psl: add support for the [:*i..j] operator
This operator is to ':' what [*i..j] is to ';'. Part of issue #51. * doc/tl/tl.tex: Document syntax, semantic, and trivial simplifications. * doc/tl/spotltl.sty: Add macros for new operators. * src/ltlast/bunop.cc, src/ltlast/bunop.hh: Implement it. * src/ltlast/multop.cc: Add some trivial simplifications. * src/ltlparse/ltlparse.yy, src/ltlparse/ltlscan.ll: Parse it. * src/ltltest/equals.test, src/ltltest/latex.test, src/tgbatest/ltl2tgba.test: Add more tests. * src/ltlvisit/randomltl.cc: Output this operator in random PSL formulas. * src/ltltest/rand.test: Adjust. * src/tgbaalgos/ltl2tgba_fm.cc: Add translation rules. * src/ltlvisit/tostring.cc: Add pretty printing code. * src/ltlvisit/simplify.cc, src/ltlvisit/snf.cc: Adjust switches. * NEWS: Mention the new operator.
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Alexandre Duret-Lutz
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17 changed files with 442 additions and 162 deletions
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@ -92,6 +92,7 @@
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\newcommand{\EQUAL}[1]{\texttt{[=#1]}}
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\newcommand{\GOTO}[1]{\texttt{[->#1]}}
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\newcommand{\PLUS}{\texttt{[+]}}
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\newcommand{\FPLUS}{\texttt{[:+]}}
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\newcommand{\eword}{\texttt{[*0]}}
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\newcommand{\Esuffix}{\texttt{<>->}}
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@ -599,14 +600,22 @@ denote arbitrary SERE.
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NLM intersection\footnotemark & $f\AND g$ \\
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concatenation & $f\CONCAT g$ \\
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fusion & $f\FUSION g$ \\
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bounded repetition & $f\STAR{\mvar{i}..\mvar{j}}$
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bounded ;-iter. & $f\STAR{\mvar{i}..\mvar{j}}$
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& $f\STAR{\mvar{i}:\mvar{j}}$
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& $f\STAR{\mvar{i} to \mvar{j}}$
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& $f\STAR{\mvar{i},\mvar{j}}$\\
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\llap{un}bounded repetition & $f\STAR{\mvar{i}..}$
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\llap{un}bounded ;-iter. & $f\STAR{\mvar{i}..}$
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& $f\STAR{\mvar{i}:}$
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& $f\STAR{\mvar{i} to}$
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& $f\STAR{\mvar{i},}$\\
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bounded :-iter. & $f\FSTAR{\mvar{i}..\mvar{j}}$
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& $f\FSTAR{\mvar{i}:\mvar{j}}$
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& $f\FSTAR{\mvar{i} to \mvar{j}}$
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& $f\FSTAR{\mvar{i},\mvar{j}}$\\
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\llap{un}bounded :-iter. & $f\FSTAR{\mvar{i}..}$
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& $f\FSTAR{\mvar{i}:}$
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& $f\FSTAR{\mvar{i} to}$
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& $f\FSTAR{\mvar{i},}$\\
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\end{tabular}
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\end{center}
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@ -657,6 +666,26 @@ $a$ is an atomic proposition.
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\text{or} & \mvar{i}>0 \land (\exists k\in\N,\,
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(\sigma^{0..k-1}\VDash f) \land (\sigma^{k..}
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\VDash f\STAR{\mvar{i-1}..}))\\
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\end{cases}\\
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\sigma\VDash f\FSTAR{\mvar{i}..\mvar{j}}& \iff
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\begin{cases}
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\text{either} & \mvar{i}=0 \land \mvar{j}=0 \land \sigma\VDash\1 \\
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\text{or} & \mvar{i}=0 \land \mvar{j}>0 \land (\exists k\in\N,\,
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(\sigma^{0..k}\VDash f) \land (\sigma^{k..}
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\VDash f\FSTAR{\mvar{0}..\mvar{j-1}}))\\
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\text{or} & \mvar{i}>0 \land \mvar{j}>0 \land (\exists k\in\N,\,
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(\sigma^{0..k}\VDash f) \land (\sigma^{k..}
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\VDash f\FSTAR{\mvar{i-1}..\mvar{j-1}}))\\
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\end{cases}\\
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\sigma\VDash f\FSTAR{\mvar{i}..} & \iff
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\begin{cases}
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\text{either} & \mvar{i}=0 \land \sigma\VDash\1 \\
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\text{or} & \mvar{i}=0 \land (\exists k\in\N,\,
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(\sigma^{0..k}\VDash f) \land (\sigma^{k..}
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\VDash f\FSTAR{\mvar{0}..}))\\
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\text{or} & \mvar{i}>0 \land (\exists k\in\N,\,
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(\sigma^{0..k}\VDash f) \land (\sigma^{k..}
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\VDash f\FSTAR{\mvar{i-1}..}))\\
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\end{cases}
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\end{align*}}
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@ -668,6 +697,29 @@ operands are Boolean formulas.
