462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\documentclass{entcs} \usepackage{entcsmacro}

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462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder% At @ Left bracket [ Backslash \

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462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder% A couple of exemplary definitions:

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder%\newcommand{\Nat}{{\mathbb N}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder%\newcommand{\Real}{{\mathbb R}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\newcommand{\COLOSS}{{\textrm CoLoSS}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\def\lastname{Hausmann and Schr\"oder}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\begin{document}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\begin{frontmatter}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \title{Optimizing Conditional Logic Reasoning within \COLOSS}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \author[DFKI]{Daniel Hausmann\thanksref{myemail}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \author[DFKI,UBremen]{Lutz Schr\"oder\thanksref{coemail}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \address[DFKI]{DFKI Bremen, SKS}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \address[UBremen]{Department of Mathematics and Computer Science, Universit\"at Bremen, Germany}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder% \thanks[ALL]{Work forms part of DFG-project \emph{Generic Algorithms and Complexity

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder% Bounds in Coalgebraic Modal Logic} (SCHR 1118/5-1)}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \thanks[myemail]{Email: \href{mailto:Daniel.Hausmann@dfki.de} {\texttt{\normalshape Daniel.Hausmann@dfki.de}}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \thanks[coemail]{Email: \href{mailto:Lutz.Schroeder@dfki.de} {\texttt{\normalshape Lutz.Schroeder@dfki.de}}}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\begin{abstract}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder The generic modal reasoner \COLOSS~covers a wide variety of logics

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder ranging from graded and probabilistic modal logic to coalition logic

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder and conditional logics, being based on a broadly applicable

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder optimisation of the reasoning strategies employed in

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \COLOSS. Specifically, we discuss strategies of memoisation and

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder dynamic programming that are based on the observation that short

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder sequents play a central role in many of the logics under

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder study. These optimisations seem to be particularly useful for the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder case of conditional logics, for some of which dynamic programming

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder even improves the theoretical complexity of the algorithm. These

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder strategies have been implemented in \COLOSS; we give a detailed

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder comparison of the different heuristics, observing that in the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder targeted domain of conditional logics, a substantial speed-up can be

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder achieved.

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\end{abstract}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\begin{keyword}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder Coalgebraic modal logic, conditional logic, automated reasoning,

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder optimisation, heuristics, memoizing, dynamic programming

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\end{keyword}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\end{frontmatter}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\section{Introduction}\label{intro}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian MaederIn recent decades, modal logic has seen a development towards semantic

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederheterogeneity, witnessed by an emergence of numerous logics that,

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederwhile still of manifestly modal character, are not amenable to

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederstandard Kripke semantics. Examples include probabilistic modal

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederlogic~\cite{FaginHalpern94}, coalition logic~\cite{Pauly02}, and

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederconditional logic~\cite{Chellas80}, to name just a few. The move away

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederfrom Kripke semantics, mirrored on the syntactical side by the failure

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederof normality, entails additional challenges for tableau and sequent

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedersystems, as the correspondence between tableaus and models becomes

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederlooser, and in particular demands created by modal formulas can no

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederlonger be discharged by the creation of single successor nodes.

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian MaederThis problem is tackled on the theoretical side by introducing the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedersemantic framework of coalgebraic modal

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederlogic~\cite{Pattinson03,Schroder05}, which covers all logics mentioned

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederabove and many more. It turns out that coalgebraic modal logic does

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederallow the design of generic reasoning algorithms, including a generic

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedertableau method originating from~\cite{SchroderPattinson09}; this

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedergeneric method may in fact be separated from the semantics and

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederdeveloped purely syntactically, as carried out

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederin~\cite{PattinsonSchroder08b,PattinsonSchroder09a}.

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian MaederGeneric tableau algorithms for coalgebraic modal logics, in particular

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederthe algorithm described in~\cite{SchroderPattinson09}, have been

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederimplemented in the reasoning tool

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\COLOSS~\cite{CalinEA09}\footnote{available under

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder \url{http://www.informatik.uni-bremen.de/cofi/CoLoSS/}}. As

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederindicated above, it is a necessary feature of the generic tableau

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedersystems that they potentially generate multiple successor nodes for a

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedergiven modal demand, so that in addition to the typical depth problem,

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederproof search faces a rather noticeable problem of breadth. The search

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederfor optimisation strategies to increase the efficiency of reasoning

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederthus becomes all the more urgent. Here we present one such strategy,

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederwhich is generally applicable, but particularly efficient in reducing

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederboth depth and branching for the class of conditional logics. We

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederexploit a notable feature of this class, namely that many of the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederrelevant rules rely rather heavily on premises stating equivalence

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederbetween formulas; thus, conditional logics are a good candidate for

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedermemoising strategies, applied judiciously to short sequents. We

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederdescribe the implementation of memoising and dynamic programming

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederstrategies within \COLOSS, and discuss the outcome of various

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedercomparative experiments.

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder\section{Generic Sequent Calculi for Coalgebraic Modal Logic}

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maeder

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian MaederCoalgebraic modal logic, originally introduced as a specification

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederlanguage for coalgebras, seen as generic reactive

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedersystems~\cite{Pattinson03}, has since evolved into a generic framework

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederfor modal logic beyond Kripke semantics~\cite{CirsteaEA09}. The basic

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederidea is to encapsulate the branching type of the systems relevant for

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederthe semantics of a particular modal logic, say probabilistic or

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedergame-theoretic branching, in the choice of a set functor, the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedersignature functor (e.g.\ the distribution functor and the games

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederfunctor in the mentioned examples), and to capture the semantics of

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maedermodal operators in terms of so-called predicate liftings. For the

462ec4b2fa3e0e788eb60dcb4aebc518298f342cChristian Maederpurposes of the present work, details of the semantics are less

relevant than proof-theoretic aspects, which we shall recall

presently. The range of logics covered by the coalgebraic approach is

extremely broad, including, besides standard Kripke and neighbourhood

semantics, e.g.\ graded modal logic~\cite{Fine72}, probabilistic modal

logic~\cite{FaginHalpern94}, coalition logic~\cite{Pauly02}, various

conditional logics equipped with selection function

semantics~\cite{Chellas80}, and many more.

