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\newcommand{\COLOSS}{{\textrm CoLoSS}}

\def\lastname{Hausmann and Schr\"oder}

\begin{document}

\begin{frontmatter}

\title{Optimizing Conditional Logic Reasoning within \COLOSS}

\author[DFKI]{Daniel Hausmann\thanksref{myemail}}

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

\address[DFKI]{DFKI Bremen, SKS}

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

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

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

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

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

\begin{abstract}

The generic modal reasoner CoLoSS covers a wide variety of logics

ranging from graded and probabilistic modal logic to coalition logic

and conditional logics, being based on a broadly applicable

coalgebraic semantics and an ensuing general treatment of modal

sequent and tableau calculi. Here, we present research into

optimisation of the reasoning strategies employed in

CoLoSS. Specifically, we discuss strategies of memoisation and

dynamic programming that are based on the observation that short

sequents play a central role in many of the logics under

study. These optimisations seem to be particularly useful for the

case of conditional logics, for some of which dynamic programming

even improves the theoretical complexity of the algorithm. These

strategies have been implemented in CoLoSS; we give a detailed

comparison of the different heuristics, observing that in the

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

achieved.

\end{abstract}

\begin{keyword}

Coalgebraic modal logic, conditional logic, automated reasoning,

optimisation, heuristics, memoizing, dynamic programming

\end{keyword}

\end{frontmatter}

\section{Introduction}\label{intro}

In recent decades, modal logic has seen a development towards semantic

heterogeneity, witnessed by an emergence of numerous logics that,

while still of manifestly modal character, are not amenable to

standard Kripke semantics. Examples include probabilistic modal

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

conditional logic~\cite{Chellas80}, to name just a few. The move

beyond Kripke semantics, mirrored on the syntactical side by the

failure of normality, entails additional challenges for tableau and

sequent systems, as the correspondence between tableaus and models

becomes looser, and in particular demands created by modal formulas

can no longer be discharged by the creation of single successor nodes.

This problem is tackled on the theoretical side by introducing the

semantic framework of coalgebraic modal

logic~\cite{Pattinson03,Schroder05}, which covers all logics mentioned

above and many more. It turns out that coalgebraic modal logic does

allow the design of generic reasoning algorithms, including a generic

tableau method originating from~\cite{SchroderPattinson09}; this

generic method may in fact be separated from the semantics and

developed purely syntactically, as carried out

in~\cite{PattinsonSchroder08b,PattinsonSchroder09a}.

Generic tableau algorithms for coalgebraic modal logics, in particular

the algorithm described in~\cite{SchroderPattinson09}, have been

implemented in the reasoning tool

\COLOSS~\cite{CalinEA09}\footnote{available under

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

indicated above, it is a necessary feature of the generic tableau

systems that they potentially generate multiple successor nodes for a

given modal demand, so that in addition to the typical depth problem,

proof search faces a rather noticeable problem of breadth. The search

for optimisation strategies to increase the efficiency of reasoning

thus becomes all the more urgent. Here we present one such strategy,

which is generally applicable, but particularly efficient in reducing

both depth and branching for the class of conditional logics. We

exploit a notable feature of this class, namely that many of the

relevant rules rely rather heavily on premises stating equivalence

between formulas; thus, conditional logics are a good candidate for

memoising strategies, applied judiciously to short sequents. We

describe the implementation of memoising and dynamic programming

strategies within \COLOSS, and discuss the outcome of various

comparative experiments.

\section{Generic Sequent Calculi for Coalgebraic Modal Logic}

Coalgebraic modal logic, originally introduced as a specification

language for coalgebras, seen as generic reactive

systems~\cite{Pattinson03}, has since evolved into a generic framework

for modal logic beyond Kripke semantics~\cite{CirsteaEA09}. The basic

idea is to encapsulate the branching type of the systems relevant for

the semantics of a particular modal logic, say probabilistic or

game-theoretic branching, in the choice of a set functor, the

signature functor (e.g.\ the distribution functor and the games

functor in the mentioned examples), and to capture the semantics of

modal operators in terms of so-called predicate liftings. For the

purposes 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

that $S$ is the set of all formulas, is a finite subset 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 $\Gamma_1,\dots,\Gamma_n$, the

\emph{premises}, and a $\Lambda(S)$-sequent $\Gamma_0$, the

\emph{conclusion}. A \emph{goal} is a set of sequents, typically

arising as the set of instantiated premises of a rule application.

