Part 4: RSDL Formal Definition
4.3 Static Semantics
4.3.7 State Machine
4.3.7.1 State Transition Graph
Abstract Syntax
State-transition-graph :: Start-node State-node-set Free-action-set
Conditions on Abstract Syntax
∀ g ∈ State-transition-graph:
| g.s-Free-action-set | = | { f.s-Connector-name | f ∈ g.s-Free-action-set } | All the <connector name>s defined in a body must be distinct.
Concrete Syntax
<state machine graph> ::
<start> {<state> | <free action>}*
Mapping to Abstract Syntax
| <state machine graph>(st, items) =>
mk-State-transition-graph(Mapping(st), { i ∈ Mapping(items).toSet: i ∈ State-node}, { i ∈ Mapping(items).toSet: i ∈ Free-action} )
4.3.7.2 Start Node
Abstract Syntax
Start-node :: Transition
Concrete Syntax
<start> :: <transition>
Mapping to Abstract Syntax
| <start>(trans) => mk-Start-node(Mapping(trans))
4.3.7.3 State Node
Abstract Syntax
State-node :: State-name
Save-signalset Input-node-set Continuous-signal-set Save-signalset = Signal-identifier-set Conditions on Abstract Syntax
∀ sn1, sn2∈ State-node: (sn1≠ sn2) ∧ (parentAS1(sn1)=parentAS1(sn2)) ⇒ (s-State-name(sn1) ≠ s-State-name(sn2))
State-nodes within a State-transition-graph must have different State-name.
Concrete Syntax
<state> = <basic state>
<basic state> ::
<state list>
{ <input part> | <save part> | <continuous signal>}*
[<state<name>]
<state list> = <state<name>+
<save part> :: <save list>
<save list> = <signal list>
Conditions on Concrete Syntax
∀ s ∈ <basic state>: length(s.s-<state list>) ≠ 1 ⇒ s.s-<name> = undefined
The optional <state name> ending a <state> may be specified only if the <state list> in the <state> consists of a single <state name> in which case it must be that <state name>.
∀ s ∈ <basic state>: s.s-<name> ≠ undefined ⇒ s.s-<name> = s.s-<state list>.head
The optional name in a definition after the ending keyword must be syntactically the same as the name following the commencing keyword.
Transformations
< <basic state>(< n > ∩ r, t, *) > provided r ≠ empty
=3=> < <basic state>(< n >, t, undefined), <basic state>(r, t, undefined) >
Multiple basic states are separated.
< b1=<basic state>(< n >, t1, *) > ∩ x ∩ < b2=<basic state>(< n >, t2, *) >
=3=> < <basic state>(< n >, t1 ∩ t2, undefined) > ∩ x Basic states with the same name are merged.
< <save part>(< s > ∩ r) > provided r ≠ empty
=3=> < <save part>(< s >), <save part>(r) >
Multiple save parts are separated.
Mapping to Abstract Syntax
| <basic state>(< name >, triggers, *) => mk-State-node(Mapping(name),
{ t ∈ Mapping(triggers).toSet: t ∈ Identifier}, { t ∈ Mapping(triggers).toSet: t ∈ Input-node}, { t ∈ Mapping(triggers).toSet: t ∈ Continuous-signal} )
| <save part>(s) => Mapping(s)
The mapping to the abstract syntax is done selecting the appropriate parts for the AS1 grammar.
4.3.7.4 Input Node
Abstract Syntax
Input-node :: Signal-identifier
[ Variable-identifier ]*
Transition Conditions on Abstract Syntax
∀ in1, in2 ∈ Input-node: parentAS1(in1)=parentAS1(in2) ∧ in1 ≠ in2 ⇒ s-Signal-identifier(in1) ≠ s-Signal-identifier(in2)
The Signal-identifiers in the Input-node-set must be distinct.
∀ in ∈ Input-node: let sd = in.s-Signal-identifier.refersto1 in
length(s-Variable-identifier-seq(in)) = length(s-Sort-name-seq(sd)) ∧
∀i ∈ 1 .. length(s-Variable-identifier-seq(in)):
let vd = in.s-Variable-identifier-seq[i].refersto1 in
sortCompatible(s-Sort-name(vd), s-Sort-name-seq(sd)[i]) endlet
endlet
The length of the list of optional Variable-identifiers must be the same as the number of Sort-names in the Signal-definition denoted by the Signal-identifier and the sorts of the variables must correspond by position to the sorts of the data items that can be carried by the signal.
