Efficient Local Reductions to Basic Modal Logic

Abstract

We present novel reductions of the propositional modal logics , , , and to Separated Normal Form with Sets of Modal Levels. The reductions result in smaller formulae than the well-known reductions by Kracht and allow us to use the local reasoning of the prover to determine the satisfiability of modal formulae in these logics. We show experimentally that the combination of our reductions with the prover performs well when compared with a specialised resolution calculus for these logics and with the b̆uilt-in reductions of the first-order prover SPASS.

C. Dixon was partially supported by the EPSRC funded RAI Hubs FAIR-SPACE (EP/R026092/1) and RAIN (EP/R026084/1), and the EPSRC funded programme Grant S4 (EP/N007565/1).

1 Introduction

The main motivation for reducing problems in one logic (the source logic) to ‘equivalent’ problems in another logic (the target logic) is to exploit results and tools for the target logic to solve theoretical or practical problems in the source logic. For propositional modal logics this approach has been researched extensively for reductions of the satisfiability problem in these logics to the satisfiability problem in ‘stronger’ logics such as first-order logic [10, 20], the second-order theory of n successors [6], simple type theory [4], and regular grammar logics [19].

An alternative approach is to reduce propositional modal logics to a ‘weaker’ logic, in particular, the basic modal logic . For extensions of with one of the axioms $\mathsf {B}$, $\mathsf {D}$, $\mathsf {alt_1}$, $\mathsf {T}$, and $\mathsf {4}$, Kracht [12] defines reduction functions of their global and local satisfiability problem to the corresponding problem in and proves their correctness. He also defines a reduction function for , the extension of with $\mathsf {5}$, to , but this reduction is incorrect as not all theorems of are theorems of . Several features of Kracht’s approach are relevant to our work. First, as is not uncommon in modal logic, he treats the modal operator $\Diamond$ as abbreviation for $\lnot \Box \lnot$, that is, $\Box$ is the only modal operator occurring in modal formulae. Second, the basic idea underlying his reduction functions is for a given modal formula $\varphi$ to generate sufficiently many instances $\varDelta$ of a modal axiom $\Lambda$ so that $\varphi$ is -satisfiable iff $\varphi \wedge \varDelta$ is -satisfiable. Third, Kracht is only concerned with preservation of the computational complexity of the satisfiability problem under consideration, as well as the preservation of other theoretical properties. For instance, the local satisfiability problem in the modal logics covered by Kracht is PSPACE-complete. So, it is sufficient to ensure that $\varDelta$ is polynomial in size with respect to $\varphi$. As Kracht himself concludes, his method offers a uniform way of transferring results about one modal logic to another, but may not be as useful for practical applications.

In [15, 16] we have introduced a new normal form for basic multi-modal logic, called Separated Normal Form with Modal Levels, , that uses labelled modal clauses. These labels refer to the level within a tree Kripke structure at which a modal clause holds. This can be seen as a compromise between approaches that label formulae with worlds at unspecified level [1, 3] and approaches that label formulae with paths [5, 23]. A combination of a normal form transformation for modal formulae and a resolution-based calculus for labelled modal clauses can then be used to decide local and global satisfiability in basic modal logic. In [17, 18] we have presented , an implementation of that calculus, together with an experimental evaluation that indicates that performs well if propositional variables are evenly spread across a wide range of modal levels within the formulae one wants to decide.

A feature of is its use of additional propositional symbols as ‘surrogates’ for subformulae of a modal formula $\varphi$. In the following we take advantage of the availability of those surrogates to provide a novel transformation from extensions of with a single one of the axioms $\mathsf {B}$, $\mathsf {D}$, $\mathsf {T}$, $\mathsf {4}$ and $\mathsf {5}$ to . Another novel aspect is that we modify the normal form so that it uses sets of modal levels as labels instead of a single modal level. In we only need a definition of a surrogate at the modal level at which the corresponding subformula occurs in $\varphi$. But in , , and , we need a definition at every reachable modal level, of which there can be many. We call the resulting normal form, Separated Normal Form with Sets of Modal Levels, .

The structure of the paper is as follows. In Section 2 we recap common concepts of propositional modal logic including its syntax and semantics. Section 3 defines and the reductions of , , , , and to . Correctness is proved in Section 4. Related work is discussed in Section 5. In Section 6 we compare the performance of a combination of our reductions and the modal-layered resolution calculus implemented in prover with resolution calculi specifically designed for the logics under consideration and with translation-based approaches built into the first-order theorem prover SPASS.

2 Preliminaries

The language of modal logic is an extension of the language of propositional logic with a unary modal operator $\Box$ and its dual $\Diamond$. More precisely, given a denumerable set of propositional symbols, $P= \{p, p_0, q, q_0, t, t_0,\ldots \}$ as well as propositional constants $\mathbf{true}$ and $\mathbf{false}$, modal formulae are inductively defined as follows: Constants and propositional symbols are modal formulae. If $\varphi$ and $\psi$ are modal formulae, then so are $\lnot \varphi$, $(\varphi \wedge \psi )$, $(\varphi \vee \psi )$, $(\varphi \rightarrow \psi )$, $\Box \varphi$, and $\Diamond \varphi$. We also assume that $\wedge$ and $\vee$ are associative and commutative operators and consider, e.g., $(p\vee (q\vee r))$ and $(r\vee (q\vee p))$ to be identical formulae. We often omit parentheses if this does not cause confusion. By we denote the set of all propositional symbols occurring in $\varphi$. This function straightforwardly extends to finite sets of modal formulae. A modal axiom (schema) is a modal formula $\psi$ representing the set of all instances of $\psi$.

