# Syntax

## Syntax †

### Definition †

• 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.
• 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 †

• In a declaration, we declare constraints with priorities between them.
• 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 $A, (B << C)$.
• 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 †

• 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 †

• The first type of lists is priority lists.
• A priority list can be denoted by an extensional notation of the following form.

$\{MP_1, MP_2, ..., MP_n\}$

• It also can be denoted by an intensionally

$\{MP | LC_1, LC_2, ..., LC_n\}$

• 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 †

• 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.

$\{RE .. RE\}$.

• $RE$ 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 †

• 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 $\{\texttt{x1, x2, x3}\}$.

#### Other Notations †

• 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 †

• 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 here.
Last-modified: 2019-06-10 (Mon) 06:16:55 (468d)