Structure-Constrained Process Graphs for the Process Semantics of Regular Expressions
PP. Bahr (Ed.): 11th International Workshop onComputing with Terms and Graphs (TERMGRAPH 2020)EPTCS 334, 2021, pp. 29–45, doi:10.4204/EPTCS.334.3 © C. GrabmayerThis work is licensed under theCreative Commons Attribution License.
Structure-Constrained Process Graphsfor the Process Semantics of Regular Expressions
Clemens Grabmayer
Gran Sasso Science InstituteViale F. Crispi, 7, 67100 L’Aquila AQ, Italy [email protected]
Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike forthe language semantics, where every regular (that is, DFA-accepted) language is the interpretationof some regular expression, there are finite process graphs that are not bisimilar to the process in-terpretation of any regular expression. For reasoning about graphs that are expressible by regularexpressions it is desirable to have structural representations of process graphs in the image of theinterpretation.For ‘1-free’ regular expressions, their process interpretations satisfy the structural property LEE(loop existence and elimination). But this is not in general the case for all regular expressions,as we show by examples. Yet as a remedy, we describe the possibility to recover the propertyLEE for a close variant of the process interpretation. For this purpose we refine the process se-mantics of regular expressions to yield process graphs with 1-transitions, similar to silent movesfor finite-state automata.
Milner [9] (1984) defined a process semantics for regular expressions as process graphs: the interpre-tation of 0 is deadlock, of 1 is successful termination, letters a are atomic actions, the operators + and · stand for choice and concatenation of processes, and (unary) Kleene star ( · ) ∗ represents iteration withthe option to terminate successfully after each pass-through. In order to disambiguate the use of reg-ular expressions for denoting processes, Milner called them ‘star expressions’ in this context. Unlikefor the standard language semantics, where every regular language is the interpretation of some regu-lar expression, there are finite process graphs that are not bisimilar to the process interpretation of anystar expression. This phenomenon led Milner to the formulation of two questions: (R) the problem ofrecognizing whether a given process graph is bisimilar to one in the image of the process semantics forstar expressions, and (A) whether a natural adaptation of Salomaa’s complete proof system for languageequivalence of regular expressions is complete for bisimilarity of process interpretations on pairs of starexpressions. While (R) has been shown to be decidable in principle, so far only partial solutions havebeen obtained for (A).For tackling these problems it is expedient to obtain structural representations of process graphs inthe image of the interpretation. The result of Baeten, Corradini, and myself [2] that the problem (R) isdecidable in principle was based on the concept of ‘well-behaved (recursive) specifications’ that linksprocess graphs with star expressions. Recently in [6, 7], Wan Fokkink and I obtained a partial solutionfor (A) in the form of a complete proof system for ‘1-free’ star expressions, which do not contain 1, butare formed with binary Kleene star iteration ( · ) ⊛ ( · ) instead of unary iteration. For this, we defined the E.g., the process graphs C (ne)1 and C (ne)2 in Ex. 2.6 on page 31 are not expressible by a star expression modulo bisimilarity. C ( e ) of all starexpressions e . However, it is the aim of this article to describe how LEE can nevertheless be made appli-cable, by stepping over to a variant C ( · ) of the process interpretation C ( · ) . In Section 3 we explain theloop existence and elimination property LEE for process graphs, and we define the concept of a ‘layeredLEE-witness’, for short a ‘LLEE-witness’ for process graphs. Hereby Section 3 is an adaptation for starexpressions that may contain 1 of the motivation of LLEE-witnesses in Section 3 in [6], which concernedthe process semantics of ‘1-free star expressions’. LLEE-witnesses arise by adding natural-number la-bels to transitions that are subject to suitable constraints. A process graph for which a LLEE-witnessexists is ‘structure constrained’, since it satisfies LEE (as guaranteed by the LLEE-witness) in contrastwith a process graph for which no LLEE-witness exists (which then does not satisy LEE).In Section 4 we explain examples that show that LEE does not hold in general for process interpreta-tions of star expressions from the full class. As a remedy, we introduce process graphs with 1-transitions(similar to silent moves for finite-state automata). In Section 5 we define the variant C ( · ) of the processgraph semantics C ( · ) such that C ( · ) yields process graphs with 1-transitions. Then we formulate andillustrate the following two properties of the variant process graph semantics C ( · ) about its relation to C ( · ) , and the structure of process graphs that it defines:(P1) C ( · ) and C ( · ) coincide up to bisimilarity (Thm. 5.9): For every star expression e , there is a func-tional bisimulation from the variant process semantics C ( e ) of e to the process semantics C ( e ) of e .Therefore the process interpretation C ( e ) of a star expression e and its variant C ( e ) are bisimilar.(P2) C ( · ) guarantees LEE -structure (Thm. 5.14): The variant process semantics C ( e ) of a star expres-sion e satisfies the loop existence and elimination property LEE.Section 6 is devoted to the proofs of these two properties of C ( e ) . There, we elaborate the proof of (P2).Also, we sketch the crucial steps of the proof of (P1), but refer to the report version [5] for the details.Finally in Section 7 we briefly report on which of the steps that we developed in [6] for obtaining so-lutions of the problems (A) and (R) for 1-free star expressions can be extended or adapted in the directionof solutions of these problems for all star expressions, and where the remaining difficulty resides.The idea to define structure-constrained process graphs via edge-labelings with constraints, on whichLLEE-witnesses are based, originated from ‘higher-order term graphs’ that can be used for represent-ing functional programs in a maximally compact, shared form (see [8, 4]). There, additional concepts(scope sets of vertices, or abstraction-prefix labelings) are used to constrain the form of term graphs.The common underlying idea with LLEE-witnesses is an enrichment of graphs that: (i) guarantees thatgraphs can be directly expressed by terms of some language, (ii) does not significantly hamper sharing ofrepresented subterms, (iii) is simple enough so as to keep reasoning about graph transformations feasible. In this section we define the process semantics of regular expressions as charts: finite labeled transitionsystems with initial states. We proceed by a sequence of definitions, and conclude by providing examples.
Definition 2.1.
We assume, for subsequent definitions implicitly, a set A whose members we call actions .The set StExp ( A ) of star expressions over (actions in) A is defined by the following grammar: e , e , e :: = | | a | e + e | e · e | e ∗ (where a ∈ A ) . .Grabmayer 31The (syntactic) star height | e | ∗ of a star expression e ∈ StExp ( A ) denotes the maximal nesting depth ofstars in e via: | | ∗ : = | | ∗ : = | a | ∗ : = | e + e | ∗ : = | e · e | ∗ : = max {| e | ∗ , | e | ∗ } , and | e ∗ | ∗ : = + | e | ∗ . Definition 2.2. A labeled transition system (LTS) with termination and actions in A is a 4-tuple h S , A , → , ↓i where S is a non-empty set of states , A is a set of action labels , → ⊆ S × A × S is the labeled transitionrelation , and ↓ ⊆ V is a set of states with immediate termination , for short, the terminating states . Insuch an LTS, we write s a −→ s for a transition h s , a , s i ∈ → , and s ↓ for a terminating state s ∈ ↓ . Definition 2.3.
The transition system specification (TSS) T ( A ) is defined by the axioms and rules:1 ↓ e i ↓ ( i ∈ { , } ) ( e + e ) ↓ e ↓ e ↓ ( e · e ) ↓ ( e ∗ ) ↓ a a −→ e i a −→ e ′ i ( i ∈ { , } ) e + e a −→ e ′ i e a −→ e ′ e · e a −→ e ′ · e e ↓ e a −→ e ′ e · e a −→ e ′ e a −→ e ′ e ∗ a −→ e ′ · e ∗ If e a −→ e ′ is derivable in T ( A ) , for e , e ′ ∈ StExp ( A ) , and a ∈ A , then we say that e ′ is a derivative of e . If e ↓ is derivable in T ( A ) , for e ∈ StExp ( A ) , then we say that e permits immediate termination . If e ↓ is notderivable in T ( A ) , then we write e .The TSS T ( A ) defines the process semantics for star expressions in StExp ( A ) in the form of the starexpressions LTS L ( StExp ( A )) : = L T ( A ) , where L T ( A ) = h StExp ( A ) , A , → , ↓i is the LTS generated by T ( A ) , that is, its set → ⊆ StExp ( A ) × A × StExp ( A ) of transitions, and its set ↓ ⊆ StExp ( A ) of verticeswith the immediate-termination property are defined via derivations in T ( A ) in the natural way.For every set S ⊆ StExp ( A ) we denote by L ( S ) the S-generated sub-LTS h V S , A , → S , ↓ S i of the starexpressions LTS L ( StExp ( A )) , that is, the sub-LTS whose states are those in S together with all starexpressions that are reachable from ones in S via paths that follow transitions of L ( StExp ( A )) . Definition 2.4. A chart is a 5-tuple C = h V , A , v s , → , ↓i such that h V , A , → , ↓i is an LTS, which we callthe LTS underlying C , with a finite set V of states, which we call vertices , and that is rooted by a specified start vertex v s ∈ V ; we call → ⊆ V × A × V the set of labeled transitions of C , and ↓ ⊆ V the set thevertices with immediate termination of C . Definition 2.5.
