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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.uni-sofia.bg
Abstract. In earlier papers we have introduced an algorithm, SQEMA, for computing first-order
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 d-persistence of formulae on which
the algorithm succeeds, and show that SQEMAsubis complete with respect to the class of complex
inductive formulae.
Keywords: SQEMA, correspondence, d-persistence, 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 first-order 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 first-order
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 first-order logic with least fix points [8, 13, 17, 20,
31, 32, 35, 36].
Because of the undecidability of the class of first-order definable modal formulae [3], the hierarchy
of effective extensions of the Sahlqvist class concerning first-order 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 (Second-Order
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 d-persistent
(respectively, locally di-persistent for the case of languages with nominals and converse modalities), and
hence canonical in the respective senses.
With respect to first-order correspondence, our approach was preceded and influenced by two earlier
developed algorithms for the elimination of second-order 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 second-order logic, attempts to eliminate all occurring existentially quan-
tified predicate variables and thus to compute a first-order 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 second-order 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 (de-Skolemization, or un-Skolemization)
is applied. That procedure is not always successful, which may lead to (sometimes unnecessary) failure
of the main algorithm. To avoid the necessity for de-Skolemization, SQEMA does not use the stan-
dard translation into the first-order 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 first-order 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 second-order logic. Further
information about SCAN, DLS, and SQEMA can be found in the recent book on second-order 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.uni-sofia.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 first-order equivalents and d-persistence), 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 first-order definable and canonical, thus implying that this
is the currently largest effective extension of the Sahlqvist class of first-order 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 meta-properties 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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4 W. 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+1-ary 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+1-ary
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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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 τ-frames. 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 first-order 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
first-order condition, a fact which will be used in the final stage of algorithm SQEMA and its extension
SQEMAsubfor obtaining the desired local first-order 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 τ-models; 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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frames; locally equivalent if they are valid at exactly the same states in all τ-general frames.
1.2. Modal algebras
For a given similarity type τ, let F = (W,{Rα}α∈τ) be a τ-frame. For any (n + 1)-place rela-
tion Rαon W, not necessarily corresponding to a given modal τ-term, we define two n-ary 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
first-order 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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1.3. Ackermann’s lemma and SQEMA, informally
The main transformation rule of the algorithm SQEMA ([6, 7]) is the so called Ackermann-Rule, the
details of which will be recalled in the next section. The Ackermann-Rule is based on the Ackermann
lemma introduced by Ackermann in [1] for elimination of second-order 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 τ-frame 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 first-order equivalents of modal formulae.
As an example, consider the formula [α]p → [α][α]p. Let F = (W,{Rα}α∈τ) be a τ-frame 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 first-order 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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(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 first-order 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 high-level description of the algorithm SQEMA and its transformation rules,
and also some of its meta-properties.
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 SQEMA-equations is called a SQEMA-system. 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 Ackermann-rule, systems
with new systems. Note that each actual elimination of a variable is achieved through an application of
the Ackermann-rule 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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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 3-rules simplify as follows:
(Monadic 2-rule)
A ∨ [α]B
[α−1]A ∨ B
(Monadic inverse 2-rule)
A ∨ [α−1]B
[α]A ∨ B
(Monadic 3-rule)
¬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 meta-properties of SQEMA
A. Correctness
A formula on which SQEMA succeeds will be called a SQEMA-formula.
Theorem 1.1. (Correctness of SQEMA, [7])
Every SQEMA-formula is locally frame-correspondent to the first-order formula returned.
B. Canonicity
For a definition of descriptive frames see e.g., [7].
A formula ϕ is locally d-persistent, 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 d-persistent if F?? ϕ whenever
F ? ϕ. Clearly, local d-persistence implies d-persistence.
Theorem 1.2. (D-persistence,[7])
1. Every SQEMA-formula in Lτis locally persistent with respect to the class of all descriptive τ-
frames.
2. Every SQEMA-formula in Lr(τ)is locally persistent with respect to the class of all reversive
descriptive τ-frames.
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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10 W. 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
(left-shift ∨-rule)
(C ∨ A) ∨ B
C ∨ (A ∨ B)
(right-shift ∨-rule)
A ∨ [γ](B1,...,Bn)
[γ−i](B1,...,Bi−1,A,Bi+1,...,Bn) ∨ Bi
(2-rule)
A ∨ [γ−i](B1,...,Bn)
[γ](B1,...,Bi−1,A,Bi+1,...,Bn) ∨ Bi
(inverse 2-rule)
¬j ∨ ?γ?(A1,...,An)
¬j ∨ ?γ?(k1,...,kn), ¬k1∨ A1,...,¬kn∨ An
(3-rule∗)
∗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.
Ackermann-rule
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 substitutions 11
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 (RCM-formulae) and their subclass of inductive complex modal formulae (ICM-formulae).
The class of the ICM-formulae 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 ICM-formulae 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 co-domain of S, and T
reverses S.
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12 W. 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 co-domain 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. Non-trivial 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
mono-modal 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 Sahlqvist-van 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 Church-Rosser-like
first-order 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 first-order 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 non-first-order
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
first-order 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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14 W. 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 sub-procedure 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 ICM-formulae
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 2nnon-equivalent 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 box-formula 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 (RCM-formula 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 RM-formula for short. Thus, RM-formulae have no (non-degenerate) 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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16W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Remark 2.1. Wecangeneralizedefinition2.2slightlybyallowingsimplificationsofthecomplexblocks.
For example, the complex block (p1∨p2)∧(p1∨¬p2)∧(¬p1∨p2) can be simplified to p1∧p2. However,
when proving theorems for RCM-formulae, we will always assume that they have not been simplified.
Let A = [α](¬B1,...,¬Bm,C1,...,Cn) be a RCM-formula and P be a propositional complex of A. We
say that P is essential for A if some of the heads of A belong to P.
Definition 2.3. The dependency digraph of an RCM-formula A = [α](¬B1,...,¬Bm,C1,...,Cn) is
the digraph G = (VA,EA), where the vertex set VA= {P1,...,Pk} is the set of all essential propositional
complexes of A, and PiEAPjholds if a complex block from the propositional complex Pioccurs in a
negative component of a formula Blfrom B1,...,Bmsuch that the head of Blis from the propositional
complex Pj. A digraph is acyclic if it contains no directed cycles or loops.
Definition 2.4. An RCM-formula A is called an inductive complex modal formula, or an ICM-formula
for short, if the dependency digraph of A is acyclic. If the dependency digraph of A has no arcs at all,
then A is called a simple ICM-formula. In the case that all propositional complexes of A are of the form
P = ??p?,?p?? (i.e., of dimension one and with the decreasing order of the elementary disjunctions), A
is called an inductive modal formula, or an IM-formula for short.
The formulae A,B in the examples from subsection 2.2 are simple ICM-formulae, while the formulae
A?,B?are inductive formulae. In order to see this for A, we rewrite it in the following box form:
?¬?D1∨ ?¬?D2∨ ?¬D3∨ ?3(D1∧ D2∧ D3).
If we denote by α the unary modal term corresponding to ?, the formula above can be presented as:
[β](¬[α]D1,¬[α]D2,¬[α]D3,?α?(D1∧ D2∧ D3)),
where β = ι4(α,α,α,α). In this form the formula is obviously a simple ICM-formula with heads
D1,D2,D3and only one complex block D1∧ D2∧ D3.