Concrete Syntax
<input part> :: <input list> <transition>
<input list> = <stimulus>+
<stimulus> :: <signal list item> [ [<variable>]+ ] Conditions on Concrete Syntax
∀ s ∈ <stimulus>: s.s-<signal list item>.refersto0 ∉ <remote variable definition>
A <signal list item> must not denote a <remote variable identifier>.
Transformations
< <input part>(< s > ∩ r, t) > provided r ≠ empty
=3=> < <input part>(< s >, t), <input part>(r, t) >
Multiple inputs are separated.
Mapping to Abstract Syntax
| <input part>(< <stimulus>(item, vars) >, trans) => mk-Input-node(Mapping(item),
if vars = undefined then empty else Mapping(vars) endif, Mapping(trans) )
4.3.7.5 Continuous Signal
Abstract Syntax
Continuous-signal :: Continuous-expression Transition
Boolean-expression = Expression
Continuous-expression = Boolean-expression Conditions on Abstract Syntax
∀ c ∈ Continuous-signal:
sortCompatible(exprSort(c.s-Continuous-expression), mk-Name(“Boolean”)) The continuous expression must be of type Boolean.
Concrete Syntax
<continuous signal> :: <continuous expression> <transition>
<continuous expression> = <Boolean<expression>
Mapping to Abstract Syntax
| <continuous signal>(ex, trans)
=> mk-Continuous-signal(Mapping(ex), Mapping(trans))
4.3.7.6 Free Action
Abstract Syntax
Free-action :: Connector-name Transition
Concrete Syntax
<label> :: <connector<name>
<free action> :: <transition> [ <connector<name> ]
Auxiliary Functions
getLabel(t: <transition>): <label> =def
if t.s-<action statement> = empty then t.s-<terminator statement>.s-<label>
else t.s-<action statement>.head.s-<label>
endif
The function getLabel extracts the first label from the transition.
Conditions on Concrete Syntax
∀ f ∈ <free action>: f.s-<transition>.getLabel ≠ undefined
If the <transition string> of the <transition> in <free action> is non-empty, the first <action statement> must have a <label> otherwise the <terminator statement> must have a <label>.
Mapping to Abstract Syntax
| <free action>(trans, *)
=> mk-Free-action(Mapping(getLabel(trans)), Mapping(trans))
4.3.8 Transition
4.3.8.1 Transition
Abstract Syntax
Transition :: Graph-node* { Terminator | Decision-node } Concrete Syntax
<transition> = <transition gen transition string> | <terminator statement>
<transition gen transition string> :: <transition string> [<terminator statement>]
<transition string> = {<action statement>}+
Conditions on Concrete Syntax
∀ t ∈ <transition gen transition string>:
t.s-<terminator statement> ≠ undefined ∨
t.parentAS0.parentAS0.parentAS0 ∈ <decision> ∨
(t.s-<transition string>.last ∈ <decision> ∧ TerminatingDecision(t.s-<transition string>.last) )
If the <terminator> of a <transition> is omitted, then the last action in the <transition> must contain a terminating <decision>, except when a <transition> is contained in a <decision>.
Transformations
t=<terminator statement>(*,*) provided t.parentAS0 ∉ <transition>
=2=> <transition gen transition string>(empty, t)
This rule unifies the two possible representations for <transition> into one. Please note, that the resulting structure would not be valid concrete syntax. However, this is remedied by the transformations for decisions, see Section 4.3.8.12.
Mapping to Abstract Syntax
| <transition gen transition string>(s, t)
=> if t = undefined then mk-Transition(Mapping(< x in s: (x ∉ <decision>) >), Mapping(s.last)) else mk-Transition(Mapping(s), Mapping(t))
endif
The mapping to the abstract syntax must handle decisions specially. Please note that due to the transformations for decision there can be only one decision within one transition string and this decision is then the last element of the sequence. In this case there will be no terminating statement in the transition and the decision is mapped to the terminating statement of the AS1 transition.