The chart interpretation C ( e ) = h V ( e ) , A , e , → , ↓i of a star expression e ∈ StExp ( A ) is the { e } -generated sub-LTS L ( { e } ) = h V { e } , A , → { e } , ↓ { e } i of L ( StExp ( A )) . Example 2.6.
In the chart illustrations below and later, we indicate the start vertex by a brown arrow ,and the property of a vertex v to permit immediate termination by emphasizing v in brown as includinga boldface ring. Each of the vertices of the charts C (ne)1 and C ( e ) below permits immediate termination,but none of the vertices of the other charts does. The charts C (ne)1 and C (ne)2 are not expressible by theprocess interpretation modulo bisimilarity, as shown by Bosscher [3] and Milner [9]. That C (ne)2 is notexpressible, Milner proved by observing the absence of a ‘loop behaviour’. That concept has inspiredthe stronger concept of ‘loop chart’ in Def. 3.1 below. For the weaker result that C (ne)1 and C (ne)2 are notexpressible by 1-free star expressions please see Remark 3.6.The chart C ( g ) middle left below is the interpretation of the star expression g = (( · a ) · g ) · g = ( c · a + a · ( b + b · a )) ∗ , and with g = ( · g ) ·
0, and g = (( · ( b + b · a )) · g ) · C ( e ) is the interpretation of e = ( a ∗ · b ∗ ) ∗ with e = (( · a ∗ ) · b ∗ ) · e , and e = ( · b ∗ ) · e . With f = a · ( + b · ) + a · ( + b · ) + a · ( + b · ) , the chart C ( f ) is the in-terpretation of f = f ∗ · f i = ( · ( + b i · ) · f ∗ ) · i ∈ { , , } ), and sink = (( · ) · f ∗ ) · That this generated sub-LTS is finite, and so that C ( e ) is indeed a chart, follows from Antimirov’s result in [1] that everyregular expression has only finitely many iterated ‘partial derivatives’, which coincide with repeated derivatives from Def. 2.3. C ( g ) g a g ac g b b a a C (ne)2 a a a a C (ne)1 ab e C ( e ) a be a b e ba f C ( f ) a a a f a a a b f a a a b f a a a b sink The chart interpretations C ( e ) and C ( f ) , which will be used later, have been constructed as expressiblevariants of the not expressible charts C (ne)1 and C (ne)2 . In particular, C ( e ) contains C (ne)1 as a subchart, and C ( f ) contains C (ne)2 as a subchart (a ‘subchart’ arises by taking a part of a chart, and picking a start ver-tex). We finally note that all of these charts with the exception of C ( e ) are bisimulation collapses. Definition 2.7.
For i ∈ { , } we consider the LTSs L i = h S i , A , → i , ↓ i i . By a bisimulation between L and L we mean a binary relation B ⊆ S × S with the properties that it is non-empty, that is, B = ∅ ,and that for every h s , s i ∈ B the following three conditions hold:(forth) ∀ s ′ ∈ S ∀ a ∈ A (cid:0) s a −→ s ′ = ⇒ ∃ s ′ ∈ S (cid:0) s a −→ s ′ ∧ h s ′ , s ′ i ∈ B ) (cid:1) , (back) ∀ s ′ ∈ S ∀ a ∈ A (cid:0) (cid:0) ∃ s ′ ∈ S (cid:0) s a −→ s ′ ∧ h s ′ , s ′ i ∈ B ) (cid:1) ⇐ = s a −→ s ′ (cid:1) , (termination) s ↓ ⇐⇒ s ↓ . For a partial function f : S ⇀ S we say that f defines a bisimulation between L and L if its graph {h v , f ( v ) i | v ∈ S } is a bisimulation between L and L . We call this graph a functional bisimulation. Definition 2.8 (bisimulation between charts) . For i ∈ { , } we consider the charts C i = h V i , A , v s , i , → i , ↓ i i .By a bisimulation between C and C we mean a binary relation B ⊆ V × V such that h v s , , v s , i ∈ B ( B relates the start vertices of C and C ), and B is a bisimulation between the underlying LTSs.We denote by C ↔ C , and say that C and C are bisimilar , if there is a bisimulation between C and C . By C → C we denote the stronger statement that there is a partial function f : V ⇀ V whosegraph {h v , f ( v ) i | v ∈ V } is a bisimulation between C and C .Each of four charts C (ne)1 , C (ne)2 , C ( g ) , and C ( f ) in Ex. 2.6 is a ‘bisimulation collapse’: by that wemean a chart for which no two subcharts that are induced by different vertices are bisimilar. But thatdoes not hold for C ( e ) in which any two subcharts that are induced by different vertices are bisimilar. The chart translation C ( g ) of g as in Ex. 2.6 satisfies the ‘loop existence and elimination’ propertyLEE that we will explain in this section. For this purpose we summarize Section 3 in [6], and in doingso we adapt the concepts defined there from 1-free star expressions to the full class of star expressions asdefined in Section 2. The property LEE is defined by a dynamic elimination procedure that analyses thestructure of a chart by peeling off ‘loop subcharts’. Such subcharts capture, within the chart interpretationof a star expression e , the behavior of the iteration of f within innermost subterms f ∗ in e . Definition 3.1.
A chart LC = h V , A , v s , → , ↓i is called a loop chart if:.Grabmayer 33(L1) There is an infinite path from the start vertex v s .(L2) Every infinite path from v s returns to v s after a positive number of transitions (and so visits v s infinitely often).(L3) Immediate termination is only permitted at the start vertex, that is, ↓ ⊆ { v s } .We call the transitions from v s loop-entry transitions , and all other transitions loop-body transitions . Aloop chart LC is a loop subchart of a chart C if it is the subchart of C rooted at some vertex v ∈ V thatis generated, for a nonempty set U of transitions of C from v , by all paths that start with a transition in U and continue onward until v is reached again (so the transitions in U are the loop-entry transitions of LC ).Both of the not expressible charts C (ne)1 and C (ne)2 in Ex. 2.6 are not loop charts: C (ne)1 violates (L3),and C (ne)2 violates (L2). Moreover, none of these charts contains a loop subchart. The chart C ( g ) inEx. 2.6 is not a loop chart either, as it violates (L2). But we will see that C ( g ) has loop subcharts.Let LC be a loop subchart of a chart C . The result of eliminating LC from C arises by removingall loop-entry transitions of LC from C , and then removing all vertices and transitions that becomeunreachable. A chart C has the loop existence and elimination property (LEE) if the procedure, startedon C , of repeated eliminations of loop subcharts results in a chart that does not have an infinite path.For the not expressible charts C (ne)1 and C (ne)2 in Ex. 2.6 the procedure stops immediately, as they donot contain loop subcharts. As both of them have infinite paths, it follows that they do not satisfy LEE.Now we consider (see below) three runs of the elimination procedure for the chart C ( g ) in Ex. 2.6.The loop-entry transitions of loop subcharts that are removed in each step are marked in bold. Each runwitnesses that C ( g ) satisfies LEE. Note that loop elimination does not yield a unique result. v a v ac v b b v a v a v b b elim v a v a v b elim v a v a v v a v ac v b b v a v c elim v a v v a v ac v b b elim Runs can be recorded, in the original chart, by attaching a marking label to transitions that get removed inthe elimination procedure. That label is the sequence number of the corresponding elimination step. Forthe three runs of loop elimination above we get the following marking labeled versions of C , respectively: v a v a [ ] c v [ ] b [ ] b v a v a [ ][ ] c v b b v a v a [ ][ ] c v b b Since all three runs were successful (as they yield charts without infinite paths), these recordings (marking-labeled charts) can be viewed as ‘LEE -witnesses’. We now will define the concept of a ‘layered LEE-wit-ness’ (LLEE-witness), i.e., a LEE-witness with the added constraint that in the recorded run of the loopelimination procedure it never happens that a loop-entry transition is removed from within the body of apreviously removed loop subchart. This refined concept has simpler properties, but is equally powerful. Confluence can be shown if a pruning operation is added that permits to drop transitions to deadlocking vertices.