Here is an example of ICM-formula which is not a simple one:
Example 2.1. C = [α](¬[β](p1∨ p2),¬[β](p1∨ ¬p2),¬[β](¬p1∨ p2),¬[γ](q,N((p1∨ p2),((p1∨
p2) ∧ (p1∨ ¬p2)),((p1∨ p2) ∧ (p1∨ ¬p2) ∧ (¬p1∨ p2)),Pos(q)) where α, β, and γ are modal terms
of respective arities, N(·,·,·) is a negative formula built from three different variables, and Pos(q) is a
positive formula built from the variable q. C has two propositional complexes, namely, P, containing
the complex variables (p1∨p2),(p1∨¬p2),(¬p1∨p2), and Q, containing only the degenerate complex
variable q. The dependency digraph has only one arc from P to Q.
Remark 2.2. Inductive complex modal formulae are obvious generalizations of inductive modal formu-
lae introduced in [17] and, under another name, in [26]. The adjective inductive comes from the fact that
their first-order equivalents can be computed by a procedure using simple induction. Recursive complex
formulae are obvious generalization of recursive modal formulae which were introduced in [17] under
the name regular formulae. It was proven in [17] (see also [8]) that regular formulae have equiva-
lents in the extension of first-order logic with least fix points which are solutions of systems of recursive
equations, hence the name recursive modal formulae.
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions17
2.4. Examples of ICM-formulae for which SQEMA fails
SQEMA succeeds on some complex modal formulae. For instance, the formula A = 3?(p1∨ p2) ∧
3?(p1∨¬p2)∧3?(¬p1∨p2) → ?3(p1∧p2) discussed in Section 2.2 is one of the examples in [6] for
which the monadic SQEMA succeeds. However, this is not always the case, as the following examples
illustrate:
Example 2.2. Consider the ICM-formula:
ϕ = [3](¬[1](p1∨ ¬p2),¬[1](¬p1∨ p2),?2?((p1∨ ¬p2),(p1∨ ¬p2) ∧ (¬p1∨ p2))),
where 1,2,3 are modal terms of arities 1, 2, and 3, respectively. Let us see that SQEMA fails on ϕ.
Step 1 Negating and moving the negation inside, we obtain
¬ϕ ≡ ?3?([1](p1∨ ¬p2),[1](¬p1∨ p2),[2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))).
Step 2 The initial system of SQEMA-equations:
??? ¬i ∨ ?3?([1](p1∨ ¬p2),[1](¬p1∨ p2),[2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))) .
Step 3 Applying the 3 rule yields:
??????????
Step 4 We choose to try and eliminate p1first. We apply the ?-rule to the second equation:
??????????
Step 5 After applying commutativity of disjunction, and then the Left-shift rule to the second equation
we obtain:
??????????
negative nor positive in the fourth equation, so the Ackermann-rule cannot be applied.
¬i ∨ ?3?(j1,j2,j3)
¬j1∨ [1](p1∨ ¬p2)
¬j2∨ [1](¬p1∨ p2)
¬j3∨ [2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))
.
¬i ∨ ?3?(j1,j2,j3)
?1−1?¬j1∨ (p1∨ ¬p2)
¬j3∨ [2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))
¬j2∨ [1](¬p1∨ p2)
.
¬i ∨ ?3?(j1,j2,j3)
([1−1]¬j1∨ ¬p2) ∨ p1
¬j2∨ [1](¬p1∨ p2)
¬j3∨ [2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))
.
Now we obtain p1separated in the second equation and negative in the third equation, but neither
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18 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Step 6 Now, we try to eliminate p2and apply the ?-rule to the third equation:
??????????
Step 7 After applying the Left-shift rule to the third equation we obtain:
??????????
Now, p2is separated in the third equation, but the Ackermann-rule cannot be applied because of the
fourth equation. The only step which can be taken is to change polarity of p1or of p2and then to try the
elimination procedure again (we invite the reader to do this). But, again, we will reach similar situation,
because the fourth equation will be neither positive nor negative with respect to p1and p2. So, the
algorithm returns FAIL.
¬i ∨ ?3?(j1,j2,j3)
([1−1]¬j1∨ ¬p2) ∨ p1
?1−1?¬j2∨ (¬p1∨ p2)
¬j3∨ [2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))
.
¬i ∨ ?3?(j1,j2,j3)
([1−1]¬j1∨ ¬p2) ∨ p1
([1−1]¬j2∨ ¬p1) ∨ p2
¬j3∨ [2]((¬p1∧ p2),(¬p1∧ p2) ∨ (p1∧ ¬p2))
.
There are many examples of ICM-formulae for which SQEMA fails for similar reasons. We mention
two more, without proof.
Example 2.3. Firstly, the formula B discussed in section 2.2:
B = [α](¬[β](¬p1∨ p2),¬[β](p1∨ ¬p2),¬[β](p1∨ p2),?γ?((¬p1∨ p2),(p1⇔ p2),(p1∧ p2))).
Secondly, the following (simplified) ICM-formula in the standard monomodal language (see also [30]):
3?(p1∨ ¬p2) ∧ 3?(¬p1∨ p2) ∧ 3?(p1∨ p2) → ?(p1∨ ¬p2) ∨ ?♦(p1↔ p2) ∨ ?♦?(p1∧ p2).
3. Complex substitutions
In this section we will prove some results that guarantee the existence of reversible substitutions for
complex formulae.
Lemma 3.1. Let p = ?p1,...,pn? be a list of different propositional variables and let D1,...,D2n be
a list of all (different) elementary disjunctions from p. Then:
(i) if i ?= j then Di∨ Dj≡ ?,
(ii)
(iii) ¬Dj≡?2n
?2n
i=1Di≡ ⊥,
i=1,i?=jDi, j = 1,...,2n,
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 19
(iv) ¬D1∨ ··· ∨ ¬Di∨ Di+1≡ Di+1, i = 1,...,2n− 1.
(v)
?{Dl: pk∈ Dl} ≡ pk, k = 1,...,n.
Proof:
Claim (i) is obvious, (ii) is a well-known, (iii) and (iv) follow from (i) and (ii) by easy Boolean manipula-
tions. For (v) note that?{Dl: pk∈ Dl} ≡ pk∨?2n−1
Lemma 3.2. (First substitution lemma , [26])
Let p = ?p1,...,pn? be a list of different propositional variables and D1,...,D2n be a fixed list of
all elementary disjunctions of them. Let A1...A2n be an arbitrary list of propositional formulae not
containing variables from p. Then, the following two conditions are equivalent:
i=1D
?
i≡ pk∨⊥ ≡ pk, where D
?
iare all elementary
disjunctions built from the variables of p different from pk, which by (ii) is equivalent to ⊥.
? ?
1. There exists a substitution S acting on the variables p1,...,pnsuch that the following equations
hold:
??????????
2. The following two conditions hold for any i,j ≤ 2n:
(a) If i ?= j then Ai∨ Aj≡ ?,
(b)
(#1)
A1
A2
...
A2n
≡
≡
S(D1)
S(D2)
...
S(D2n).
≡
?2n
i=1Ai= ⊥.
Moreover, if the condition 2 is fulfilled, then the substitution S is uniquely determined by the equations
?
An example of how to solve a system like (#1) is given in Section 2.2.
(#2)
S(pk) =
{Al| pk∈ Dl,l ≤ 2n},k = 1,...,n.
Lemma 3.3. (Second substitution lemma , [26])
Let p = ?p1,...,pn?, be a sequence of different propositional variables, let D1,...,D2n be a sequence
of all elementary disjunctions of these variables, and let q = ?q1,...,q2n−1? be a sequence of proposi-
tional formulae not containing variables from p. Then:
1. There exists a substitution S (depending on the given order of the disjunctions Di) acting only on
the variables p1,...,pnand satisfying the following conditions:
??????????
(∗)
q1
q1∧ q2
...
q1∧ q2∧ ... ∧ q2n−1
≡
≡
S(D1)
S(D1∧ D2)
...
S(D1∧ D2∧ ... ∧ D2n−1).
