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grammars.tex
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% !TEX root=main.tex
%% Language %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newmacro{G-Language}{
\begin{grammar}
Expressions
& e &::= & \lambda x:\tau.\ e & – abstraction \\
& &\mid& e_1\ e_2 & – application \\
& &\mid& x & – variable \\
& &\mid& s & – symbol \\
& &\mid& c & – constant \\
\addlinespace
& &\mid& u\ e_1 & – unary operation \\
& &\mid& e_1\ o\ e_2 & – binary operation \\
& &\mid& \If{e_1}{e_2}{e_3} & – conditional \\
\addlinespace
& &\mid& \unit & – unit \\
& &\mid& \tuple{e_1, e_2} & – pair \\
& &\mid& \Fst e & – first projection \\
& &\mid& \Snd e & – second projection \\
\addlinespace
& &\mid& \Ref e & – reference \\
& &\mid& !e & – dereference \\
& &\mid& e_1 := e_2 & – assignment \\
% & &\mid& e_1; e_2 & – sequence \\
& &\mid& l & – location \\
\addlinespace
& &\mid& p & – pretask \\
\addlinespace
Constants
& c &::= & B & – boolean \\
& &\mid& I & – integer \\
& &\mid& S & – string \\
\addlinespace
Unary operations
& u &::= & \lnot & – logical \\
& &\mid& - & – numerical \\
& &\mid& \Len & – sequential \\
\addlinespace
Binary operations
& o &::= & \land \mid \lor & – logical \\
& &\mid& < \mid \le \mid \equiv \mid \nequiv \mid \ge \mid > & – equational \\
& &\mid& + \mid - \mid \times \mid / & – numerical \\
& &\mid& \pp & – sequential \\
\end{grammar}
}
\newmacro{G-Language-Compact}{
\begin{grammar}
\noalign{Expressions}
\addlinespace
& \sime &::= & \lambda x:\tau.\ \sime \Mid \sime_1\ \sime_2 & – abstraction, application \\
& & \mid & x \Mid c \Mid \unit & – variable, constant, unit \\
& & \mid & u\ \sime_1 \Mid \sime_1\ o\ \sime_2 & – unary, binary operation \\
& & \mid & \If{\sime_1}{\sime_2}{\sime_3} & – conditional \\
& & \mid & \tuple{\sime_1, \sime_2} \Mid \Fst \sime \Mid \Snd \sime & – pair, projections \\
& & \mid & [\ ]_\beta \Mid \sime_1 :: \sime_2 & – nil, cons \\
& & \mid & \Head\ \sime \Mid \Tail\ \sime & – first element, list tail \\
& & \mid & \Ref \sime \Mid !\sime \Mid \sime_1 := \sime_2 \Mid l & – references, location \\
& & \mid & \tilde{p} \Mid \highlight{s} & – pretask, symbol \\
\\
\noalign{Constants}
\addlinespace
& c& ::= & B \Mid I \Mid S & – boolean, integer, string \\
\\
\noalign{Unary Operations}
\addlinespace
& \highlight{u} &::= & \lnot \Mid - \Mid \Len \Mid \Uniq & – not, negate, length, unique \\
\\
\noalign{Binary Operations}
\addlinespace
& \highlight{o} &::= & < \Mid \le \Mid \equiv \Mid \nequiv \Mid \ge \Mid > & – equational \\
& & \mid & + \Mid - \Mid \times \Mid / & – numerical \\
& & \mid & \land \Mid \lor & – conjunction, disjunction \\
& & \mid & \pp \Mid \in & – append, elementhood \\
\end{grammar}
}
\newmacro{G-Pretasks}{
\begin{grammar}
Pretasks
& \tilde{p} &::= & \Edit \sime & – valued editor \\
