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

\begin{document}

\begin{frontmatter}

\title{Optimizing Conditional Logic (Draft)}

\author[Bremen]{Daniel Hausmann\thanksref{ALL}\thanksref{myemail}}

\author[Bremen]{Lutz Schr\"oder\thanksref{ALL}\thanksref{coemail}}

\address[Bremen]{Deutsches Zentrum f\"ur k\"unstliche Intelligenz (DFKI)\\ Universit\"at Bremen, Germany}

\thanks[ALL]{This work is supported by DFG-project no1234}

\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}

We describe a specific optimization of the decision algorithm

of conditional logic. The general framework

of this optimization is the setting of coalgebraic modal logic,

the actual implementation is an extension to

the generic satisfiability solver for modal logics CoLoSS.

\end{abstract}

\begin{keyword}

Coalgebraic modal logic, conditional logic,

satisfiability / provability, optimization by memoizing

\end{keyword}

\end{frontmatter}

\section{Introduction}\label{intro}

Introduce here :)

Structure of the paper: First we give a short introduction to

the theory of coalgebraic modal logics, the general abstract

setting in which this paper takes place. Then we describe the

satisfiability (and provability) solving algorithm and its

optimization with regards to conditional logics. Further, we

analyze the quality of this optimization by relating

the structure of the input formula to the amount of work which may

be saved by using the proposed optimization. Furthermore,

we compare our tool with other model checkers for conditional logic

and finally we discuss to which extent the described

optimization could be used for other modal logics than just

conditional logic.

\section{Coalgebraic conditional logic}

Some theory here...

Needed from this section: coalgebraic modal logic etc, notion of the set of

formulae of a modal logic, provability / satisfiability (weak completeness?

see ~\ref{rem:satProv}).

Some (general) words about conditional logic?

Daniel to-do-list (in other sections): Memoizing vs. dynamic programming (cite Vardi).

Complexity of algorithm(s) (cite lutz+co. paper).

\section{The algorithm}

\subsection{The generic sequent calculus}

According to the introduced framework, 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.

\begin{definition}

A \emph{Sequent} $\Gamma$ is a set of formulae. If the provability

of a sequent is yet to be decided, it is called an \emph{open sequent}.

If the provability of a sequent has allready been shown or refuted,

the sequent is called a \emph{treated sequent}. A \emph{premise}

$\Lambda$ is a set of open sequents.

\end{definition}

\begin{definition}

\noindent A \emph{rule} $r=(\Lambda,\Gamma)$ consists of a premise

and an open sequent (usually called \emph{conclusion}).

\end{definition}

\noindent The generic sequent calculus is given by a set of rules

$\mathcal{R}_{sc}$ which consists 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 rule $\mathcal R^m_{sc}$. 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}. For now, we consider the modal

rule $\mathcal R^m_{sc}$ of the standard modal logic \textbf{K}

(as given by Figure~\ref{fig:modalK}). Note again that

it is possible to treat different modal logics with the same generic

calculus just by appropriately instantiating the modal rule.

\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}

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.

\begin{remark}

Due to $\ldots$ %what?

, the \emph{satisfiability} of a formula $\phi$ amounts to the provability

of its negation $\neg \phi$.

\label{rem:satProv}

\end{remark}

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

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

sequent $\Gamma_0 = \{\phi\}$ from it. Set i=0.\\

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

$\Gamma_i$. Let $\Lambda_i$ denote the set of the resulting open

premises.\\

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

sequent $\Gamma_i$ as proven. If $\Lambda_i$ contains no premise at

all, mark the open sequent $\Gamma_i$ as refuted.\\

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

premises and choose any premise from this set. Add all $\Gamma^j_i$

from this premise to the set of open sequents and continue with

Step 2.

\end{upshape}

\label{alg:seq}

\end{alg}

\end{algorithm}

\begin{proposition}

\begin{upshape}

Algorithm~\ref{alg:seq} is sound and complete w.r.t. provability. Furthermore,

Algorithm~\ref{alg:seq} has complexity X.

\end{upshape}

\end{proposition}

\begin{proof}

...

\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. Note however, that we restrict ourselves to the

examplary conditional logic \textbf{CKCEM} for the remainder of this section

(slightly adapted versions of the optimization will work for other conditional logics).

\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}

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 optimization}

\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 may allow us

to decide provability (and satisfiability) of formulae more efficiently

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

(namely from the so-called \emph{pre-proving} function and the actual proving

function):

\begin{algorithm}[h]

\begin{alg}

\begin{upshape}

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

input. Set $\Gamma_0 = \{\phi\}$, $i=0$.\\

Step 2: Try to apply all rules $r\in \mathcal{RO}_{sc}$

(supplying $\mathcal{RO}^m_{sc}$ with knowledge $(\mathcal{K},eval)$)

to any open sequent $\Gamma_i$. Let

$\Lambda_i$ denote the set of the resulting premises.\\

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

sequent $\Gamma_i$ as proven. If $\Lambda_i$ contains no premise at

all, mark the open sequent $\Gamma_i$ as refuted.\\

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

premises and choose any premise from this set. Add all $\Gamma^j_i$

from this premise to the set of open sequents and continue with

Step 2.

\label{alg:optSeq}

\end{upshape}

\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} ($\phi$, $(\mathcal{K},eval)$).