Syntactically, logics are parametrised by the choice of a \emph{modal

similarity type} $\Lambda$, i.e.\ a set of modal operators with

associated finite arities. This choice determines the set of formulas

$\phi,\psi$ via the grammar

\begin{equation*}

\phi, \psi ::= p \mid

\phi \land \psi \mid \lnot \phi \mid \hearts(\phi_1, \dots, \phi_n)

\end{equation*}

where $\hearts$ is an $n$-ary operator in $\Lambda$. Examples are

$\Lambda=\{L_p\mid p\in[0,1]\cap\mathbb{Q}\}$, the unary operators

$L_p$ of probabilistic modal logic read `with probability at least

$p$'; $\Lambda=\{\gldiamond{k}\mid k\in\Nat\}$, the operators

$\gldiamond{k}$ of graded modal logic read `in more than $k$

successors'; $\Lambda=\{[C]\mid C\subseteq N\}$, the operators $[C]$

of coalition logic read `coalition $C$ (a subset of the set $N$ of

agents) can jointly enforce that'; and, for our main example here,

$\Lambda=\{\Rightarrow\}$, the \emph{binary} modal operator

$\Rightarrow$ of conditional logic read e.g.\ `if \dots then normally

\dots'.

Coalgebraic modal logic was originally limited to so-called

\emph{rank-$1$ logics} axiomatised by formulas with nesting depth of

modal operators uniformly equal to $1$~\cite{Schroder05}. It has since

been extended to the more general non-iterative

logics~\cite{SchroderPattinson08d} and to some degree to iterative

logics, axiomatised by formulas with nested

modalities~\cite{SchroderPattinson08PHQ}. The examples considered here

all happen to be rank-$1$, so we focus on this case. In the rank-$1$

setting, it has been shown~\cite{Schroder05} that all logics can be

axiomatised by \emph{one-step rules} $\phi/psi$, where $\phi$ is

purely propositional and $\psi$ is a clause over formulas of the form

$\hearts(a_1,\dots,a_n)$, where the $a_i$ are propositional

variables. In the context of a sequent calculus, this takes the

following form~\cite{PattinsonSchroder08b}.

\begin{definition}

If $S$ is a set (of formulas or variables) then $\Lambda(S)$ denotes

the set $\lbrace \hearts(s_1, \dots, s_n) \mid \hearts \in \Lambda

\mbox{ is $n$-ary, } s_1, \dots, s_n \in S \rbrace$ of formulas

comprising exactly one application of a modality to elements of

$S$. An \emph{$S$-sequent}, or just a \emph{sequent} in case

$S=\Form(\Lambda)$, is a finite set of elements of $S \cup \lbrace

\neg A \mid A \in S \rbrace$. Then, a \emph{one-step rule}

$\Gamma_1,\dots,\Gamma_n/\Gamma_0$ over a set $V$ of variables

consists of $V$-sequents $\Gamma_1,\dots,\Gamma_n$ and a

$\Lambda(S)$-sequent $\Gamma_0$.

\end{definition}

\noindent A given set of one-step rules then induces an instantiation

of the \emph{generic sequent calculus}~\cite{PattinsonSchroder08b}

which is given by a set of rules $\mathcal{R}_{sc}$ consisting of the

finishing and the branching rules $\mathcal{R}^b_{sc}$ (i.e. rules

with no premise or more than one premise), the linear rules

$\mathcal{R}^l_{sc}$ (i.e. rules with exactly one premise) and the

modal rules $\mathcal R^m_{sc}$, i.e.\ the given one-step rules. The

finishing and the branching rules are presented in

Figure~\ref{fig:branching} (where $\top=\neg\bot$ and $p$ is an atom),

the linear rules are shown in Figure~\ref{fig:linear}. So far, all

these rules are purely propositional. As an example for a set of modal

one-step rules, consider the modal rules $\mathcal R^m_{sc}$ of the

standard modal logic \textbf{K} as given by Figure~\ref{fig:modalK}.

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| c c c |}

\hline

& & \\[-5pt]

(\textsc {$\neg$F}) \inferrule{ }{\Gamma, \neg\bot} &

(\textsc {Ax}) \inferrule{ }{\Gamma, p, \neg p} &

(\textsc {$\wedge$}) \inferrule{\Gamma, A \\ \Gamma, B}{\Gamma, A\wedge B} \\[-5pt]

& & \\

\hline

\end{tabular}

\end{center}

\caption{The finishing and the branching sequent rules $\mathcal{R}^b_{sc}$}

\label{fig:branching}

\end{figure}

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| c c |}

\hline

& \\[-5pt]

(\textsc {$\neg\neg$})\inferrule{\Gamma, A}{\Gamma, \neg\neg A} &

(\textsc {$\neg\wedge$}) \inferrule{\Gamma, \neg A, \neg B}{\Gamma, \neg(A\wedge B)} \\[-5pt]

& \\

\hline

\end{tabular}

\end{center}

\caption{The linear sequent rules $\mathcal{R}^l_{sc}$}

\label{fig:linear}

\end{figure}

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{K}})\inferrule{\neg A_1, \ldots , \neg A_n, A_0}

{\Gamma, \neg \Box A_1,\ldots,\neg \Box A_n, \Box A_0 } \\[-5pt]

\\

\hline

\end{tabular}

\end{center}

\caption{The modal rule of \textbf{K}}

\label{fig:modalK}

\end{figure}

This calculus has been shown to be complete under suitable coherence

assumptions on the rule set and the coalgebraic

semantics~\cite{PattinsonSchroder08b}, provided that the set of rules

\emph{absorbs cut and contraction} in a precise sense. We say that a

formula is \emph{provable} if it can be derived in the relevant

instance of the sequent calculus; the algorithms presented below are

concerned with the \emph{provability problem}, i.e.\ to decide whether

a given sequent is provable. This is made possible by the fact that

the calculus does not include a cut rule, and hence enables automatic

proof search. For rank-1 logics, proof search can be performed in

$\mathit{PSPACE}$ under suitable assumptions on the representation of

the rule set~\cite{SchroderPattinson09,PattinsonSchroder08b}. The

corresponding algorithm has been implemented in the system

CoLoSS~\cite{CalinEA09} which remains under continuous

development. Particular attention is being paid to finding heuristic

optimisations to enable practically efficient proof search, as

described here.