Definition 3.2. An entry/body-labeling of a chart C = h V , A , v s , → , ↓i is a chart b C = h V , A × N , v s , ˆ → , ↓i where ˆ → ⊆ A × N arises from → by adding, for each transition τ = h v , a , v i ∈ → , to the action label a of τ a marking label n ∈ N , yielding b τ = h v , h a , n i , v i ∈ ˆ → . In an entry/body-labeling we call transitionswith marking label 0 body transitions , and transitions with marking labels in N + entry transitions .By an entry/body-labeling of an LTS L = h S , A , → , ↓i we mean an LTS c L = h S , A × N , ˆ → , ↓i whereˆ → ⊆ S × ( A × N ) × S arises from → by adding marking labels in N to the action labels of transitions.We define the following designations and concepts for LTSs, but will use them also for charts. Let c L be an entry/body-labeling of L , and let v and w be vertices of L and c L . We denote by v → bo w that there is a body transition v h a , i −−−→ w in c L for some a ∈ A , and by v → [ n ] w , for n ∈ N + that thereis an entry-transition v h a , n i −−−→ w in c L for some a ∈ A . By the set E ( c L ) of entry-transition identifiers we denote the set of pairs h v , n i ∈ V × N + such that an entry-transition → [ n ] departs from v in c L . For h v , n i ∈ E ( c L ) , we define by C c L ( v , n ) the subchart of L with start vertex v s that consists of the verticesand transitions which occur on paths in L as follows: any path that starts with a → [ n ] entry-transitionfrom v , continues with body transitions only (thus does not cross another entry-transition), and haltsimmediately if v is revisited.The three recordings obtained above of the loop elimination procedure for the chart C ( g ) in Ex. 2.6indicate entry/body-labelings by signaling the entry-transitions but neglecting body-step labels 0. Definition 3.3.
Let L = h V , A , → , ↓i be an LTS. A LLEE -witness (a layered LEE -witness ) of L is anentry/body-labeling c L of L that satisfies the following three properties:(W1) Body-step termination:
There is no infinite path of → bo transitions in L .(W2) Loop condition:
For all h s , n i ∈ E ( c L ) , C c L ( s , n ) is a loop chart.(W3) Layeredness:
For all h s , n i ∈ E ( c L ) , if an entry-transition t → [ m ] t ′ departs from a state t = s of C c L ( s , n ) , then its marking label m satisfies m < n .The condition (W2) justifies to call an entry-transition in a LLEE-witness a loop-entry transition . For aloop-entry transition → [ m ] with m ∈ N + , we call m its loop level .For a chart C = h V , A , v s , → , ↓i , we define a LLEE -witness b C analogously as an entry/body-labe-ling b C of C with the properties (W1), (W2), and (W3) with C and b C for L and c L , respectively. Example 3.4.
The three entry/body-labelings of the chart C ( g ) in Ex. 2.6 that we have obtained asrecordings of runs of the loop elimination procedure are LLEE-witnesses of C ( g ) , as is easy to verify. Proposition 3.5.
If a chart C has a LLEE -witness, then it satisfies
LEE .Proof.
Let b C be a LLEE-witness of a chart C . Repeatedly pick an entry-transition identifier h v , n i ∈ E ( b C ) with n ∈ N + minimal, remove the loop subchart that is generated by loop-entry transitions of level n from v (it is indeed a loop by condition (W2) on b C , noting that minimality of n and condition (W3) on b C ensure the absence of departing loop-entry transitions of lower level), and perform garbage collection.Eventually the part of C that is reachable by body transitions from the start vertex is obtained. Thissubchart does not have an infinite path due to condition (W1) on b C . Therefore C satisfies LEE.The condition (W2) on a LLEE-witness b C of a chart C requires the loop structure defined by b C tobe hierarchical. This permits to extract a star expression ˜ e from b C (defined in [6, 7]) that expresses C inthe sense that C ( ˜ e ) ↔ C holds, intuitively by unfolding the underlying chart C to the syntax tree of ˜ e . Remark 3.6.
In [6, 7] we established a connection between charts that have a LLEE-witness (and hencesatisfy LEE) and charts that are expressible by 1-free star expressions (that is, star expressions without 1,.Grabmayer 35and with binary star iteration instead of unary star iteration). By saying that a chart C ‘is expressible’ wemean here that C is bisimilar to the chart interpretation C ( ˜ e ) of some star expression ˜ e . Now Cor. 6.10in [6, 7] states that if a chart is expressible by a 1-free star expression then its bisimulation collapse hasa LLEE-witness, and thus satisfies LEE. This statement entails that neither of the charts C (ne)1 and C (ne)2 in Ex. 2.6 is expressible by a 1-free star expression, because both are bisimulation collapses, and neitherof them satisfies LEE, as we have already observed above. The chart interpretations C ( e ) of e , and C ( f ) of f in Ex. 2.6 do not satisfy LEE, contrasting with C ( g ) .For C ( e ) we find the following run of the loop elimination procedure that successively eliminates thetwo loop subcharts that are induced by the cycling transitions at e and e : e C ( e ) a be a b e ba ea be b e ba elim e C ′′ a be b e a elim The resulting chart C ′′ does not contain loop subcharts any more, because taking, for example, a transi-tion from e to e as an entry-transition does not yield a loop subchart, because in the induced subchartimmediate termination is not only possible at the start vertex e but also in the body vertex e , in con-tradiction to (L3). But while C ′′ does not contain a loop subchart any more, it still has an infinite trace.Therefore it follows that C ( e ) does not satisfy LEE.In order to see that C ( f ) does not satisfy LEE, we can consider a run of the loop elimination pro-cedure that successively removes the cyclic transitions at f , f , and f . After these removals a variantof the not expressible chart C (ne)2 is obtained that still describes an infinite behavior, but that does notcontain any loop subchart. The latter can be argued analogously as for C (ne)2 , namely that for all choicesof entry-transitions between f , f , and f the loop condition (L2) fails. We conclude that C ( f ) doesnot satisfy LEE.The reason for this failure of LEE is that, while the syntax trees of star expressions can provide anested loop-chart structure, this is not guaranteed by the specific form of the TSS T . Execution of aniteration g ∗ in an expression g ∗ · h leads eventually, in case that termination is reachable in g , to an iteratedderivative ( · g ∗ ) · h . Also, as in the examples above, an iterated derivative ( ˜ g · g ∗ ) · h with ˜ g ↓ may bereached. In these cases, continued execution will bypass the initial term g ∗ · h , and either proceed withanother execution of the iteration to ( g ′ · g ∗ ) · h , where g ′ is a derivative of g , or take a step into the exitto h ′ , where h ′ is a derivative of h . In both cases the execution does not return to the initial term g ∗ of theexecution, as would be required for a loop subchart at g ∗ to arise in accordance with loop condition (L2). A remedy for the frequent failure of LEE for the chart translation of star expressions can consist in theuse of ‘1-transitions’. Such transitions may be used to create a back-link to an expression g ∗ · h from aniterated derivative ( ˜ g · g ∗ ) · h with ˜ g ↓ (where ˜ g is an iterated derivative of g ) that is reached by a descentof the execution into the body of g . This requires an adapted refinement of the TSS T from page 31.6 Structure-Constrained Process Graphs for the Process Semantics ofRegular ExpressionsWhile different such refinements are conceivable, our choice is to introduce 1-transitions most sparingly,making sure that every 1-transition can be construed as a backlink in an accompanying LLEE-witness.In particular we want to create transition rules that facilitate a back-link to an expression g ∗ after theexecution has descended into g reaching ˜ g · g ∗ with ˜ g ↓ . In order to distinguish a concatenation expression˜ g · g ∗ that arises from the descent of the execution into an iteration g ∗ from other concatenation expres-sions we introduce a variant operation ∗ . The rules of the refined TSS should guarantee that in the exam-ple the reached iterated derivative of g ∗ is a ‘stacked star expression’ G ∗ g ∗ where G is itself a stackedstar expression that denotes an iterated derivative of g . If now G is also a star expression ˜ g with ˜ g ↓ , thenthe expression G ∗ g ∗ of the form ˜ g ∗ g ∗ should permit a 1-transition that returns to g ∗ .This intuition guided the definition of the rules of the TSS T ( ∗ ) in Def. 5.3 below, starting fromthe adaptation of the rule for steps from iterations e ∗ , and the rule that creates 1-transition backlinksto iterations e ∗ from stacked expressions E ∗ e ∗ with E ↓ . The ‘stacked product’ ∗ has the followingfeatures: E ∗ e ∗ never permits immediate termination; for defining transitions it behaves similarly asconcatenation · except that a transition from E ∗ e ∗ into e ∗ when E permits immediate termination nowrequires a 1-transition to e ∗ first. The formulation of these rules of T ( ∗ ) led to the tailor-made set ofstacked star expressions as defined below. Definition 5.1.