\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}

\noindent This calculus has been shown to be complete under suitable

coherence assumptions on the rule set and the coalgebraic semantics,

provided that the set of rules \emph{absorbs cut and contraction} in a

precise sense~\cite{PattinsonSchroder08b}. 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 $\textbf{CK}$~\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 $\textbf{K}$ 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 formulas, which is easily

instantiated to specific logic by just implementing the relevant modal

rules.

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 just

keeps trying to apply all of the sequent rules in $R_{sc}$ to it,

giving precedence to the linear rules. Below, we refer to a sequent

as \emph{open} if it has not yet been checked for provability. Under

suitable tractability assumptions on the rule set as

in~\cite{SchroderPattinson09,PattinsonSchroder08b}, this algorithm

realises the theoretical upper bound $\mathit{PSPACE}$. It is the

starting point of the proof search algorithms employed in CoLoSS,

being essentially a sequent reformulation of the algorithm described

in~\cite{CalinEA09}; optimisations of this algorithm are the subject

of the present work.

%\begin{algorithm}[h]

%\fbox{\parbox{\myboxwidth}{

\begin{alg}(Decide provability of a sequent $\Gamma$)

\begin{upshape}

\begin{algenumerate}

\item Try all possible applications of rules from $R_{sc}$ to $\Gamma$,

giving precedence to linear rules. For every such rule application,

perform the following steps, and answer `provable' in case these steps

succeed for one of the rule applications.

\item Let $\Lambda$ denote the set of premises arising from the rule

application.

\item Check recursively that every sequent in $\Lambda$ is provable.

\end{algenumerate}

\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}

\noindent 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 using Algorithm~\ref{alg:seq} 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

$2^n-1$ new premises for a sequent with $n$ top-level modal

operators.

\begin{example}\label{ex:cond}

For the sequent $\Gamma=\{(A_0\Rightarrow B_0),

(A_1\Rightarrow B_1), (A_2\Rightarrow B_2)\}$, there are $2^3=8$

possible instances of the rule to be tried, corresponding to the

non-empty subsequents of the goal; the premise to be checked for

$I\subseteq\{1,2,3\}$ consists of $A_i=A_j$ for $i,j\in I$, and

$\{B_i\mid i\in I\}$.

\end{example}

\noindent 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 allows for a significant reduction

both of the number of rules to be tried and of the number of premises

to be actually proved for each rule.

\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}

\noindent 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 is 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, as illustrated next.

\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 sequents together with a function

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

a \emph{knowledge base}.

\end{definition}

\noindent We may now construct an optimized algorithm which allows us

to decide provability (and satisfiability) of formulas 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}\label{alg:opt}

(Decide provability of $\phi$ using the knowledge

base $(\mathcal{K},\eval)$)

{\upshape

\begin{algenumerate}

\item\label{step:look-up} If $\Gamma\in\mathcal{K}$, answer `provable' if

$\eval(\Gamma)=\top$, else `unprovable'. Otherwise:

\item\label{step:rule} Try all possible applications of rules from

$\mathcal{RO}_{sc}$ to $\Gamma$, giving precedence to linear rules,

where $\mathcal{RO}_{sc}$ is an optimised set of rules taking into

account the knowledge base, as explained below. For every such rule

application, perform the following steps, and answer `provable' in

case these steps succeed for one of the rule applications.

\item Let $\Lambda$ denote the set of premises arising from the rule

application.

\item Check recursively that every sequent in $\Lambda$ is provable.