A literal is either a propositional symbol or its negation; the set of literals is denoted by $L$. We denote by $\lnot l$ the complement of the literal $l\in L$, that is, $\lnot l$ denotes $\lnot p$ if l is the propositional symbol p, and $\lnot l$ denotes p if l is the literal $\lnot p$. A modal literal is either $\Box l$ or $\Diamond l$, where $l \in L$.

A (normal) modal logic is a set of modal formulae which includes all propositional tautologies, the axiom schema $\Box {(\varphi \rightarrow \psi )} \rightarrow (\Box {\varphi } \rightarrow \Box {\psi })$, called the axiom $\mathsf {K}$, is closed under modus ponens (if $\vdash {} \varphi$ and $\vdash {} \varphi \rightarrow \psi$ then $\vdash {}\psi$) and the rule of necessitation (if $\vdash {}\varphi$ then $\vdash {} \Box {\varphi }$).

  is the weakest modal logic, that is, the logic given by the smallest set of modal formulae constituting a normal modal logic. By we denote an extensions of by a set $\Sigma$ of axioms.

The standard semantics of modal logics is the Kripke semantics or possible world semantics. A Kripke frame $F$ is an ordered pair $\langle W, R \rangle$ where $W$ is a non-empty set of worlds and $R$ is a binary (accessibility) relation over $W$. A Kripke structure $M$ over $P$ is an ordered pair $\langle F,V \rangle$ where $F$ is a Kripke frame and the valuation V is a function mapping each propositional symbol in $P$ to a subset V(p) of $W$. We say $M= \langle F,V \rangle$ is based on the frame $F$. A rooted Kripke structure is an ordered pair $\langle M,w_0 \rangle$ with $w_0\in W$. To simplify notation, in the following we write $\langle W, R, V \rangle$ and $\langle W, R, V,w_0 \rangle$ instead of $\langle \langle W, R \rangle , V \rangle$ and $\langle \langle \langle W, R \rangle , V \rangle ,w_0 \rangle$, respectively.

Satisfaction (or truth) of a formula at a world $w$ of a Kripke structure $M= \langle W, R, V \rangle$ is inductively defined by:

If $\langle M, w \rangle \models \varphi$ holds then $M$ is a model of $\varphi$, $\varphi$ is true at w in $M$ and $M$ satisfies $\varphi$. A modal formula $\varphi$ is satisfiable iff there exists a Kripke structure $M$ and a world w in $M$ such that $\langle M, w \rangle \models \varphi$. A modal formula $\varphi$ is globally true or valid in a Kripke structure $M$ if it is true at all worlds of $M$; it is valid if it is valid in all Kripke structures.

In the following we are interested in extensions of with the axiom schemata shown in Table 1. Each of these axiom schemata defines a class of Kripke frames where the accessibility relation $R$ satisfies the first-order property stated in the table. Given a normal modal logic L with corresponding class of frames $\mathfrak {F}$, we say a modal formula $\varphi$ is L-satisfiable iff there exists a frame $F\in \mathfrak {F}$, a valuation V and a world $w_0\in F$ such that $\langle F,V,w_0 \rangle \models \varphi$.

Table 1. Modal axioms and relational frame properties

A path rooted at $w$ of length k, $k\ge 0$, in a frame $F = \langle W,R \rangle$ is a sequence $\vec {w} = (w_0,w_1,\ldots ,w_k)$ where for every i, $1\le i\le k$, $w_{i-1}\mathbin {R}w_i$. We say that the path $(w_0,w_1,\ldots ,w_k)$ connects $w_0$ and $w_k$. For a path $\vec {w} = (w_0,\ldots ,w_k)$ and world $w_{k+1}$ with $w_k\mathbin {R}w_{k+1}$, $\vec {w}\circ w_{k+1}$ denotes the path $(w_0,\ldots ,w_k,w_{k+1})$. A path $(w_0)$ of length 0 is identified with its root $w_0$. We denote the set of all paths rooted at a world $w_0$ in $F$ by $\vec {F}[w_0]$ and the set of all paths by $\vec {F}$. The function maps every path $\vec {w} = (w_0,\ldots ,w_k)$ to its terminal world $w_k$ while the function maps every path $\vec {w} = (w_0,w_1,\ldots ,w_k)$ to its length k.

A rooted Kripke structure $M = \langle W,R,V,w_0 \rangle$ is a rooted tree Kripke structure iff R is a tree, that is, a directed acyclic connected graph where each node has at most one predecessor, with root $w_0$. It is a rooted tree Kripke model of a modal formula $\varphi$ iff $\langle W,R,V,w_0 \rangle \models \varphi$. In a rooted tree Kripke structure with root $w_0$ for every world $w_k\in W$ there is exactly one path $\vec {w}$ connecting $w_0$ and $w_k$; the modal level of $w_k$ (in M), denoted by , is given by .

Let $F = \langle W, R \rangle$ be a Kripke frame with $w\in W$. The unravelling $F^u[w]$ of $F$ at $w$ is the frame $\langle \vec {W}, \vec {R} \rangle$ where:

• $\vec {W} = \vec {F}[w]$ is the set of all rooted paths at $w$ in $F$;

• for all $\vec {v},\vec {w} \in \vec {W}$, if $\vec {w} = \vec {v}\circ w$ for some $w\in W$, then $\vec {v}\mathbin {\vec {R}}\vec {w}$.

