A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity
AA Complete Proof System for1-Free Regular Expressions Modulo Bisimilarity
Clemens Grabmayer
Department of Computer ScienceGran Sasso Science InstituteL’Aquila, Italy [email protected]
Wan Fokkink
Department of Computer ScienceVrije Universiteit AmsterdamAmsterdam, The Netherlands [email protected]
Abstract
Robin Milner (1984) gave a sound proof system for bisimi-larity of regular expressions interpreted as processes: BasicProcess Algebra with unary Kleene star iteration, deadlock0, successful termination 1, and a fixed-point rule. He askedwhether this system is complete. Despite intensive researchover the last 35 years, the problem is still open.This paper gives a partial positive answer to Milner’s prob-lem. We prove that the adaptation of Milner’s system overthe subclass of regular expressions that arises by droppingthe constant 1, and by changing to binary Kleene star itera-tion is complete. The crucial tool we use is a graph structureproperty that guarantees expressibility of a process graphby a regular expression, and is preserved by going over froma process graph to its bisimulation collapse.
Keywords regular expressions, process algebra, bisimilar-ity, process graphs, complete proof system
Regular expressions, introduced by Kleene [17], are widelystudied in formal language theory, notably for string search-ing [29]. They are constructed from constants 0 (no strings),1 (the empty string), and a (a single letter) from some alpha-bet; binary operators ` and ¨ (union and concatenation); andthe unary Kleene star ˚ (zero or more iterations).Their interpretations are Kleene algebras with as prime ex-ample the algebra of regular events, the language semanticsof regular expressions, which is closely linked with deter-ministic finite state automata. Aanderaa [1] and Salomaa[24] gave complete axiomatizations for the language seman-tics of regular expressions, with a non-algebraic fixed-pointrule that has a non-empty-word property as side condition.Krob [20] gave an infinitary, and then Kozen [18] a finitaryalgebraic axiomatization involving equational implications.Regular expressions also received significant attentionin the process algebra community [5], where they are in-terpreted modulo the bisimulation process semantics [22].Robin Milner [21] was the first to study regular expressionsin this setting, where he called them star expressions. Herethe interpretation of 0 is deadlock, 1 is (successful) termi-nation, a is an atomic action, and ` and ¨ are alternative Report version . and sequential composition of two processes, respectively.Milner adapted Salomaa’s axiomatization to obtain a soundproof system for this setting, and posed the (still open) ques-tion whether this axiomatization is complete, meaning thatif the process graphs of two star expressions are bisimilar,then they can be proven equal.Milner’s axiomatization contains a fixed-point rule, whichis inevitable because due to the presence of 0 the underly-ing equational theory is not finitely based [25, 26]. Bergstra,Bethke, and Ponse [4] studied star expressions without 0and 1, replaced the unary by the binary Kleene star f , whichrepresents an iteration of the first argument, possibly eventu-ally followed by the execution of the second argument. Theyobtained an axiomatization by basically omitting the axiomsfor 0 and 1 as well as the fixed-point rule from Milner’s ax-iomatization, and adding Troeger’s axiom [30]. This purelyequational axiomatization was proven complete in [9, 11].A sound and complete axiomatization for star expressionswithout unary Kleene star, but with 0 and 1 and a unary per-petual loop operator ˚ e ˚ ¨ v in which (1) there is an infinite path from v , (2) each infi-nite path eventually returns to v , and (3) termination is notpermitted. A graph is said to satisfy LLEE (Layered LoopExistence and Elimination) if repeatedly eliminating the en-try transitions of a loop, and performing garbage collection,leads to a graph without infinite paths. LLEE offers a gen-eralization (and more elegant definition) of the notion of awell-behaved specification. a r X i v : . [ c s . L O ] A p r eport version Clemens Grabmayer and Wan Fokkink Our completeness proof roughly works as follows (formore details see Sect. 4). Let e and e be star expressionsthat have bisimilar graphs process graph interpretations д and д . We show that д and д satisfy LLEE. We moreoverprove that LLEE is preserved under bisimulation collapse.And we construct for each graph that satisfies LLEE a starexpression that corresponds to this graph, modulo bisimilar-ity. In particular such a star expression f can be constructedfor the bisimulation collapse of д and д . We show that both e and e can be proven equal to f , by a pull-back over thefunctional bisimulations from the bisimulation collapse backto д and д . This yields the desired completeness result.In our proof, the minimization of terms (and thereby of theassociated process graphs) in the left-hand side of a binaryKleene star modulo bisimilarity is partly inspired by [8, 10].Interestingly, we will be able to use as running example theprocess graph interpretation of the star expression that at theend of [10] is mentioned as problematic for a completenessproof. Our crucial use of witnesses for the graph propertyLLEE borrows from the representation of cyclic λ -terms [15]as structure-constrained term graphs, as used for definingand implementing maximal sharing in the λ -calculus with letrec [16] (see also [13]).The completeness result for star expressions with 0 butwithout 1 and with the binary Kleene star settles a natu-ral question. We are also hopeful that the property LLEEprovides a strong conceptual tool for approaching Milner’slong-standing open question regarding the class of all starexpressions. The presence of 1-transitions in graphs presentsnew challenges, such as that LLEE is not always preservedunder bisimulation collapse. In order to be able to still workwith this concept, we will need workarounds.This is a report version of the article [14] in the proceed-ings of the conference LICS 2020. It was compiled from thesubmission version, containing a technical appendix. Please see the appendix for details of proofs that have beenomitted or that are only sketched.
In this section we define star expressions, their process se-mantics as ‘charts’, the proof system
BBP for bisimilarity oftheir chart interpretations, and provable solutions of charts.
Definition 2.1.
Given a set A of actions , the set StExp p A q of star expressions over A is generated by the grammar: e :: “ | a | p e ` e q | p e ¨ e q | p e f e q (with a P A ).0 represents deadlock (i.e., does not perform any action), a an atomic action, ` alternative and ¨ sequential composition,and f the binary Kleene star. Note that 1 (for empty steps)is missing from the syntax. ř ki “ e i is defined recursively as0 if k “ e if k “
1, and p ř k ´ i “ e i q ` e k if k ą star height | e | f of a star expression e P StExp p A q de-notes the maximum number of nestings of Kleene stars in e : it is defined by | | f : “ | a | f : “ | f ` д | f : “ | f ¨ д | f : “ max t| f | f , | д | f u , and | f f д | f : “ max t| f | f ` , | д | f u . Definition 2.2.
By a (finite sink-termination) chart C weunderstand a 5-tuple x V , ‘ , v s , A , T y where V is a finite setof vertices , ‘ is, in case ‘ P V , a special vertex with no out-going transitions (a sink) that indicates termination (in case ‘ R V , the chart does not admit termination), v s P V zt ‘ u is the start vertex , A is a set actions , and T Ď V ˆ A ˆ V theset of transitions . Since A can be reconstructed from T , wewill frequently keep A implicit, denote a chart as a 4-tuple x V , ‘ , v s , T y . A chart is start-vertex connected if every vertexis reachable by a path from the start vertex. This property canbe achieved by removing unreachable vertices (‘garbage col-lection’). We will assume charts to be start-vertex connected.In a chart C , let v P V and U Ď T be a set of transitionsfrom v . By the x v , U y -generated subchart of C we mean thechart C “ x V , ‘ , v , A , T y with start vertex v where V isthe set of vertices and T the set of transitions that are onpaths in C from v that first take a transition in U , and then,until v is reached again, continue with other transitions of C .We use the standard notation v a ÝÑ v in lieu of x w , a , w y P T . Definition 2.3.
Let C i “ x V i , ‘ , v s , i , T i y for i P t , u betwo charts. A bisimulation between C and C is a relation B Ď V ˆ V that satisfies the following conditions:( start ) v s , B v s , (it relates the start vertices),and for all v , v P V with v B v :( forth ) for every transition v a ÝÑ v in C there is a transi-tion v a ÝÑ v in C with v B v ,( back ) for every transition v a ÝÑ v in C there is a transi-tion v a ÝÑ v in C with v B v ,( termination ) v “ ‘ if and only if v “ ‘ .If there is a bisimulation between C and C , then we write C Ø C and say that C and C are bisimilar . If a bisimula-tion is the graph of a function, we say that it is a functional bisimulation. We write C Ñ C if there is a functional bisim-ulation between C and C . Definition 2.4.