≡
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20 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
2. The conditions (∗) uniquely determine S (up to Boolean equivalence) and S can be effectively
computed from them.
3. The substitution S also satisfies the following conditions:
(∗∗)
??????????
q1
q2
...
q2n−1
|=
|=
S(D1)
S(D2)
...
S(D2n−1).
|=
In [26] lemma 3.3 was proven by an application of Lemma 3.2 and the formulae defining S are given
in that proof. The proof contains the following fact, which we formulate now explicitly as Lemma 3.4,
from which Lemma 3.3 can be easily derived.
Lemma 3.4. Let the assumptions of Lemma 3.3 be fulfilled, and let S be a substitution acting on the
variables from the list p. Also let A1= q1, A2= ¬q1∨ q2, ..., Ai+1= ¬q1∨ ... ∨ ¬qi∨ qi+1, ...,
A2n = ¬q1∨ ... ∨ ¬q2n−1.
1. The following two conditions (a) and (b) are equivalent:
??????????
(b)
... ...
A2n
≡
2. If S satisfies condition (a) then it is determined by the following formulae
?
Proof:
(Sketch)
(a)
q1
q1∧ q2
...
q1∧ q2∧ ... ∧ q2n−1
A1
≡
A2
≡
≡
≡
S(D1)
S(D1∧ D2)
...
S(D1∧ D2∧ ... ∧ D2n−1).
≡
??????????
S(D1)
S(D2)
S(D2n).
S(pk) =
{Al: pk∈ Dl,l ≤ 2n},k = 1,...,n.
(S(pk))
1. An idea for the proof of (a) ⇒ (b) is given by some examples in Section 2.2, using Lemma 3.1.
The implication (b) ⇒ (a) can be proved by direct Boolean calculations on the left hand side and
then using Lemma 3.1 for the right hand side.
2. It is easy to see that the formulae Aisatisfy the conditions of Lemma 3.2, so we may apply it.
From that lemma and (1) we obtain (2).
? ?
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 21
Lemma 3.5. Let S be the substitution from Lemma 3.4 with the assumption that the propositional vari-
ables in q = ?q1,...,q2n−1? are different. Then S is reversible with domain p = ?p1,...,pn? and
co-domain q. In particular, the substitution T, defined by putting T(qi) = Di, i = 1,...,2n− 1,
reverses S.
Proof:
The explicit definition of S from Substitution lemma 3.4 is
?
Then, by Lemma 3.1 we obtain
S(pk) =
{¬q1∨ ... ∨ ¬ql−1∨ ql| pk∈ Dl,l ≤ 2n},k = 1,...,n.
T(S(pk))=
T(
?
?
{¬D1∨ ... ∨ ¬Dl−1∨ Dl| pk∈ Dl,l ≤ 2n}
{Dl| pk∈ Dl,l ≤ 2n} = pk.
{¬q1∨ ... ∨ ¬ql−1∨ ql| pk∈ Dl,l ≤ 2n})
=
=
?
? ?
Remark 3.1. Lemmas 3.3 and 3.5 together guarantee the existence of reversible substitutions. The
substitutions of the form S from Lemma 3.3 will be used in SQEMAsub. Let us note that S depends
on the given list of the variables p = ?p1,...,pn? and the sequence ?D1,...,D2n? of the elementary
disjunctions built from p. Another feature of the specific substitution S is that its codomain contains
2n− 1 variables where n is the number of variables of the domain of S. So, S produces formulae with
exponentially many more variables than the input formula. Let us also mention that we cannot claim that
the substitution T from Lemma 3.5 (which reverses S) is reversible.
4.The translation θ
The following theorem is the main technical statement in this section.
Theorem 4.1. (Translation Theorem)
For every RCM-formula A there exists an effectively computable translation θ such that the following
hold:
(i) θ(A) is an RM-formula, and if A is an ICM-formula, then θ(A) is an inductive modal formula.
(ii) A is locally equivalent to θ(A).
Proof:
Definition of the translation θ: The translation θ has a very simple definition – it replaces all complex
variables in A with new propositional variables (“new” here means “not appearing in A”), replacing
different complex variables by different new variables.
Consequently, θ transforms all complex blocks into positive formulae, hence it transforms A into
a recursive modal formula θ(A). Furthermore, if A is an ICM-formula, then it is easy to see that the
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22 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
dependency digraph of θ(A) does not contain cycles, and therefore θ(A) is an inductive formula. Thus,
(i) is proved.
(ii) Since the propositional complexes in A are disjoint, we may realize the translation θ step-by-step,
substituting the complex variables one by one (in any arbitrary order). More precisely, we will consider a
‘partial’ translation θPfor each propositional complex P, by just substituting in A only the occurrences
of complex variables from P. So, it is sufficient to prove that A is locally equivalent to θP(A).
Let F = (W,{Rα}α∈MTτ,W) be a general τ-frame (for the corresponding definition see [2] or [7]),
x ∈ W and A a be an RCM-formula of type τ. We have to prove that A is locally valid at x iff θ(A) is.
First, note that A can be obtained from θP(A) by ordinary substitution, so the direction from θP(A) to
A is immediate.
For the direction from A to θP(A) we proceed as follows. Let p = ?p1,...,pn? and let D(p) =
?D1,...,D2n−1? be the string of complex variables built from p. For the translation we need a string of
different new variables ?q1,...,q2n−1?. Then, by the Substitution lemma 3.3 there exists a substitution
S satisfying (∗) and (∗∗). (To be more precise, we should denote S by SP, but we keep the notation S
for simplicity).
Internal Lemma. If Q is a complex positive or negative atom then S(Q) ≡ θP(Q).
The proof is by a simple induction on the formation of Q and by application of the conditions (∗) in
the Substitution lemma 3.3.
Since A is an RCM-formula it has the following form A = [α](¬B1,...,¬Bk,C1,...,Cl). Then, ap-
plying S and θPto A we obtain:
S(A) = [α](¬S(B1),...,¬S(Bk),S(C1),...,S(Cl))
and
θP(A) = [α](¬θP(B1),...,¬θP(Bk),θP(C1),...,θP(Cl)).
By the Internal Lemma , for all Ci, i ≤ l, we have that S(Ci) ≡ θP(Ci), and therefore
θP(A) ≡ [α](¬θP(B1),...,¬θP(Bk),S(C1),...,S(Cl)).
Using (1) we have to show that (F,x) ? [α](¬B1,...,¬Bk,C1,...,Cl) implies
(F,x) ? [α](¬θP(B1),...,¬θP(Bk),S(C1),...,S(Cl)). Suppose for the sake of contradiction that this
is not true, i.e.,
(1)
(F,x) ? [α](¬B1,...,¬Bk,C1,...,Cl)
(2)
but that
(F,x) ?? [α](¬θP(B1),...,¬θP(Bk),S(C1),...,S(Cl)).
Then, there is a pointed model (M,x) over F such that
(M,x) ? [α](¬θP(B1),...,¬θP(Bk),S(C1),...,S(Cl)).
(3)
Page 23
W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 23
Then, there exist y1,...,yk,z1,...,zl∈ W such that
Rαxy1...ykz1...zl,
(4)
(M,yi) ? θP(Bi),i = 1,...,k,
(5)
and
(M,zj) ? S(Cj),
N) be any of the headed boxes Bi, 1 ≤ i ≤ k, with head D and
j = 1,...,l.
(6)
Let B = [β](D,N1,...,Nm) = [β](D,− →
negative part− →
N.
Case 1: D is not a complex variable from D(p). Then θP(D) = S(D) = D. Also, by the Internal
Lemma we have S(− →
N). Consequently S(B) ≡ θP(B). So in this case
(M,yi) ? S(Bi).