4.3.8.2 Graph Node
Abstract Syntax
Graph-node = Task-node
| Output-node
| Create-request-node
| Set-node
| Reset-node
Concrete Syntax
<action statement> :: [<label>] <action 1>
<action 1> =
<task> | <output> | <create request> | <decision> | <set> | <reset> | <export> | <import>
Transformations
< a, <action statement>(l, *) > ∩ r provided l ≠ undefined =2=> < a >
and
a.parentAS0 => <transition gen transition string>(a.parentAS0.s-<action statement>,
<terminator statement>(undefined, <join>(l.s-<name>))) and
let p = parentAS0ofKind(a,<free action> ∪ <state>) in
< p > => < p, <free action>(r, undefined) >
If a <label> is not the first label of a <transition string>, the <transition string> is split into two parts. All <action statements> preceding the <label> are preserved in the original transition, which is terminated with a <join> to the <label>. All action statements following <label> are copied to a new <free action>, which starts with the
<label>.
Mapping to Abstract Syntax
| <action statement>(*, x) => Mapping(x)
4.3.8.3 Task
Abstract Syntax
Task-node = Assignment
Assignment :: Variable-identifier Expression Conditions on Abstract Syntax
∀ a ∈ Assignment: a.s-Variable-identifier.refersto1 ∈ Variable-definition In an Assignment, the identifier must be a variable identifier.
∀ a ∈ Assignment: let d = a.s-Variable-identifier.refersto1 in sortCompatible(exprSort(s-Expression(a)), s-Sort-name(d)) endlet
In an Assignment, the sort of the Expression must be equal to the sort of the Variable-identifier.
Concrete Syntax
<task> :: <textual task body>
<textual task body> = <assignment>
<assignment> :: <variable> <expression>
<variable> = <variable<identifier>
Mapping to Abstract Syntax
| <task>(<assignment>(var, expr)) => mk-Assignment(Mapping(var), Mapping(expr))
4.3.8.4 Output
Abstract Syntax
Output-node :: Signal-identifier
[ Expression ]*
[ Signal-destination ]
Signal-destination = Expression
Conditions on Abstract Syntax
∀ n ∈ Output-node: let sd = in.s-Signal-identifier.refersto1 in length(s-Expression-seq(n)) = length(s-Sort-name-seq(sd)) ∧
∀i ∈ 1 .. length(s-Expression-seq(n)):
sortCompatible(exprSort(s-Expression-seq(n)[i]), s-Sort-name-seq(sd)[i]) endlet
The length of the list of optional Expressions must be the same as the number of Sort-names in the Signal-definition denoted by the Signal-identifier. Each Expression must be sort compatible to the corresponding (by position) Sort-name in the Signal-definition.
∀ o ∈ Output-node: sortCompatible(exprSort(o.s-Signal-destination), mk-Name(“Pid”)) In an output, the destination must be of sort Pid.
∀ o ∈ Output-node: o.s-Signal-identifier.refersto1 ∈ Signal-definition In an output, the identifier must be a signal identifier.
Concrete Syntax
<output> :: <output body>
<output body> :: <output body gen identifier>+ <communication constraints>
<output body gen identifier> :: <signal<identifier> [<actual parameters>]
<actual parameters> :: <actual parameter list>
<actual parameter list> = [<expression>]+
<communication constraints> = [ <destination> ]
<destination> = <expression>
Transformations
< <action statement>(l, <output>(<output body>(< i > ∩ r, C))) > provided r ≠ empty
=3=> < <action statement>(l, <output>(<output body>(< i >, C) ) ), <action statement>(undefined, <output>(<output body>(r, C) ) ) >
If several pairs of <signal identifier> and <actual parameters> are specified in an <output body>, this is derived syntax for specifying a sequence of <output>s in the same order as specified in the original <output body>, each containing a single pair of <signal identifier> and <actual parameters>. The to <destination> clause is repeated in each of the <output>s.