Let $F=\langle W, R \rangle$ and $F'=\langle W', R' \rangle$ be two Kripke frames. A function $f :W\mapsto W'$ is a p-morphism (or a bounded morphism) from $F$ to $F'$ if the following holds:

• if $v\mathbin {R}w$, then $f(v)\mathbin {R'}f(w)$.

• if $f(u)\mathbin {R'}w$, then there exists $v\in W$ s.t. $f(v)=w$ and $u\mathbin {R}v$.

Analogously for Kripke models. For $F = \langle W,R \rangle$, $M' = \langle F,V',w_0 \rangle$, and $M = \langle F^u[w_0],V,(w_0) \rangle$, the function is a p-morphism from $M$ to $M'$.

When considering local satisfiability, the following holds (see, [8]):

Theorem 1

Let $\varphi$ be a modal formula. Then $\varphi$ is -satisfiable iff there is a finite rooted tree Kripke structure $M=\langle F,V,w_0 \rangle$ such that $\langle M, w_0 \rangle \models \varphi$.

Table 2. Rewriting Rules for Simplification

For the normal form transformation presented in the next section we assume that any modal formula $\varphi$ has been simplified by exhaustively applying the rewrite rules in Table 2 and is in Negation Normal Form (NNF), that is, a formula where only propositional symbols are allowed in the scope of negations. We say that such a formula is in simplified NNF.

3 Layered Normal Form with Sets of Levels

A formula to be tested for satisfiability is first transformed into a normal form called Separated Normal Form with Sets of Modal Levels, , whose language extends that of modal logic with labels consisting of sets of modal levels. Informally, we write $S: \varphi$, where S is a set of natural numbers, to denote that a formula $\varphi$ is true at modal levels $ml\in S$. We write instead of $\mathbb N:\varphi$.

We introduce some notation that will be used in the following. Let $S^+= \{l + 1\in \mathbb N\mid l \in S\}$, $S^- = \{ l - 1 \in \mathbb N\mid l \in S\}$, and $S^\ge = \{n \mid n \ge \min (S)\}$, where $\min (S)$ is the least element in S. Note that the restriction of the elements being in $\mathbb N$ implies that $S^-$ cannot contain negative numbers.

The labels in work as a kind of weak universal operator, allowing us to talk about formulae that are satisfied at all worlds in a given set of modal levels. Formally, we restrict ourselves to rooted tree Kripke structures $M = \langle W,R,V,w_0 \rangle$ and if S is a set of modal levels, then by M[S] we denote the set of worlds that are at a modal level in S, that is, . The satisfaction of labelled formulae in a rooted tree Kripke structure M is then defined as follows:

If $M\models S: \varphi$, then we say that $S: \varphi$ holds in $M$. Note that if $S=\emptyset$, then $M\models S: \varphi$ trivially holds. For a set $\varPhi$ of labelled formulae, $M\models \varPhi$ iff $M\models S:\varphi$ for every $S:\varphi$ in $\varPhi$, and we say $\varPhi$ is -satisfiable.

A labelled modal formula is then an clause iff it is of one of the following forms:

• Literal clause      $S: \bigvee _{b=1}^{r} l_b$

• Positive modal clause      $S: l' \rightarrow \Box l$

• Negative modal clause      $S: l' \rightarrow \Diamond l$

where $S \subseteq \mathbb N$ and l, $l'$, $l_b$ are propositional literals with $1 \le b \le r$, $r \in \mathbb N$. Positive and negative modal clauses are together known as modal clauses. We regard a literal clause as a set of literals, that is, two clauses are the same if they contain the same set of literals.

We assume that the set $P$ of propositional symbols is partitioned into two infinite sets Q and T such that for every modal formula $\psi$ we have and there exists a propositional symbol $t_\psi \in T$ uniquely associated with $\psi$.

Given a modal formula $\varphi$ in simplified NNF and , then we can obtain a set $\varPhi _L$ of clauses in such that $\varphi$ is L-satisfiable iff $\varPhi _L$ is -satisfiable as $\varPhi _L =\{\{ 0 \}: t_\varphi \}\cup \rho _{L}(\{0\}:t_\varphi \rightarrow \varphi )$, where $\rho _{L}$ is defined as follows:

where $\eta$ and $\delta _{L}$ are defined as follows:

and functions $P_{L}$, $\varDelta _{L}$ are defined as shown in Table 3. The function $\eta$ maps a propositional literal $\psi$ to itself while it maps every other modal formula $\psi$ to a new propositional symbol $t_\psi \in T$ uniquely associated with $\psi$. We call $t_\psi$ the surrogate of $\psi$ or simply a surrogate. The functions and introduce additional propositional symbols, called supplementary propositional symbols, $t_{\Box \lnot t_{\Box \psi }}\in T$ and $t_{\Diamond t_{\Box \psi }}\in T$, respectively, that do not correspond to subformulae of the formula we are transforming.