\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 every combination

of any 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 may 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}

Two conditional antecedents $A$ and $B$ are called \emph{connected (in $\phi$)} if

they both are the antecedent of a top-level modality.

Furthermore, we define sub-antecedents $C$ and $D$ of connected antecedents $A$ and $B$

to be connected to each other, if they appear at the same position in a chain

of conditional antecedents (formally: $A=(\ldots(C\Rightarrow a_n)\ldots \Rightarrow a_0)$,

$B=(\ldots(D\Rightarrow b_n)\ldots \Rightarrow b_0)$).

Two conditional antecedents $A$ and $B$ are called \emph{independent (in $\phi$)} if

they occur at a different modal depth (even though they may have modal nestings the same depth).

The relevance of two antecedents with regard to each other is said to be

\emph{pending} if they occur at the same depth in $\phi$ but each

in the consequent of a different modal operator.

\end{definition}

Since two independent 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 and pending conditional antecedents. By pre-proving only

the equivalences of the connected antecedents, we can be assured that we are

always as efficient as Algorithm~\ref{alg:optSeq}. However, it would still

be necessary to treat the equivalence of those initially pending antecedents which

turn out to be connected during the execution of Algorithm~\ref{alg:optSeq}.

This would not allow us to completely seperate to treatment of equivalences

(achieved by Algorithm~\ref{alg:preprove}) and the actual sequent calculus (realized by

Algorithm~\ref{alg:optSeq}).

If the pending antecedents are also included in the pre-proving, it is 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 later turn out to

be independent.

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 or if their

relevance to each other is pending; only then it treats 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} ($\phi$, $(\mathcal{K},eval)$).

\label{alg:optPreprove}

\end{upshape}

\end{alg}

\end{algorithm}

\begin{proposition}

\begin{upshape}

Algorithm~\ref{alg:optPreprove} is sound and complete w.r.t. provability.

Algorithm~\ref{alg:optPreprove} has complexity X. Furthermore, Algorithm~\ref{alg:optPreprove}

allows us, to completely seperate the pre-proving of equivalences and

the actual sequent calculus.

\end{upshape}

\end{proposition}

\begin{proof}

...

\end{proof}

\section{Generalized optimization}

As previously mentioned, the demonstrated optimization 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 optimization 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 optimization may be applied}

\label{fig:modalOpt}

\end{figure}

We are now able to 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 $S_1$ 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} ($\phi$, $(\mathcal{K},eval)$).

\end{upshape}

\label{alg:genOptPreprove}

\end{alg}

\end{algorithm}

Thus it is valid to use this new Algorithm~\ref{alg:genOptPreprove} 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 may be instances of the generic optimization:

\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{ {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}

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 optimization 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{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{More information and a web-interface to the tool are

available at \verb|http://www.informatik.uni-bremen.de/cofi/CoLoSS/|}). CoLoSS provides the general coalgebraic framework in which a 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 CoLoSS matching function for conditional logic was hence adapted in order to realize

the different optimizations, 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 optimizations, we devise three classes

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

difficulty w.r.t. the different algorithms:

Let $p_i$ denote atoms in the following.

\begin{itemize}

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

inqevuilant 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 perfomance of Algorithm~\ref{alg:preprove} whenever

a formula contains many modal antecedents which appear at different depths. Since Algorithm~\ref{alg:preprove}

does not check whether pairs of modal antecedents are independent or connected, it should perform considerably worse

than Algorithm~\ref{alg:optPreprove} which only attempts to prove the logical equivalence of formulae which are not

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

Figure~\ref{fig:benchBloat}.

\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 contains formulae which contain logically (but not sytactically!) equivalent antecedents.

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

of the considered formula. Hence the optimized should perform considerably better than the unoptimized

Algorithm~\ref{alg:seq}. A comparison of the different algorithms w.r.t. formulae \verb|conjunct(|$i$\verb|)| is depicted in

Figure~\ref{fig:benchConjunct}.

\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|)| = $X^i_1\vee\ldots\vee X^i_i$\\

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

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

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

\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 antecedent allows

the proving algorithm to reach all conclusions $c_i$, and only the combined sequent

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

conclusion) is provable. The algorithms perform as expected (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 & 0,008s & 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{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 & $>>$600,000s & 1,152s & 1,161s\\

7 & $>>$600,000s & 8,746s & 8,667s\\

8 & $>>$600,000s & 74,805s & 75,595s\\

9 & $>>$600,000s & 730,589s & 742,357s\\

\hline

\end{tabular}

\end{center}

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

\label{fig:benchConjunct}

\end{figure}

\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 & $>>$600,000s & 0,054s & 0,060s\\

8 & $>>$600,000s & 0,079s & 0,089s\\

9 & $>>$600,000s & 0,122s & 0,140s\\

\hline

\end{tabular}

\end{center}

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

\label{fig:benchExplode}

\end{figure}

The tests reveal that a significant increase of performance may be obtained through

the proposed

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

in CoLoSS) has a comparable performance to

other conditional logic provers (such as CondLean). 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 optimization

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{Summary}

\begin{problem}

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

\end{problem}

\section{Bibliographical references}\label{references}

\begin{thebibliography}{10}\label{bibliography}

\bibitem{cy} Civin, P., and B. Yood, \emph{Involutions on Banach

algebras}, Pacific J. Math. \textbf{9} (1959), 415--436.

\end{thebibliography}

\end{document}