For every star expression e P StExp p A q the chart interpretation C p e q “ x V p e q , ‘ , e , A , T p e qy of e is thechart with start vertex e that is specified by iteration via thefollowing transition rules, which form a transition systemspecification (TSS), with e , e , e , e P StExp p A q , a P A : a a ÝÑ ‘ e i a ÝÑ ξ p i “ , q e ` e a ÝÑ ξe a ÝÑ e e ¨ e a ÝÑ e ¨ e e a ÝÑ ‘ e ¨ e a ÝÑ e e a ÝÑ e e f e a ÝÑ e ¨ p e f e q e a ÝÑ ‘ e f e a ÝÑ e f e e a ÝÑ ξe f e a ÝÑ ξ Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version with ξ P StExp p A q ‘ : “ StExp p A q Y t ‘ u , where ‘ indicatessink termination. If e a ÝÑ ξ can be proved, ξ is called an a -de-rivative , or just derivative , of e . The set V p e q Ď StExp p A q ‘ consists of the iterated derivatives of e . To see that C p e q isfinite, Antimirov’s result [2], that a regular expression hasonly finitely many iterated derivatives, can be adapted.We say that a star expression e P StExp p A q is normed ifthere is a path of transitions from e to ‘ in C p e q . aac bb a C p e q v a v ac v b b C p e q aaca b bac aa C p e q Example 2.5.
By the rules in Def. 2.4, e : “ a ¨ e with e : “p c ¨ a ` a ¨ p b ` b ¨ a qqq f C p e q as above, with v : “ e , v : “ e and v : “ p b ` b ¨ a q ¨ e . This chartis the bisimulation collapse of the charts C p e q and C p e q of star expressions e : “ p a ¨ pp a ¨ p b ` b ¨ a qq f c qq f e : “ a ¨ pp c ¨ a ` a ¨ p b ¨ a ¨ pp c ¨ a q f a qq f b q f q . Bisimula-tions between C p e q and C p e q , and between C p e q and C p e q are indicated by the broken lines. The chart C p e q was con-sidered problematic in [10]. Example 2.6.
The left chart below does not admit termi-nation. The right chart is a double-exit graph with the sinktermination vertex ‘ at the bottom. ‘ aab cab a ca a These charts are not bisimilar to chart interpretations of starexpressions. For the left chart this was shown by Milner [21],and for the right chart by Bosscher [6].
Definition 2.7.
The proof system
BBP or the class of starexpressions has the axioms (B1)–(B6), (BKS1), (BKS2), theinference rules of equational logic, and the rule RSP f : p B1 q x ` y “ y ` x p B2 q p x ` y q ` z “ x ` p y ` z qp B3 q x ` x “ x p B4 q p x ` y q ¨ z “ x ¨ z ` y ¨ z p B5 q p x ¨ y q ¨ z “ x ¨ p y ¨ z qp B6 q x ` “ x p B7 q ¨ x “ p BKS1 q x ¨ p x f y q ` y “ x f y p BKS2 q p x f y q ¨ z “ x f p y ¨ z qp RSP f q x “ p y ¨ x q ` zx “ y f z By e “ BBP e we denote that e “ e is derivable in BBP . BBP is a finite ‘implicational’ proof system [28], becauseunlike in Salomaa’s and Milner’s systems for regular ex-pressions with 1 the fixed-point rule does not require anyside-condition to ensure ‘guardedness’.
Definition 2.8.
For a chart C “ x V , ‘ , v s , A , T y , a provablesolution of C is a function s : V z t ‘ u Ñ StExp p A q such that: s p v q “ BBP ´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ s p w j q ¯ (for all v P V zt ‘ u )holds, given that the union of ␣ v a i ÝÑ ‘ ˇˇ i “ , . . . , m ( and ␣ v b j ÝÑ w j ˇˇ j “ , . . . , n , w j ‰ ‘( is the set of transitionsfrom v in C . We call s p v s q the principal value of s . Proposition 2.9 (uses
BBP -axioms (B1)–(B7), (BKS1)) . Forevery e P StExp p A q , the identity function id V p e q : V p e q Ñ V p e q Ď StExp p A q , e ÞÑ e , is a provable solution of the chartinterpretation C p e q of e .Proof (Idea). Each e in StExp p A q is the BBP -provable sum ofexpressions a and a ¨ e over all a P A for a -derivatives ‘ and e , respectively, of e . This ‘fundamental theorem of differential calculus for star expressions’ implies, quitedirectly, that id V p e q is a provable solution of C p e q . □ As preparation for the definition of the central concept of‘LLEE-witness’, we start with an informal explanation of thestructural chart property ‘LEE’. It is a necessary condition fora chart to be the chart interpretation of a star expression. LEEis defined by a dynamic elimination procedure that analysesthe structure of the graph by peeling off ‘loop subcharts’.Such subcharts capture, within the chart interpretation of astar expression e , the behaviour of the iteration of f withininnermost subterms f f f in e . (A weaker form of ‘loop’ byMilner [21], which describes the behavior of general iterationsubterms, is not sufficient for our aims.) Definition 3.1.
A chart L “ x V , ‘ , v s , T y is a loop chart if:(L1) There is an infinite path from the start vertex v s . Rutten [23] used this name for an analogous result on infinite streams[23]. The first author [12], and Kozen and Silva [19, 27] used it for theprovable synthesis of regular expressions from their Brzozowski derivatives.The result here can be viewed as stating the provable synthesis of regularexpressions from their partial derivatives (due to Antimirov [2]).3 eport version Clemens Grabmayer and Wan Fokkink (L2) Every infinite path from v s returns to v s after a positivenumber of transitions (and so visits v s infinitely often).(L3) V does not contain the vertex ‘ .In such a loop chart we call the transitions from v s loop-entrytransitions , and all other transitions loop-body transitions .Let C be a chart. A loop chart L is called a loop subchart of C if L is the x v , U y -generated subchart of C for some vertex v of C , and a set U of transitions of C that depart from v (sothe transitions in U are the loop-entry transitions of L ).Note that the two charts in Ex. 2.6 are not loop charts:the left one violates (L2), and the right one violates (L3).Moreover, none of these charts contains a loop subchart.While the chart C p e q in Ex. 2.5 is not a loop chart either, asit violates (L2), we will see that it has loop subcharts.Let L be a loop subchart of a chart C . Then the resultof eliminating L from C arises by removing all loop-entrytransitions of L from C , and then removing all vertices andtransitions that get unreachable. We say that a chart C hasthe loop existence and elimination property (LEE) if the pro-cess, started on C , of repeated eliminations of loop subchartsresults in a chart that does not have an infinite path.For the charts in Ex. 2.6 the procedure stops immediately,as they do not contain loop subcharts. Since both of themhave infinite paths, it follows that they do not satisfy LEE.We consider three runs of the elimination procedure forthe chart C p e q in Ex. 2.5. The loop-entry transitions of loopsubcharts that are removed in each step are marked in bold. v a v ac v b b v a v c elim v a v elim v a v ac v b b elim 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 elim Each run witnesses that C satisfies LEE. Note that loop elim-ination does not yield a unique result. Runs can be recordedby attaching, in the original chart, to transitions that getremoved in the elimination procedure as marking label thesequence number of the appertaining elimination step. For Confluence, and unique normalization, can be shown if a pruning opera-tion is added that permits to drop transitions to deadlocking vertices. the three runs of loop elimination above we get the followingmarking labeled versions of C , respectively: v a v a r s c v r s b r s b v a v a r sr s c v b b v a v a r sr s c v b b Since all three runs were successful (as they yield chartswithout infinite paths), these recordings (marking-labeledcharts) can be viewed as ‘LEE -witnesses’. We now will definea concept of a ‘layered LEE-witness’ (LLEE-witness), i.e., aLEE-witness with the added constraint that in the formulatedrun of the loop elimination procedure it never happens thata loop-entry transition is removed from within the body ofa previously removed loop subchart. This refined concepthas simpler properties, and it will fit our purpose.Before introducing ‘LLEE-witnesses’, we first define chartlabelings that mark transitions in a chart as ‘(loop-)entry’and as ‘(loop-)body’ transitions, but without safeguardingthat these markings refer to actual loops.
Definition 3.2.