% & &\mid& \View e & – valued read-only editor \\
& &\mid& \Enter \beta & – unvalued editor \\
& &\mid& \Update \sime & – shared editor \\
% & &\mid& \Watch e & – shared read-only editor \\
\addlinespace
& &\mid& e_1 \Then e_2 & – step \\
& &\mid& e_1 \Next e_2 & – user step \\
\addlinespace
& &\mid& e_1 \And e_2 & – composition \\
\addlinespace
& &\mid& e_1 \Or e_2 & – choice \\
& &\mid& e_1 \Xor e_2 & – user choice \\
\addlinespace
& &\mid& \Fail & – fail task \\
\end{grammar}
}
\newmacro{G-Pretasks-Compact}{
\begin{grammar}
\noalign{Pretasks}
\addlinespace
& \tilde{p} &::= & \Edit \sime \Mid \highlight{\Enter \beta} \Mid \Update \sime & – editors: valued, unvalued, shared \\
& &\mid& \sime_1 \Then \sime_2 \Mid \sime_1 \Next \sime_2 & – steps: internal, external \\
& &\mid& \Fail \Mid \sime_1 \And \sime_2 & – fail, composition \\
& &\mid& \sime_1 \Or \sime_2 \Mid \sime_1 \Xor \sime_2 & – choice: internal, external\\
\end{grammar}
}
\newmacro{G-Types}{
\begin{grammar}
Types
& \tau &::= & \tau_1 \to \tau_2 & – function type \\
& &\mid& \Reference \tau & – reference type \\
& &\mid& \Task \tau & – task type \\
& &\mid& \beta & – basic type \\
% & &\mid& \alpha & – universal type \\
Basic types
&\beta &::= & \tau_1 \times \tau_2 & – product type \\
& &\mid& \text{List} \beta & - list type\\
& &\mid& \Unit & – unit type \\
& &\mid& \Bool & – boolean type \\
& &\mid& \Int & – integer type \\
& &\mid& \String & – string type \\
\end{grammar}
}
\newmacro{G-Types-Compact}{
\begin{grammar}
\noalign{Types}
\addlinespace
& \tau & ::= & \tau_1 \to \tau_2 \Mid \beta & – function, basic \\
& & \mid & \Reference \tau \Mid \Task \tau & – reference, task \\
\\
\noalign{Basic types}
\addlinespace
& \beta & ::= & \highlight{\beta_1 \times \beta_2 \Mid \List \beta} \Mid \Unit & – product, list, unit \\
& & \mid & \Bool \Mid \Int \Mid \String & – boolean, integer, string \\
\end{grammar}
}
\newmacro{G-Values}{
\begin{grammar}
Values
& v &::= & \lambda x:\tau.\ e & – abstraction \\
& &\mid& c & – constant \\
& &\mid& l & – location \\
\addlinespace
& &\mid& s & – symbol \\
& &\mid& u\ v & – symbolic unary operation \\
& &\mid& v_1\ o\ v_2 & – symbolic binary operation \\
\addlinespace
& &\mid& \tuple{v_1, v_2} & – pair value \\
& &\mid& \unit & – unit \\
\addlinespace
& &\mid& t & – task \\
\end{grammar}
}
\newmacro{G-Values-Compact}{
\begin{grammar}
\noalign{Values}
\addlinespace
& \simv & ::= & \lambda x:\tau.\ \sime \Mid \tuple{\simv_1, \simv_2} \Mid \unit & – abstraction, pair, unit \\
& & \mid & [\ ]_\beta \Mid \simv_1 :: \simv_2 \Mid c & - nil, cons, constant \\
& & \mid & l \Mid \simt \Mid \highlight{s} & – location, task, symbol \\
& & \mid & \highlight{u\ \simv \Mid \simv_1\ o\ \simv_2} & – unary/binary operation \\
\end{grammar}
}
\newmacro{G-Tasks}{
\begin{grammar}
Tasks
& t &::= & \Edit v & – valued editor \\