Case 2: The head D of B is a complex variable from D(p) and D = Dmfor some m ≤ 2n− 1.
Then θP(Dm) = qm, and consequently θP(Bi) = [β](qm,θP(− →
have
(M,yi) ? [β](qm,S(− →
We will show in this case that (M,yi) ? [β](S(Dm),S(− →
exist t1,...,tρ(β)such that
N) ≡lθP(− →
(7)
N)) = [β](qm,S(− →
N)). So, by (5), we
N)).
(8)
N)). Suppose that this is not true. Then there
Rβyit1...tρ(β),
(9)
and
(M,t1) ? S(Dm),
(10)
and for all 1 < j ≤ ρ(β),
(M,tj) ? S(Nj).
(11)
By the Substitution lemma 3.3, condition (∗∗), we have qm|= S(Dm), and by (10) we obtain
(M,t1) ? qm.
Then by (9), (12) and (11) we obtain (M,yi) ? [β](qm,S(− →
have that (M,yi) ? [β](S(Dm),S(− →
(12)
N)), which contradicts (8). Therefore, we
N)).
Thus, in both cases we have that
(M,yi) ? S(Bi),i = 1,...k.
(13)
Now by (4), (6) and (13) we obtain (M,x) ? S([α](¬B1,...,¬Bk,C1,...,Cl)), which contradicts
(2), since local validity is preserved by substitutions.
? ?
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24 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
As a corollary we obtain the following important generalization of the Sahlqvist theorem for ICM-
formulae:
Theorem 4.2. (Sahlqvist theorem for ICM-formulae)
Every ICM-formula is locally first-order definable and locally d-persistent. Moreover, its local first-order
correspondent can be effectively computed.
Proof:
By theorem 4.1 every ICM-formula A is locally equivalent to the inductive formula θ(A). By Corollary
60 of [17], every inductive formula B is locally first-order definable and locally d-persistent and the
local correspondent of B can be effectively computed. The claim now follows since local equivalence
preserves local first-order definability and local d-persistence. The second claim of the theorem follows
since θ is an effective translation.
? ?
5.Complex normal forms and the translation Σ
5.1. The substitution σ
The translation θ(A), which we defined in the previous section, is the simplest one that translates an
RCM-formula A directly into an RM-formula θ(A) with the property that if A is an ICM-formula then
θ(A) is an inductive modal formula. θ(A) is realized by a sequence of translations θPcorresponding to
all propositional complexes P of A, i.e., θ(A) = θP1(θP2(...(θPn(A))...)), where P1,...,Pnare the
propositional complexes of the formula A. The proof of the translation theorem (Theorem 4.1) shows
that A and θP(A) are locally equivalent, which implies that A and θ(A) are locally equivalent, too. In
fact, the proof uses the substitutions SP, the existence of which is established by the second substitution
lemma (Lemma 3.3), such that SP(A) and θP(A) are locally equivalent.
This motivates the introduction of a substitution σ, obtained by applying consecutively all substitu-
tions of the type SPto A for all different propositional complexes of A, namely:
σ(A) = SP1(SP2(...(SPn(A))...)),
So the translation theorem implies that A, σ(A), and θ(A) are locally equivalent. The difference between
θ and σ is that θ is not a substitution, while σ is a reversible substitution which guarantees the local
equivalence between A and σ(A) for arbitrary formula A. Another difference between θ and σ is that
the image of an ICM-formula under θ is always an inductive modal formula, while its image under σ is
generally not in the format of inductive formulae. However, the latter can always be post-processed into
an inductive formula. Let us illustrate this post-processing on the formula from example 2.1:
C = [α](¬[β](p1∨ p2),¬[β](p1∨ ¬p2),¬[β](¬p1∨ p2),¬[γ](q,N((p1∨ p2),((p1∨ p2) ∧ (p1∨
¬p2)),((p1∨ p2) ∧ (p1∨ ¬p2) ∧ (¬p1∨ p2)),Pos(q)).
Let P be the propositional complex corresponding to the three complex variables D1= (p1∨ p2),
D2 = (p1∨ ¬p2) and D3 = (¬p1∨ p2), in this order. To define SP we need three new variables
q1,q2,q3. Using the corresponding formulae from Lemma 3.4 we obtain: SP(p1) = q1∧(¬q1∨q2) and
SP(p2) = q1∧ (¬q1∨ ¬q2∨ ¬q3).
By applying this substitution to C we do not automatically obtain an inductive formula, but using the
properties of SPfrom Substitution lemmas 3.3 and 3.4 we obtain the following equivalences:
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions25
SP(D1) ≡ q1,
SP(D2) ≡ ¬q1∨ q2,
SP(D3) ≡ ¬q1∨ ¬q2∨ q3,
SP(D1∧ D2) ≡ q1∧ q2,
SP(D1∧ D2∧ D3) ≡ q1∧ q2∧ q3.
Using these simplifications we obtain that SP(C)(= σ(C)) is semantically equivalent to the formula:
C?= [α](¬[β]q1,¬[β](¬q1∨q2),¬[β](¬q1∨¬q2∨q3),¬[γ](q,N(q1,(q1∧q2),(q1∧q2)∧q3)),Pos(q)).
This is an inductive formula with 6 arcs in the dependency digraph: q1−→ q2, q1−→ q3, q1−→ q,
q2−→ q3, q2−→ q, and q3−→ q. The result of the application of θ to C, however, is different:
C??= [α](¬[β]q1,¬[β]q2,¬[β]q3,¬[γ](q,N(q1,(q1∧ q2),(q1∧ q2) ∧ q3)),Pos(q)).
C??is an inductive formula, simpler than C?. It differs from C?only in the heads. By the translation
theorem 4.1, C?and C??are locally equivalent and hence, although they are different inductive formulae,
they define the same first-order conditions.
The above discussion leads to the conclusion that in the extension of SQEMA to SQEMAsubit is
bettertousesubstitutionslikeSP, becausetheyarereversibleandthereforewillguaranteethecorrectness
of the extension. But, we would have to modify SP(and consequently, σ) in order to be able to obtain
inductive formulae without the need for additional post-processing, in the way that, for instance,
θ does. One way to do this is to make it possible to apply σ not to variables, but to some Boolean
subformulae, forinstance, tothecomplexvariablesandcomplexblocksforwhichtheSubstitutionlemma
3.3 and Lemma 3.4 guarantee a better result. For instance, in the example above, the better result for
SP(D1) is q1and for SP(D1∧ D2) it is q1∧ q2.
To this end, in order for such extended σ (denoted later on by Σ) to be applicable to arbitrary for-
mulae, we have to transform the Boolean sub-formulae of the input of SPinto some ‘complex normal
form’, which is the topic of the next subsection.
5.2.Complex normal forms
By a Boolean formula we mean any formula of the classical propositional language.
Let B be a subformula of a modal formula A. This means that B may have several occurrences in
A. Note that any two such occurrences are either identical or disjoint. The positions in A where the
occurrences of B are placed will be called the locations of B in A. If C is a subformula of A and B
is a subformula of C, we say that C is a proper extension of B if B is a proper subformula of C. An
occurrence of a Boolean subformula B at a given location in a modal formula A is called a maximal
Boolean subformula of A at this location if B has no proper Boolean extensions that occurs at that
location. For example, in the formula A = ?(¬(p ∧ q) ∨ 3(p ∧ q)) ∨ (r ∧ s) the Boolean subformula
(p ∧ q) has two occurrences. It is not maximal at the first location, because it has as a proper Boolean
extension ¬(p ∧ q) which occurs there, but it is maximal at the second location.