Let C “ x V , v s , ‘ , A , T y be a chart. An en-try/body-labeling ˆ C “ x V , v s , ‘ , A ˆ N , p T y of C is a chart thatarises from C by adding, for each transition τ “ x v , a , v y P T , to the action label a of τ a marking label α P N , yielding p τ “ x v , x a , α y , v y P p T . In such an entry/body-labeling wecall transitions with marking label 0 body transitions , andtransitions with marking labels in N ` entry transitions .Let ˆ C be an entry/body-labeling of C , and let v and w be vertices of C and ˆ C . We denote by v Ñ bo w that thereis a body transition v x a , y ÝÝÝÑ w in ˆ C for some a P A , andby v Ñ r α s w , for α P N ` that there is an entry transition v x a , α y ÝÝÝÑ w in ˆ C for some a P A . We will use α , β , γ , . . . for marking labels in N ` of entry transitions. By the set E p ˆ C q of entry transition identifiers we denote the set of pairs x v , α y P V ˆ N ` such that an entry transition Ñ r α s departsfrom v in ˆ C . For x v , α y P E p ˆ C q , we define by C ˆ C p v , α q thesubchart of C with start vertex v s that consists of the verticesand transitions which occur on paths in C as follows: theystart with a Ñ r α s entry transition from v , continue withbody transitions only, and halt immediately if v is revisited. Definition 3.3. A LLEE-witness ˆ C of a chart C is an entry/body-labeling of C that satisfies the following properties:(W1) There is no infinite path of Ñ bo transitions from v s .(W2) For all x v , α y P E p ˆ C q , (a) C ˆ C p v , α q is a loop chart, and(b) ( layeredness ) from no vertex w ‰ v of C ˆ C p v , α q there departs in ˆ C an entry transition Ñ r β s with β ě α . Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
The stipulation in (W2)(a) justifies to call entry transitionsin a LLEE-witness a loop-entry transition . For a loop-entrytransition Ñ r β s with β P N ` , we call β its loop level .A chart is a LLEE-chart if it has a LLEE-witness.
Example 3.4.
The three labelings of the chart C p e q in Ex. 2.5that arose as recordings of runs of the loop elimination pro-cedure can be viewed as entry/body-labelings of that chart.There, and below, we dropped the body labels of transitions,and instead only indicated the entry labels in boldface to-gether with their levels. By checking conditions (W1) and(W2),(a)-(b), it is easy to verify that these entry/body-labe-lings are LLEE-witnesses. In fact it is not difficult to establishthat every LLEE-witness of C p e q in Ex. 2.5 is of either of thefollowing two forms, with marking labels α , β , γ , δ , ϵ P N ` : v a v a r β sr α s c v b b v a v a r γ s c v r δ s b r ϵ s b (with γ ă δ , ϵ )We now argue that LLEE-witnesses guarantee the prop-erty LEE. Let ˆ C be a LLEE-witness of a chart C . Repeatedlypick an entry transition identifier x v , α y with α P N ` mini-mal, remove the loop subchart that is generated by loop-entrytransitions of level α from v (it is indeed a loop by (W2)(a),and minimality of α and (W2)(b) ensure the absence of de-parting loop-entry transitions of lower level), and performgarbage collection. Eventually the part of C that is reach-able by body transitions from the start vertex is obtained.This subchart does not have an infinite path due to (W1).Therefore C indeed satisfies LEE, as witnessed by ˆ C .The property LEE and the concept of LLEE-witness areclosely linked with the process semantics of star expressions.In fact, we now define a labeling of the TSS in Def. 2.4 thatpermits to define, for every star expression e , an entry/body-labeling of the chart interpretation C p e q of e , which can thenbe recognized as a LLEE-witness of C p e q .We refine the TSS rules in Def. 2.4 as follows: A bodylabel is added to transitions that cannot return to the starexpression in their left-hand side. The rule for transitionsinto the iteration part e of an iteration e f e is split into thecases where e is normed or not. Only in the normed casecan e f e return to itself, and then a loop-entry transitionwith the star height | e | f of e as its level is created. Definition 3.5.
For every e P StExp p A q , we define the en-try/body-labeling y C p e q of the chart interpretation C p e q of e in analogy with C p e q by using the following transition rulesthat refine the rules in Def. 2.4 by adding marking labels: a a ÝÑ bo ‘ e i a ÝÑ l ξ i P t , u e ` e a ÝÑ bo ξ r s a r s ac bb a z C p e q a r s a r s cb b z C p e q a r s a r s ca r s b ba r s c aa z C p e q Figure 1.
LLEE-witness entry/body-labelings as defined byDef. 3.5 for the chart interpretations of e , e , and e in Ex. 2.5. e a ÝÑ l e e ¨ e a ÝÑ l e ¨ e e a ÝÑ bo ‘ e ¨ e a ÝÑ bo e e a ÝÑ l e if e is normed e f e a ÝÑ r| e | f ` s e ¨ p e f e q e a ÝÑ l e if e is not normed e f e a ÝÑ bo e ¨ p e f e q e a ÝÑ bo ‘ e f e a ÝÑ r| e | f ` s e f e e a ÝÑ l ξe f e a ÝÑ bo ξ for l P t bo u Y tr α s | α P N ` u , where we employed notationdefined in Def. 2.4 for writing marking labels as subscripts. Example 3.6.
In Fig. 1 we depict the entry/body-labelings,as defined in Def. 3.2, for star expressions e , e , and e inEx. 2.5. It is easy to verify that these labelings are LLEE-wit-nesses of the charts C p e q , C p e q , and C p e q in Ex. 2.5, resp.. Proposition 3.7.
For every e P StExp p A q , the entry/body-la-beling y C p e q of C p e q is a LLEE -witness of C p e q . For a binary relation R , let R ` and R ˚ be its transitiveand transitive-reflexive closures. u Ñ l v denotes that thereis a transition u a ÝÑ l v for an a P A , and in proofs (butnot pictures) u Ñ v denotes that u Ñ l v for some label l . By u ÝÝÝÑ t p w q l v we denote that u Ñ l v and v ‰ w (this transitionavoids target w ). Likewise, u ÝÝÝÑ t p w q v denotes that u ÝÝÝÑ t p w q l v for some label l . By scc p u q we denote the strongly connectedcomponent (scc) to which u belongs. Definition 3.8.
Let ˆ C be a LLEE-witness of chart C . If thereis a path v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ bo w , then we write v α ñ w . (Notethat v α ñ w holds if and only if w is a vertex ‰ v of theloop chart C ˆ C p v , α q that is generated by the Ñ r α s entry tran-sitions at v in C .) We write v ñ w and say that v descendsin a loop to w if v α ñ w for some α P N ` . eport version Clemens Grabmayer and Wan Fokkink We write w ü v (or v ý w ), and say that w loops backto v , if v ñ w Ñ ` bo v . The loops-back-to relation ü totallyorders its successors (see Lem. 3.9, (vi)). Therefore we definethe ‘direct successor relation’ d ü of ü as follows: We write w d ü v (or v d ý w ), and say that w directly loops back to v ,if w ü v and for all u with w ü u either u “ v or v ü u . Lemma 3.9.
The relations Ñ bo , ñ , ü , d ü as defined by a LLEE -witness ˆ C on a chart C satisfy the following properties:(i) There are no infinite Ñ bo paths (so no Ñ bo cycles).(ii) If scc p u q “ scc p v q , then u ñ ˚ v implies v ü ˚ u .(iii) If v ñ w and ␣p w ü q , then w is not normed.(iv) scc p u q “ scc p v q if and only if u ü ˚ w and v ü ˚ w forsome vertex w .(v) ü ˚ is a partial order with the least-upper-bound prop-erty: if a nonempty set of vertices has an upper boundwith respect to ü ˚ , then it has a least upper bound.(vi) ü is a total order on ü -successor vertices: if w ü v and w ü v , then v ü v or v “ v or v ü v .(vii) If v d ü u and v d ü u for distinct v , v , then there isno vertex w such that both w ü ˚ v and w ü ˚ v . After having introduced LLEE-charts as our crucial auxiliaryconcept, we now sketch the completeness proof. In doingso we need to anticipate four results that will be developedin the next two sections: (C)
The bisimulation collapse of aLLEE-chart is again a LLEE-chart. (E)
From every LLEE-charta provable solution can be extracted. (S)
All provable solu-tions of LLEE-charts are provably equal. (P)
All provablesolutions can be pulled back from the target to the sourcechart of a functional bisimulation.Then completeness of
BBP can be argued as follows. Giventwo bisimilar star expressions e and e , obtain their chartinterpretations C p e q and C p e q , which are LLEE-charts dueto Prop. 3.7. By Prop. 2.9, e and e are principal values ofprovable solutions of C p e q and C p e q . These charts have thesame bisimulation collapse C . By ( C , Thm. 6.9), C is again aLLEE-chart. Use ( E , Prop. 5.5) to build a provable solution s of C ; let its principal value be e . Apply ( P , Prop. 5.1) to trans-fer s backwards over the functional bisimulations to obtainprovable solutions s and s of C p e q and C p e q , respectively.By construction, s and s have the same principal value e as s . Finally, by using ( S , Prop. 5.8), e and e are both provablyequal to e . Hence, e “ BBP e “ BBP e .In his completeness proof for regular expressions in formallanguage theory, Salomaa [24] argued ‘upwards’ from twoequivalent regular expressions to a larger regular expressionthat can be homomorphically collapsed onto both of them.In contrast, our proof approach forces us ‘downwards’ to thebisimulation collapse, because in the opposite direction theproperty of being a LLEE-chart may be lost. Example 4.1.