Our running example used in the presentation of our optimisation

strategies below is conditional logic as mentioned above. The most

basic conditional logic is $\mathit{CK}$~\cite{Chellas80} (we shall

consider a slightly extended logic below), which is characterised by

assuming normality of the right-hand argument of the non-monotonic

conditional $\Rightarrow$, but only replacement of equivalents in the

left-hand arguments. Absorption of cut and contraction requires a

unified rule set consisting of the rules

\begin{equation*}

\infrule{A_0=A_1;\;\dots;\; A_n=A_0\quad \neg B_1,\dots,\neg B_1,B_0}

{\neg(A_1\Rightarrow B_1),\dots,\neg(A_n\Rightarrow B_n),(A_0\Rightarrow B_0)}

\end{equation*}

with $A=C$ abbreviating the pair of sequents $\neg A,C$ and $\neg

C,A$. This illustrates two important points that play a role in the

optimisation strategies described below. First, the rule has a large

branching degree both existentially and universally -- we have to

check that there \emph{exists} a rule proving some sequent such that

\emph{all} its premises are provable, and hence we have to guess one

of exponentially many subsequents to match the rule and then attempt

to prove linearly many premises; compare this to linearly many rules

and a constant number of premises per rule (namely, $1$) in the case

of $K$ shown above. Secondly, many of the premises are short

sequents. This will be exploited in our memoisation strategy. We note

that \emph{labelled} sequent calculi for conditional logics have been

designed previously~\cite{OlivettiEA07} and have been implemented in

the CondLean prover~\cite{OlivettiPozzato03}; contrastingly, our

calculus is unlabelled, and turns out to be conceptually simpler. The

comparison of the relative efficiency of labelled and unlabelled

calculi remains an open issue for the time being.

\section{The Algorithm}

According to the framework introduced above, we now devise a generic

algorithm to decide the provability of formulae. By instantiating the

generic algorithm to a specific modal logic (simply by using the

defining modal rule of this logic), it is possible to obtain an

algorithm to decide provability of formulae of this specific modal

logic.

\noindent

Algorithm~\ref{alg:seq} makes use of the sequent rules in the following manner:

In order to show the \emph{provability} of a formula $\phi$, the

algorithm starts with the sequent $\{\phi\}$ and tries to apply all

of the sequent rules $r\in R_{sc}$ to it

\footnote{The actual haskell

implementation of the algorithm first tries to apply the linear rules

from $\mathcal{R}^l_{sc}$ and afterwards it tries to apply the other

rules from $\mathcal{R}^b_{sc}$ and $\mathcal{R}^m_{sc}$.}.

% explain what 'applying' means

The result of this one step of application of the rules will be a set

of premises $P$. The proving function is then recursively started on

each sequent of every premise in this set. If there is any premise

whose sequents are all provable, the application of the according rule

is justified and hence the algorithm has succeeded. Below, we refer to

a sequent as \emph{open} if it has not yet been checked for

provability, and otherwise as \emph{treated}.

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

Step 1: Take the input formula $\phi$, construct the initial set

of premises (containing just one premise with just one sequent)

$S = \{\{\phi\}\}$ from it.\\

Step 2: Try to apply all rules $r\in R_{sc}$ to any open sequent

$\Gamma\in P$ from any premise $P\in S$. Let $\Lambda$ denote the set

of the resulting premises.\\

Step 3: If $\Lambda$ contains the empty premise, mark the open

sequent $\Gamma$ as proved (and if $P$ contains no more open

sequents, mark $P$ as proved). If $\Lambda$ contains no premise at

all, mark the open sequent $\Gamma$ and the premise $P$ as refuted.\\

Step 4: Else, add all premises from $\Lambda$ to the set of

premises (i.e. $S=S\cup\Lambda$). If there are any remaining open

sequents, continue with Step 2.\\

Step 5: If all premises resulting from the application of all rules

from $R_{sc}$ to $\phi$ are marked as proved, finish with result \verb|True|

else finish with result \verb|False|.

\end{upshape}

\label{alg:seq}

\end{alg}

\end{algorithm}

\begin{proposition}

\begin{upshape}

Let $\mathcal{R}_{sc}$ be strictly one-step complete, closed under contraction,

and PSPACE-tractable. Then Algorithm~\ref{alg:seq} is sound and complete w.r.t. provability

and it is in PSPACE.

\end{upshape}

\end{proposition}

\begin{proof}

We just note, that Algorithm~\ref{alg:seq} is an equivalent implementation of the algorithm proposed

in~\cite{SchroderPattinson09}. For more details, refer to ~\cite{SchroderPattinson09}, Theorem 6.13.