Let A be a set whose members we call actions . The set StExp ( ∗ ) ( A ) of stacked starexpressions over (actions in) A is defined by the following grammar: E :: = e | E · e | E ∗ e ∗ (where e ∈ StExp ( A ) ) . Note that the set
StExp ( A ) of star expressions would arise again if the clause E ∗ e ∗ were dropped.The star height | E | ∗ of stacked star expressions E is defined by adding | E · e | ∗ : = max {| E | ∗ + | e | ∗ } ,and | E ∗ e ∗ | ∗ : = max {| E | ∗ + | e ∗ | ∗ } to the defining clauses for star height of star expressions.The projection function π : StExp ( ∗ ) ( A ) → StExp ( A ) is defined by interpreting ∗ as · by the clauses: π ( E · e ) : = π ( E ) · e , π ( E ∗ e ∗ ) : = π ( E ) · e ∗ , and π ( e ) : = e , for all E ∈ StExp ( ∗ ) ( A ) , and e ∈ StExp ( A ) . Definition 5.2.
By a labeled transition system with termination, actions in A and empty steps (a 1-LTS) we mean a 5-tuple h S , A , , → , ↓i where S is a non-empty set of states , A is a set of (proper) action labels ,1 / ∈ A is the specified empty step label , → ⊆ S × A × S is the labeled transition relation , where A : = A ∪ { } is the set of action labels including 1, and ↓ ⊆ V is a set of states with immediate termination .Note that then h S , A , → , ↓i is an LTS. In such a 1-LTS, we call a transition in → ∩ ( S × A × S ) (labeled bya proper action in A ) a proper transition , and a transition in → ∩ ( S × { } × S ) (labeled by the empty-stepsymbol
1) a 1 -transition . Reserving non-underlined action labels like a , b , . . . for proper actions, we useunderlined action label symbols like a for actions labels in the set A that includes the label 1. Definition 5.3.
The transition system specification T ( ∗ ) ( A ) has the following axioms and rules, where1 / ∈ A is an additional label (for representing empty steps), a ∈ A , a ∈ A : = A ∪ { } , stacked star ex-pressions E , E , E ′ , E ′ , E ′ ∈ T ( ∗ ) ( A ) , and star expressions e , e , e ∗ , e ∗ ∈ StExp ( A ) (here and below wehighlight in red transitions that may involve 1-transitions):1 ↓ e i ↓ ( i ∈ { , } ) ( e + e ) ↓ e ↓ e ↓ ( e · e ) ↓ ( e ∗ ) ↓ a a −→ e i a −→ E ′ i ( i ∈ { , } ) e + e a −→ E ′ i E a −→ E ′ E · e a −→ E ′ · e e ↓ e a −→ E ′ e · e a −→ E ′ e a −→ E ′ e ∗ a −→ E ′ ∗ e ∗ E a −→ E ′ E ∗ e ∗ a −→ E ′ ∗ e ∗ e ↓ e ∗ e ∗ −→ e ∗ .Grabmayer 37If E a −→ E ′ is derivable in T ( ∗ ) ( A ) , for E , E ′ ∈ StExp ( A ) , and a ∈ A , then we say that E ′ is a subderivative of E . The TSS T ( ∗ ) ( A ) defines the variant process semantics for stacked star expressions in StExp ( ∗ ) ( A ) asthe stacked star expressions 1-LTS L ( StExp ( ∗ ) ( A )) : = L T ( ∗ ) ( A ) , where L T ( ∗ ) ( A ) = h StExp ( ∗ ) ( A ) , A , → , ↓i is the T ( ∗ ) ( A ) , that is, its transitions → ⊆ StExp ( ∗ ) ( A ) × A × StExp ( ∗ ) ( A ) , and itsimmediately terminating vertices ↓ ⊆ StExp ( ∗ ) ( A ) are defined via derivations in T ( A ) in the natural way.For sets S ⊆ StExp ( ∗ ) ( A ) , S -generated sub-1-LTSs are defined analogously as for L ( StExp ( A )) . Definition 5.4.
A 1 -chart is a (rooted) 1-LTS h V , A , , v s , → , ↓i such that h V , A , , → , ↓i is a 1-LTS whosestates we refer to as vertices , and where v s ∈ V is called the start vertex of the 1-chart. Definition 5.5.
The 1 -chart interpretation C ( e ) = h V ( e ) , A , , e , → e , ↓ e i of a star expression e ∈ StExp ( A ) is the e -rooted version of the { e } -generated sub-1-LTS L ( { e } ) = h V { e } , A , , → { e } , ↓ { e } i of L ( StExp ( A )) .In order to link the 1-LTS L ( StExp ( ∗ ) ( A )) to the LTS L ( StExp ( A )) , we need to take account of the se-mantics of 1-transitions as empty steps (see Vrancken [10]). For this, we define ‘induced transitions’ ((( · ]]] −→ ,and ‘induced termination’ ↓ ( ) as follows: E ((( a ]]] −→ E ′ holds if there is a sequence of 1-transitions from E tosome e E from which there is an a -transition to E ′ (the asymmetric notation ((( · ]]] −→ is intended to reflect thisasymmetry), and E ↓ ( ) holds if there is a sequence of 1-transitions from E to some e E with e E ↓ . Definition 5.6.
Let L = h S , A , , −→ , ↓i be a 1-LTS. By the induced LTS of L , and the LTS induced by L ,we mean the LTS L ( · ]( · ]( · ] = h S , A , ((( · ]]] −→ , ↓ ( ) i where ((( · ]]] −→ ⊆ S × A × S is the induced transition relation , and ↓ ( ) ⊆ V is the set of vertices with induced termination that are defined as follows, for all s , s ′ ∈ S and a ∈ A :(ind-1) s ((( a ]]] −→ s ′ holds if s = s −→ s −→ . . . −→ s n a −→ s ′ , for some s , . . . , s n ∈ V and n ∈ N (we then say thatthere is an induced transition between s , s ′ ∈ S with respect to L ),(ind-2) s ↓ ( ) holds if s = s −→ s −→ . . . −→ s n ∧ s n ↓ ( ) , for some s , . . . , s n ∈ V and n ∈ N (then we say that s has induced termination with respect to L ). Definition 5.7.
The induced chart C ( · ]( · ]( · ] = h V , A , v s , ((( · ]]] −→ , ↓ ( ) i of a 1-chart C = h V , A , , v s , → , ↓i is definedanalogously as the induced LTS of a 1-LTS in Def. 5.6 with the induced transition relation ((( · ]]] −→ definedanalogously to (ind-1), and the set of vertices ↓ ( ) with induced termination defined analogously to (ind-2). Lemma 5.8.
The projection function π defines a bisimulation between the induced LTS L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] of the stacked star expressions 1-LTS StExp ( ∗ ) ( A ) and the star expressions LTS L ( StExp ( A )) . With this lemma, the proof of which we outline in Section 6, we will be able to prove the followingconnection between the chart interpretation and the 1-chart interpretation of a star expression.
Theorem 5.9. C ( e ) ( · ]( · ]( · ] → C ( e ) holds for all e ∈ StExp ( A ) , that is, there is a functional bisimulation fromthe induced chart of the -chart interpretation of a star expression e to the chart interpretation of e. For the construction of LLEE-witnesses for the 1-chart interpretation C ( · ) we need to distinguish‘normed + ’ stacked star expressions that permit an induced-transition path of positive length to an ex-pression with induced termination from those that do not enable such a path. This property slightlystrengthens normedness of expressions, which means the existence of a path to an expression with im-mediate termination (or equally, an arbitrary-length induced-transition path to an expression with in-duced termination). Definition 5.10.
Let E ∈ StExp ( ∗ ) ( A ) . We say that E is normed + (and E is normed ) if there is E ′ ∈ StExp ( ∗ ) ( A ) such that E ((( · ]]] −→ + E ′ and E ′ ↓ ( ) in L ( StExp ( A )) ( · ]( · ]( · ] (resp., E → ∗ E ′ and E ′ ↓ in L ( StExp ( A )) ).8 Structure-Constrained Process Graphs for the Process Semantics ofRegular ExpressionsThese properties permit inductive definitions, and therefore they are easily decidable. Also, a stackedstar expression is normed + if and only if it enables a transition to a normed stacked star expression.Now we define, similarly as we have done so for 1-free star expressions in [6, 7], a refinement ofthe TSS T ( ∗ ) into a TSS that will supply entry/body-labelings for LLEE-witnesses, by adding markinglabels to the rules of T ( ∗ ) . In particular, body labels are added to transitions that cannot return to theirsource expression. The rule for transitions from an iteration e ∗ is split into the case in which e is normed + or not. Only if e is normed + can e ∗ return to itself after a positive number of steps, and then a (loop-)entry-transition with the star height | e ∗ | ∗ of e ∗ as its level is created; otherwise a body label is introduced. Definition 5.11.
