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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") #noattach #mathjax #norelated * 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].