\end{proof}

\subsection{The conditional logic instance}

The genericity of the introduced sequent calculus allows us to easiliy

create instantiations of Algorithm~\ref{alg:seq} for a large variety

of modal logics: By setting $\mathcal R^m_{sc}$ for instance to the

rule shown in Figure~\ref{fig:modalCKCEM} (where $A_a = A_b$ is

shorthand for $A_a\rightarrow A_b\wedge A_b\rightarrow A_a$), we

obtain an algorithm for deciding provability (and satisfiability) of

conditional logic. We restrict ourselves to the examplary conditional

logic \textbf{CKCEM} for the remainder of this section; slightly

adapted versions of the optimisation will work for other conditional

logics. \textbf{CKCEM} is characterised by the additional axioms of

\emph{conditional excluded middle} $(A\Rightarrow

B)\lor(A\Rightarrow\neg B)$, which to absorb cut and contraction is

integrated in the rule for \textbf{CK} as follows.

\begin{figure}[h!]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{CKCEM}})\inferrule{A_0 = \ldots = A_n \\ B_0,\ldots, B_j,\neg B_{j+1},\ldots,\neg B_n}

{\Gamma, (A_0\Rightarrow B_0),\ldots,(A_j\Rightarrow B_j),

\neg(A_{j+1}\Rightarrow B_{j+1}),\ldots,\neg(A_n\Rightarrow B_n) } \\[-5pt]

\\

\hline

\end{tabular}

\end{center}

\caption{The modal rule $\textbf{CKCEM}$ of conditional logic}

\label{fig:modalCKCEM}

\end{figure}

It has been shown in~\cite{PattinsonSchroder09a} that the complexity

of \textbf{CKCEM} is $\mathit{coNP}$, using a dynamic programming

approach in the spirit of~\cite{Vardi89}; in fact, this was the

original motivation for exploring the optimisation strategies pursued

here.

In the following, we use the notions of \emph{conditional antecedent} and

\emph{conditional consequent} to refer to the parameters of the modal operator of

conditional logic.

In order to decide whether there is an

instance of the modal rule of conditional logic which can be applied to

the actual current sequent, it is necessary to create a preliminary premise

for each possible combination of equalities of all the premises of the

modal operators in this sequent. This results in $2^n$ new premises for

a sequent with $n$ top-level modal operators.

\begin{example}

Consider the sequent $\Gamma=\{(A_0\Rightarrow B_0),(A_1\Rightarrow B_1),

(A_2\Rightarrow B_2)\}$. By instantiating the modal rule, several

new premises

such as $\{\{A_0=A_1=A_2\},\{B_0,B_1,B_2\}\}$ or $\{\{A_0=A_2\},\{B_0,B_2\}\}$

may be obtained. Only after the equalities in the first sequent of each

of these premises have been shown (or refuted), it is clear whether the

current instantiation of the modal rule is actually leading to a possibly

provable premise.

\label{ex:cond}

\end{example}

It seems to be a more intelligent approach to first

partition the set of all antecedents of the top-level modal operators in the

current sequent into equivalence classes with respect to logical equality.

This partition does not only allow for a reduction of the number of newly

created open premises in each modal step,

but it also allows us to seperate the two parts of the conditional modal step

(i.e. the first part, showing the necessary equalities, and the second

part, showing that the resulting sequent - consisting only of the appropriate

conditional consequences - is actually provable itself).

\begin{example}

Consider again the sequent from Example~\ref{ex:cond}. By using the examplary

knowledge, that $A_0=A_1$, $A_1\neq A_2$ and $A_0\neq A_2$, it is immediate,

that there are just two reasonable instations of the modal rule, leading

to the two premises $\{\{B_0,B_1\}\}$ and $\{\{B_2\}\}$. For the

first of these two premises, note that it is not necessary to show the

equivalence of $A_0$ and $A_1$ again.

\end{example}

\begin{remark}

In the case of conditional logic, observe the following: Since the

modal antecedents that appear in a formula are not being changed by any

rule of the sequent calculus, it is possible to extract all possibly

relevant antecedents of a formula even \emph{before} the actual sequent

calculus iswe applied. This allows us to first compute the equivalence

classes of all the relevant antecedents and then feed this knowledge into

the actual sequent calculus:

\end{remark}

\section{The Optimisation}

\begin{definition}

A conditional antecedent of \emph{modal nesting depth} $i$ is a

conditional antecedent which contains at least one antecedent of

modal nesting depth $i-1$ and which contains no antecedent

of modal nesting depth greater than $i-1$. A

conditional antecedent of nesting depth 0 is an antecedent

that does not contain any further modal operators.

Let $p_i$ denote the set of all conditional antecedents of modal

nesting depth $i$. Further, let $prems(n)$ denote the set of all

conditional antecedents of modal nesting depth at most $n$ (i.e.

$prems(n)=\bigcup_{j=1..n}^{} p_j$).

Finally, let $depth(\phi)$ denote the maximal modal nesting in

the formula $\phi$.

\end{definition}

\begin{example}

In the formula $\phi=((p_0\Rightarrow p_1) \wedge ((p_2\Rightarrow p_3)\Rightarrow p_4))

\Rightarrow (p_5\Rightarrow p_6)$,

the premise $((p_0\Rightarrow p_1) \wedge ((p_2\Rightarrow p_3)\Rightarrow p_4))$ has a

nesting depth of 2 (and not 1), $(p_0\Rightarrow p_1)$ and $(p_2\Rightarrow p_3)$ both

have a nesting depth of 1, and finally $p_0$, $p_2$ and $p_4$ all have a nesting depth of 0.

Furthermore, $prems(1)=\{p_0,p_2,p_4,(p_0\Rightarrow p_1),(p_2\Rightarrow p_3)\}$

and $depth(\phi)=3$.

\end{example}

\begin{definition}

A set $\mathcal{K}$ of treated sequents together with a function

$eval:\mathcal{K}\rightarrow \{\top,\bot\}$ is called \emph{knowledge}.

\end{definition}

We may now construct an optimized algorithm which allows us

to decide provability (and satisfiability) of formulae more efficiently

in some cases. The optimized algorithm is constructed from two functions

(namely from the actual proving function and from the so-called

\emph{pre-proving} function):

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

Step 1: Take the input formula $\phi$ and the knowledge $(\mathcal{K},eval)$,

construct the initial set of premises (containing just one premise with just

one sequent) $S = \{\{\phi\}\}$ from it.\\

Step 2: Try to apply all rules $r\in R_{sc}$ (supplying $\mathcal{RO}^m_{sc}$

with knowledge $(\mathcal{K},eval)$) to any open sequent

$\Gamma\in P$ from any premise $P\in S$. Let $\Lambda$ denote the set

of the resulting premises.\\

Step 3: If $\Lambda$ contains the empty premise, mark the open

sequent $\Gamma$ as proved (and if $P$ contains no more open

sequents, mark $P$ as proved). If $\Lambda$ contains no premise at

all, mark the open sequent $\Gamma$ and the premise $P$ as refuted.\\

Step 4: Else, add all premises from $\Lambda$ to the set of

premises (i.e. $S=S\cup\Lambda$). If there are any remaining open

sequents, continue with Step 2.\\

Step 5: If all premises resulting from the application of all rules

from $R_{sc}$ to $\phi$ are marked as proved, finish with result \verb|True|

else finish with result \verb|False|.