Let B1,...,Bm, be the list of all maximal Boolean subformulae at their respective locations, listed
in the order of their occurrence in A. The formulae B1,...,Bmwill be called the Boolean blocks of
A. Then we can regard A as being built from its Boolean blocks, and write A = A(B1,...,Bm). Note
that all Bihave different locations and are disjoint, so replacement of each block with some formula
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26 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
will not destroy the other blocks. The list of Boolean blocks in the above example A is B1= ¬(p ∧ q),
B2= (p ∧ q), B3= (r ∧ s) and, accordingly, A = ?(B1∨ 3B2) ∨ B3.
Note that each occurrence of a variable in a formula A = A(B1,...,Bm) is in one of its Boolean
blocks B1,...Bm. Further, note that we can partition the set of Boolean blocks of a given formula A
into disjoint non-empty clusters, called neighbourhood classes, satisfying the following conditions:
(1) formulae from different groups have no common variables, and
(2) each group is minimal with this property.
This partitioning corresponds to the equivalence relation obtained as the transitive closure of the
relation of sharing a common variable between Boolean blocks. It also divides the set of variables in A
into disjoint sets.
For instance, the neighbourhood classes for the above example A are I = {B1,B2} and II = {B3},
where the variables for the group I are {p,q} while those for the group II are {r,s}.
Let A = A(B1,...Bm) be a formula with a list of Boolean blocks B1,...Bm, and let them be
divided into neighbourhood classes C1,...,Ck. Let C be any neighbourhood class and let pC =
?p1,...,pn? be a fixed sequence of all different variables occurring in the formulae from C and let
D(pC) be the set of all different elementary disjunctions built from the sequence pCin which the vari-
ables are ordered as in pC. Now, in each set D(pC) we fix an order of the elementary disjunctions
?D1,...,D2n? and denote the resulting vector by− − − − →
sitional complex P(− − − − →
D2n−1?.
Further, we can replace each formula Bifrom C by its conjunctive normal form B?
junctions from D(pC), so that the disjuncts in this normal form are ordered as in− − − − →
by replacing every Boolean block Biof A with its respective conjunctive normal form B?
formula A?called a complex normal form of A corresponding to the sequence− − − − − →
Propositional complexes Pi = P(− − − − →
D(pCi)), i = 1...,k are called propositional complexes of A?.
So, A has many complex normal forms, all of which are equivalent to A, and the difference between
them is only in the order of the elementary disjunctions in the conjunctive normal forms of its Boolean
blocks. This order is inessential for the formula A but is essential for the substitutions of the form SPi
corresponding to each propositional complex Pioccurring in the substitution σ. The following lemma is
immediate:
D(pC). Note that− − − − →
D∗(pC)? with− − − − − →
D(pC) also determines the propo-
D∗(pC) = ?D1,D1∧D2,...,D1∧D2∧...∧
D(pC)) = ?− − − − →
D(pC),− − − − − →
iusing the dis-
D(pC). In this way,
iwe obtain a
D(pC1),...,− − − − − →
D(pCk).
Lemma 5.1. If A is a non-simplified RCM-formula, then A is itself in a complex normal form.
5.3.The translation Σ
Let A be a modal formula in a complex normal form and let P1,...,Pkbe the propositional complexes
of A. Note that each P from the above sequence is determined by the corresponding vector− − − − →
?D1,...,D2n?, where n is the number of the propositional variables from pC. With each P we associate
a sequence of new propositional variables ?q1,...,q2n? such that the variables associated to P1,...,Pk
are all different. Then we consider the unique substitutions SPi, i = 1,...,k, guaranteed to exist by the
Substitution lemma 3.3. By the definition of the substitution σ introduced in Section 5.1 we have:
D(pC) =
σ(A) = SP1(SP2(...(SPk(A))...)).
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 27
Note that σ coincides with the composition SP1◦...◦SPk. Note also that the order of the components in
the above composition is inessential, because different propositional complexes of A have disjoint sets
of propositional variables and each of the components SPiacts only on the Boolean blocks built from
the variables in Pi. Because the SPiare reversible substitutions we immediately obtain the following
lemma.
Lemma 5.2. Let A be a modal formula and A?a complex normal form of A. Then the formulae A, A?
and σ(A?) are locally equivalent.
Now, we will transform σ component-wise into a new translation Σ, which, when applied to ICM-
formulae, will, without any post-processing, produce inductive modal formulae.
The idea is the following. Let SP be a fixed component of σ and let q = ?q1,...,q2n? be the
sequence of the new variables associated to P. As a substitution SPacts on an arbitrary formula A ho-
momorphically. So, let A = A(B1,...,Bm), where B1,...,Bmare its Boolean blocks in the respective
conjunctive normal form. Then we have:
SP(A) = A(SP(B1),...,SP(Bm))
and
σ(A) = SP1(SP2(...(SPk(A))...)).
We want to define Σ to act on A per propositional complex, like σ does, i.e., we first want to define
the translations ΣP depending on each propositional complex P of A and then define Σ(A), again in
analogy to the definition of σ, as
Σ(A) =defΣP1(ΣP2(...ΣPk(A))...),
where P1,...,Pkare all propositional complexes of A.
Definition 5.1. (The translation Σ)
Definition of ΣP(A). Let P be any of the propositional complexes of A. First we define ΣP on the
Boolean blocks of A.
Case 1: If B is a Boolean block of A not containing variables from P, then ΣP(B) = B.
Case 2: B is a Boolean block of A built over variables from P.
Subcase 2.1: B is in the form Di, for some 1 ≤ i ≤ 2n. Then, in accordance with Lemmas
3.3 and 3.4, we define ΣP(D1) = q1, and for 1 < i < 2n, put ΣP(Di) = ¬q1∨ ... ∨
¬qi−1∨ qi. Finally in this case we put ΣP(D2n) = ¬q1∨ ... ∨ ¬q2n−1.
Subcase 2.2: B is in the form D1∧ D2∧ ... ∧ Di, for some 1 ≤ i ≤ 2n− 1. Then, again
in accordance with Lemmas 3.3 and 3.4, we define ΣP(B) = q1∧ q2∧ ...qi. If i = 2n
then we put ΣP(B) = ⊥.
Subcase 2.3: B falls in neither of the previous two subcases. Then B = Di1∧ ... ∧ Dij,
j > 1, and in this case we put ΣP(B) = ΣP(Di1) ∧ ... ∧ ΣP(Dij), considering
conjuncts as in case 2.1.
Lastly we define ΣP(A) =defA(ΣP(B1),...,ΣP(Bm)).
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28 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Definition of Σ(A) Σ(A) is given by means of the translations ΣPi, i = 1,...,k, as
Σ(A) = ΣP1(ΣP2(...ΣPk(A)...).
Note that Σ can act only on modal formulae in complex normal form. Also, the result Σ(A) depends not
only on A itself but it also depends on the propositional complexes of A ( which are determined during
the construction of A), and on the new propositional variables associated to the propositional complexes
of A. So, when we write Σ(A) we will also have in mind all this additional information needed to
compute Σ(A). The following lemma lists some useful properties of Σ.
Lemma 5.3. Let A be a modal formula and A?be a complex normal form of A. Then:
(i) σ(A?) ≡ Σ(A?).
(ii) Let A?= A(A1,...,Ai), where all subformulae A1,...,Aiare composed from Boolean blocks
of A?. Then Σ(A?) = A(Σ(A1),...,Σ(Ai)).
(iii) A, A?, and Σ(A?) are locally equivalent.
Proof:
The proofs of conditions (i) and (ii) are straightforward, because Σ is defined so as to capture the prop-
erties of σ on the complex variables and complex atoms. As for (iii), by Lemma 5.2, A, A?, and σ(A?)
are locally equivalent. By (i) σ(A?) and Σ(A?) are (semantically) equivalent, which implies the local
equivalence of A, A?and Σ(A?).
? ?