Mapping to Abstract Syntax
| <output>(<output body>(< <output body gen identifier>(id, par) >, dest) )
=> mk-Output-node(Mapping(id),
if par=undefined then empty else Mapping(par) endif, Mapping(dest))
4.3.8.5 Create
Abstract Syntax
Create-request-node :: Agent-identifier
Conditions on Abstract Syntax
∀ c ∈ Create-request-node: c.s-Agent-identifier.refersto1 ∈ Agent-definition In a create, the identifier must be an agent identifier.
Concrete Syntax
<create request> :: <create body>
<create body> = <identifier>
Mapping to Abstract Syntax
| <create request>(id) => mk-Create-request-node(Mapping(id))
4.3.8.6 Set
Abstract Syntax
Set-node :: Time-expression Timer-identifier
Time-expression = Expression
Conditions on Abstract Syntax
∀ s ∈ Set-node: sortCompatible(exprSort(s.s-Time-expression), mk-Name(“Time”)) In a set node, the expression must be of sort Time.
Concrete Syntax
<set> :: <set clause>+
<set clause> :: <Time<expression> <timer<identifier>
Conditions on Concrete Syntax
∀ s ∈ <set clause>: s.s-<identifier>.refersto0 ∈ <timer definition>
In a set clause, the identifier must be a timer identifier.
Transformations
< <action statement>(l, <set>(< i > ∩ r)) > provided r ≠ empty
=3=> < <action statement>(l, <set>(< i >) ), <action statement>(undefined, <set>(r) ) >
A <set> may contain several <set clause>s. This is derived syntax for specifying a sequence of <set>s, one for each <set clause> such that the original order in which they were specified in <set> is retained.
Mapping to Abstract Syntax
| <set>(< <set clause>(ex, id) >) => mk-Set-node(Mapping(ex), Mapping(id))
4.3.8.7 Reset
Abstract Syntax
Reset-node :: Timer-identifier
Concrete Syntax
<reset> :: <reset clause>+
<reset clause> = <timer<identifier>
Conditions on Concrete Syntax
∀ r ∈ <reset>: ∀ c ∈ r.s-<reset clause>: c.s-<identifier>.refersto0 ∈ <timer definition>
In a reset, the identifier must be a timer identifier.
Transformations
< <action statement>(l, <reset>(< i > ∩ r)) > provided r ≠ empty
=3=> < <action statement>(l, <reset>(< i >) ), <action statement>(undefined, <reset>(r) ) >
A <reset> may contain several <reset clause>s. This is derived syntax for specifying a sequence of <reset>s, one for each <reset clause> such that the original order in which they were specified in <reset> is retained.
Mapping to Abstract Syntax
| <reset>(< id >) => mk-Reset-node(Mapping(id))
4.3.8.8 Terminator
Abstract Syntax
Terminator = Nextstate-node
| Stop-node
| Join-node
Concrete Syntax
<terminator statement> :: [<label>] <terminator 2>
<terminator 2> = <nextstate> | <join> | <stop>
Mapping to Abstract Syntax
| <terminator statement>(*, t) => Mapping(t)
4.3.8.9 Nextstate
Abstract Syntax
Nextstate-node :: State-name
Conditions on Abstract Syntax
∀nn ∈ Nextstate-node: nn.s-State-name.referstoName1 ≠ undefined
The State-name specified in a nextstate must be the name of a state within the same State-transition-graph.
Concrete Syntax
<nextstate> :: <nextstate body>
<nextstate body> = <state<name>
Mapping to Abstract Syntax
| <nextstate>(n) => mk-Nextstate-node(Mapping(n))
4.3.8.10 Stop
Abstract Syntax
Stop-node :: ()
Concrete Syntax
<stop> = stop
Mapping to Abstract Syntax
| stop => mk-Stop-node()
4.3.8.11 Join
Abstract Syntax
Join-node :: Connector-name
Conditions on Abstract Syntax
∀jn ∈ Join-node: jn.s-Connector-name.referstoName1 ≠ undefined
The Connector-name in a join must be the name of a free action within the same State-transition-graph.