The TSS c T ( ∗ ) ( A ) has the following rules, where l ∈ { bo } ∪ { [ n ] | n ∈ N + } :1 ↓ e i ↓ ( i ∈ { , } ) ( e + e ) ↓ e ↓ e ↓ ( e · e ) ↓ ( e ∗ ) ↓ a a −→ bo e i a −→ l E ′ i ( i ∈ { , } ) e + e a −→ bo E ′ i E a −→ l E ′ E · e a −→ l E ′ · e e ↓ e a −→ l E ′ e · e a −→ bo E ′ e a −→ l E ′ ( e normed + ) e ∗ a −→ [ | e ∗ | ∗ ] E ′ ∗ e ∗ e a −→ l E ′ ( e not normed + ) e ∗ a −→ bo E ′ ∗ e ∗ E a −→ l E ′ E ∗ e ∗ a −→ l E ′ ∗ e ∗ e ↓ e ∗ e ∗ −→ bo e ∗ The entry/body-labeling c L ( StExp ( ∗ ) ( A )) = h StExp ( A ) , A , · −→ · , ↓i of the stacked star expressions 1-LTS L ( StExp ( ∗ ) ( A )) is defined as the LTS generated by c T ( ∗ ) ( A ) , with ↓ ⊆ StExp ( ∗ ) ( A ) as set of terminatingvertices, and with set · −→ · ⊆ StExp ( ∗ ) ( A ) × ( A × N ) × StExp ( ∗ ) ( A ) of transitions, where A = A ∪ { } . Definition 5.12.
For every star expression e ∈ StExp ( A ) we denote by d C ( e ) the entry/body-labeling thatis the chart formed as the e -rooted sub-LTS generated by { e } of the entry/body-labeling c L ( StExp ( ∗ ) ( A )) .For this entry/body-labeling we will show in Section 6 that it recovers the property LEE for thestacked star expressions 1-LTS, and as a consequence, for the 1-chart interpretation of star expressions. Lemma 5.13. c L ( StExp ( ∗ ) ( A )) is an entry/body-labeling, and indeed a LLEE -witness, of L ( StExp ( ∗ ) ( A )) . Theorem 5.14.
For every e ∈ StExp ( A ) , the entry/body-labeling d C ( e ) of C ( e ) is a LLEE -witness of C ( e ) .Hence the -chart interpretation C ( e ) of a star expression e ∈ StExp ( A ) satisfies the property LEE . Example 5.15.
We consider the chart interpretations C ( e ) and C ( f ) for the star expressions e and f inEx. 2.6 for which we saw in Section 4 that LEE fails. We first illustrate the 1-chart interpretations C ( e ) of e together with the entry/body-labeling d C ( e ) of C ( e ) . The dotted transitions indicate 1-transitions. Theexpressions at non-initial vertices of C ( e ) are E ′ = (( ∗ a ∗ ) · b ∗ ) ∗ e , E = ( a ∗ · b ∗ ) ∗ e , E = b ∗ ∗ e ,and E ′ = ( ∗ b ∗ ) ∗ e , which are obtained as iterated derivatives via the TSS T ( ∗ ) . Furthermore, wedepict the induced chart C ( e ) ( · ]( · ]( · ] of C ( e ) , and its relationship to the chart interpretation C ( e ) of e via theprojection function π that defines a functional bisimulation (which we indicate via arrows ). e C ( e ) , d C ( e ) a [ ] b [ ] E a [ ] b E b [ ] E ′ E ′ e C ( e ) ( · ]( · ]( · ] a bE E E ′ ba E ′ a b e C ( e ) a be a b e ba ππ ππ π .Grabmayer 39The transitions of the induced chart C ( e ) ( · ]( · ]( · ] of the 1-chart interpretation C ( e ) of e correspond to paths in C ( e ) that start with a (potentially empty) 1-transition path and have a final proper action transition, whichalso provides the label of the induced transition. For example the b -transition from E ′ to E ′ in C ( e ) ( · ]( · ]( · ] arises as the induced transition in C ( e ) that is the path that consists of the 1-transitions from E ′ to E , andfrom E to e , followed by the final b -transition from e to E ′ . The vertices with immediate terminationin C ( e ) ( · ]( · ]( · ] are all those that permit 1-transition paths in C ( e ) to vertices with immediate termination.Therefore in C ( e ) only e needs to permit immediate termination in order to get induced termination in C ( e ) ( · ]( · ]( · ] also at all other vertices (like in C ( e ) ). Now clearly the projection function π that maps e e ,and E i , E ′ i e i for i ∈ { , } defines a bisimulation from C ( e ) ( · ]( · ]( · ] to C ( e ) . Provided that the unreachablevertices E and E in C ( e ) ( · ]( · ]( · ] are removed by garbage collection, π defines an isomorphism. What ismore, the entry/body-labeling d C ( e ) of C ( e ) can be readily checked to be a LLEE-witness of C ( e ) . f C ( f ) [ C ( f ) a [ ] a [ ] a [ ] F b F b F b Sink f C ( f ) ( · ]( · ]( · ] a a a F a a a b F a a a b F a a a b sink f C ( f ) a a a f a a a b f a a a b f a a a b sink ππ ππ π Above we have illustrated the 1-chart interpretation C ( f ) of f together with the entry/body-labeling [ C ( f ) of C ( f ) , the induced chart C ( f ) ( · ]( · ]( · ] of C ( f ) , and its connection via π to the chart interpretation C ( f ) of f : The stacked star expressions at non-initial vertices in C ( f ) are F i = (( · ( + b i · )) ∗ f ∗ ) · i ∈ { , , } , and Sink = (( · ) ∗ f ∗ ) ·
0. Here the projection function π maps f f , Sink sink , and F i f i for i ∈ { , , } , defining a functional bisimulation from C ( f ) ( · ]( · ]( · ] to C ( f ) , which is a isomorphism.The entry/body-labeling [ C ( f ) of C ( f ) can again readily be checked to be a LLEE-witness of C ( f ) .Thus we have verified the joint claims of Thm. 5.9 and of Thm. 5.14 for the two examples e and f ofstar expressions from Ex. 2.6 for which, as we saw in Section 4, LEE fails for their chart interpretations C ( e ) and C ( f ) : the property LEE can be recovered for the 1-chart interpretations C ( e ) of e , and C ( f ) of f (by Thm. 5.14), and also, the induced charts C ( e ) ( · ]( · ]( · ] of C ( e ) , and C ( f ) ( · ]( · ]( · ] of C ( f ) map to the originalprocess interpretations C ( e ) of e , and C ( f ) of f , respectively, via a functional bisimulation (by Thm. 5.9). In this section we provide the proofs of the properties (P1) and (P2) (see in the Introduction) of thevariant process semantics C ( · ) in relation to the process semantics C ( · ) . In Section 6.1 we first sketchthe crucial steps that are necessary for proving Lem. 5.8, but refer to the report version [5] for moredetails of the argument. From Lem. 5.8 we prove Thm. 5.9, and in this way demonstrate property (P1).In Section 6.2 we provide the details of the proofs of Lem. 5.13, and of Thm. 5.14, thereby demonstratingthe property (P2). C ( · ) and C ( · ) For Lem. 5.8 we need to show that the projection function π : StExp ( ∗ ) ( A ) → StExp ( A ) defines a bisimu-lation between the induced LTS L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] of the stacked star expressions 1-LTS L ( StExp ( ∗ ) ( A )) L ( StExp ( A )) . We show this by using statements that relate derivability inthe TSS T ( A ) that defines transitions in L ( StExp ( A )) to derivability in a TSS that defines transitions inthe induced LTS L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] . For this we define a TSS T ( ∗ )( · ]( · ]( · ] ( A ) as an extension of T ( ∗ ) ( A ) by rulesthat produce induced transitions and induced termination. We also formulate two easy lemmas about T ( ∗ )( · ]( · ]( · ] ( A ) , where the first one states that T ( ∗ )( · ]( · ]( · ] ( A ) defines L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] . Definition 6.1.