Now, we are ready to establish the following important proposition.
Proposition 5.1. Let A be an ICM-formula. Then:
(i) A is in a complex normal form and Σ(A) is an inductive formula.
(ii) A and Σ(A) are locally equivalent.
Proof:
(i) Let A be an ICM-formula. Then A = [α](¬B1,...,¬Bk,C1,...,Cl), where the Biare the
headed boxes of A and the Cjare the positive components of A. By Lemma 5.1 A is in a complex
normal form. Then we may apply Σ to A, and by Lemma 5.3(ii) we obtain:
Σ(A) = [α](¬Σ(B1),...,¬Σ(Bk),Σ(C1),...,Σ(Cl)).
Since all C1,...Clare built from complex atoms of the form D1∧ D2∧ ... ∧ Dj, taken from the
propositional complexes of A, then Σ acts on them exactly as θ. The same holds for the negative parts
of the headed boxes B1,...,Bk. The only difference between the actions of Σ and θ on headed boxes
is how they act on their heads. If Di+1is the head of some headed box B = [β](Di+1,N1,...,Nj)
and Di+1is a complex variable from some propositional complex P, then θ(Di+1) = qi+1, while
Σ(Di+1) = ¬q1∨ ¬q2∨ ... ∨ ¬qi∨ qi+1. Observe that now Σ(B) = [β](¬q1∨ ¬q2∨ ... ∨ ¬qi∨
Page 29
W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 29
qi+1,Σ(N1),...,Σ(Nj)), which is also a headed box with a head qi+1. Indeed, this can be seen after
composing β with the disjunction (which is a box modality) ¬q1∨ ¬q2∨ ... ∨ ¬qi∨ qi+1. Then ¬q1,
¬q2,...,¬qigo into the negative part of the headed box and qi+1becomes a head. So, we see that Σ(A) is
an RM-formula in which all complex variables are of dimension 1.
In order to prove that Σ(A) is an inductive formula, it remains to show that its dependency digraph
has no cycles. Let A be an ICM-formula, G be the dependency digraph of A and denote by Σ(G) the
dependency digraph of Σ(A). By definition, the vertices of G are the essential propositional complexes
of A, and those of Σ(G) are the heads of Σ(A). Let P and P?be two vertices of G with corresponding
dimensions n and n?and let a be an arc with source P and target P?. Note that P and P?are different,
otherwise G will have a loop. Let q = ?q1,...,q2n−1? and q?= ?q?
variables needed by Σ for P and P?, respectively. Some of these new variables occur in Σ(A) as heads,
and we call them the ‘real heads’; the rest will be called ‘potential heads’. The ideal case is when all
these variables are real heads. Let us now see what kind of arcs correspond to the arc a in the graph
Σ(G). Note that the positive components of Σ(A) and the negative components of the headed boxes are
composed from blocks of the form r1∧...∧ri, where all conjuncts are real or potential heads of Σ(A).
Having this in mind we will associate to the arc a the following sets of arcs from Σ(G).
Inherited arcs. Let qibe a real head which occurs in the negative part of a headed box with a
real head q?
corresponding to the arc a.
New arcs. Since each head Dibecomes a real head qiafter the transformation Σ, namely Σ(Di) =
¬q1∨ ... ∨ ¬qi−1∨ qiwith additional negative parts ¬q1∨ ... ∨ ¬qi−1, then for every i > 1, qimay
receive new arcs starting from real heads from the sequence q1,...,qi−1with targets in qi. Similarly
for the real heads q?
analogs in G, and that is why we call them new arcs.
Thus, we have seen that the inherited and the new arcs are the only arcs in Σ(G) related to the arc a.
Note that it is possible that each of these sets to be empty, as the following example shows.
Let D1= p ∨ q, D2= p ∨ ¬q, D3= ¬p ∨ q and let A be the following ICM-formula: ¬[α]D2∨
¬[β](r,¬D1). A has two propositional complexes, P1, of dimension 2, built from the variables p,q, and
P2, of dimension 1, built from the variable r. Both complexes are essential, and the dependency digraph
contains only one arc a from P1to P2. Then Σ(A) is the following inductive formula: ¬[α](¬q1∨q2)∨
¬[β](r,¬q1) with real heads q2and r. One can see that the dependency digraph of Σ(A) has no arcs at
all. So, the sets of inherited and the new arcs of a are empty.
It is easy to see that the following facts are true.
1,...,q2n?−1? be the strings of new
j. Then in Σ(G) there exists an arc from qito q?
j. All such arcs are called inherited arcs
i— they, too, may receive arcs from real heads q?
j, with j < i. These arcs have no
Fact 1. (i) If there is a new arc in Σ(G) from qito qjthen i < j.
(ii) If there exists a path consisting only of new arcs then the vertices of this path are real heads qi
from Σ(G) corresponding to one essential propositional complex of the formula A.
Fact 2. (i) If there exists an inherited arc in Σ(G) from qito q?
(ii) Let qi1−→ qi2−→ ... −→ qikbe a path of new arcs in Σ(G) with vertices corresponding to the
essential complex P of A; let q?
essential complex P?of A and let qik−→ q?
from P to P?in G.
jthen there is an arc in G from P to P?.
j1−→ q?
j2−→ ... −→ q?
j1be an inherited arc from qikto q?
jk?be a path of new arcs corresponding to the
j1. Then there is an arc
The rest of the proof of (i) follows from the following
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30 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
Internal lemma. The digraph Σ(G) has no cycles.
Proof of the Internal lemma. Suppose that there is a cycle C in Σ(G). We proceed to arrive at a
contradiction.
Case 1: All arcsofC are newarcs. Then it iseasy to seeby Fact 1(ii)that the vertices ofC = qi1−→
qi2−→ ... −→ qik−→ qi1consists of real heads corresponding to the new variables of one essential
propositional complex P of the formula A. Then by Fact 1(i) we obtain i1< i2< ... < ik< i1. This
implies i1< i1which is a contradiction.
Case 2: The cycle C contains inherited arcs. Consider the following 3 sub-paths of C: qi1−→
qi2−→ ... −→ qikof new arcs, the inherited arc qik−→ q?
q?
P and P?in G. Going further consider the 3 sub-paths q?
q?
exists an arc between the corresponding propositional complexes P?and P??in G. Reasoning in this way
we obtain by induction a path P −→ P?−→ P??−→ ... −→ P which is a cycle in G. Since G has no
cycles, this is the intended contradiction — the lemma is proved.
(ii) is an immediate corollary of Lemma 5.3(iii).
j1, and the new arcs q?
j1−→ q?
j2−→ ... −→
jk?. By Fact 2(ii) this implies that there exists an arc between the corresponding propositional complexes
j1−→ q?
ik??. Again by fact 2(ii) this implies that there
j2−→ ... −→ q?
jk?, the inherited arc
jk?−→ q??
m1, and the new arcs q??
m1−→ q??
m2−→ ... −→ q??
? ?
6. The algorithm SQEMAsub
In this section we give an informal introduction to SQEMAsub, followed by a more formal description.
We illustrate the algorithm with some examples and prove some theorems about its meta-properties.
6.1.The algorithm SQEMAsub, informally
In Example 2.2 we considered the formula
ϕ = [3](¬[1](p1∨ ¬p2),¬[1](¬p1∨ p2),?2?((p1∨ ¬p2),(p1∨ ¬p2) ∧ (¬p1∨ p2))),
on which SQEMA fails. The reason for this failure was that, in any attempt to eliminate the variables p1
and p2one by one, the algorithm was unable to apply the Ackermann-rule since one of the equations was
neitherpositivenornegativewithrespecttothechosenvariable. Thisexampleindicatesaweaknessofthe
Ackermann lemma and suggests that it could be strengthened in order to handle such cases. One possible
solution would be to strengthen the Ackermann-rule. Indeed, in [28, 29, 30] some generalizations of the
Ackermann lemma, by means of which one can eliminate several variables at once, were considered.