The picture below highlights why we can-not adopt Salomaa’s proof strategy of linking two language-equivalent regular expressions via the product of the DFAsthey represent. The bisimilar LLEE-charts C and C are in-terpretations of p a ¨ p a ` b q ` b q f p b ¨ p a ` b q ` a q f C and ˆ C are LLEE-wit-nesses). But their product C is a not a LLEE-chart; it is of theform of one the not expressible charts from Ex. 2.6. Yet theircommon bisimulation collapse C , the chart interpretationof p a ` b q f
0, is a LLEE-chart with LLEE-witness ˆ C . r s a r s b r s a b v v a b C , ˆ C C , ˆ C r s b w r s a w ba C , ˆ C x v , w y a b x v , w y ba x v , w y a b C ÑÑÑ Ñ
In view of C Ð C Ñ C this also shows that LLEE-chartsare not closed under converse functional bisimilarity Ð . In this section we develop the results (E) , (S) , and (P) asmentioned in Sect. 4. We start with the statement (P) . Proposition 5.1 (requires
BBP -axioms (B1), (B2), (B3)) . Let ϕ : V Ñ V be a functional bisimulation between charts C and C . If s : V z t ‘ u Ñ StExp p A q is a provable solution of C , then s ˝ ϕ : V z t ‘ u Ñ StExp p A q is a provable solution of C with the same principal value as s .Proof (Idea). The bisimulation clauses make it possible todemonstrate the condition for s ˝ ϕ to be a provable solutionof C at w by using the condition for the provable solution s of C at ϕ p w q , together with the axioms p B1 q , p B2 q , p B3 q . □ We now turn to proving results (E) and (S) from Sect. 4.We show that from every chart C with LLEE-witness ˆ C aprovable solution s ˆ C of C can be extracted. Intuitively, theextraction process follows a run of the loop-elimination pro-cedure on C , guided by the LLEE-witness ˆ C . All loop sub-charts that are generated by the loop-entry transitions froma vertex v are removed in a row. Extraction synthesizes astar expression e whose behavior captures the eliminatedloop subcharts of v and their previously eliminated innerloop subcharts, and that will later be part of an iteration We repeatedly pick vertices v in the remaining LLEE-witness with entrystep level | v | en (see in the text below) minimal.6 Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version expression e f e in the solution value at v . This idea moti-vates an inside-out extraction process that works with partialsolutions, and eventually builds up a provable solution of C .In particular, we inductively define ‘relative extracted so-lutions’ t ˆ C p w , v q for vertices v and w where w is in a loopsubchart C ˆ C p v , α q at v , for some α P N ` , that is, v α ñ w .Hereby t ˆ C p w , v q captures the part of the behavior in C from w until v is reached. Then we define the from ˆ C ‘extractedsolution’ s ˆ C p v q at v by using the relative solutions t ˆ C p w j , v q for all targets w j of loop-entry transitions from v to definethe iteration part e of the extracted solution s ˆ C p v q “ e f e at v . We start with a preparation.Let ˆ C be a LLEE-witness, and let v be a vertex of ˆ C . By the entry step level | v | en of v we mean the maximum loop levelof a loop-entry transition in ˆ C that departs from v , or 0 if noloop-entry transition departs from v . By the body step norm ∥ v ∥ bo of v we mean the maximal length of a body transitionpath in C from v (well-defined by Lem. 3.9, (i)). Lemma 5.2.
For all vertices v , w in a chart C with LLEE -wit-ness ˆ C it holds (for the concepts as defined with respect to ˆ C ):(i) v Ñ bo w ñ ∥ v ∥ bo ą ∥ w ∥ bo ,(ii) v ñ w ñ | v | en ą | w | en . Definition 5.3.
Let ˆ C be a LLEE-witness of a chart C . Thenthe relative extraction function of ˆ C is defined inductively as: t ˆ C : tx w , v y | v , w P V z t ‘ u , v ñ w u Ñ StExp p A q , t ˆ C p w , v q : “ ´´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b i ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯¯ , provided that w has loop-entry transitions t w a i ÝÑ r α i s w | i “ , . . . , m u Y t w b j ÝÑ r β j s w j | j “ , . . . , n ^ w j ‰ w u andbody transitions t w c i ÝÑ bo v | i “ , . . . , p u Y t w d j ÝÑ bo u j | j “ , . . . , q ^ u j ‰ v u . Hereby the induction proceeds on x | v | en , ∥ w ∥ bo y with the lexicographic order ă lex on N ˆ N :For t ˆ C p w j , w q we have x | w | en , ∥ w j ∥ bo y ă lex x | v | en , ∥ w ∥ bo y due to | w j | en ă | v | en , which follows from v ñ w by Lem. 5.2,(ii). For t ˆ C p u j , v q we have x | v | en , ∥ u j ∥ bo y ă lex x | v | en , ∥ w ∥ bo y due to ∥ u j ∥ bo ă ∥ w ∥ bo , which follows from w Ñ bo u j byLem 5.2, (i).The extraction function of ˆ C is defined by: s ˆ C : V z t ‘ u Ñ StExp p A q , s ˆ C p w q : “ ´´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯¯ , with induction on ∥ w ∥ bo , provided that w has loop-entrytransitions t w a i ÝÑ r α i s w | i “ , . . . , m u Y t w b j ÝÑ r β j s w j | j “ , . . . , n ^ w j ‰ w u and body transitions t w c i ÝÑ bo ‘ | i “ , . . . , p u Y t w d j ÝÑ bo u j | j “ , . . . , q ^ u j ‰ ‘ u . For s ˆ C p u j q the induction hypothesis holds due to (cid:13)(cid:13) u j (cid:13)(cid:13) bo ă ∥ w ∥ bo ,which follows from w Ñ bo u j by Lem. 5.2, (i). Lemma 5.4 (uses the
BBP -axioms (B1)–(B6), (BKS2), but notthe rule RSP f ) . In a chart C with LLEE -witness ˆ C , if v ñ w ,then s ˆ C p w q “ BBP t ˆ C p w , v q ¨ s ˆ C p v q . Proposition 5.5 (uses the
BBP -axioms (B1)–(B6), (BKS1),(BKS2), but not the rule RSP f ) . For every
LLEE -witness ˆ C ofa chart C , the extraction function s ˆ C is a provable solution of C . The proof of Lem. 5.4 proceeds by induction on ∥ w ∥ bo ; noinduction is needed for the proof of Prop. 5.5 (cf. appendix). Example 5.6.
Left in Fig. 2 we illustrate the extraction of aprovable solution for the LLEE-witness ˆ C “ z C p e q in Ex. 3.6of the chart C “ C p e q in Ex. 2.5. In order to obtain theprincipal value s ˆ C p v q of the extracted solution s ˆ C , its defini-tion is expanded. It recurs on s ˆ C p v q , and then on t ˆ C p v , v q and t ˆ C p v , v q . After computing those star expressions byusing the definition of t ˆ C , the principal value can be obtainedby substitution. The star expressions s ˆ C p v q and s ˆ C p v q areobtained similarly. For readability we have simplified thearising terms on the way by using the equality 0 f x “ BBP x (which follows by p B1 q , p B6 q , p B7 q , and (BKS1)). Lemma 5.7 (uses the
BBP -axioms (B1)–(B6), and the ruleRSP f ) . If v ñ w , then s p w q “ BBP t ˆ C p w , v q ¨ s p v q for everyprovable solution s of a chart C with LLEE -witness ˆ C . Proposition 5.8 (uses the
BBP -axioms (B1)–(B6), and therule RSP f ) . Let s and s be provable solutions of a LLEE -chart.Then s p w q “ BBP s p w q for all vertices w ‰ ‘ . For the proof of this proposition, see Fig. 3. The proof ofLem. 5.7 (see in the appendix) proceeds by the same inductionmeasure as we used for the relative extraction function.
Example 5.9.