The LTS L T ( ∗ )( · ]( · ]( · ] ( A ) = h StExp ( ∗ ) ( A ) , A , ↓ ( ) , ((( · ]]] −→i is generated by derivations in the TSS T ( ∗ )( · ]( · ]( · ] ( A ) that in addition to the axioms and rules of T ( ∗ ) ( A ) also contains the following rules: E ↓ E ↓ ( ) E −→ e E e E ↓ ( ) E ↓ ( ) E a −→ E ′ E ((( a ]]] −→ E ′ E −→ e E e E ((( a ]]] −→ E ′ E ((( a ]]] −→ E ′ Lemma 6.2. L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] = L T ( ∗ ) ( A ) . Lemma 6.3.
The following rules are admissible for T ( ∗ )( · ]( · ]( · ] (they can be eliminated from T ( ∗ )( · ]( · ]( · ] derivations): E ↓ ( ) e ↓ ( E · e ) ↓ ( ) E ↓ ( ) ( E ∗ e ∗ ) ↓ ( ) E ((( a ]]] −→ E ′ E · e ((( a ]]] −→ E ′ · e E ((( a ]]] −→ E ′ E ∗ e ∗ ((( a ]]] −→ E ′ ∗ e ∗ E ↓ ( ) e ((( a ]]] −→ E ′ E · e ((( a ]]] −→ E ′ E ↓ ( ) e ∗ ((( a ]]] −→ E ′ E ∗ e ∗ ((( a ]]] −→ E ′ For the proofs of these two lemmas, please see the report version [5]. Next we formulate and outlinethe proof of a crucial lemma that relates derivability statements in T ( ∗ )( · ]( · ]( · ] ( A ) with derivability statements in T ( ∗ ) ( A ) in a way that will enable us to show that the projection function π defines a bisimulation betweenthe LTSs generated by these two TSSs. Lemma 6.4.
For all E , E ′ ∈ StExp ( ∗ ) ( A ) , e ′ ∈ StExp ( A ) , and a ∈ A the following statements hold concern-ing derivability in the TSS T ( ∗ )( · ]( · ]( · ] ( A ) and derivability in the TSS T ( A ) (we drop A from their designations): ⊢ T ( ∗ )( · ]( · ]( · ] E ↓ ( ) = ⇒ ⊢ T π ( E ) ↓ , (1) ⊢ T ( ∗ )( · ]( · ]( · ] E ((( a ]]] −→ E ′ = ⇒ ⊢ T π ( E ) a −→ π ( E ′ ) , (2) ⊢ T ( ∗ )( · ]( · ]( · ] E ↓ ( ) ⇐ = ⊢ T π ( E ) ↓ , (3) ∃ E ′ ∈ StExp ( ∗ ) (cid:2) π ( E ′ ) = e ′ ∧ ⊢ T ( ∗ )( · ]( · ]( · ] E ((( a ]]] −→ E ′ (cid:3) ⇐ = ⊢ T π ( E ) a −→ e ′ . (4) Proof outline (for the full proof, see the report version [5]).
For all E , E ′ ∈ StExp ( ∗ ) ( A ) , e ′ ∈ StExp ( A ) ,and a ∈ A the following implications hold from derivability in the TSS T ( ∗ ) ( A ) to derivability in theTSS T ( A ) (again we drop A from their designations): ⊢ T ( ∗ ) E ↓ = ⇒ E ∈ StExp ( A ) ∧ ⊢ T E ↓ , (5) ⊢ T ( ∗ ) E a −→ E ′ = ⇒ ⊢ E π ( E ) a −→ π ( E ′ ) , (6) ⊢ T ( ∗ ) E −→ e E = ⇒ (cid:0) ⊢ T π ( e E ) ↓ = ⇒ ⊢ T π ( E ) ↓ (cid:1) , (7) ⊢ T ( ∗ ) E −→ e E = ⇒ (cid:0) ⊢ T π ( e E ) a −→ e ′ = ⇒ ⊢ T π ( E ) a −→ e ′ (cid:1) . (8)Statements (5) and (6) can be established by rule inspection, statements (7) and (8) can be shown byinduction on the structure of E .Then statement (1) of the lemma can be shown from (5) and (7), and statement (2) from (6) and (8).Statements (3) and (4) of the lemma can be shown by means of a proof by induction on the structure of E that makes crucial use of the admissible rules of T ( ∗ )( · ]( · ]( · ] in Lem. 6.3..Grabmayer 41 Proof of Lem. 5.8.
By transferring the four statements of Lem. 6.4 from the TSSs T ( ∗ )( · ]( · ]( · ] ( A ) and T ( A ) totheir generated LTSs L T ( ∗ )( · ]( · ]( · ] ( A ) and L T ( A ) , and by using L T ( ∗ )( · ]( · ]( · ] ( A ) = L ( StExp ( ∗ ) ( A )) from Lem. 6.2, and L T ( A ) = L ( StExp ( A )) from Def. 5.3 we obtain the following statements, for all stacked star expressions E , E ′ ∈ StExp ( ∗ ) ( A ) , star expressions e ′ ∈ StExp ( A ) , and actions a ∈ A : E ((( a ]]] −→ E ′ in L ( T ( ∗ ) ( A )) ( · ]( · ]( · ] = ⇒ π ( E ) a −→ π ( E ′ ) in L ( StExp ( A )) , (9) ∃ E ′ ∈ StExp ( ∗ ) (cid:2) π ( E ′ ) = e ′ ∧ E ((( a ]]] −→ E ′ in L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] (cid:3) ⇐ = π ( E ) a −→ e ′ in L ( StExp ( A )) , (10) E ↓ ( ) in L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] ⇐⇒ π ( E ) ↓ in L ( StExp ( A )) . (11)Hereby (9) follows from (2), (10) from (4), and (11) from (1) and (3). These three statements witness theforth, the back, and the termination condition in Def. 2.7 for the graph of the projection function π , whichalso is non-empty. Therefore π defines a bisimulation between L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] and L ( StExp ( A )) . Proof of Thm. 5.9.
Let e ∈ StExp ( A ) be a star expression. By Def. 5.5, the 1-chart interpretation C ( e ) of e with start vertex e is the { e } -generated sub-1-chart of the stacked star expressions 1-LTS L ( StExp ( ∗ ) ( A )) .It follows by the definition of induced transitions in Def. 5.6 that the induced chart C ( e ) ( · ]( · ]( · ] of C ( e ) is the { e } -generated subchart of the induced LTS L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] of L ( StExp ( ∗ ) ( A )) . On the other hand, thechart interpretation C ( e ) of e is the { e } -generated subchart of the star expressions LTS L ( StExp ( A )) byDef. 2.5. Now since, due to Lem. 5.8, the graph of the projection function π defines a bisimulation from L ( StExp ( ∗ ) ( A )) ( · ]( · ]( · ] to the star expressions LTS L ( StExp ( A )) which due to π ( e ) = e relates e with itself, itfollows that the restriction of π to the { e } -generated part C ( e ) ( · ]( · ]( · ] of L ( StExp ( A )) ( · ]( · ]( · ] defines a bisimulationto the { e } -generated part C ( e ) of L ( StExp ( A )) . From this we conclude that C ( e ) ( · ]( · ]( · ] → C ( e ) holds. C ( · ) We now turn to proving Lem. 5.13, which states that the entry/body-labeling c L ( StExp ( ∗ ) ( A )) fromDef. 5.11 is a LLEE-witness for the stacked star expression 1-LTS L ( StExp ( ∗ ) ( A )) . As a preparationfor verifying that the properties of a LLEE-witness are fulfilled, we prove a number of technical state-ments in the two lemmas below. For this, we introduce the set AppCxt ( A ) of applicative contexts ofstacked star expressions over A by which we mean the set of contexts that are defined by the grammar: C [ · ] :: = (cid:3) | C [ · ] · e | C [ · ] ∗ e ∗ (where e ∈ StExp ( A ) ) . In view of Def. 2.1 every stacked star expression E ∈ StExp ( ∗ ) ( A ) can be parsed uniquely as of the form E = C [ e ] for some star expression e ∈ StExp ( A ) , and an applicative context C [ · ] ∈ AppCxt ( A ) . Lemma 6.5.