Intuitively, is based on the following consideration: Take a world w in a Kripke structure $M$ with a symmetric accessibility relation $R$. If there exists a world v with $w\mathbin {R}v$ such that $\langle M, v \rangle \models \Box \psi$, then $\langle M, w \rangle \models \psi$. Now, take the contrapositive of that statement: If $\langle M, w \rangle \not \models \psi$, then for every world v with $w\mathbin {R}v$, $\langle M, v \rangle \not \models \Box \psi$. Equivalently, $\langle M, w \rangle \models \psi$ or $\langle M, w \rangle \models \Box \lnot \Box \psi$. This is expressed by the formula $\eta (\psi )\vee t_{\Box \lnot t_{\Box \psi }}$. For , the formula $t_{\Diamond t_{\Box \psi }}\rightarrow \Box t_{\Diamond t_{\Box \psi }}$ expresses an instance of axiom schema , $\Diamond \varphi \rightarrow \Box \Diamond \varphi$, with $\varphi = \Box {\psi }$, i.e., $\Diamond \Box \psi \rightarrow \Box \Diamond \Box \psi$. The contrapositive of axiom schema $\mathsf {5}$ is $\Diamond \Box \varphi \rightarrow \Box \varphi$, equivalent to $\lnot \Diamond \Box \varphi \vee \Box \varphi$. For $\varphi =\psi$ this is expressed by the formula $\lnot t_{\Diamond t_{\Box \psi }}\vee t_{\Box \psi }$. For the formula $\lnot t_{\Diamond t_{\Box \psi }}\rightarrow \Box \lnot t_{\Box \psi }$, consider $\lnot \Diamond \Box \psi$. By duality of $\Box$ and $\Diamond$, this is equivalent to $\lnot \lnot \Box \lnot \Box \psi$ and $\Box \lnot \Box \psi$. So, $\lnot \Diamond \Box \psi \rightarrow \Box \lnot \Box \psi$ in every normal modal logic, not only . The remaining labelled formulae introduced by and ensure that supplementary propositional symbols are defined. For the remaining logics the additional clauses are also based directly on the axiom schemata.

Table 3. Transformation of $\Box$-formulae in modal logic L

To simplify presentation in the following, we define a function $\eta _f$ as follows:

$$\begin{aligned} \eta _f(\varphi _1\wedge \varphi _2)&= \eta (\varphi _1)\wedge \eta (\varphi _2)&\eta _f(\varphi _1\vee \varphi _2)&= \eta (\varphi _1)\vee \eta (\varphi _2) \\ \eta _f(\Box \varphi )&= \Box \eta (\varphi )&\eta _f(\Diamond \varphi )&= \Diamond \eta (\varphi ) \end{aligned}$$

and we treat the two clauses $S: \lnot t_{\psi _1\wedge \psi _2} \vee \eta (\psi _1)$ and $S: \lnot t_{\psi _1\wedge \psi _2} \vee \eta (\psi _2)$ resulting from the normal form transformation of $\psi _1\wedge \psi _2$ as a single ‘clause’ $S: \lnot t_{\psi _1\wedge \psi _2} \vee \eta _f(\psi _1\wedge \psi _2)$. We also interchangeably write $S:\lnot t_{\Box \psi }\vee \eta _f(\Box \psi )$ for $S:t_{\Box \psi }\rightarrow \eta _f(\Box \psi )$ and, analogously, $S:\lnot t_{\Diamond \psi }\vee \eta _f(\Diamond \psi )$ for $S:t_{\Diamond \psi }\rightarrow \eta _f(\Diamond \psi )$. We then call any clause of the form $S:\lnot t_\psi \vee \eta _f(\psi )$ a definitional clause.

Definition 1

Let $\varPhi$ be a set of clauses. We say $t_\psi \in T$ occurs at level ml in $\varPhi$ iff either

• (a) there exists a clause $S:\vartheta$ in $\varPhi$ with $ml\in S$ such that $\vartheta$ is a propositional formula and $t_\psi$ occurs positively in $\vartheta$, or

• (b) there exists a clause $S:t_{\Box \psi }\rightarrow \Box t_\psi$ in $\varPhi$ with $ml-1\in S$, or

• (c) there exists a clause $S:t_{\Diamond \psi }\rightarrow \Diamond t_\psi$ in $\varPhi$ with $ml-1\in S$.

Definition 2

Let $\varPhi$ be a set of clauses. Then $\varPhi$ is definition-complete iff for every $t_\psi \in T$ and every level ml, if $t_\psi$ occurs at level ml in $\varPhi$ then there exists a clause $S: \lnot t_\psi \vee \eta _f(\psi )$ in $\varPhi$ with $ml\in S$.

Theorem 2

Let . Then $\varPhi _L =\{\{ 0 \}: t_\varphi \}\cup \rho _{L}(\{0\}:t_\varphi \rightarrow \varphi )$ is definition-complete.

Proof

By induction over the computation of $\varPhi _L$. It is straightforward to see that the transformation of labelled formulae $S: t\rightarrow (\psi _1\wedge \psi _2)$ and $S: t\rightarrow (\psi _1\vee \psi _2)$ only introduces surrogates at levels in S and $\varDelta _L$ then adds definitional clauses for those surrogates. The transformation of a labelled formula $S: t_{\Diamond \psi }\rightarrow \Diamond \psi$ may introduce a surrogate at levels in $S^+$ and $\delta _L(S^+,\psi )$ then adds definitional clauses for those surrogates. The transformation of a labelled formula $S: t_{\Box \psi }\rightarrow \Box \psi$ depends on the logic L. We can see that for every level at which a new surrogate occurs in $P_L(S: t_{\Box \psi }\rightarrow \Box \psi )$, then $\varDelta _L(S: t_{\Box \psi }\rightarrow \Box \psi )$ contains a definitional clause for it at that level.