\end{upshape}

\label{alg:optSeq}

\end{alg}

\end{algorithm}

Algorithm~\ref{alg:optSeq} is very similar to Algorithm~\ref{alg:seq},

however, it uses a modified

set of rules $\mathcal{RO}_{sc}$ and it also allows for the

input of some knowledge $(\mathcal{K},eval)$ in the form of a set of sequents

together with an evaluation function. This knowledge is passed on to

the modified modal rule of conditional logic, which makes appropriate use of it.

$\mathcal{RO}_{sc}$ is obtained from $\mathcal{R}_{sc}$ by replacing

the modal rule from Figure~\ref{fig:modalCKCEM} with the modified modal

rule from Figure~\ref{fig:modalCKCEMm}.

\begin{figure}[h!]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{CKCEM}$^m$})\inferrule{\bigwedge{}_{i,j=\{1..n\}}{eval(A_i=B_j)=\top}

\\ B_0,\ldots, B_j,\neg B_{j+1},\ldots,\neg B_n}

{\Gamma, (A_0\Rightarrow B_0),\ldots,(A_j\Rightarrow B_j),

\neg(A_{j+1}\Rightarrow B_{j+1}),\ldots,\neg(A_n\Rightarrow B_n) } \\[-5pt]

\\

\hline

\end{tabular}

\end{center}

\caption{The modified modal rule \textbf{CKCEM}$^m$ of conditional logic}

\label{fig:modalCKCEMm}

\end{figure}

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

Step 1: Take a formula $\phi$ as input. Set $i=0$, $\mathcal{K}_0=\emptyset$, $eval_0=\emptyset$.\\

Step 2: Generate the set $prems_i$ of all conditional antecedents of $\phi$

of nesting depth at most $i$. If $i<depth(\phi)$ continue

with Step 3, else set $\mathcal{K}=\mathcal{K}_{i-1}, eval=eval_{i-1}$ and continue with Step 4.\\

Step 3: Let $eq_i$ denote the set of all equalities $A_a = A_b$ for different

formulae $A_a,A_b\in prems_i$. Compute

Algorithm~\ref{alg:optSeq} ($\psi$, $(\mathcal{K}_i,eval_i)$) for all $\psi\in eq_i$.

Set $\mathcal{K}_{i+1} = eq_i$, set $i = i + 1$. For each equality $\psi\in eq_i$,

set $eval_{i+1}(\psi)=\top$ if the result of Algorithm~\ref{alg:optSeq} was \verb+True+

and $eval_{i+1}(\psi)=\bot$ otherwise. Continue with Step 2.\\

Step 4: Call Algorithm~\ref{alg:optSeq} ($\phi$, $(\mathcal{K},eval)$) and return its result

as result.

\label{alg:preprove}

\end{upshape}

\end{alg}

\end{algorithm}

Algorithm~\ref{alg:preprove} first computes the knowledge $(\mathcal{K},eval)$ about specific

subformulae of $\phi$ and then finally checks for provability of

$\phi$ (using this knowledge): In order to show the equivalence of

two conditional antecedents of nesting depth at most $i$, we assume,

that the equalities $\mathcal{K}_{i}$ between modal antecedents of nesting depth less

than $i$ have allready been computed and the result is stored in $eval_i$; hence,

two antecedents are equal, if their equivalence is provable by

Algorithm~\ref{alg:optSeq} using only the knowledge $(\mathcal{K}_{i},eval_i)$.

\subsection{Treatment of needed equivalences only}

Since Algorithm~\ref{alg:preprove} tries to show the logical equivalence of any combination

of two conditional antecedents that appear in $\phi$, it will have worse completion

time than Algorithm~\ref{alg:seq} on many formulae:

\begin{example}

Consider the formula

\begin{quote}

$\phi=(((p_0\Rightarrow p_1)\Rightarrow p_2)\Rightarrow p_4)\vee

(((p_5\Rightarrow p_6)\Rightarrow p_7)\Rightarrow p_8)$.

\end{quote}

Algorithm~\ref{alg:preprove} will not only

try to show the necessary equivalences between the pairs

$(((p_0\Rightarrow p_1)\Rightarrow p_2), ((p_5\Rightarrow p_6)\Rightarrow p_7))$,

$((p_0\Rightarrow p_1), (p_5\Rightarrow p_6))$ and $(p_0,p_5)$, but it will

also try to show equivalences between any two conditional antecedents (e.g. $(p_0,

(p_5\Rightarrow p_6))$), even though these equivalences will not be needed

during the execution of Algorithm~\ref{alg:optSeq}.

\end{example}

Based on this observation it is possible to assign a category to each pair of

antecedents that appear in it:

\begin{definition}

The \emph{paths (in $\phi$)} to a conditional antecedent $\psi$ describe the orders

of modal arguments through which $\psi$ is reached, when starting from the root

of $\phi$:

The path to a top-level antecedent is just $\{1\}$. If $\psi$ does not appear as

antecedent on the topmost level of $\phi$, the path to it is $\{1\}$ prepended to the set

of paths to $\psi$ in any top-level conditional antecedent of $\phi$ together with $\{0\}$

prepended to the set of paths to $\psi$ in any top-level conditional consequent of $\phi$.

\end{definition}

\begin{example}

Consider the formula $\phi=(p_0\Rightarrow p_2)\Rightarrow ((p_0\Rightarrow p_1)\Rightarrow p_3)$. Then the path to

$(p_0\Rightarrow p_2)$ is $\{1\}$, whereas the path to $(p_0\Rightarrow p_1)$ is $\{01\}$. The paths

to $p_0$ are $\{11,011\}$.