In the right half of Fig. 2 we prove that anarbitrary provable solution s of LLEE-chart C “ C p e q inEx. 2.5 with LLEE-witness ˆ C “ z C p e q in Ex. 3.6 is provablyequal to the extracted solution s ˆ C of C . Crucially, the definingconditions for s as a provable solution of C are expandedalong the loop at v . The loop behavior obtained is the sameas that which is used in the definition of s ˆ C p v q . By applyingthe fixed-point rule RSP f we can then deduce BBP -provableequality of s p v q and s ˆ C p v q . By using the solution conditionsfor s again, provable equality is then transferred to v and v . In this section we establish the remaining result (C) fromSect. 4 that is crucial for the completeness proof: that thebisimulation collapse of a LLEE-chart is again a LLEE-chart.This result is achieved by a step-wise construction of abisimulation collapse. Pairs of bisimilar vertices w and w eport version Clemens Grabmayer and Wan Fokkink s ˆ C p v q : “ f p a ¨ s ˆ C p v qq“ BBP a ¨ s ˆ C p v q“ BBP a ¨ p c ¨ a ` a ¨ p b ` b ¨ a qq f s ˆ C p v q : “ p c ¨ t ˆ C p v , v q ` a ¨ t ˆ C p v , v qq f “ BBP p c ¨ a ` a ¨ p b ` b ¨ a qq f t ˆ C p v , v q : “ f a “ BBP at ˆ C p v , v q : “ f p b ` b ¨ t ˆ C p v , v qq“ BBP b ` b ¨ as ˆ C p v q : “ f p b ¨ s ˆ C p v q ` b ¨ s ˆ C p v qq“ BBP b ¨ s ˆ C p v q ` b ¨ p a ¨ s ˆ C p v qq“ BBP p b ` b ¨ a q ¨ s ˆ C p v q“ BBP p b ` b ¨ a q ¨ pp c ¨ a ` a ¨ p b ` b ¨ a qq f q v a v r s a r s c v b b C , ˆ C s p v q “ (sol) BBP a ¨ s p v q ( (sol) meansuse of ‘is provable solution’) s p v q “ (sol) BBP c ¨ s p v q ` a ¨ s p v q“ (sol) BBP c ¨ p a ¨ s p v qq ` a ¨ p b ¨ s p v q ` b ¨ s p v qq“ (sol) BBP c ¨ p a ¨ s p v qq ` a ¨ p b ¨ s p v q ` b ¨ p a ¨ s p v qqq“ BBP p c ¨ a ` a ¨ p b ` b ¨ a qq ¨ s p v q ` ó applying RSP f s p v q “ BBP p c ¨ a ` a ¨ p b ` b ¨ a qq f “ BBP s ˆ C p v q (see in the derivation on the left) ó s p v q “ (sol) BBP a ¨ s p v q “ BBP a ¨ s ˆ C p v q “ (sol) BBP s ˆ C p v qó s p v q “ (sol) BBP b ¨ s p v q ` b ¨ s p v q“ BBP b ¨ s ˆ C p v q ` b ¨ s ˆ C p v q “ (sol) BBP s ˆ C p v q Figure 2.
Left: the process of extracting the provable solution s ˆ C of a chart C from an LLEE-witness ˆ C of C as in the middle.Right: steps for showing that an arbitrary provable solution s of C is BBP -provably equal to the extracted solution s ˆ C . Proof (of Prop. 5.8).
Let ˆ C be a LLEE-witness of a chart C . Let s be a provable solution of C . We have to show that s p w q “ BBP s ˆ C p w q for all w ‰ ‘ . For this, let w ‰ ‘ . The derivation below is based on the set representation of transitions from w in ˆ C asformulated in the definition of s ˆ C p w q . The first derivation step uses that s is a provable solution of C and axioms p B1 q , p B2 q ,and p B3 q , the second step uses Lem. 5.7 in view of w ñ w j for j “ , . . . , n , and the third step uses axioms p B4 q , p B5 q , and p B6 q . s p w q “ BBP ´´ m ÿ i “ a i ¨ s p w q ¯ ` ´ n ÿ j “ b j ¨ s p w j q ¯¯ ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s p u j q ¯¯ “ BBP ´´ m ÿ i “ a i ¨ s p w q ¯ ` ´ n ÿ j “ b j ¨ ` t ˆ C p w j , w q ¨ s p w q ˘¯¯ ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s p u j q ¯¯ “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ ` b j ¨ t ˆ C p w j , w q ˘¯¯ ¨ s p w q ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s p u j q ¯¯ In view of this derived provable equality for s p w q , we can now apply the rule RSP f in order to obtain: s p w q “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ ” s ˆ C p w q In this last step we have used the definition of s ˆ C p w q . □ Figure 3.
Proof of Prop. 5.8.are collapsed one at a time, whereby the incoming transitionsof w are redirected to w . The crux is to take care, and toprove, that the resulting chart has again a LLEE-witness. Definition 6.1.
Let C be a chart, with vertices w and w .The connect- w -through-to- w chart C p w q w of C is obtainedby redirecting all incoming transitions at w over to w , and,if w is the start vertex of C , making w the new start vertex;in this way w gets unreachable, and it is removed with otherunreachable vertices to obtain a start-vertex connected chart. Let ˆ C be an entry/body-labeling of C . Then we define theentry/body-labeling ˆ C p w q w of C p w q w as follows: every transi-tion in C p w q w that was already a transition τ in C inherits itsmarking label from τ in ˆ C ; and every transition in C p w q w thatarises as the redirection τ w to w of a transition τ to w in C such that τ w does not coincide with a transition alreadyin C inherits its marking label from τ in ˆ C . Lemma 6.2. If w Ø w in C , then C p w q w Ø C . Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
While the connect-through operation of bisimilar verticesin a chart thus results in a bisimilar chart, its application toa LLEE-witness (an entry/body-labeling) does not need toyield a LLEE-witness again: the property LEE may be lost.
Example 6.3.
Consider the LLEE-witness ˆ C in the middlebelow. The unspecified action labels are assumed to facili-tate that w and w are bisimilar. Hence also p w and p w arebisimilar. Bisimilarity is indicated by the broken lines. Theconnect- w -through-to- w chart on the left is not a LLEE-chart, because it does not satisfy LEE: after the loop subchartinduced by the downwards transition from p w is eliminated,and garbage collection is done, the remaining chart withoutthe dotted transitions still has an infinite path; yet it does notcontain another loop subchart, because each infinite path canreach ‘ without returning to its source. An example of thisis the red path from p w via w and p w to ‘ . In ˆ C , the bisim-ilar pair w , w progresses to the bisimilar pair p w , p w . Theconnect- p w -through-to- p w chart on the right is a LLEE-chart,as witnessed by the entry/body-labeling ˆ C p p w q p w . C p w q w p w p w ‘ w ˆ C ‘ r s p w w r s p w w ‘ ˆ C p p w q p w r s p w w C p w q w ÐSS C p I q p p w q p w This illustrates that bisimilar pairs of vertices must be se-lected carefully, to safeguard that the connect-through con-struction preserves LLEE. The proposition below expressesthat a pair of distinct bisimilar vertices can always be se-lected in one of three mutually exclusive categories. Later,three LLEE-preserving transformations I, II, and III will bedefined for each of these categories.
Proposition 6.4.
If a
LLEE -chart C is not a bisimulationcollapse, then it contains a pair of bisimilar vertices w , w that satisfy, for a LLEE -witness of C , one of the conditions:(C1) ␣p w Ñ ˚ w q ^ p ñ w ñ w is not normed q ,(C2) w ü ` w ,(C3) D v P V ` w d ü v ^ w ü ` v ˘ ^ ␣p w Ñ ˚ bo w q . Condition (C1) requires that w and w are in differentscc’s, as there is no path from w to w . The additional provisoin (C1) constrains the pair in such a way that if both arenormed, then w must be outside of all loops (otherwisethe connect- w -through-to- w operation does not preserveLLEE-charts, see Ex. 6.3); its asymmetric formulation helps toavoid the assumption of bisimilarity in Prop. 6.8 below. The two other conditions concern the situation that w and w are in the same scc. While in (C2) w and w are comparable(but different) by the loops-back-to relation ü ˚ , they areincomparable in (C3). In the situation that w , w loop backto the same vertex v , but w directly loops back to v , (C3)also demands that no body step path exists from w to w (otherwise the connect- w -through-to- w construction doesnot preserve LLEE-charts, see an example in the appendix).In the proof of Prop. 6.4 we progress, from a given pairof distinct bisimilar vertices, repeatedly via transitions, atone side picking loop-back transitions, over pairs of distinctbisimilar vertices, until one of the conditions (C1) , (C2) ,(C3) is met. We will use a subset of the body transitions in aLLEE-witness. By a loop-back transition , written as u Ñ lb v ,we mean a transition u Ñ bo v that stays within an scc, thatis, scc p u q “ scc p v q . The loops-back-to norm ∥ u ∥ min lb of u is themaximal length of a Ñ lb path from u (which is well-definedby Lem. 3.9, (i) and chart finiteness). Note that ∥ u ∥ min lb “ u does not loop back (denoted by ␣p u ü q ). Proof of Prop. 6.4.