4 Correctness

Due to space constraints we only prove the correctness of the transformation for . We first state several lemmata that are used in the correctness proofs for all logics.

Lemma 1

Let $\varPhi$ be a set of definitional clauses such that every $t_\psi$ occurring in $\varPhi$ is an element of T and all other propositional symbols occurring in $\varPhi$ are in Q. Let $M=\langle W, R, V, w_0 \rangle$ be a rooted Kripke structure. Let $\langle \vec {W},\vec {R} \rangle$ be the unravelling of $\langle W, R \rangle$ at $w_0$. Let $\vec {M}= \langle \vec {W}, \vec {R}, \vec {V_\Sigma },(w_0) \rangle$ be a Kripke structure such that

• for every propositional symbol $p\in Q$, and

• $\vec {V_\Sigma }(t_\psi ) = \{\vec {w}\in \vec {W}\mid \langle \vec {M}, \vec {w} \rangle \models \psi \}$ for every surrogate .

Then $\vec {M}\models \varPhi$.

Lemma 2

Let $\varphi$ be a L-satisfiable modal formula in simplified NNF where L is a normal modal logic and let . Let $M=\langle W, R, V, w_0 \rangle$ be a rooted model of $\varphi$. Let $\langle \vec {W},\vec {R} \rangle$ be the unravelling of $\langle W,R \rangle$ at $w_0$. Let $\vec {M} = \langle \vec {W}, \vec {R}, \vec {V},(w_0) \rangle$ be a Kripke structure such that

• for every propositional symbol , and

• $\vec {V}(t_\psi ) = \{\vec {w}\in \vec {W}\mid \langle \vec {M}, \vec {w} \rangle \models \psi \}$ for every surrogate .

Then $\vec {M}\models \varPhi$.

Lemma 3

Let $M=\langle W, R, V, w_0\rangle$ be a rooted Kripke structure. Let $\langle \vec {W},\vec {R} \rangle$ be the unravelling of $\langle W, R \rangle$ at $w_0$. Let $\vec {M}= \langle \vec {W}, \vec {R}, \vec {V_\Sigma },(w_0)\rangle$ where for every propositional symbol $p\in Q$.

Then for every modal formula $\psi$ over Q and for every world $\vec {w}\in \vec {W}$, $\langle \vec {M}, \vec {w} \rangle \models \psi$ iff .

Lemma 4

Let $\varphi$ be a modal formula in simplified NNF. Let . Let be a definition-complete set of clauses, let $M = \langle W,R,V,w_0 \rangle$ be a tree model of $\varPhi$ and let $M' = \langle W,R',V,w_0 \rangle$ be such that

• (4a) $R\subseteq R'$;

• (4b) for every modal clause $S: t_{\Box \psi }\rightarrow \Box \eta (\psi )$ in $\varPhi$ and every world $w\in M[S]$, $\langle M', w \rangle \models t_{\Box \psi }\rightarrow \Box \eta (\psi )$;

• (4c) for every modal clause $S: t_{\Box \psi }\rightarrow \Box t_\psi$ in $\varPhi$ and all worlds $v,w\in W$, if (i) $w\in M[S]$ and (ii) $w\mathbin {R'}v$ then (iii) there exists a clause $S':\lnot t_\psi \vee \eta _f(\psi )$ in $\varPhi$ with $v\in M[S']$.

Then $\langle M', w_0 \rangle \models \varphi$.

Theorems 3 and 4 now state the correctness of our transformation for .

Theorem 3

Let $\varphi$ be a modal formula in simplified NNF. Let . If $\varphi$ is -satisfiable, then $\varPhi _B$ is -satisfiable.

Proof

The main idea is to show that given a rooted model of $\varphi$, then a small variation of its unravelling is a rooted tree model of $\varPhi _B$.

Let $M=\langle W, R, V, w_0 \rangle$ be a rooted model of $\varphi$ with $\langle M, w_0 \rangle \models \varphi$ and symmetric relationship $R$. Let $\langle \vec {W},\vec {R} \rangle$ be the unravelling of $\langle W,R \rangle$ at $w_0$. Let where

• for every propositional symbol ,

• for every surrogate introduced by rewriting, and

• for every supplementary propositional symbol $t_{\Box \lnot t_{\Box \psi }}$ introduced in the normal form transformation of a labelled formula $S:t_{\Box \psi }\rightarrow \Box \psi$.

Note that is well-defined as for every surrogate $t_\psi \in T$, $\psi$ only contains propositional symbols in Q. Let .

We now consider the clauses occurring in and show that they hold in . By Lemma 2 it follows that . Also, all definitional clauses in are true in by Lemma 1.

Next consider clauses of the form

where $t_{\Box \psi }$ is a surrogate for $\Box \psi$. These are not in . We show both are true in . We do so by first considering that $t_{\Box \lnot t_{\Box \psi }}$ is true at a world and then that it is false.

• Case (a): Let with . Clearly, . Also, by definition of , . So, for every $\vec {v}\in \vec {W}$ with $\vec {w}\mathbin {\vec {R}}\vec {v}$, . As $t_{\Box \psi }$ is a surrogate for $\Box \psi$, by definition of $\vec {V_B}$, $\vec {v}\not \in \vec {V_B}(t_{\Box \psi })$ and . Thus, and, by the semantics of implication, .