\end{example}

\begin{definition}

Let $A$ and $B$ be two conditional antecedents. $A$ and $B$ are called \emph{connected (in $\phi$)} if

at least one path to $A$ is also a path to $B$ (and hence vice-versa). If no path to $A$ is a path to $B$,

the two antecedents are said to be \emph{independent}.

\end{definition}

Since two independent conditional antecedents will never appear in the scope of the

same application of the modal rule, it is in no case necessary to show (or

refute) the logical equivalence of independent conditional antecedents. Hence it suffices to focus

our attention to the connected conditional antecedents. It is then obvious that

any possibly needed equivalence and its truth-value are allready included in $(K,eval)$

when the main proving is induced. On the other hand, we have to be aware that it

may be the case, that we show equivalences of antecedents which are in fact not needed

(since antecedents may indeed be connected and still it is possible that they never appear together in an application

of the modal rule - this is the case whenever two preceeding antecedents are not logically equivalent).

As result of these considerations, we devise Algorithm~\ref{alg:optPreprove},

an improved version of Algorithm~\ref{alg:preprove}. The only difference is

that before proving any equivalence, Algorithm~\ref{alg:optPreprove} checks

whether the current pair of conditional antecedents is actually connected;

only then does it treat the equivalence. Hence independent pairs of antecedents

remain untreated.

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

Step 1: Take a formula $\phi$ as input. Set $i=0$, $\mathcal{K}_0=\emptyset$, $eval_0=\emptyset$.\\

Step 2: Generate the set $prems_i$ of all conditional antecedents of $\phi$

of nesting depth at most $i$. If $i<depth(\phi)$ continue

with Step 3, else set $\mathcal{K}=\mathcal{K}_{i-1}, eval=eval_{i-1}$ and continue with Step 4.\\

Step 3: Let $eq_i$ denote the set of all equalities $A_a = A_b$ for different and not independent

pairs of formulae $A_a,A_b\in prems_i$. Compute

Algorithm~\ref{alg:optSeq} ($\psi$, $(\mathcal{K}_i,eval_i)$) for all $\psi\in eq_i$.

Set $\mathcal{K}_{i+1} = eq_i$, set $i = i + 1$. For each equality $\psi\in eq_i$,

set $eval_{i+1}(\psi)=\top$ if the result of Algorithm~\ref{alg:optSeq} was \verb+True+

and $eval_{i+1}(\psi)=\bot$ otherwise. Continue with Step 2.\\

Step 4: Call Algorithm~\ref{alg:optSeq} ($\phi$, $(\mathcal{K},eval)$) and return its result

as result.

\label{alg:optPreprove}

\end{upshape}

\end{alg}

\end{algorithm}

\section{Implementation}

The proposed optimized algorithms have been implemented (using the programming

language Haskell) as part of the generic coalgebraic modal logic satisfiability

solver (\COLOSS\footnote{As already mentioned above, more information about \COLOSS,

a web-interface to the tool and

the tested benchmarking formulae can be found at \url{http://www.informatik.uni-bremen.de/cofi/CoLoSS/}}).

\COLOSS~provides the general coalgebraic framework in which the generic

sequent calculus is embedded. It is easily possible to instantiate this generic sequent

calculus to specific modal logics, one particular example being conditional logic.

The matching function for conditional logic in \COLOSS~was hence adapted in order to realize

the different optimisations (closely following Algorithms~\ref{alg:seq},~\ref{alg:preprove} and

~\ref{alg:optPreprove}), so that \COLOSS~now provides an efficient algorithm for

deciding the provability (and satisfiability) of conditional logic formulae.

\subsection{Comparing the proposed algorithms}

In order to show the relevance of the proposed optimisations, we devise several classes

of conditional formulae. Each class has a characteristic general shape, defining its

complexity w.r.t. different parts of the algorithms and thus exhibiting specific

advantages or disadvantages of each algorithm:

\begin{itemize}

\item The formula \verb|bloat(|$i$\verb|)| is a full binary tree of depth $i$ (containing $2^i$ pairwise logically

inequivalent atoms and $2^i-1$ modal antecedents):

\begin{quote}

\verb|bloat(|$i$\verb|)| = $($\verb|bloat(|$i-1$\verb|)|$)\Rightarrow($\verb|bloat(|$i-1$\verb|)|)\\

\verb|bloat(|$0$\verb|)| = $p_{rand}$

\end{quote}

Formulae from this class should show the problematic performance of Algorithm~\ref{alg:preprove} whenever

a formula contains many modal antecedents which appear at different depths. A comparison of the different

algorithms w.r.t. formulae \verb|bloat(|$i$\verb|)| is depicted in Figure~\ref{fig:benchBloat}.

Since Algorithm~\ref{alg:preprove} does not check whether pairs of modal antecedents are independent or connected,

it performs considerably worse than Algorithm~\ref{alg:optPreprove} which only attempts to prove the logical

equivalence of formulae which are not independent. Algorithm~\ref{alg:seq} has the best performance in this

extreme case, as it only has to consider pairs of modal antecedents which actually appear during the course

of a proof. This is the price to pay for the optimisation by dynamic programming.

\end{itemize}

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| l | r | r | r |}

\hline

$i$ & Algorithm~\ref{alg:seq} & Algorithm~\ref{alg:preprove} & Algorithm~\ref{alg:optPreprove} \\

\hline

1 & 0.008s & 0.009s & 0.010s\\

2 & 0.008s & 0.011s & 0.010s\\

3 & 0.009s & 0.028s & 0.014s\\

4 & 0.010s & 0.233s & 0.022s\\

5 & 0.011s & 2.840s & 0.087s\\

6 & 0.013s & 33.476s & 0.590s\\

7 & 0.019s & 402.239s & 4.989s\\

\hline

\end{tabular}

\end{center}

\caption{Results for bloat($i$)}

\label{fig:benchBloat}

\end{figure}

\begin{itemize}

\item The formula \verb|conjunct(|$i$\verb|)| is just an $i$-fold conjunction of a specific formula $A$:

\begin{quote}

\verb|conjunct(|$i$\verb|)| = $A_1\wedge\ldots\wedge A_i$\\

$A=(((p_1\vee p_0)\Rightarrow p_2)\vee((p_0\vee p_1)\Rightarrow p_2))\vee\neg(((p_0\vee p_1)\Rightarrow p_2)\vee((p_1\vee p_0)\Rightarrow p_2))$)

\end{quote}

This class consists of formulae which contain logically (but not sytactically) equivalent antecedents.

