulae
were introduced.
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14W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
of the complex variables. This special structure of the complex formulae is needed to guarantee their
translation into inductive formulae by means of the substitutions T,S, respectively T?,S?.
The question of how to find such suitable substitutions arises. For instance, the definitions of T and
T?are obvious, but how to find S and S?? We will show later that substitutions like S and S?can be
effectively computed from the form of the given complex formula as a solution of a special system of
Boolean equations corresponding to that formula. In order to give some preliminary intuition we present
that system of equations for the case of the second example. Looking at the formula B?we see that the
substitution S?should satisfy the following equations:
S?(D3) ≡ q1, S?(D3∧ D2) ≡ q1∧ q2, S?(D3∧ D2∧ D1) ≡ q1∧ q2∧ q3.
In this system S?is an unknown substitution, which has to be extracted from the given equations. Using
the fact that S?should be a substitution, the system can be transformed equivalently to the following one
which has to be solved with respect to S?(p1),S?(p2):
???????
equation, and do the same with the second and third equations) this system can be transformed into the
following equivalent one:
???????
on the right in the second and third equations and simplifying, one can obtain S?(p1) ≡ ¬q1∨(q2∧q3);
then, by taking the conjunctions on the left and on the right in the first and third equations, one can obtain
S?(p2) ≡ q1∧ (¬q2∨ q3) — just what was expected.
The system for the substitution T?is the following one:
q1≡ (¬S?(p1) ∨ S?(p2))
q1∧ q2≡ (¬S?(p1) ∨ S?(p2)) ∧ (S?(p1) ∨ ¬S?(p2))
q1∧ q2∧ q3≡ (¬S?(p1) ∨ S?(p2)) ∧ (S?(p1) ∨ ¬S?(p2)) ∧ (S?(p1) ∨ S?(p2))
By easy Boolean manipulations (negate both part of the first equation and add disjunctively to the second
(1)
q1≡ ¬S?(p1) ∨ S?(p2)
¬q1∨ q2≡ S?(p1) ∨ ¬S?(p2)
¬q1∨ ¬q2∨ q3≡ S?(p1) ∨ S?(p2)
,
(2)
Now, (2) can be easily solved with respect to S?(p1) and S?(p2): by taking conjunctions on the left and
T?(q1) ≡ D1, T?(q1∧ q2) ≡ D1∧ D2, T?(q1∧ q2∧ q3) ≡ D1∧ D2∧ D3.
An obvious solution of this system with respect to T?(q1),T?(q2),T?(q3) is: T?(q1) = D1, T?(q2) = D2
and T?(q3) = D3.
To conclude this informal discussion, let us mention that the polyadic version of SQEMA from [7]
does not succeed on the formula B, but it succeeds on its equivalent inductive formula B?(because
SQEMA succeeds on all inductive formulae). So, the intuitive idea for the extension of SQEMA is to
supply it with a subprocedure based on substitutions like S and S?, whence the name of this extension:
“SQEMA with substitutions”, hereafter denoted SQEMAsub.
2.3.Formal definitions of the classes of RCM and ICMformulae
Let p be a propositional variable. Denote p0=def¬p and p1=defp. Let p = ?p1,...,pn? be a list of
different variables. Formulae of the form D = pi1
1∨...∨pin
n(in this order of the variables) will be called
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions15
elementary disjunctions. If p = ?p1? then we have only two (degenerate) elementary disjunctions, p1
and ¬p1. There are of course exactly 2nnonequivalent elementary disjunctions of p = ?p1,...,pn?.
Sometimes we will consider a fixed order of all elementary disjunctions: D1,...,D2n. If we consider
{0,1}strings (i1,...,in) as binary codes of the integers then there are two natural orderings of the
sequence of Di’s: the increasing order – starting from (0,...,0) = 0 and ending with (1,...,1) =
2n− 1, and the decreasing order which is just the opposite of the increasing order.
Let p = ?p1,...,pn? and let the elementary disjunctions built from p have the same order of the
variables. Let D(p) = ?D1,...,D2n−1? be a fixed sequence of different elementary disjunctions (the
lastoneD2n ismissing), andletD∗(p) = ?D1,D1∧D2,...,D1∧···∧D2n−1?. Thepair?D(p),D∗(p)?
will be called a propositional complex (of dimension n). The disjunctions in D(p) will be called
complex variables (of dimension n), and the elements from D∗(p) will be called complex blocks (of
dimension n). We say that the a complex ?D(p),D∗(p)? is disjoint from a complex ?D(q),D∗(q)? if
the strings p and q do not share variables.
As we have seen in section 2.2, elementary disjunctions Dican be used to “code” in some sense
ordinary variables. This suggests that one could see the elementary disjunctions Dias a new kind of
“complex” variable.
Let Σ be a set of pairwise disjoint propositional complexes. Let A(p1,...,pk) be a formula and
B1,...,Bkbe a list of complex blocks from Σ. Then A(B1,...,Bk) is called a complex atom in
Σ. If pioccurs in A(p1,...,pk) only positively (negatively) then Bioccurs in A(B1,...,Bk) positively
(negatively). If A(p1,...,pk) is a positive (negative) formula then A(B1,...,Bk) is a positive (negative)
complex atom in Σ. A complex essentially boxformula in Σ is any formula of the type
B = [β](D,N1,...,Nm) = [β](D,− →
N),
where β is a modal term of arity m+1,− →
D is a complex variable from Σ.2A formula of this type is called a headed complex box, where D is
the head of B and− →
N is the negative part of B.
Note that in ¬B = ¬[β](D,N1,...,Nm) the head D has a negative occurrence and all complex
blocks in N1,...,Nmhave positive occurrences. Also note that β can be a composed modal term. If, for
instance, α and β are unary terms, then the formula [α]p ∨ [β]q can be represented as [γ](p,q), where γ
is the following composition γ = ι2(α,β).
Recall that a constant formula is a formula not containing propositional variables.
N = N1,...,Nmis a string of negative complex atoms in Σ and
Definition 2.2. A recursive complex modal formula (RCMformula for short) in Σ is any constant
formula or a formula A = [α](¬B1,...,¬Bm,C1,...,Cn) where B1,...,Bnare complex essentially
box formulae in Σ, C1,...,Cnare positive complex atoms in Σ, and where both m and n may be zero.
The formulae Bi, 1 ≤ i ≤ m, are called the headed boxes of A, while the formulae Cj, 1 ≤ j ≤ n, are
called the positive components of A. Note that all heads in A have only negative occurrences and all
complex blocks (other than heads) in A have only positive occurrences.
InthecasethatallpropositionalcomplexesofAareoftheformP = ??p?,?p??(i.e., ofdimensionone
and with the decreasing order of the elementary disjunctions), A is called a recursive modal formula,
or RMformula for short. Thus, RMformulae have no (nondegenerate) complex variables.
2Here D need not be only in the first argument place, but we put it first for simplicity of notation.
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