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regardless of the value of $f$ and $g$. For instance
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$a\STAR{}\FUSION b\STAR{}$ is actually equivalent to
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$a\STAR{}\CONCAT\sere{a\ANDALT b}\CONCAT b\STAR{}$.
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\item The $\FSTAR{\mvar{i}..}$ and $\FSTAR{\mvar{i}..\mvar{j}}$ operators are
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iterations of the $\FUSION$ operator just like
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The $\STAR{\mvar{i}..}$ and $\STAR{\mvar{i}..\mvar{j}}$ are
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iterations of the $\CONCAT$ operator. More graphically:
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\begin{align*}
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f\STAR{\mvar{i}..\mvar{j}} &=
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\underbrace{f\CONCAT f\CONCAT \ldots \CONCAT f}_{\text{between $\mvar{i}$ and $\mvar{j}$ copies of $f$}} &
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f\FSTAR{\mvar{i}..\mvar{j}} &=
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\underbrace{f\FUSION f\FUSION \ldots \FUSION f}_{\text{between $\mvar{i}$ and $\mvar{j}$ copies of $f$}}\\
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\intertext{with the convention that}
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f\STAR{0..0} &= \eword &
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f\FSTAR{0..0} &= \1
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\end{align*}
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\item The $\FSTAR{\mvar{i}..}$ and $\FSTAR{\mvar{i}..\mvar{j}}$
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operators are not defined in PSL. While the bounded iteration can
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be seen as syntactic sugar on $\FUSION$, the unbounded version
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really is a new operator.
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$\FSTAR{1..}$, for which we define the $\FPLUS$ syntactic sugar
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below, actually corresponds to the $^\oplus$ operator introduced
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by~\citet{dax.09.atva}. With this simple addition, it is possible
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to define a subset of PSL that expresses exactly the
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stutter-invariant $\omega$-regular languages.
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\end{itemize}
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\subsection{Syntactic Sugar}
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@ -687,24 +739,28 @@ it for output. $b$ must be a Boolean formula.
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\begin{align*}
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f\STARALT &\equiv f\STAR{0..}\\
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f\STAR{} &\equiv f\STAR{0..} &
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f\PLUS{} &\equiv f\STAR{1..} &
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f\FSTAR{} &\equiv f\FSTAR{0..} &
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f\EQUAL{} &\equiv f\EQUAL{0..} &
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f\GOTO{} &\equiv f\GOTO{1..1} \\
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f\STAR{..} &\equiv f\STAR{0..} &
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&&
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f\FSTAR{..} &\equiv f\FSTAR{0..} &
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f\EQUAL{..} &\equiv f\EQUAL{0..} &
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f\GOTO{..} &\equiv f\GOTO{1..} \\
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f\STAR{..\mvar{j}} &\equiv f\STAR{0..\mvar{j}} &
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&&
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f\FSTAR{..\mvar{j}} &\equiv f\FSTAR{0..\mvar{j}} &
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f\EQUAL{..\mvar{j}} &\equiv f\EQUAL{0..\mvar{j}} &
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f\GOTO{..\mvar{j}} &\equiv f\GOTO{1..\mvar{j}} \\
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f\STAR{\mvar{k}} &\equiv f\STAR{\mvar{k}..\mvar{k}} &
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&&
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f\FSTAR{\mvar{k}} &\equiv f\FSTAR{\mvar{k}..\mvar{k}} &
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f\EQUAL{\mvar{k}} &\equiv f\EQUAL{\mvar{k}..\mvar{k}} &
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f\GOTO{\mvar{k}} &\equiv f\GOTO{\mvar{k}..\mvar{k}} \\
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\STAR{} &\equiv \1\STAR{0..} &
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\PLUS{} &\equiv \1\STAR{1..} \\
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\STAR{\mvar{k}} &\equiv \1\STAR{\mvar{k}..\mvar{k}} &
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f\PLUS{} &\equiv f\STAR{1..} &
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f\FPLUS{} &\equiv f\FSTAR{1..}
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\end{align*}
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\begin{align*}
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\STAR{\mvar{k}} &\equiv \1\STAR{\mvar{k}..\mvar{k}} &
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\STAR{} &\equiv \1\STAR{0..} &
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\PLUS{} &\equiv \1\STAR{1..}
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\end{align*}
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\subsection{Trivial Identities (Occur Automatically)}
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@ -720,6 +776,14 @@ $b_1$, $b_2$ are assumed to be Boolean formulas.
