Backup diff of Syntax vs current(No. 4)

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#author("2017-03-17T18:00:40+09:00","default:Uedalab","Uedalab")
#author("2019-06-10T15:16:55+09:00","default:Uedalab","Uedalab")
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#mathjax
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* Syntax [#r2da58d9]
- The abstract syntax of HydLa is given in the following BNF.
#ref(./HydLa_BNF.png, center);
- A HydLa program consists of definitions  of constraints and declarations of constraint hierarchies.
** Definition [#s649ac18]
- In a definition, we define a named constraint or a named constraint hierarchies with arguments using the operator " ''<=>'' ".
If the definition has no arguments, we can omit parentheses.
INIT <=> y = 0 & y' = 0. // definition of a constraint for the initial state of a ball.
FALL <=> [](y'' = -10).   // definition of a constraint for falling of the ball.
BOUNCE(x) <=> [](x- = 0 => x' = -x'-).  // definition of  a constraint for bouncing of the ball.
BALL{FALL << BOUNCE}. // definition of a constraint hierarchy for the ball.
- $Constraints$ allow conjunctions of constraints and implications.
- The antecedents of implications are called $guards$.
- " ''[]'' " denotes a temporal operator which means that the constraint always holds from the time point at which the constraint is enabled.
- " [] " denotes a temporal operator which means that the constraint always holds from the time point at which the constraint is enabled.
- Each variable is denoted by a string starting with lower case($vname$).
- The notation $vname′$ means the derivative of $vname$, and $vname−$ means the left-hand limit of $vname$.

** Declaration of Constraint Hierarchies [#oc2bb594]
- In a declaration, we declare constraints with priorities between them.
- The operator ''$<<$'' describes a weak composition of constraints. For example,
- The operator "$<<$" is a concrete notation of the operator "$\ll$" and it describes a weak composition of constraints. For example,
$A << B$ means that the constraint A is weaker than B.
-If we declare a constraint without ''$<<$'', it means that there is no priority about the constraint.
- The operator ''$<<$'' has a higher precedence than ''$,$'', that is, $A, B << C$  is equivalent to $A, (B << C)$.
-If we declare a constraint without "$<<$", it means that there is no priority about the constraint.
- The operator "$<<$" has a higher precedence than "$,$", that is, $A, B << C$  is equivalent to $A, (B << C)$.
- The unit of constraints
that is declared with a priority is called a module or a constraint module.
- Meaning of a HydLa program is a set of trajectories that satisfy maximal consistent sets of candidate constraint modules sets at each time point.
- Each candidate constraint set must satisfy conditions below:~

$\forall M_1, M_2((M_1 \ll M_2 \land M_1 \in {MS} \,) \Rightarrow M_2 \in {MS}\,)$
$\forall M (\neg \exists ( R \ll M ) \Rightarrow R \in {MS} \,)$

** List Comprehension [#eea12551]
- The set of some variables can be used.
- The syntax of the set is some way.
- In modeling of hybrid systems, we often come across necessity to introduce
multiple similar objects.
- HydLa allows a list comprehention to easily describe models with multiple objects.
- A list can be defined by the operator "$:=$", like $\texttt{X := {x0..x9}.}$
- There are a two types of lists: priority lists and expression lists.
*** Priority List [#rb9d66d1]
- The first type of lists is priority lists.
-- A priority list can be denoted by an extensional notation of the following form.
>
set_name := set .
$\{MP_1, MP_2, ..., MP_n\}$
<
- Example of Defining
-- It also can be denoted by an intensionally
>
~ es := {e1, e2, e3, e4}.
~ ten := { i | i in {0..9} }.
~ xten := {x1..x10} .
$\{MP | LC_1, LC_2, ..., LC_n\}$
<
- The calculation of sets
-- For example, $\{\texttt{INIT(i)}\ |\ \texttt{i in \{1,2,3,4\}}\}$ is equivalent to $\{\texttt{INIT(1),INIT(2),INIT(3),INIT(4)}\}$
-- If a HydLa program includes declarations of priority lists, the elements of the lists are expanded, that is, a declaration of $\{\texttt{A, B, C}\}$
is equivalent to $\texttt{A, B, C}$.
*** Expression List [#c1744645]
- The second of list is expression lists, that is, lists of arithmetic expressions.
--  We can denote an expression list in an extensional or intensional notation
as well as a priority list.
-- In addition, we can use range expressions in the following form.
>
~  union := xten or es .
~  intersection := ten and xten .
$\{RE .. RE\}$.
<
- The element of sets
>
~ set[i] : The i'th element of the "set" (element is start 0)
~ sum(set) : The sum of the "set"
~ |set| : The number of the "set"
<
- Examples:
>
~  { constarint(set[i]) | i in {1..|set|} }
~  { (constarint1(set[i]) , constarint1(set[i]) ) << constraint2(set[i], set[j]) | i in {1..|set|-1} , j in {i+1..|set|} }
~  { constraint(x) | x in set }
<
-- $RE$ is an arithmetic expression without variables or an arithmetic expression with a variable whose name terminates with a number such as x0 and y1.
*** Example of Lists [#b3820221]
- An expression list $\{1*2+1..5\}$ is equivalent to $\{3,4,5\}$.
- An expression list $\{\texttt{j | i in \{1,2\}, j in \{i+1..4\}}\}$ is equivalent to $\{2,3,4,3, 4\}$.
- An expression list $\{\texttt{x1..x3}\}$ is equivalent to $\{\texttt{x1, x2, x3}\}$.

** Assert Statement[#m5d52a83]
- The condition(ask) is described in the assert sentence.
- The condition(ask) must be always satisfied in the simulation of HydLa program.
- In the condition(ask), the always operator("[]") can't be used. But the condition is always judged for all the time of simulation.
- The condition is judged within simulating time and is not judged over simulating time.
>
ASSERT{ ask }.
<
*** Other Notations [#mc029821]
- The n-th $\texttt{(n > 0)}$ element of a list $\texttt{L}$ can be accessed by $\texttt{L[n]}$.
-- The index allows an arbitrary expression that results in an integer.
- The size of a list $\texttt{L}$ is denoted by $\texttt{|L|}$, which can be used as a constant value in a HydLa program.
- $\texttt{sum(L)}$ is a syntactic sugar of the sum of the elements in an expression list $\texttt{L}$

** Tips [#jb66d45b]
- As the special values , Napier's constant "E" and Pi "Pi"(the ratio of circumference to diameter) can be used too.

*** Special function [#w69ae32d]
- Trigonometric functions
- Napier's constant "E" and the ratio of circumference to diameter "Pi" also can be used as constant values.
- The following trigonometric functions are available.
>
~ sin(x)
~ cos(x)
~ tan(x)
~ asin(x)
~ acos(x)
~ atan(x)
<
- Detailed syntax and semantics can be found [http://www.ueda.info.waseda.ac.jp/~matsusho/public/dissertation.pdf here].