Particularly, a generalized Ackermann lemma was given which is sufficient to eliminate all ordinary
variables of one given complex variable in an ICM-formula at once, and finally to find the corresponding
first-order frame condition. Now, in step 7 of example 2.2 both variables are ready to be simultaneously
eliminated according to (a rule based upon) one such generalization of Ackermann lemma. Following
this route would, however, require a serious reconstruction of the algorithm SQEMA. Moreover, all
theorems relating to SQEMA, e.g., its correctness and the canonicity of the reducible formulae, would
have to be re-proved for the new algorithm so obtained.
That is why we opt for another, rather more modular approach, based on the method of reversible
substitutions. We have already shown how, with this method, each ICM-formula can be transformed by
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 31
a suitable substitution-like transformation (viz. the transformation Σ developed in section 5.3) into an
inductive formula. We know from [7] that SQEMA succeeds on all inductive formulae, so the main idea
is to introduce an extension of SQEMA, called SQEMAsub, composed of two subprograms. The original
(polyadic) algorithm SQEMA itself constitutes the first of these subprograms, while the second, to be
called SUB, deals with the substitution-like transformation Σ. SQEMAsubwill inherit all the important
properties of SQEMA, viz. the local first-order definability and d-persistence (hence, canonicity) of the
formulae reducible by it. Moreover, it will succeed not only on all inductive formulae, but also on all
ICM-formulae.
Let us now describe informally how SQEMAsubworks. The input formula A is first sent to the sub-
program SQEMA. If SQEMA succeeds it reports SUCCESS and outputs the corresponding first-order
condition of A. If SQEMA does not succeed then SQEMAsubruns SUB to find (nondeterministically)
a complex normal form A?of the input formula A, then applies Σ to A?, and then send the result back to
SQEMA. This cycle is repeated until SQEMA succeeds or SUB cannot produce any new complex nor-
mal form of A. If all (finitely many) possible complex normal forms have been generated and SQEMA
has not succeeded on any of them, then SQEMAsubreports FAIL and halts.
6.2.Description of SQEMAsub
Here is a formal description of SQEMAsub.
Algorithm SQEMAsub(ϕ) This is the main body of the algorithm. It takes as input an Lτ-formula ϕ,
for which it either returns a local first-order frame correspondent, or reports failure.
(1) Call SQEMA(ϕ).
If SQEMA(ϕ) returns a first-order formula F
then return F and terminate,
else if SQEMA(ϕ) returns FAIL, proceed to step 2.
(2) Call procedure SUB(ϕ).
If SUB(ϕ) returns a first-order formula F
then return F and terminate,
else if SUB(ϕ) returns, return FAIL and terminate.
Procedure SUB(ϕ) This procedure takes as input an Lτ-formula ϕ, for which it either returns a local
first-order frame correspondent, or reports failure.
(1) Initialize procedure Complex normal form with ϕ.
(2) Repeat
(2.1) Request a complex normal form ϕ?of ϕ from Complex normal from. Procedure
Complex normal fromreturnssuchacomplexnormalform, togetherwithanassociated
list
StringDVar = ?StringDVar(C1),...,StringDVar(Cl)?
encoding the propositional complexes P1,P2,...,Plof ϕ?, and a list
StringNewVar = ?StringNewVar(C1),...,StringNewVar(Cl)?
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32W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
of strings of new variables.
If procedure Complex normal from returns ‘DONE’
then return FAIL and terminate,
else proceed to (2.2).
(2.2) Call procedure Translation Σ(ϕ?,StringDVar,StringNewVar).
This procedure returns a L-formula ϕ??.
(2.3) Call SQEMA(ϕ??).
If SQEMA(ϕ) returns a first-order formula F
then return F and terminate.
Procedure Complex normal form This procedure is initialized with a formula A, for which it then
successively produces all possible complex normal forms, as requested.
Initialization The procedure initializes with a formula A by performing the following four steps:
(1.1) Determine the list of Boolean blocks B1,...,Bkof A and represent A as built from
these blocks: A = A(B1...,Bk).
(1.2) Partition the set of Boolean blocks into neighbourhood classes C1,...,Cland deter-
mine the sets of variables Var(Ci), i = 1,...,l, where each set Var(Ci) is considered
as a string of different variables, in a fixed order.
(1.3) For each set of variables Var(C) = ?p1,...,pn? produce all elementary disjunctions
DVar(C) over Var(C) and order the variables in each disjunction in the same way as the
order of the variables in Var(C).
(1.4) ForeachsetofvariablesVar(C) = ?p1,...,pn?chooseanewsetofvariablesNewVar(C)
with cardinality 2n, such that all these sets, and the sets Var(C), are pairwise disjoint.
Complex normal forms When a complex normal form for the formula A initialized with is re-
quested, the following steps are performed:
(1.5.1) Choose nondeterministically a new order of the elementary disjunctions in each set
DVar(C) and produce the string StringDVar(C) = ?D1,...,D2n? according to the
chosen order.
(1.5.2) ReorderthevariablesinthesetNewVar(C)andproducesthestringStringNewVar(C)
in order to obtain a one-one correspondence between the strings StringDVar(C) and
StringNewVar(C). (If Diis the i-th element of StringDVar(C) then the i-th element of
StringNewVar(C) will be denoted by qi.)
If the new order cannot be generated, i.e., all possible orders have been attempted
then return DONE,
else proceeds to (1.6.1).
(1.6.1) For each Boolean block B of A produce its conjunctive normal form B?as follows:
choose the neighbourhood class C such that B ∈ C and define the conjunctive nor-
mal form by the elementary disjunctions from the string StringDVar(C), such that the
conjuncts follow the order of StringDVar(C).
(1.6.2) Produce the complex normal form A?of A, replacing each Boolean block B of A by
its conjunctive normal form B?, i.e., A?= A(B?
1,...,B?
k).
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions33
(1.6.3) Return the triple (A?,StringDVar,StringNewVar).
Remark 6.1. Each time (1.5.1) is called, it produces a new order of the elementary disjunc-
tions of all sets DVar(C) = ?D1,...,D2n? and (1.5.2) produces the corresponding order of
the new variables in the sets NewVar(C). The execution of the next steps of the procedure
depends on that new order. Note also that the number N of possible orders of the sets of
disjunctions is finite and can be obtained as N =?l
Procedure Translation Σ This procedure receives a triple (A?,StringDVar,StringNewVar), such
as is produced by procedure Complex normal form, as input. This triple contains all the
needed data to compute the new modal formula Σ(A?).
Proceed recursively top-down:
Σ(A?) = A(Σ(B?
lows:
Choose the neighbourhood class C such that B?∈ C.
Choose the corresponding string StringNewVar(C).
Compute Σ(B?) according to the following rules (see the definition of Σ in Section 5.3):
(2.1) If B?is in the form Di, i = 1,...,2n− 1, then Σ(D1) = q1, for 1 < i < 2n,
Σ(Di) = ¬q1∨ ... ∨ ¬qi−1∨ qi, and Σ(D2n) = ¬q1∨ ... ∨ ¬q2n−1.
(2.2) If B?is in the form D1∧D2∧...∧Di, i = 1,...,2n−1, then Σ(B?) = q1∧q2∧...qi,
and Σ(B?) = ⊥ if i = 2n.
(2.3) If B is in neither of these forms, then B = Di1∧ Di2...Dij, j > 1, and in this case
we put Σ(B) = Σ(Di1) ∧ ... ∧ Σ(Dij), considering conjuncts as in (2.1).