• Case (b): Let with . Clearly, by the semantics of implication, . Also, by definition of $\vec {V_B}$, $\vec {w}\not \in \vec {V_B}(t_{\Box \lnot t_{\Box \psi }})$ implies which in turn implies . So, there exists $\vec {v}\in \vec {W}$ with $\vec {w}\mathbin {\vec {R}}\vec {v}$ and . Since is a p-morphism from to M, . Since $R$ is symmetric, we also have and by construction of , for we have $\vec {v}\mathbin {\vec {R}}\vec {u}$. Since , . As is a p-morphism and and since , . By Lemma 3, from we obtain . If $\psi$ is a literal, then $\eta (\psi ) = \psi$ and $\langle M, \vec {w} \rangle \models \eta (\psi )$. If $\psi$ is not a literal, then $\eta (\psi ) = t_\psi$ and from , by definition of , and . So, $\langle M, \vec {w} \rangle \models \eta (\psi )\vee t_{\Box \lnot t_{\Box \psi }}$.

Thus, in both cases, for arbitrary , $\eta (\psi )\vee t_{\Box \lnot t_{\Box \psi }}$ and $t_{\Box \lnot t_{\Box \psi }} \rightarrow \Box \lnot t_{\Box \psi }$ and therefore Clauses (1) and (2) are true in .

Theorem 4

Let $\varphi$ be a modal formula in simplified NNF. Let . If is -satisfiable, then $\varphi$ is -satisfiable.

Proof

The main idea is to show that given a rooted tree model of , its symmetric closure is a rooted model of $\varphi$.

Let $M = \langle W,R,V,w_0 \rangle$ be a rooted tree model of . Let be a structure such that

• (a) is the symmetric closure of $R$, that is, is the smallest relation on $W$ such that and for every $v,w\in W$, implies ;

• (b) for every propositional symbol.

Let . We show that satisfies the three preconditions of Lemma 4. By Lemma 4 this in turn implies that .

• Condition 4a holds as .

• For Condition 4b let be a modal clause in . Then also contains the additional clauses and . Let $w\in M[S]$. We have to show that  . Assume . As this implies $\langle M, w \rangle \models t_{\Box \psi }$. Let $v\in W$ such that .

  Case (a): Assume $w\mathbin {R}v$. As $\langle M, w \rangle \models t_{\Box \psi }$ and $\langle M, w \rangle \models t_{\Box \psi }\rightarrow \Box \eta (\psi )$, we have $\langle M, w \rangle \models \Box \eta (\psi )$. As $w\mathbin {R}v$, $\langle M, v \rangle \models \eta (\psi )$. As $\eta (\psi )$ is a literal and we obtain . So, .

  Case (b): Assume v is not reachable from w via $R$. Then was introduced by the symmetric closure operation on $R$ and we must have $v\mathbin {R}w$. That is, v is a $R$-predecessor of w and from $w\in M[S]$ it follows that $v\in M[S^-]$. So, and . From $v\mathbin {R}w$, $\langle M, w \rangle \models t_{\Box \psi }$ and (8), it follows that $\langle M, v \rangle \models \lnot t_{\Box \lnot t_{\Box \psi }}$. This together with (7) implies $\langle M, v \rangle \models \eta (\psi )$. As $\eta (\psi )$ is a literal and we obtain . So, .

  Case (a) and Case (b) together show that Property (6) holds.

• For Condition 4c let be in , $v,w\in W$, (i.e., $w\in M[S]$) and . We need to show that there exists a clause $S':\lnot t_\psi \vee \eta _f(\psi )$ in with $v\in M[S']$.

  As in the previous case implies either $w\mathbin {R}v$ or $v\mathbin {R}w$. In the first case while in the second case .

  As contains Clause (9), $t_\psi$ occurs at level $ml+1$ in . By definition of , also contains the clause . As $ml\in S$, $ml-1\in S^-$ and therefore $t_\psi$ also occurs at level $ml-1$ in . By Theorem 2, is definition-complete, so there must be a clause $S':\lnot t_\psi \vee \eta _f(\psi )$ in such that $ml+1$ and $ml-1$ in $S'$.

Theorem 5

Let $\varphi$ be a modal formula in simplified NNF, , and $\varPhi _L = \{\{ 0 \}: t_\varphi \}\cup \rho _{L}(\{0\}:t_\varphi \rightarrow \varphi )$. Then $\varphi$ is -satisfiable iff $\varPhi _{L}$ is -satisfiable.

5 Comparison With Related Work

The approaches most closely related to ours are Kracht’s reductions of normal modal logics to basic modal logic [11, 12], the global modal resolution calculus [14], and Schmidt and Hustadt’s axiomatic translation principle for translations of normal modal logics to first-order logic [24].

The first significant difference to our approach is that Kracht’s reductions and the axiomatic translation exclude the modal operator $\Diamond$ from the language and only consider the modal operator $\Box$.