If E → l E ′ in c L ( StExp ( ∗ ) ( A )) , then also C [ E ] → l C [ E ′ ] , for each l ∈ { bo } ∪ { [ n ] | n ∈ N + } .Proof. By induction on the structure of applicative contexts, using the rules for · and ∗ of c T ( ∗ ) ( A ) . Lemma 6.6. (a) Every maximal → bo path from C [ E ∗ e ∗ ] in c L ( StExp ( ∗ ) ( A )) , where E ∈ StExp ( ∗ ) ( A ) , e ∈ StExp ( A ) , and C [ · ] ∈ AppCxt ( A ) , is of either of the following two forms:(i) C [ E ∗ e ∗ ] = C [ E ∗ e ∗ ] → bo C [ E ∗ e ∗ ] → bo . . . → bo C [ E n ∗ e ∗ ] → bo . . . is finite or infinite withE , E , . . . , E n , . . . ∈ StExp ( ∗ ) ( A ) such that E → bo E → bo . . . → bo E n → bo . . . ,(ii) C [ E ∗ e ∗ ] = C [ E ∗ e ∗ ] → bo C [ E ∗ e ∗ ] → bo . . . → bo C [ E n ∗ e ∗ ] → bo C [ e ∗ ] → bo . . . for n ∈ N ,E , E , . . . , E n ∈ StExp ( ∗ ) ( A ) with E → bo E → bo . . . → bo E n , E n ↓ , and E n ∗ e ∗ → bo e ∗ . (b) Every maximal → bo path from E · e in c L ( StExp ( ∗ ) ( A )) , for E ∈ StExp ( ∗ ) ( A ) , and e ∈ StExp ( A ) , wherenow no filling in an applicative context is permitted, is of either of the following two forms:(i) E · e = E · e → bo E · e → bo . . . → bo E n · e → bo . . . is finite or infinite with stacked star expres-sions E , E , . . . , E n , . . . ∈ StExp ( ∗ ) ( A ) such that E → bo E → bo . . . → bo E n → bo . . . ,(ii) E · e = E · e → bo E · e → bo . . . → bo E n · e → bo E ′ → bo . . . with n ∈ N , and stacked star ex-pressions E , E , . . . , E n , F ∈ StExp ( ∗ ) ( A ) such that E → bo E → bo . . . → bo E n , E n ↓ and e → l E ′ .Proof. We first argue for statement (b). Due to the rules for · in c T ( ∗ ) it holds for every step E · e → bo G either (1) E → bo E , and G = E · e , for some E , or (2) E ↓ , and e → l F , and G = F , for some F and amarking label l . In case (2) we have recognized any maximal path E · e → bo G → bo . . . as of the form (ii)of item (b) for n =
0. In case (1) such a maximal path is parsed as E · e → bo E · e = G → bo . . . . Thenwe can use the same argument again for the second step E · e → bo G of that path. So by parsing a(finite or infinite) maximal path E · e → bo G → bo . . . → bo G n → bo . . . with this argument over its stepsfrom left to right, we either discover at some finite stage that one of its steps is of case (2), and then canconclude that the path is of form (ii), or we only encounter steps of case (1) and thereby parse the pathas of form (i).We turn to statement (a). We observe that any first step C [ E ∗ e ∗ ] → bo G can, in view the TSS rulesfor · and ∗ and since E ∗ e ∗ does not terminate immediately, only arise from a step E ∗ e ∗ → bo F viacontext filling as C [ E ∗ e ∗ ] → bo C [ F ] = G . Therefore it suffices to show the statement for (b) for theempty context C = (cid:3) . For this restricted statement we can argue analogously as for item (a). Here weuse that due to the TSS rules for ∗ it holds for every step E ∗ e ∗ → bo G that either (1) E → bo E , and G = E ∗ e , for some E , or (2) E ↓ , and E ∗ e ∗ → bo e ∗ , and G = e ∗ . By parsing the steps of a maximal → bo path E ∗ e ∗ → bo G → bo . . . → bo G n → bo . . . from left to right we either encounter a step of case (2), andthen find that the path is of form (ii), or otherwise we find that the path is of form (i) in statement (a). Lemma 6.7.
The following statements hold for paths of transitions in c L ( StExp ( A )) :(i) There are no infinite → bo paths in c L ( StExp ( A )) .(ii) If E ∈ StExp ( ∗ ) ( A ) is normed, then E → ∗ bo f for some f ∈ StExp ( A ) with f ↓ .(iii) If E → [ n ] E ′ with n > and E , E ′ ∈ StExp ( ∗ ) ( A ) , then E = C [ e ∗ ] , e normed + , e → l E ′ , E ′ = C [ E ′ ∗ e ∗ ] , and n = | e | ∗ + , for some e ∈ StExp ( A ) , C [ · ] ∈ AppCxt ( A ) , and e ′ ∈ StExp ( ∗ ) ( A ) .(iv) Neither → bo and → [ n ] steps in c L ( StExp ( A )) , where n ≥ , increase the star height of expressions.Proof. For statement (i) we prove that there is no infinite → bo path from any F ∈ StExp ( ∗ ) ( A ) , by structuralinduction on F ∈ StExp ( ∗ ) ( A ) . For F = F = F a −→ l F ′ in the TSS c T ( ∗ ) ( A ) . For F = a , where a ∈ A , the single step possible is F a −→ bo
1, which cannot be extended by any further steps. For F = e + e it holds, due to the TSS rulefor + , that every → bo path F → bo F ′ → bo . . . from F gives rise to also a path e i → bo F ′ → bo . . . from e i ,for i ∈ { , } . Since by induction hypothesis the → bo path from e i cannot be infinite, this follows also forthe → bo path from F . Now we let F = E · e for E ∈ StExp ( ∗ ) ( A ) and e ∈ StExp ( A ) . Then every maximal → bo path from F is of the form (i) or of the form (ii) in by Lem. 6.6, (b). So it either has the length ofa → bo path from E , or 1 plus the finite length of a → bo path from E , plus the length of a → bo path from e . Since by induction hypothesis every → bo path from E or from e is finite, it follows that every → bo path from F must have finite length as well. For the remaining case F = E · e ∗ , for E ∈ StExp ( ∗ ) ( A ) and e ∈ StExp ( A ) , we can argue analogously by using Lem. 6.6, (a). This concludes the proof of item (i).Item (ii) can be shown by induction on the structure of E ∈ StExp ( ∗ ) ( A ) . We only consider the non-trivial case in which E = E ∗ e ∗ , for E normed. We have to show that there is f ∈ StExp ( A ) with E → ∗ bo f .Grabmayer 43and f ↓ . We first observe that unless E ↓ holds, any step from E = E ∗ e ∗ must, due to the TSS rules, be ofthe form E ∗ e ∗ → l E ′ ∗ e ∗ for some E ′ ∈ StExp ( ∗ ) ( A ) such that also E → l E ′ . Therefore the assumptionthat E is normed implies that also E is normed. Then we can apply the induction hypothesis for E inorder to obtain a path E → ∗ bo e ′ and e ′ ↓ for some e ′ ∈ StExp ( A ) . From this we get E ∗ e ∗ → ∗ bo e ′ ∗ e ∗ by Lem. 6.5, and e ′ ∗ e ∗ → bo e ∗ due to e ′ ↓ . Consequently we find E = E ∗ e ∗ → ∗ bo e ∗ and e ∗ ↓ , and henceby letting f : = e ∗ ∈ StExp ( A ) we have obtained the proof obligation E → ∗ bo f and f ↓ in this case.Statement (iii) can be proved by induction on the depth of derivations in the TSS c T ( ∗ ) . This proofemploys two crucial facts. First, every entry-step is created by a step e ∗ → [ n ] E ′ · e ∗ with e → l E ′ and n = | e ∗ | ∗ = | e | ∗ +
1, for some e ∈ StExp ( A ) , E ′ ∈ StExp ( ∗ ) ( A ) , and marking label l . And second, entry-steps are preserved under contexts (cid:3) · e and (cid:3) ∗ e ∗ , and hence are preserved under applicative contexts.Statement (iv) can be proved by induction on the depth of derivations in the TSS c T ( ∗ ) . Proof of Lem. 5.13.