As $i$ increases, so does the amount of appearances of identical modal antecedents in different positions

of the considered formula. A comparison of the different algorithms w.r.t. formulae \verb|conjunct(|$i$\verb|)| is depicted in

Figure~\ref{fig:benchConjunct}. It is obvious that the optimized algorithms perform considerably better than the unoptimized

Algorithm~\ref{alg:seq}. The reason for this is, that Algorithm~\ref{alg:seq} repeatedly proves equivalences between the same

pairs of modal antecedents. The optimized algorithms on the other hand are equipped with knowledge about the modal antecedents,

so that these equivalences have to be proved only once. However, even the runtime of the optimized algorithms is exponential in $i$,

due to the exponentially increasing complexity of the underlying propositional formula. Note that the use of propositional taulogies (such as

$A \leftrightarrow (A\wedge A) $ in this case) would help to greatly reduce the computing time for \verb|conjunct(|$i$\verb|)|.

Optimisation of propositional reasoning is not the scope of this paper though, thus we devise the following examplary class of formulae

(for which propositional tautologies would not help).

\end{itemize}

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| l | r | r | r |}

\hline

$i$ & Algorithm~\ref{alg:seq} & Algorithm~\ref{alg:preprove} & Algorithm~\ref{alg:optPreprove} \\

\hline

1 & 0.012s & 0.013s & 0.012s\\

2 & 0.021s & 0.015s & 0.014s\\

3 & 0.681s & 0.021s & 0.021s\\

4 & 131.496s & 0.048s & 0.048s\\

5 & $>$600.000s & 0.199s & 0.201s\\

6 & $\gg$600.000s & 1.152s & 1.161s\\

7 & $\gg$600.000s & 8.746s & 8.667s\\

8 & $\gg$600.000s & 74.805s & 75.595s\\

9 & $\gg$600.000s & $>$600.000s & $>$600.000s\\

\hline

\end{tabular}

\end{center}

\caption{Results for conjunct($i$)}

\label{fig:benchConjunct}

\end{figure}

\begin{itemize}

\item The formula \verb|explode(|$i$\verb|)| contains equivalent but not syntactically

equal and interchangingly nested modal antecedents of depth at most $i$:

\begin{quote}

\verb|explode(|$i$\verb|)| = $X^i_1\vee\ldots\vee X^i_i$\\

$X^i_1=(A^i_1\Rightarrow(\ldots(A^i_i\Rightarrow (c_1\wedge\ldots\wedge c_i))\ldots))$\\

$X^i_j=(A^i_{j\bmod i}\Rightarrow(\ldots(A^i_{(j+(i-1))\bmod i}\Rightarrow \neg c_j)\ldots))$\\

$A^i_j=p_{j \bmod i}\wedge\ldots\wedge p_{(j+(i-1)) \bmod i}$

\end{quote}

This class contains complex formulae for which the unoptimized algorithm should not be

efficient any more: Only the combined knowledge about all appearing modal antecedents $A^i_j$ allows

the proving algorithm to reach all modal consequents $c_n$, and only the combined sequent

$\{(c_1\wedge\ldots\wedge c_i),\neg c_1,\ldots,\neg c_i\}$ (containing every appearing

consequent) is provable. For formulae from this class (specifically designed to show the

advantages of optimization by dynamic programming),

the optimized algorithms vastly outperform the unoptimized algorithm (see Figure~\ref{fig:benchExplode}).

\end{itemize}

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| l | r | r | r |}

\hline

$i$ & Algorithm~\ref{alg:seq} & Algorithm~\ref{alg:preprove} & Algorithm~\ref{alg:optPreprove} \\

\hline

1 & 0.009s & 0.008s & 0.009s\\

2 & 0.011s & 0.010s & 0.010s\\

3 & 0.028s & 0.012s & 0.014s\\

4 & 0.268s & 0.018s & 0.018s\\

5 & 4.555s & 0.025s & 0.027s\\

6 & 10.785s & 0.035s & 0.039s\\

7 & $\gg$600.000s & 0.054s & 0.060s\\

8 & $\gg$600.000s & 0.079s & 0.089s\\

9 & $\gg$600.000s & 0.122s & 0.140s\\

\hline

\end{tabular}

\end{center}

\caption{Results for explode($i$)}

\label{fig:benchExplode}

\end{figure}

The tests were conducted on a Linux PC (Dual Core AMD Opteron 2220S (2800MHZ), 16GB RAM).

It is obvious that a significant increase of performance may be obtained through

the proposed optimisations. In general, the implementation of the proposed algorithms (realized

as a part of \COLOSS) has a comparable performance to

other conditional logic provers (such as CondLean~\cite{OlivettiEA07}). However, the authors of other

provers did not publish their benchmarking sets of formulae, so that a direct

comparison was not possible. Since other provers do not implement the optimisation

strategies proposed in this paper, it is reasonable to assume, that the optimized

implementation in \COLOSS~outperforms other systems at least for the

considered examplary formulae \verb|explode(|i\verb|)| and \verb|conjunct(|i\verb|)|.

\section{Generalized optimisation}

As previously mentioned, the demonstrated optimisation is not restricted to the

case of conditional

modal logics.

\begin{definition}

If $\Gamma$ is a sequent, we denote the set of all arguments of

top-level modalities from $\Gamma$ by $arg(\Gamma)$.