Concrete Syntax
<join> :: <connector<name>
Mapping to Abstract Syntax
| <join>(n) => mk-Join-node(Mapping(n))
4.3.8.12 Decision
Abstract Syntax
Decision-node :: Decision-question
Decision-answer-set
[ Else-answer ]
Decision-question = Expression
Decision-answer :: Constant-expression-set Transition
Else-answer :: Transition
Conditions on Abstract Syntax
∀ da ∈ Decision-answer: ∀ ce ∈ da.s-Constant-expression-set:
sortCompatible(exprSort(ce), exprSort(s-Decision-question(da.parentAS1)))
The Constant-expressions of the Decision-answers must be sort compatible to the sort of the Decision-question.
∀ da1 ∈ Decision-answer: ∀ da2 ∈ Decision-answer:
da1≠ da2∧ parentAS1(da1) = parentAS1(da2) ⇒ { evalExpr(e) | e ∈ da1.s-Constant-expression } ∩ { evalExpr(e) | e ∈ da2.s-Constant-expression } = ∅
The Constant-expressions of the Decision-answers must be mutually exclusive.
Concrete Syntax
<decision> :: <question> <decision body>
<decision body> :: <answer part>+ [<else part>]
<answer part> :: <answer> [<transition>]
<answer> = <constant expression>+
<else part> :: [<transition>]
<question> = <expression>
Auxiliary Functions
findContinueLabel(x: DefinitionAS0): <name> =def
if x ∈ <transition gen transition string> ∧ x.s-<terminator statement> ≠ undefined ∧ x.s-<terminator statement>.s-<label> = undefined ∧
x.s-<terminator statement>.s-<terminator 2> ∈ <join>
then x.s-<terminator statement>.s-<terminator 2>.s-<name>
else findContinueLabel(x.parentAS0) endif
The function findContinueLabel computes the continuation label after a decision within a transition string.
TerminatingDecision(d: <decision>): BOOLEAN =def
(∀ a ∈ d.s-<answer part>: TerminatingTransition(a.s-<transition>) ) ∧
(d.s-<else part> = undefined ∨ TerminatingTransition(d.s-<else part>.s-<transition>) )
A <decision> is a terminating decision, if each <answer part> and <else part> in its <decision body> is a terminating <answer part> or <else part> respectively.
TerminatingTransition(t: <transition>): BOOLEAN =def t ∈ <terminator statement> ∨
t.s-<terminator statement> ≠ undefined ∨
(let d= t.s-<action statement>.last in d ∈ <decision> ∧ TerminatingDecision(d) endlet)
An <answer part> or <else part> in a decision is a terminating <answer part> or <else part> respectively if it contains a <transition> where a <terminator statement> is specified, or contains a <transition string> whose last
<action statement> contains a terminating decision.
Transformations
<else part>(undefined) =2=> <else part>(<transition gen transition string>(empty, undefined))
<answer part>(a, undefined) =2=>
<answer part>(a, <transition gen transition string>(empty, undefined))
These first two transformations are used to insert an empty transition instead of an undefined one. This empty transition will be filled with a terminator within the step below (inserting terminating actions into the transition).
t=<transition gen transition string>(a, undefined)
provided a.last ∉ <decision> ∧ t.parentAS0.parentAS0.parentAS0 ∈ <decision> ∧ t.findContinueLabel ≠ undefined
=2=> <transition gen transition string>(a,
<terminator statement>(undefined,<join>(findContinueLabel(t))))
If a <decision> is not terminating then it is derived syntax for a <decision> wherein all not terminating <answer part>s and the <else part> if not terminating have inserted at the end of their <transition> a <join> to the first
<action statement> following the decision or if the decision is the last <action statement> in a <transition string>
to the following <terminator statement>.
< d =<decision>(*, *), <action statement>(undefined, a) > provided ¬ TerminatingDecision(d) =2=> < d, <action statement>(newName(undefined), a) >
<transition gen transition string>(str, <terminator statement>(undefined, t)) provided str.last ∈ <decision> ∧ ¬ str.last.TerminatingDecision
=2=> <transition gen transition string>(str, <terminator statement>(newName(undefined), t)) The rules above insert a new label after a non-terminating decision.
Mapping to Abstract Syntax
| <decision>(ex, <decision body>(ans, els))
=> mk-Decision-node(Mapping(ex), Mapping(ans).toSet, Mapping(els))