We pick distinct bisimilar vertices u , u .First we consider the case scc p u q ‰ scc p u q . Without lossof generality, suppose ␣p u Ñ ˚ u q . We progress to a pairof vertices where (C1) holds, using induction on ∥ u ∥ min lb .In the base case, ∥ u ∥ min lb “
0, it suffices to show that it isnot possible that both ñ u holds and u is normed, becausethen we can define w “ u and w “ u , and are done.Therefore suppose, toward a contradiction, that ñ u holdsand u is normed. Then u is normed, too, since u and u arebisimilar. Also ␣p u ü q follows from ∥ u ∥ min lb “
0, whichsays that there are no loops-back-to steps from u . So weget that ñ u , ␣p u ü q , and u is normed. This contradictsLemma 3.9, (iii). In the induction step, ∥ u ∥ min lb ą u Ñ lb u and (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb for some u . Since u Ø u ,we have u Ñ u and u Ø u for some u . Since u Ñ lb u ,by definition, u and u are in the same scc. Hence u Ñ ˚ u .This implies ␣p u Ñ ˚ u q , for else u Ñ u Ñ ˚ u Ñ ˚ u ,which contradicts the assumption ␣p u Ñ ˚ u q . Since u Ø u and ␣p u Ñ ˚ u q and (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb , by inductionthere exists a bisimilar pair w , w for which (C1) holds.Now let scc p u q “ scc p u q . Then by Lem. 3.9, (iv), u ü ˚ v and u ü ˚ v for some v . By Lem. 3.9, (v) we pick v as theleast upper bound of u , u with regard to ü ˚ . If u “ v , then u ü ` u , so (C2) holds for w “ u and w “ u . If u “ v ,then likewise (C2) holds for w “ u and w “ u . Now let u , u ‰ v . Since v is the least upper bound, u ü ˚ v d ü v d ý v ý ˚ u for distinct v , v P V . There cannot be acycle of body transitions, so ␣p v Ñ ˚ bo v q or ␣p v Ñ ˚ bo v q .By symmetry it suffices to consider ␣p v Ñ ˚ bo v q . Summa-rizing, u ü ˚ v d ü v d ý v ý ˚ u and ␣p v Ñ ˚ bo v q . Forthis situation we use induction on ∥ u ∥ min lb . If u “ v , then u d ü v ; taking w “ u and w “ u , (C3) holds. So we canassume u ü ` v d ü v . Pick a transition u Ñ lb u with eport version Clemens Grabmayer and Wan Fokkink (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb ; by definition, scc p u q “ scc p u q . Since u Ø u , there is a transition u Ñ u with u Ø u for some u . If scc p u q ‰ scc p u q , then as before we can find bisimilar w , w for which (C1) holds. Now let scc p u q “ scc p u q , so u , u , u , u are in the same scc. Since u ü ` v and u Ñ u ,either u “ v or v ñ ` u . Moreover, scc p u q “ scc p u q “ scc p v q , so by Lem. 3.9, (ii), u ü ˚ v . Since u ü ˚ v , wecan distinguish two cases (for illustrations for each of thesubcases, see the appendix). Case 1: u ü ` v . Since u Ñ u , either u “ v or v ñ ` u . Moreover, scc p u q “ scc p u q “ scc p v q , soby Lem. 3.9, (ii), u ü ˚ v . Hence, u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q , and (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb .We apply the induction hypothesis to obtain a bisim-ilar pair w , w for which (C1) , (C2) , or (C3) holds.Below we illustrate both of the cases in which u Ñ u is a loop-entry transition, or a body transition. vv v { bo u u lb u u u r α s bo u s e i n d . h y p . u s e i n d . h y p . Case 2: u “ v . We distinguish two cases. Case 2.1: u Ñ r α s u . Then either u “ u or u ñ ` u . Moreover, scc p u q “ scc p u q , so by Lem. 3.9, (ii), u ü ˚ u , and hence u ü ˚ v . Thus we have ob-tained u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q .Due to (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb , we can apply the inductionhypothesis again. Case 2.2: u Ñ bo u . Then ␣p v Ñ ˚ bo v q together with v “ u Ñ bo u and u Ñ ˚ bo v (because u ü ˚ v )imply u ‰ u . We distinguish two cases. Case 2.2.1: u “ v . Then u ü ˚ v d ü v “ u , i.e., u ü ` u , so we are done, because (C2) holds for w “ u and w “ u . Case 2.2.2: u ‰ v . By Lem. 3.9, (ii), u ü ` v . Hence, u ü ˚ v d ü v for some v . Since v “ u Ñ bo u ü ˚ v and ␣p v Ñ ˚ bo v q , it follows that ␣p v Ñ ˚ bo v q .So u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q .Due to (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb , we can apply the inductionhypothesis again.This exhaustive case analysis concludes the proof. □ Now we define, for LLEE-witnesses ˆ C of a LLEE-chart C ,and for bisimilar vertices w , w in C , in each of the threecases (C1) , (C2) , or (C3) of Prop. 6.4 a transformation of ˆ C into an entry/body-labeling of the connect- w -through-to- w chart C p w q w that can be shown to be a LLEE-witness again.We number the transformations for (C1) , (C2) , and (C3) as I , II , and III , respectively. Each transformation makes useof the connect-through construction for entry/body-labelings as defined in Def. 6.1. Additionally, in each transformationan adaptation of labels of transitions is performed, to avoidviolations of LLEE-witness properties. In transformationsI and III the adaptation is performed before connecting w through to w , and is needed to guarantee that layerednessis preserved; in transformation II it is performed right aftereliminating w , and avoids the creation of body step cycles.The level adaptations for the three transformations are: L I Let m “ max t β : there is a path w Ñ ˚ ¨ Ñ r β s in ˆ C u .In loop-entry transitions u Ñ r α s v for which there is apath v Ñ ˚ w in C , replace α by an α with α “ α ` m .This increases the labels of loop-entry transitions thatdescend to w in ˆ C to a higher level than the loop labelsreachable from w . L II Since w ü ` w , there exists a p w with w ü ˚ p w d ü w . Let γ be the maximum loop level among the loop-entries at w in ˆ C . (Note that since w ü ` w , there isat least one such transition.) Turn the body transitionsfrom p w into loop-entry transitions with loop label γ . L III
Let γ be a loop label of maximum level among the loop-entry transitions at v in ˆ C . (Note that since w ü v ,there is at least one such transition.) Turn the looplabels of the loop-entry transitions from v into γ .Each of these transformations ends with a clean-up step : ifthe loop-entry transitions from a vertex with the same looplabel no longer induce an infinite path (due to the removalof w ), then they are changed into body transitions. Example 6.5.
The LLEE-witness on the left in Fig. 4 is re-duced in three transformation steps to a LLEE-witness ofthe chart C p e q in Ex. 2.5. Broken lines are between bisimilarvertices. In step one, a transformation I, the start state v isconnected through to the bisimilar vertex v , whereby v becomes the start vertex; note that there is no path from v to v , and no vertex descends into a loop to v . In step two,a transformation II, v is connected through to the bisimilarvertex v ; note that v ü ` v . In step three, a transformationIII, the start vertex v is connected through to the bisimi-lar vertex v , whereby v becomes the start vertex; notethat v d ü v and v ü ` v and there is no body steppath from v to v . By the loop level adaptation L III , all loopentries from v get level 3. The final step is an isomorphicdeformation. Only the left and right charts depict actions. Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version a v r s a r s c v a v r s b b v a v r s c a v a v r sr s v v r s v v r s v v p I q p v q v r s v r s v r s v v p II q p v q v r s v r s r s v v p III q p v q v v a v a r s c v r s b r s b Figure 4.
Three connect-through-steps according to the transformations I, II, and III from the LLEE-witness on the left, and afinal isomorphic deformation, leading to the LLEE-witness on the right. For clarity, we neglected action labels in the middle.The following examples provide more illustrations of thetransformations II and III. Similarly as Ex. 6.3 does so fortransformation I and (C1), they also show that the conditions(C2) and (C3) mark rather sharp borders between whether,on a given LLEE-witness, a connect-through operation ispossible while preserving LLEE, or not.
Example 6.6.