Return the obtained formula Σ(A?).
i=12ni! where niis the cardinality of the
set Var(Ci), for i = 1,...,l.
1),...,Σ(B?
k)), computing Σ(B?) for each component B?= B?
i, as fol-
6.3. Examples
We now illustrate SQEMAsubwith two examples.
Example 6.1. Let ϕ := 32(p → q) ∧ 32(q → p) → 32(p ↔ q). When ϕ is given to SQEMAsub
as input, it is first passed to the subroutine SQEMA. As the reader can check, SQEMA will fail on
ϕ, as it will be unable to solve for either p or q. Thus ϕ is passed to procedure SUB. SUB calls
and initializes Complex normal form with ϕ. When Complex normal form initializes with ϕ, step
(1.1) determines the boolean blocks B1 = (p → q), B2 = (q → p) and B3 = (p ↔ q) of ϕ and
represents ϕ as ϕ = 32B1∧ 32B2→ 32B3. Step (1.2) determines that C1= {B1,B2,B3} and that
Var(C1) = ?p,q?. Step (1.3) determines that DVar(C1) = ?(p∨q),(p∨¬q),(¬p∨q),(¬p∨¬q)?. Step
(1.4) determines NewVar(C1) = ?q1,q2,q3,q4?.
WhenSUBrequestsacomplexnormalformofϕfromComplexnormalform, thelatterprocedurewill,
in response to one of these requests, return a triple (A?,StringDVar,StringNewVar) with A?= 32(¬p∨
q)∧32(p∨¬q) → 32((¬p∨q)∧(p∨¬q)), StringDVar = ??(¬p∨q),(p∨¬q),(p∨q),(¬p∨¬q)??,
and StringNewVar = ??q1,q2,q3,q4??. SUB passes this triple to Translation Σ, which produces and
returns the formula 32q1∧32(¬q1∨q2) → 32(q1∧q2). This is an inductive formula, so when SUB
passes it to SQEMA, the latter subroutine will successfully compute a first-order frame correspondent
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34 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
for it, namely Rxy ∧ Rxz → ∃u(Rxu ∧ ∀v(Ruv → (Ryv ∧ Rzv))). Thus SQEMAsubwill return this
first-order formula and terminate successfully.
Example 6.2. Consider the formula ϕ2= [2](¬[1](¬[1]p ∨ p),p ∧ [1]⊥). As was shown in example
2.4 in [7], SQEMA fails on this formula, despite its being locally first-order definable. Thus, when
SQEMAsubis run on this formula, the initial execution of SQEMA on it will fail. Note that ϕ2is already
in complex normal form and that, in fact, this is its only complex normal form. Thus, passing it to
Complex normal form and sending the result to Translation Σ will not produce a different formula.
Hence SQEMAsubwill fail on ϕ2.
6.4.
Theorem 6.1. (Correctness)
If SQEMAsubsucceeds on an input formula A, then A is a local frame-correspondent of the returned
first-order condition.
Properties of SQEMAsub
Proof:
Let F(x) be the first-order formula returned by SQEMAsubwhen run on A. If the subroutine SQEMA
succeeded on A without the procedure SUB being called, then the claim follows from Theorem 1.1. On
the other hand, if SUB was called, then it returned a formula Σ(A?) such that A?is a complex normal
form of A, which was then passed again to SQEMA, which, in turn, returned F(x). Again by theorem
1.1, F(x) is a local first-order equivalent of the formula Σ(A?). By Lemma 5.3 we have that A, A?, and
Σ(A) are locally equivalent, which implies that F(x) is a local first-order equivalent of A.
? ?
Theorem 6.2. (Canonicity)
1. If SQEMAsubsucceeds on an Lτ-formula A, then A is locally persistent with respect to the class
of all descriptive τ-frames.
2. If SQEMAsubsucceeds on an Lr(τ)-formula A, then A is locally persistent with respect to the class
of all reversive descriptive τ-frames.
Proof:
We will prove case 1; case 2 is analogous. If SQEMAsubsucceeds on A without calling the subprogram
SUB, then it must be the case that SQEMA succeeds on A, and hence the result follows from Theorem
1.2. On the other hand, if SUB is called, then at some stage subroutine SQEMA succeeds on a formula
Σ(A?), such that A?is a complex normal form of A. Then, again by Theorem 1.2 Σ(A?) is locally
persistent with respect to the class of all descriptive τ-frames. By Lemma 5.3 A, A?and Σ(A) are
locally equivalent, which implies that A is locally persistent with respect to the class of all descriptive
τ-frames.
? ?
Theorem 6.3. (Completeness)
SQEMAsubsucceeds on all ICM-formulae.
Proof:
Let A be an ICM-formula and suppose that the subprogram SQEMA does not succeed on A. Hence
procedure SUB is called. Since A is itself in a complex normal form, A will at some stage be passed to
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W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions 35
procedure Translation Σ without change, and hence Σ(A) will be sent to SQEMA. By proposition 5.1,
Σ(A) is an inductive formula. Then, by theorem 1.3, the subprogram SQEMA will succeed on Σ(A),
and hence SQEMAsubwill succeed on A.
? ?
7. Concluding remarks
We have explored the use of reversible substitutions for transforming a modal formula into a locally
equivalent Sahlqvist-like one, for which a suitable adaptation of the algorithm SQEMA can compute a
localfirst-orderequivalentandproved-persistence, andhencecanonicity. Inparticular, wehaveextended
the algorithm SQEMA with a special module for computing such substitutions and have shown that the
resulting algorithm SQEMAsubsucceeds on all inductive complex formulae, thus extending essentially
the scope of SQEMA. An implementation of SQEMAsubis currently being developed.
The algorithm SQEMAsubdoes not exhaust the potential of the method of reversible substitutions.
Finding a suitable substitution, however, is generally a rather non-trivial task and little is known about
its applicability beyond the class of complex inductive formulae. Furthermore, even if one can guess a
suitable substitution for a given formula, the question if it is reversible seems difficult, and at present
we do not know if it is algorithmically decidable. Nevertheless the problem of finding broader effective
classes of reversible substitutions which may extend the applicability of SQEMA seems sensible, and
we formulate it as one of the important open problems. In the present version SQEMAsubtransforms
the input formula into formula with an exponentially greater number of propositional variables, which
is a rather expensive task. One of our future plans is to modify SQEMAsubsuch that the subprogram
SUB preserves the number of variables of the input formula. Another open problem is the comparison of
SQEMAsubwith other algorithms like SCAN and DLS and even with SQEMA: we know that SQEMA
fails on some ICM-formulae and the problem is to find a large subclass of ICM-formulae to which
SQEMA succeeds and to see if the whole class of ICM-formulae can be reduced by suitable reversible
substitutions to this subclass. As for SCAN we know that it succeeds on all Sahlqvist formulae [18] and
also on some ICM-formulae (for instance the formula A from Section 2.2) but it is not known whether,
for instance, SCAN is complete for the inductive (ordinary, or complex) modal formulae. In [6] there
are examples of formulae for which SQEMA succeeds but neither SCAN nor DLS does, which show
that SQEMA, and hence SQEMAsubare incomparable with SCAN and DLS in terms of the scope of
application.
Acknowledgements
Thanks are due to the anonymous referees for the very careful reading of the paper and pointing out some
inaccuracies, and especially for their useful suggestions to make some technical parts more readable and
intuitive. The work of Dimiter Vakarelov is supported by the contract IM/1510 with the Bulgarian
Ministry of Science and Education. Also, part of this research was done during his visit to Johannesburg,
funded by the University of Johannesburg and the National Research Foundation of South Africa, which
have also supported financially the research of the first two authors.
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36 W. Conradie, V. Goranko, D. Vakarelov/SQEMA with substitutions
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