\end{algenumerate}

}

\label{alg:optSeq}

\end{alg}

%\end{algorithm}

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

Algorithm~\ref{alg:seq} but relies on the knowledge base passed to it

and moreover uses a modified set of rules $\mathcal{RO}_{sc}$. The set

of rules $\mathcal{RO}_{sc}$ makes appropriate use of the knowledge

base. It 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}. The point is that the premises

$A_0=\dots=A_n$ are replaced by side conditions representing lookup in

the knowledge base. This improves on standard memoising as embodied by

the lookup operation in step~\ref{step:look-up} of the above algorithm

in that existential branching over potentially applicable rules is

reduced: the rule does not even match the target sequent unless the

equivalence premises are already in the knowledge base. This is still

a complete system due to the way memoising is organised, as explained

below.

\begin{figure}[h!]

\begin{center}

\begin{tabular}{| c |}

\hline

\\[-5pt]

(\textsc {\textbf{CKCEM}$^m$})\inferrule{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]

\\

\hfill ($\bigwedge{}_{i,j\in\{1..n\}}{\eval(A_i=A_j)=\top}$)\\

\hline

\end{tabular}

\end{center}

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

\label{fig:modalCKCEMm}

\end{figure}

%\begin{algorithm}[h]

\noindent The knowledge base used in Algorithm~\ref{alg:opt} is

computed in Algorithm~\ref{alg:preprove}. The algorithm proceeds by

dynamic programming with stages corresponding to modal nesting depth,

in the spirit of~\cite{Vardi89}. Thus, in order to show the

equivalence of two conditional antecedents of nesting depth at most

$i$, we assume that the equivalences $\mathcal{K}_{i}$ between modal

antecedents of nesting depth less than $i$ have already 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 base $(\mathcal{K}_{i},\eval_i)$.

\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

formulas $A_a,A_b\in \prems_i$. 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 `provable'

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 return

value as result.

\label{alg:preprove}

\end{upshape}

\end{alg}

%\end{algorithm}

\subsection{Treatment of Requisite 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 formulas:

\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 requisite 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 formulas $A_a,A_b\in \prems_i$. 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 `provable'

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

return value 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 formulas 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 formulas.

\subsection{Comparing the Proposed Algorithms}

\label{sec:bench}

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

of conditional formulas. 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}

formulas 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. formulas \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 formulas 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 formulas 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. formulas \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 formulas

(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 formulas 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

formulas 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 in the generic

reasoner \COLOSS has at least comparable performance as dedicated

conditional logic provers such as CondLean; a direct comparison is

presently made difficult by the fact that the benchmarking formulas

used to evaluate CondLean are not listed explicitly

in~\cite{OlivettiPozzato03}.

\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 formulas

$A_a,A_b\in args_i$. 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

`provable' 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 return value

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 formulas,

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]

\hspace{75pt}(\textsc {\textbf{Opt}$^m$}) \inferrule{ \mathcal{S} }

{ \Gamma }\hspace{75pt}\vspace{3pt}\\[-5pt]

\hfill ($\bigwedge{}_{i\in\{1..n\}}{\eval(S_i)=\top}$) \\

\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}

The according modified version of this rule is as follows:

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$^m_{Cong}$}) \inferrule{ {} }

{ \Box A = \Box B }

\end{center}

\end{quote}

with the side-condition $\eval(A=B)=\top$.

\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}

The according modified version of this rule is as follows:

\begin{quote}

\begin{center}

(\textsc {\textbf{Opt}$^m_{Mon}$}) \inferrule{ {} }

{ \Box A \rightarrow \Box B }

\end{center}

\end{quote}

with the side-condition $\eval(A\rightarrow B)=\top$.

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{Conclusion}

We presented (from a practical point of view) two optimisations for reasoning

in conditional logic:

\begin{itemize}

\item The first optimisation makes use of the concept of dynamic programming

in order to seperate the two tasks that showing validity of formulas in conditional

logic consists of: The first task of proving equivalences of antecedents and the second task

of ordinary sequent proving. This substantially decreases the branching breadth

of the resulting sequent calculus.

\item The second proposed optimisation introduces a strategy to reduce the amount

of pairs of antecedents whose equivalence has to be considered. This is achieved

by distinguishing between connected and independent pairs of modal arguments.

\end{itemize}

When both optimisations are applied at the same time, a significant increase in

performance of the sequent algorithm for conditional logic can be observed.

This was shown in

Section~\ref{sec:bench} by considering the results of benchmarking a Haskell

implementation (which forms part of the generic proving tool \COLOSS) of the

optimised algorithms.

It remains as an open question whether the gain in perfomance which is obtained

by optimising the algorithm for conditional logic may be transferred to other

logics by making use of the generic optimisation strategy as described in the

last section.

\bibliographystyle{myabbrv}

\bibliography{coalgml}

\end{document}