For the LLEE-witness ˆ C below in the middle,the chart C p w q w on the left has no LLEE-witness. C p w q w ÐSS C p II q p w q w w p w u C p w q w w r s r s p w r s uw ˆ C p w r s r s uw ˆ C p w q w It does not satisfy LEE: it has no loop subchart, since fromeach of its three vertices an infinite path starts that doesnot return to this vertex; from p w this path, drawn in red,cycles between u and w . Transformation II applied to thepair w , w (instead of w , w ) in ˆ C yields the entry/body-la-beling ˆ C p w q w where p w Ñ bo w is turned into p w Ñ r s w . Asthe pair w , w satisfies (C2) , the proof of Prop. 6.8 ensuresthat this labeling, drawn on the right, is a LLEE-witness. Example 6.7.
In the LLEE-witness ˆ C below in the middle, w , w ü ` v and there is no body step path from w to w ,but (C3) does not hold for the pair w , w due to ␣p w d ü v q .The chart C p w q w on the left has no LLEE-witness. It does notsatisfy LEE: the downwards loop-entry transition from p w can be eliminated, and then two more arising loop-entrytransitions from v ; the remaining chart of solid arrows hasno further loop subchart, because from each of its verticesan infinite path starts that does not return to this vertex.In ˆ C , loop-entry transitions from v have the same looplabel, so the preprocessing step of transformation III is void.The bisimilar pair w , w progresses to the bisimilar pair p w , p w in ˆ C , for which (C3) holds because p w d ü v ý p w and ␣p p w Ñ ˚ bo p w q . Transformation III applied to this pairyields the labeling ˆ C p p w q p w on the right. In the proof of Prop.6.8 it is argued that this is guaranteed to be a LLEE-witness.The remaining two bisimilar pairs can be eliminated by oneor by two further applications of transformation III. v C p w q w p w p w w v ˆ C p w r sr s r sr sr s w p w r s w v ˆ C p p w q p w r sr sr s w p w r s w C p w q w ÐSS C p III q p p w q p w Proposition 6.8.
Let C be a LLEE-chart. If a pair x w , w y ofvertices satisfies (C1), (C2), or (C3) with respect to a LLEE -wit-ness of C , then C p w q w is a LLEE -chart.Proof.
Let ˆ C be a LLEE-witness. For vertices w , w such that(C1) , (C2) , or (C3) holds, transformation I, II, or III, respec-tively, produces an entry/body-labeling ˆ C p w q w . We prove fortransformation I that this is a LLEE-witness, and refer to theappendix with regard to transformations II, and III. eport version Clemens Grabmayer and Wan Fokkink We first argue it suffices to show that each of the transfor-mations produces, before the final clean-up step, a labelingthat satisfies the LLEE-witness conditions, except possible vi-olations of loop property (L1) in (W2)(a). Such violations canbe removed from a loop-labeling while preserving the otherLLEE-witness conditions. To show this, suppose (L1) is vio-lated in some C ˆ C p u , α q . Then u Ñ r α s but ␣p u Ñ r α s ¨ Ñ ˚ bo u q .Let ˆ C be the result of removing this violation by chang-ing the α -loop-entry transitions from u into body transi-tions. No new violation of (L1) is introduced in ˆ C . (W1) and(W2)(a), (L2), are preserved in ˆ C because an introduced in-finite body step path in ˆ C would be a body step cycle thatstems from a path u Ñ r α s u Ñ ˚ bo u in ˆ C . (W2)(b) mightonly be violated by a path w ÝÝÝÑ t p w q r β s ¨ ÝÝÝÑ t p w q ˚ bo u ÝÝÝÝÑ t p w , u q bo u ÝÝÝÝÑ t p w , u q ˚ bo ¨ Ñ r γ s with β ď γ in ˆ C where u Ñ bo u stemsfrom u Ñ r α s u in ˆ C ; then β ą α ą γ by layeredness of ˆ C ; so(W2)(b) is preserved. Analogously we find that also (W2)(a),(L3) is preserved, because ‘ is never in C ˆ C p u , α q .To show the correctness of transformation I, consider ver-tices w and w with (C1) . We show that the result ˆ C p w q w oftransformation I before the clean-up step satisfies the LLEE-witness properties, except for possible violations of (L1).To verify (W1) and part (L2) of (W2)(a), it suffices to showthat ˆ C p w q w does not contain body step cycles. The originalloop-labeling ˆ C is a LLEE-witness, so it does not containbody step cycles. Since the level adaptation step does notturn loop-entry steps into body steps, body step cycles couldonly arise in the step connecting w through to w . Supposesuch a body step cycle arises. Then there must be a transition u Ñ bo w in ˆ C (which is redirected to w in ˆ C p w q w ) and apath w Ñ ˚ bo u in ˆ C . But then w Ñ ˚ bo u Ñ bo w in C , whichcontradicts (C1) that there is no path from w to w . Hence(W1) and part (L2) of (W2)(a) hold for ˆ C p w q w .Now we verify part (L3) of (W2)(a) in ˆ C p w q w . Consider apath u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w in ˆ C . Then u ‰ w , and u ñ w .It suffices to show that then ␣p w Ñ ` ‘ q in C . But this isguaranteed, because otherwise w were normed, and due to u ñ w we would have a contradiction with condition (C1) .Finally we show that (W2)(b) is preserved in ˆ C p w q w byboth the level adaptation and the connect-through step. First,since in the level adaptation step all adapted loop labelsare increased with the same value m , a violation of (W2)(b)would arise by a path u Ñ r α s ¨ Ñ ˚ bo ¨ Ñ r β s v in ˆ C where looplabel β is increased while α is not. But such a path cannotexist. Since β is increased, there is a path v Ñ ˚ w in C .But then there is a path u Ñ r α s ¨ Ñ ` v Ñ ˚ w in ˆ C , whichimplies that also α is increased in the level adaptation step.Second, a violation of (W2)(b) in the connect-through stepwould arise from paths u Ñ r α s ¨ Ñ ˚ bo w and w Ñ ˚ bo ¨Ñ r β s inˆ C with α ď β . However, in view of the path u Ñ r α s ¨ Ñ ˚ w , the loop label α was increased with m in the level adaptationstep . On the other hand, in view of (C1) that there is no pathfrom w to w in C , w is unreachable at the end of the path w Ñ ˚ ¨ Ñ r β s . Hence this loop label β was not increased inthe level adaptation step. So it is guaranteed that for such apair of paths in ˆ C p w q w always α ą β .We conclude that the result of transformation I is again aLLEE-witness. □ Theorem 6.9.
The bisimulation collapse of a
LLEE -chart isagain a
LLEE -chart.Proof.
Given a LLEE-chart C , repeat the following step: basedon a LLEE-witness pick, by Prop. 6.4, bisimilar vertices w and w with (C1), (C2), or (C3), and then connect w throughto w , obtaining by Prop. 6.8 a LLEE-chart bisimilar to C ,due to Lem. 6.2. Hence the bisimulation collapse of C , whichis reached eventually, is a LLEE-chart. □ We mention that by using a refinement of the interpreta-tion TSS (that avoids creating concatenations e ¨ e where e is not normed, in favor of using just e ) and a refinementof the extraction procedure (that ensures an eager use of theright distributive law (B4) of ¨ over ` ) this theorem can bestrengthened: the bisimulation collapse of a LLEE-chart isthe chart interpretation of some star expression. which thenis a LLEE-chart by Prop. 2.9. This can be proved by showingthat, on collapsed LLEE-charts, (refined) chart interpretationis the converse of (refined) solution extraction. Corollary 6.10.
If a chart is expressible by a star expressionmodulo bisimilarity, then its collapse is a
LLEE -chart.
The converse statement holds as well. But this corollarydoes not hold for star expressions with 1 and unary star. Forexample, with respect to the TSS for the process interpreta-tion of star expressions from this class, see e.g. [3], the expres-sion e : “ ppp ¨ a ˚ q ¨ p b ¨ c ˚ qq ¨ e with e : “ p a ˚ ¨ p b ¨ c ˚ qq ˚ has the following interpretation, where e : “ p ¨ c ˚ q ¨ e : e a b e bca This is a chart in the extended sense in which immediatetermination is permitted at arbitrary vertices. It is a bisimu-lation collapse that does not satisfy LEE, taking into accountthat in the definition of ‘loop’ for charts in the extended sense(L3) needs to be changed to exclude immediate terminationfor vertices in a loop chart other than the start vertex.
That bisimulation collapse preserves LLEE was the last build-ing block in the proof of the desired completeness result.
Theorem 7.1.