In order to present Kracht’s approach, we need some additional notions. Let , , and |S| denote the set of all subformulae of $\varphi$, the maximum nesting of modal operators in $\varphi$, and the cardinality of the set S, respectively. Let $\Diamond ^0\psi =\Box ^0\psi =\Box ^{<1}\psi = \psi$, $\Box ^{<n+1}\psi = (\psi \wedge \Box \Box ^{<n}\psi )$, $\Box ^{n+1}\psi =\Box \Box ^n\psi$, and $\Diamond ^{n+1}\psi =\Diamond \Diamond ^n\psi$. We can then define a reduction function $\rho ^\mathsf {K}_L$ for a normal modal logic L in as follows:

where Kracht shows that $\varphi$ is L-satisfiable iff $\rho ^\mathsf {K}_L(\varphi )$ is $\mathsf {K}{}{}$-satisfiable. There are three differences to our approach. First, $P^\mathsf {K}_L(\varphi )$ will include an axiom instance for every occurrence of a subformula $\lnot \Box \psi$, equivalent to $\Diamond \lnot \psi$, in $\varphi$. In contrast, our approach requires no logic specific treatment of such subformulae. Second, the use of $\Box ^{<n} P^\mathsf {K}_L(\varphi )$ in $\rho ^\mathsf {K}_L$ means that the axiom instance is available at every modal level. This means, for example, that for $\vartheta _1 = \Diamond ^{100}(\lnot p\wedge \Box p)$, the formula contains the axiom instance $\Box p\rightarrow p$ over 100 times, although it is only required at the level at which $\Box p$ occurs. Third, this is further compounded if the formula $\psi$ in $\Box \psi$ is itself a complex formula. We try to avoid that by using a surrogate propositional symbol $t_\psi$ instead, but this will only have a positive effect if the definitional clauses for $t_\psi$ do not have to be repeated.

Table 4. Inference rules in [14] for (EUC1 and EUC2).

The global modal resolution (GMR) calculus operates on clauses, that is, clauses of the form

$$ \textstyle \Box ^*(\mathbf{start}\rightarrow \bigvee _{b=1}^{r} l_b) \textstyle \Box ^*(\mathbf{true}\rightarrow \bigvee _{b=1}^{r} l_b) \Box ^* (l' \rightarrow \Box l) \Box ^* (l' \rightarrow \lnot \Box l) $$

where l, $l'$, $l_b$ are propositional literals with $1 \le b \le r$, $r \in \mathbb N$, and $\Box ^*$ is the universal operator. The calculus has specific inference rules for normal modal logics such as , , , , . Table 4 shows the two additional rules for , the only logic for which there are rules for both $\Box$ and $\lnot \Box \lnot$, i.e., $\Diamond$. These inference rules can be seen to perform an ‘on-the-fly’ computation of a transformation. Note that the clauses produced by differ from those produced by GMR for . Implicitly, our results here also show that it should be possible to eliminate EUC1 from the GMR calculus.

For the axiomatic translation, we only present the function $P^\mathsf {RS}_L$ that computes the logic dependent first-order clausal formulae that are part of the overall translation.

The predicate symbols $Q_\psi$ correspond to our surrogate symbols $t_\psi$. The clausal formulae used in the treatment of and are translations of the clauses we use (or vice versa). and are handled in a different way as the first-order clausal formulae refer directly the accessibility relation and can therefore more easily express the transfer of information to a predecessor world. The universal quantification over worlds also means that the constraints expressed by the formulae hold at all modal levels without the need of any repetition.

In Section 6 we will also use the relational and semi-functional translation of modal logics to first-order logic combined with structural transformation to clause normal form. In both approaches $\Box \psi$ is translated as $\forall x y (\lnot Q_{\Box \psi }(x)\vee \lnot R(x,y)\vee Q_\psi$, while $\Diamond \psi$ becomes $\forall x\exists y (\lnot Q_{\Diamond \psi }(x)\vee R(x,y))$ and $\forall x\exists \alpha (\lnot Q_{\Diamond \psi }(x)\vee R(x,[x\alpha ]))$ in the relational and semi-functional translation, respectively. Then, depending on the modal logics, further formulae representing the semantic properties of the accessibility R are added. For the relational translation these will simply be the formulae in the fourth column of Table 1. The semi-functional translation uses collections of partial accessibility function in addition to the accessibility relation. A predicate $\mathrm {def}$ is used to represent on which worlds a partial accessibility function is defined. For each modal logic there is then again a background theory consisting of formulae over $\mathrm {def}$ and R that represents the properties of the underlying accessibility relation which is added to the translation of a formula. For example, for the background theory is: $\forall x y\forall \alpha \beta ((\lnot \mathrm {def}(x)\vee \mathrm {def}(y))\wedge (\lnot \mathrm {def}(\mathrm {w_0})\vee R(\mathrm {w_0},[\mathrm {w_0}\alpha ]))\wedge (\lnot \mathrm {def}(x)\vee \lnot \mathrm {def}(y)\vee R([x\alpha ],[y\beta ]))),$ where $\mathrm {w_0}$ is a constant representing the root world in a rooted Kripke structure.

6 Evaluation

We have compared the performance of the following approaches: (i) the combination of our reductions with the modal-layered resolution (MLR) calculus for clauses [15] implemented in the modal theorem prover , with three different refinements for resolution inferences on labelled propositional clauses (ii) the global modal resolution (GMR) calculus, also implemented in , with three different refinements for resolution inferences on propositional clauses (iii) the combinations of the relational and semi-functional translation of modal logics to first-order logic with ordered first-order resolution implemented in the first-order theorem prover SPASS. In total this gives us eight different approaches to compare. The axiomatic translation is currently not implemented in SPASS. Other provers, such as LEO-III [26], LWB [9], MleanCoP [21], do not have built-in support for the full range of logics considered here. LoTREC 2.0 [7] supports all the logics, but is not intended as automatic theorem prover.