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f\STAR{\mvar{i}..\mvar{j}}\STAR{\mvar{k}..\mvar{l}} &\equiv f\STAR{\mvar{ik}..\mvar{jl}}\text{~if~}i(k+1)\le jk+1 \\
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f\STAR{0}&\equiv \eword &
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f\STAR{1}&\equiv f\\
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b\FSTAR{0..\mvar{j}} &\equiv \1 &
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b\FSTAR{\mvar{i}..\mvar{j}} &\equiv b \text{~if~}i>0 \\
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\eword\FSTAR{0..\mvar{j}} &\equiv \1&
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\eword\FSTAR{\mvar{i}..\mvar{j}} &\equiv \0\text{~if~}i>0 \\
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&&
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f\FSTAR{\mvar{i}..\mvar{j}}\FSTAR{\mvar{k}..\mvar{l}} &\equiv f\FSTAR{\mvar{ik}..\mvar{jl}}\text{~if~}i(k+1)\le jk+1 \\
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f\FSTAR{0}&\equiv \1 &
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f\FSTAR{1}&\equiv f\text{~if~}\varepsilon\nVDash f\\
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\end{align*}
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\noindent
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@ -758,20 +822,19 @@ The following rules are all valid with the two arguments swapped.
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f\AND f &\equiv f&
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f\ANDALT f &\equiv f &
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f\OR f &\equiv f&
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&&
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f\FUSION f&\equiv f\FSTAR{2}&
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f\CONCAT f&\equiv f\STAR{2}\\
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b_1 \AND b_2 &\equiv b_1\ANDALT b_2 &
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&&
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&&
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b_1:b_2 &\equiv b_1\ANDALT b_2 &
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f\STAR{\mvar{i}..\mvar{j}}\CONCAT f&\equiv f\STAR{\mvar{i+1}..\mvar{j+1}}\\
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&&
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&&
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&&
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&&
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\mathllap{f\STAR{\mvar{i}..\mvar{j}}}\CONCAT f\STAR{\mvar{k}..\mvar{l}}&\equiv f\STAR{\mvar{i+k}..\mvar{j+l}}\\
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b_1:b_2 &\equiv b_1\ANDALT b_2
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\end{align*}
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\begin{align*}
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f\STAR{\mvar{i}..\mvar{j}}\CONCAT f&\equiv f\STAR{\mvar{i+1}..\mvar{j+1}} &
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f\STAR{\mvar{i}..\mvar{j}}\CONCAT f\STAR{\mvar{k}..\mvar{l}}&\equiv f\STAR{\mvar{i+k}..\mvar{j+l}}\\
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f\FSTAR{\mvar{i}..\mvar{j}}\FUSION f&\equiv f\FSTAR{\mvar{i+1}..\mvar{j+1}} &
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f\FSTAR{\mvar{i}..\mvar{j}}\FUSION f\FSTAR{\mvar{k}..\mvar{l}}&\equiv f\FSTAR{\mvar{i+k}..\mvar{j+l}}
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\end{align*}
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\section{SERE-LTL Binding Operators}
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The following operators combine a SERE $r$ with a PSL
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