That c L ( StExp ( ∗ ) ( A )) is an entry/body-labeling of L ( StExp ( ∗ ) ( A )) is a consequenceof the fact that the rules of the TSS c T ( ∗ ) ( A ) are marking labeled versions of the rules of the TSS T ( ∗ ) ( A ) .It remains to show that c L : = c L ( StExp ( ∗ ) ( A )) is a LLEE-witness of the stacked star expressions1-LTS L = L ( StExp ( ∗ ) ( A )) . Instead of verifying the LLEE-witness conditions (W1), (W2), (W3), weestablish the following four equivalent conditions that are understood to be universally quantified overall E , E , F , F ∈ StExp ( ∗ ) ( A ) , and n , m ∈ N with n , m > E → [ n ] E = ⇒ E → [ n ] · → ∗ bo E ,(LLEE-2) → bo is terminating from E ,(LLEE-3) E −−→ t ( E ) [ n ] · −−→ t ( E ) ∗ bo F = ⇒ F (the premise means that F is in C c L ( E , n ) such that F = E ),(LLEE-4) E −−→ t ( E ) [ n ] · −−→ t ( E ) ∗ bo F → [ m ] F = ⇒ n > m ,where −−→ t ( E ) [ n ] and −−→ t ( E ) bo mean → [ n ] and → bo steps, respectively, that avoid E as their targets. Hereby(LLEE-2) obviously implies (W1). For each entry identifier h E , n i ∈ E ( c L ) it is not difficult to check that(LLEE-1), (LLEE-2), and (LLEE-3) imply that C c L ( E , n ) satisfies the loop properties (L1), (L2), and(L3), respectively, to obtain (W2). Finally, (LLEE-4) is an easy reformulation of the condition (W3).Now (LLEE-2) is guaranteed by Lem. 6.7, (i). It remains to verify the other three conditions above.For reasoning about transitions in c L we employ easy properties that follow from the rules of c T ( ∗ ) ( A ) .For showing (LLEE-1), we suppose that E → [ n ] E for some E , E ∈ StExp ( ∗ ) ( A ) . We have to show E → [ n ] · → ∗ bo E . By Lem. 6.7, (iii), E → [ n ] E implies that E = C [ e ∗ ] , e normed + , n = | e | ∗ +
1, for some e ∈ StExp ( A ) and C [ · ] ∈ AppCxt ( A ) . Since e is normed + , and because no 1-transitions can emanate from e , there must be some normed G ∈ StExp ( ∗ ) ( A ) , and a ∈ A such that e a −→ l G holds. We pick G accordingly.Together with n = | e | ∗ + e ∗ → [ n ] G ∗ e ∗ . Since G is normed, by Lem. 6.7, (ii) there are k ∈ N , G , . . . , G k ∈ StExp ( ∗ ) ( A ) such that G = G → bo . . . → bo G k and G k ↓ . Then G k ∗ e ∗ → bo e ∗ , and,by Lem. 6.5, E = C [ e ∗ ] → [ n ] C [ G ∗ e ∗ ] → bo . . . → bo C [ G k ∗ e ∗ ] → bo C [ e ∗ ] = E . Hence E → [ n ] · → ∗ bo E .For showing (LLEE-3), we suppose that E → [ n ] E → bo . . . → bo E k = F for E , E , . . . , E k , F ∈ StExp ( ∗ ) ( A ) with E , . . . , E k , F = E , and k ∈ N , k ≥
1. We have to show that F . By applying Lem. 6.7, (iii), E → [ n ] E implies that E = C [ e ∗ ] , E = C [ G ∗ e ∗ ] , e ∗ → [ n ] G ∗ e ∗ , for some e ∈ StExp ( A ) , G ∈ StExp ( ∗ ) ( A ) , and C [ · ] ∈ AppCxt ( A ) . As now the considered path is of the form E = C [ e ∗ ] → [ n ] E = C [ G ∗ e ∗ ] → bo E → bo . . . → bo E k = F , it follows from Lem. 6.6, (a), due to E , . . . , E k = E = C [ e ∗ ] that this path must be ofform (i) there: E = C [ e ∗ ] → [ n ] E = C [ G ∗ e ∗ ] → bo E = C [ G ∗ e ∗ ] → bo . . . → bo E k = C [ G k ∗ e ∗ ] = F for some G , G , . . . , G k ∈ StExp ( ∗ ) ( A ) with G = G . From the form C [ G k ∗ e ∗ ] of F we can now con-clude that F , since a stacked star expression that is not also a star expression does not permit immediatetermination.4 Structure-Constrained Process Graphs for the Process Semantics ofRegular ExpressionsFor showing (LLEE-4), we suppose that E → [ n ] E → bo . . . → bo E k = F → [ m ] F for some k , m , n ∈ N , k , m , n ≥
1, and E , E , . . . , E k , F , F ∈ StExp ( ∗ ) ( A ) with E , . . . , E k = E . We have to show that n > m .Arguing analogously as for (LLEE-3) with Lem. 6.7, (iii), and Lem. 6.6, (a), we find that the path is ofthe form E = C [ e ∗ ] → [ n ] E = C [ G ∗ e ∗ ] → bo E = C [ G ∗ e ∗ ] → bo . . . → bo E k = C [ G k ∗ e ∗ ] = F → [ m ] F for some G , G , . . . , G k ∈ StExp ( ∗ ) ( A ) such that n = | e | ∗ + e → l G and G → bo G → bo . . . → bo G k − = F → [ m ] F . Due to Lem. 6.7, (iv), it follows that n = | e | ∗ + > | e | ∗ ≥ | G | ∗ ≥ | G | ∗ ≥ . . . ≥ | G k | ∗ . Sincedue to the loop step C [ G k ∗ e ∗ ] → [ m ] F can only originate from G k , we conclude by Lem. 6.7, (iii), that | G k | ∗ ≥ m . From what we have obtained before, we conclude n > m .By having verified (LLEE-1)–(LLEE-4) we have shown that c L ( StExp ( A )) is a LLEE-witness. Proof of Thm. 5.14.
Since c L ( StExp ( ∗ ) ( A )) is an entry/body-labeling of L ( StExp ( ∗ ) ( A )) by Lem. 5.13, itfollows that d C ( e ) , the e -rooted sub-1-chart of c L ( StExp ( ∗ ) ( A )) , is an entry/body-labeling of C ( e ) , the e -rooted subchart of the LTS L ( StExp ( ∗ ) ( A )) . Now LLEE-witnesses are preserved under taking generatedsubcharts, which can be concluded directly from the alternative characterization of LLEE-witnesses(LLEE-1)–(LLEE-4) in the proof of Lem. 5.13. It follows that d C ( e ) is a LLEE-witness of C ( e ) . As shown in [6, 7], process graphs (charts) that are defined by a TSS formulation (similar to Def. 2.3)of Milner’s process interpretation for ‘1-free star expressions’ satisfy the loop existence and eliminationproperty LEE. But for process graph interpretations of arbitrary star expressions this is not the case ingeneral. We provided counterexamples for this fact in Section 4. By the development in Section 5 and theproofs in Section 6, however, we showed that LEE can still by made applicable for star expressions fromthe full class: we defined a variant C ( · ) of the process semantics C ( · ) such that C ( · ) guarantees LEE.The solution of the axiomatization problem (A) for 1-free star expressions in [6, 7] is based ontransferring ‘provable solutions’ between bisimilar charts in order to show that they are provably equal.A provable solution is a function from vertices of a chart to star expressions that provably satisfies, ineach vertex v of the chart, a correctness condition that requires that the solution value at v is provablyequal to the sum of expressions a i · w i over all transitions h v , a i , w i i from v , plus 1 if v ↓ holds. We usedsix lemmas: (I) the chart interpretation C ( e ) of a 1-free star expression e is a LLEE-chart; (SI) every1-free star expression e is the start value of a provable solution of its chart interpretation C ( e ) ; (C) thebisimulation collapse of a LLEE-chart is again a LLEE-chart; (E) from every LLEE-chart C a provablesolution of C can be extracted; (P) all provable solutions can be pulled back from the target to the sourcechart of a functional bisimulation to obtain a provable solution of the source chart; (SE) all provablesolutions of a LLEE-chart are provably equal. From these parts it follows that if C ( e ) ↔ C ( e ) holdsfor 1-free star expressions e and e , then a proof of e = e in the axiomatization can be constructedfrom a solution of the bisimulation collapse C of C ( e ) and C ( e ) , where C is a LLEE-chart dueto (I) and (C) . A solution of the recognizability problem (R) for charts concerning expressibility by1-free star expressions can be obtained by using (C) and showing that LEE can be tested for chartsin polynomial time.Concerning the extension of this approach to the set of all star expressions we can report the follow-ing. The lemmas (E) , (P) , and (SE) can be extended (with technical efforts) to versions (E) , (P) , and (SE) that apply to 1-charts and LLEE-1-charts. While (I) does not hold for all star expressions (see Sec-tion 4), we obtained as adaptation in Section 5: (I) the 1-chart interpretation C ( e ) of a star expression e is a LLEE-1-chart. An analogue version (SI) of (SI) can also be shown. The remaining obstacle for ex-.Grabmayer 45tending the approach to all star expressions is that a generalization of (C) to LLEE-1-charts is not straight-forward. Acknowledgement.
I want to thank the reviewers for their close reading, observations, and suggestionsof structural improvements. I am thankful to Luca Aceto for comments on a draft. The idea to defineLLEE-witnesses from TSSs was suggested by Wan Fokkink, with whom I worked it out for 1-free starexpressions in [6, 7]. That inspired me for this generalization to the full class of star expressions.
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