The modal-layered resolution calculus operates on clauses, that is, clauses of the form

$$ \textstyle ml: \bigvee _{b=1}^{r} l_b\qquad ml: l' \rightarrow \Box l\qquad ml: l' \rightarrow \Diamond l $$

where and l, $l'$, $l_b$ are propositional literals with $1 \le b \le r$, $r \in \mathbb N$. In the implementation of the reductions presented in Section 3, we take a clause $S:\psi$ simply as an abbreviation of the set of clauses $\{ml:\psi \mid ml\in S\}$. Note that this also means that we will have to repeat similar resolution inferences for different modal levels.

   [13] implements the reductions presented in Section 3 as well as a normal form transformation of modal formulae to sets of clauses. It implements both the MLR and the GMR calculus. Resolution inferences between (labelled) propositional clauses can either be unrestricted (cplain option), restricted by an ordering (cord option), that is, clauses can only be resolved on their maximal literals with respect to an ordering chosen by the prover in such a way to preserve completeness, restricted to negative resolution (cneg option), that is, one of the premises in an inference has to be a negative clause, or restricted to positive resolution. We do not include the last option in our evaluation as it typically performs worse. also implements a range of simplification rules that are applied to modal formulae before their transformation to normal form. Of those we have enabled pure literal elimination (early_ple option), simplification using the Box Normal Form [22] and Prenex Normal Form (bnfsimp and prenex options) [17]. For clause processing, unit resolution and pure elimination are enabled (unit, lhs_unit, and ple options).

SPASS 3.9 [27, 28] supports automated reasoning in extended modal logics, including all logics considered here, PDL-like modal logics as well as description logics. It includes eight different translations of modal logics to first-order logic. In our evaluation we have used the relational translation and the semi-functional translation. For the local satisfiability problem in to , for the relational translation we have added the first-order frame properties given in Table 1 while for the semi-functional translation we have added the background theories devised by Nonnengart [20]. For the transformation to first-order clausal form, we have enabled renaming of quantified subformulae. The only inference rules used are ordered resolution and ordered factoring, the reduction rules used are condensing, backward subsumption and forward subsumption. For the relational and semi-functional translation for , , , and we thereby obtain a decision procedure, while for the other logics we do not. For and , the fragment of first-order clausal logic corresponding to the semi-functional translation of modal formula and their background theories is decidable by ordered resolution with selection [25]. However, the non-trivial ordering and selection function required is not currently implemented in SPASS.

Table 5. Experimental results on LWB benchmark collection

For our evaluation we have chosen the LWB basic modal logic benchmark collection [2], with 20 formulae in each of 18 parameterised classes. For , all formulae in 9 classes are satisfiable while all formulae in the other 9 classes are unsatisfiable. In their negation normal form, 63% of modal operators are $\Box$ and 37% are $\Diamond$ operators. We have used the collection for each of the six logics. If a formula is unsatisfiable in then it remains unsatisfiable in the other five logics, while the opposite is not true. As we move to logics other than , it is also no longer the case that all formulae in a class have the same satisfiability status.

The third column in Table 5 indicates the total number of satisfiable and unsatisfiable formulae for each logic. In the last two lines of the table we sum up the results for all logics. The last eight columns in the table show how many formulae each of the approaches were able to solve with a time limit of 100 CPU seconds for each formula. Benchmarking was performed on a PC with an AMD Ryzen 5 5600X CPU @ 4.60GHz max and 32GB main memory using Fedora release 33 as operating system.

As we can see, the new reductions combined with the modal-layered resolution (MLR) calculus and ordered resolution refinement (cord) perform best, achieving the highest number of solved formulae in 8 out of 12 individual categories in the table, on two of those equal with the global modal resolution (GMR) calculus. On 3 categories, GMR outperfoms MLR. On both satisfiable and unsatisfiable formulae in this can be seen as evidence that ‘on-the-fly’ transformation offers a (slight) advantage over our approach given that the additional clauses hold universally in both approaches. For SPASS we see a clear advantage of the semi-functional translation over the relational one, on both satisfiable and unsatisfiable formulae.

7 Conclusion and Future Work

We have presented new reductions of propositional modal logics , , , , to Separated Normal Form with Sets of Modal Levels. We have shown experimentally that these reductions allow us to reason effectively in these logics.

The obvious next step is to consider extensions of the basic modal logic with combinations of the axioms $\mathsf {B}$, $\mathsf {D}$, $\mathsf {T}$, $\mathsf {4}$, and $\mathsf {5}$. Unfortunately, a simple combination of the reductions for each of the axioms is not sufficient to obtain a satisfiability-preserving reduction for the such modal logics. An example is the simple formula $\lnot p\wedge \Diamond \Diamond \Box p$ which is -unsatisfiable. If we define

that is, is the union of and , then the clause set obtained from is -satisfiable. The same issue also occurs in the axiomatic translation of modal logics to first-order logic where the translation for is not simply the combination of the translations for and [24, Theorem 5.6]. We are currently exploring solutions to this problem.

Regarding practical applications, it would be advantageous to have an implementation of a calculus that operates directly clauses. This would greatly reduce the number of performed on satisfiable formulae and simplify proof search in general. Again, such an implementation is future work.