A \emph{short sequent} is a sequent which consists of just one formula which

itself is a propositional formula over a fixed maximal number of modal arguments

from $arg(\Gamma)$. In the following, we fix the maximal number of modal arguments

in short sequents to be 2.

\end{definition}

The general method of the optimisation then takes the following form:

Let $S_1,\ldots S_n$ be short sequents and assume that there is

a (w.r.t the considered modal logic) sound instance of the generic rule which

is depicted in Figure~\ref{fig:modalOpt} (where $\mathcal{S}$ is a set of any

sequents).

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{Opt}}) \inferrule{ S_1 \\ \ldots \\ S_n \\ \mathcal{S} }

{ \Gamma } \\[-5pt]

\\

\hline

\end{tabular}

\end{center}

\caption{The general rule-scheme to which the optimisation may be applied}

\label{fig:modalOpt}

\end{figure}

Then we devise a final version (Algorithm~\ref{alg:genOptPreprove}) of the

optimized algorithm: Instead of considering only equivalences of conditional antecedents

for pre-proving, we now extend our attention to any short sequents over any modal arguments.

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

Step 1: Take a formula $\phi$ as input. Set $i=0$, $\mathcal{K}_0=\emptyset$, $eval_0=\emptyset$.\\

Step 2: Generate the set $args_i$ of all modal arguments of $\phi$

which have nesting depth at most $i$. If $i<depth(\phi)$ continue

with Step 3, else set $\mathcal{K}=\mathcal{K}_{i-1}, eval=eval_{i-1}$ and continue with Step 4.\\

Step 3: Let $seq_i$ denote the set of all short sequents of form $S_i$ (where $S_i$ is a sequent

from the premise of rule (\textbf{Opt})) over at most two formulae

$A_a,A_b\in args_i$. Compute Algorithm~\ref{alg:optSeq} ($\psi$, $(\mathcal{K}_i,eval_i)$) for all

$\psi\in seq_i$. Set $\mathcal{K}_{i+1} = seq_i$, set $i = i + 1$. For each short sequent

$\psi\in seq_i$, set $eval_{i+1}(\psi)=\top$ if the result of Algorithm~\ref{alg:optSeq} was

\verb+True+ and $eval_{i+1}(\psi)=\bot$ otherwise. Continue with Step 2.\\

Step 4: Call Algorithm~\ref{alg:optSeq} ($\phi$, $(\mathcal{K},eval)$) and return its result

as result.

\end{upshape}

\label{alg:genOptPreprove}

\end{alg}

\end{algorithm}

This new Algorithm~\ref{alg:genOptPreprove} may then be used to decide provability of formulae,

where the employed ruleset has to be extended by the generic modified rule given by Figure~\ref{fig:modModalOpt}.

\begin{figure}[!h]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{Opt}$^m$}) \inferrule{ eval(S_1)=\top \\ \ldots \\ eval(S_n)=\top \\ \mathcal{S} }

{ \Gamma } \\[-5pt]

\\

\hline

\end{tabular}

\end{center}

\caption{The general optimized rule}

\label{fig:modModalOpt}

\end{figure}

\begin{example}

The following two cases are instances of the generic optimisation:

\begin{enumerate}

\item (Classical modal Logics / Neighbourhood Semantics) Let $\Gamma = \{\Box A = \Box B\}$,

$n=1$, $S_1=\{A=B\}$ and $\mathcal{S}=\emptyset$. Algorithm~\ref{alg:genOptPreprove}

may be then applied whenever the following congruence rule is sound in the considered

logic:\\

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$_{Cong}$}) \inferrule{ A=B }

{ \Box A = \Box B }

\end{center}

\end{quote}

\vspace{10pt}

The according modified version of this rule is as follows:\\

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$^m_{Cong}$}) \inferrule{ {eval(A=B)=\top} }

{ \Box A = \Box B }

\end{center}

\end{quote}

\item (Monotone modal logics) By setting $\Gamma = \{\Box A \rightarrow \Box B\}$,

$n=1$, $S_1=\{A\rightarrow B\}$ and $\mathcal{S}=\emptyset$, we may instantiate

the generic algorithm to the case of modal logics which are monotone w.r.t. their

modal operator. So assume the following rule to be sound in the considered modal

logic:\\

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$_{Mon}$}) \inferrule{ A\rightarrow B }

{ \Box A \rightarrow \Box B }

\end{center}

\end{quote}

\vspace{10pt}

The according modified version of this rule is as follows:\\

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$^m_{Mon}$}) \inferrule{ {eval(A\rightarrow B)=\top} }

{ \Box A \rightarrow \Box B }

\end{center}

\end{quote}

In the case that (\textbf{Opt}$_{Mon}$) is the only modal rule in the

considered logic (i.e. the case of plain monotone modal logic), all the

prove-work which is connected to the modal operator is shifted to the

pre-proving process. Especially matching with the modal rules

$\mathcal{RO}^m_{sc}$ becomes a mere lookup of the value of $eval$.

This means, that all calls of the sequent algorithm Algorithm~\ref{alg:optSeq}

correspond in complexity just to ordinary sat-solving of propositional logic.

Furthermore, Algorithm~\ref{alg:optSeq} will be called $|\phi|$ times. This

observation may be generalized:

\end{enumerate}

\label{ex:neighMon}

\end{example}

\begin{remark}

In the case that all modal rules of the considered logic are instances of

the generic rule (\textbf{Opt}) with $P=\emptyset$ (as seen in Example~\ref{ex:neighMon}),

the optimisation does not only allow for a reduction of computing time, but

it also allows us to effectively reduce the sequent calculus to a sat-solving

algorithm.

Furthermore, the optimized algorithm will always be as efficient as the

original one in this case (since every occurence of short sequents over $arg(\Gamma)$

which accord to the current instantiation of the rule (\textbf{Opt}) will

have to be shown or refuted even during the course of the original algorithm).

\end{remark}

\section{Summary}

\begin{problem}

Finish your paper and get it to your Program Chairman on time!

\end{problem}

\bibliographystyle{myabbrv}

\bibliography{coalgml}

\end{document}