% & &\mid& \View v & – valued read-only editor \\
& &\mid& \Enter \tau & – unvalued editor \\
& &\mid& \Update l & – shared editor \\
% & &\mid& \Watch l & – shared read-only editor \\
\addlinespace
& &\mid& t_1 \Then e_2 & – step \\
& &\mid& t_1 \Next e_2 & – user step \\
\addlinespace
& &\mid& t_1 \And t_2 & – composition \\
\addlinespace
& &\mid& t_1 \Or t_2 & – choice \\
& &\mid& e_1 \Xor e_2 & – user choice \\
\addlinespace
& &\mid& \Fail & – fail task \\
\end{grammar}
}
\newmacro{G-Tasks-Compact}{
\begin{grammar}
\noalign{Tasks}
\addlinespace
& \simt & ::= & \Edit \simv \Mid \highlight{\Enter \beta} \Mid \Update l & – editors \\
& & \mid & \simt_1 \Then \sime_2 \Mid \simt_1 \Next \sime_2 & – steps \\
& & \mid & \Fail \Mid \simt_1 \And \simt_2 & – fail, combination \\
& & \mid & \simt_1 \Or \simt_2 \Mid \sime_1 \Xor \sime_2 & – choices \\
\end{grammar}
}
\newmacro{G-Inputs}{
\begin{grammar}
Symbolic inputs
& i & ::=& $s$ & – symbolic action \\
& &\mid& \First i & – pass to first \\
& &\mid& \Second i & – pass to second
\end{grammar}
}
\newmacro{G-Inputs-Compact}{
\begin{grammar}
\noalign{Symbolic inputs}
\addlinespace
& \simi & ::=& \tilde{a} \Mid \First \simi \Mid \Second \simi & – symbolic action, to first, to second \\
\\
\noalign{Symbolic actions}
\addlinespace
& \tilde{a} & ::=& \highlight{s} & – symbol \\
& &\mid& \Continue \Mid \Left \Mid \Right & – continue, go left, go right \\
\end{grammar}
}
\newmacro{G-CInputs}{
\begin{grammar}
Concrete inputs
& i & ::=& a & – concrete action \\
& &\mid& \First i & – pass to first \\
& &\mid& \Second i & – pass to second \\
Concrete actions
& a & ::=& c & – constant \\
& &\mid& \Continue & – continue with next task \\
& &\mid& \Left & – go left \\
& &\mid& \Right & – go right \\
\end{grammar}
}
\newmacro{G-CInputs-Compact}{
\begin{grammar}
\noalign{Concrete inputs}
\addlinespace
& i & ::=& a \Mid \First i \Mid \Second i & – action, to first, to second \\
\\
\noalign{Concrete actions}
\addlinespace
& a & ::=& c & – constant \\
& &\mid& \Continue \Mid \Left \Mid \Right & – continue, go left, go right \\
\end{grammar}
}
\newmacro{G-Predicates}{
\begin{grammar}
Predicates
& \phi &::= & c & – constant \\
& &\mid& s & – symbol \\
& &\mid& \Continue & – continue\\
& &\mid& \Left & – go left \\
& &\mid& \Right & – go right \\
& &\mid& u\ \phi & – symbolic unary operation \\
& &\mid& \phi_1\ o\ \phi_2 & – symbolic binary operation \\
\end{grammar}
}
\newmacro{G-Predicates-Compact}{
\begin{grammar}
\noalign{Path conditions}
\addlinespace
& \highlight{\phi} &::= & c \Mid s & – constant, symbol\\
% & &\mid& \Continue \Mid \Left \Mid \Right & – continue, go left, go right\\
& &\mid& \unit \Mid \tuple{\phi_1, \phi_2} & – unit, pairs \\
& &\mid& [\ ]_\beta \Mid \phi_1 :: \phi_2 & – nil, cons \\
& &\mid& u\ \phi \Mid \phi_1\ o\ \phi_2 & – symbolic unary/binary operation \\
\end{grammar}
}