The proof system
BBP is complete with respectto the bisimulation semantics of star expressions, that is, with Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version respect to bisimilarity of charts that interpret star expressionswithout and with binary Kleene star f .Proof. The proof steps were already explained in Sect. 4. □ Example 7.2.
The bisimilar LLEE-charts C and C in Ex. 4.1have p a ¨ p a ` b q ` b q f p b ¨ p a ` b q ` a q f C has prin-cipal solution p a ` b q f
0. Then p a ¨ p a ` b q ` b q f “ BBP p a ` b q f “ BBP p b ¨ p a ` b q ` a q f Example 7.3.
Revisiting the star expressions e , e in Ex. 2.5with bisimilar chart interpretations C p e q and C p e q , we canapply our proof in order to show that e “ BBP e . C p e q and C p e q have provable solutions with principal values e and e by Prop. 2.9. As C p e q and C p e q are LLEE-charts by Prop. 3.7with LLEE-witnesses z C p e q and z C p e q , their bisimulation col-lapse C is a LLEE-chart by Thm. 6.9. We take here the morefamiliar ˆ C , but could also take the one obtained in Fig. 4.We saw in Fig. 2 that ˆ C has a provable solution with princi-pal value s ˆ C p v q “ a ¨ pp c ¨ a ` a ¨ p b ` b ¨ a qq f q . Then byProp. 5.1 and Prop. 5.8 it follows that e “ BBP s ˆ C p v q “ BBP e . r s a r s ac bb a C p e q , z C p e q a r s a r s cb b C , ˆ C a r s a r s ca r s b ba r s c aa C p e q , z C p e q We have shown that Milner’s axiomatization, tailored tostar expressions without 1 and with f , is complete in bisimu-lation semantics. At the core of our proof is the graph struc-ture property LLEE, which characterizes the process graphsthat can be expressed by star expressions without 1 and with f as charts whose bisimulation collapse is a LLEE-chart.Completeness of BBP covers completeness of the the-ory
BPA ω ` RSP ω of perpetual loop iteration p¨q ω [10] in thesense that the latter result can be shown by our means, or bya faithful interpretation e ω ÞÑ e f BPA ω ` RSP ω in BBP .Completeness of
BBP can be extended, also by means ofa faithful interpretation, to cover star expressions with 0,1, and ˚ , but with a syntactic restriction on terms directlyunder a ˚ : that they can be rewritten to star expressions withonly ’harmless’ occurrences of 1. This is analogous to thesituation that the completeness result from [9, 11] for starexpressions without 0 and 1 and with f was extended in [7]to a setting with 1 (but not 0) and ˚ , where a generalizedversion of the non-empty-word property is disallowed for terms directly under a ˚ . With the interpretation approach,also the result in [7] can be obtained from the one in [9, 11].The main future goal is to solve Milner’s problem entirelyby extending our result to the full class of star expressions. Acknowledgments
We thank Alban Ponse for his suggestion to consider com-pleteness of Milner’s axiomatization for the fragment with-out 1, and Luca Aceto for encouragement, comments on theexposition, and facilitating a visit of the second author toGSSI, from which this paper developed. Also, we thank thereviewers for detailed remarks and suggestions on how toimprove the positioning of our completeness result.
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A Appendix: supplements, more proof details, and omitted proofs
A.1 Proofs in Section 2: PreliminariesProposition (= Proposition 2.9, uses
BBP -axioms (B1)–(B7), (BKS1)) . For every e P StExp p A q , the identity function id V p e q : V p e q Ñ V p e q Ď StExp p A q , e ÞÑ e , is a provable solution of the chart interpretation C p e q of e . In the proof of this proposition we will use the following definition concerning ‘action derivatives’, and the subsequentlemma. That statement can be viewed as the ‘fundamental theorem of differential calculus for star expressions’ which says thatevery star expressions can be reassembled by a form of ‘integration’ from its action derivatives. In this context ‘differentiation’follows the definition of action derivatives in Definition 2.4 (corresponding to Antimirov’s concept of ‘partial derivative’ in[2]), and ‘integration’ means sum formation over products of pairs x a , ξ y for actions a and a -derivatives ξ . Definition A.1.
For star expressions e P StExp p A q we define the set A Bp e q of action derivatives of e as follows: A Bp e q : “ ␣ x a , ξ y ˇˇ a P A , ξ P StExp p A q ‘ , e a ÝÑ ξ ( . Lemma A.2.
Every e P StExp p A q can be provably reassembled from its action derivatives as: e “ BBP ´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯ , (A.1) provided that A Bp e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y ( . (A.2) Proof.
We start by noting that we need to show (A.1), for all e P StExp p A q , only for one list representation of A Bp e q of the form(A.2). This is because then (A.1) follows also for all other list representations of A Bp e q the form (A.2). Indeed, the axioms (B1),(B2), and (B3) of BBP (the
ACI -axioms for associativity, commutativity, and idempotency of ` ) can be used to permute andduplicate summands as well as to remove duplicates of summands in sums (A.1) according to permutations, duplications, andremoval of duplicates in list representations of A Bp e q of the form (A.2).We proceed by induction on the structure of star expressions in StExp p A q . For performing the induction step, we distinguishthe five cases of productions in the grammar in Definition 2.1. Case 1: e ” e does not enable any transitions, and hence A Bp e q “ ∅ . We find the provable equality: e ” “ BBP ` BBP ) . This is of the form as in (A.1) with m “ n “ A Bp e q “ ∅ as a list representation of the form (A.2). Case 2: e ” a for some a P A .Then according to the TSS in Definition 2.4 the expression e enables precisely one transition, an a -transition to ‘ . Hencethe set of action derivatives of e consists only of one element: A Bp e q “ tx a , ‘ yu . (A.3)We find the provable equality: e “ BBP a ` BBP ) . The right-hand side is of the form (A.1) with m “ a “ a and n “ A Bp e q as alist representation of the form (A.2). Case 3: e ” e ` e .Since every star expression has only finitely many derivatives, each of which is either ‘ or a star expression, we mayassume that the sets of action derivatives of the constituent expressions e and e of e ` e have list representations: A Bp e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y ( , A Bp e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y ( . (A.4)Then it follows from the form of the TSS rules in Definition 2.4 concerning sums of star expressions that the sets of actionderivatives of e ` e is the union of the sets of action derivatives of e , and of e . By permuting the action derivativeswith tick to the front, this union has the list representation: A Bp e ` e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y , x b , e y , . . . , x b n , e n y ( . (A.5) eport version Clemens Grabmayer and Wan Fokkink Now we can argue as follows to reassemble e ` e from its action derivatives: e ” e ` e “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ` ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ (by the induction hypothesis, using representation (A.5)) “ BBP ´´´ m ÿ i “ a i ¯ ` ´ m ÿ i “ a i ¯¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ` ´ n ÿ j “ b j ¨ e j ¯ . (by axioms (B2) and (B1))Since ACI is a subsystem of
BBP , this chain of provably equalities is one in
BBP . It demonstrates, together withapplications of the axiom (B2) that are needed to bring each of the subexpressions of the two outermost summands intoa form with association of summation subterms to the left, that e satisfies (A.1) when we construe A Bp e q in (A.5) as a listrepresentation of the form (A.2) with m “ m ` m and n “ n ` n . Case 4: e ” e ¨ e .As argued in the previous case, we may assume that the action derivatives of e are of the form: A Bp e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y ( . (A.6)Then it follows from the forms of the two rules in the TSS in Definition 2.4 concerning transitions from expressions withconcatenation as their outermost symbol that the set of action derivatives of e ¨ e has a list representation of the form: A Bp e ¨ e q “ ␣ x a , e y , . . . , x a m , e y , x b , e ¨ e y , . . . , x b n , e n ¨ e y ( . (A.7) Case 4.1: m , n ą e ¨ e as follows: e ” e ¨ e “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ¨ e (by the induction hypothesis,using representation (A.6)) “ BBP ´ m ÿ i “ a i ¨ e ¯ ` ´ n ÿ j “ p b j ¨ e j q ¨ e ¯ (by axiom (B4)) “ BBP ´ m ÿ i “ a i ¨ e ¯ ` ´ n ÿ j “ b j ¨ p e j ¨ e q ¯ (by axiom (B5)) “ BBP ` ´´ m ÿ i “ a i ¨ e ¯ ` ´ n ÿ j “ b j ¨ p e j ¨ e q ¯¯ (by axiom (B6))This chain of provable equalities demonstrates, together with applications of the axiom (B2) that are needed to bringeach of the subexpressions of the right outermost summands into a form with association of summation subterms tothe left, that e satisfies (A.1) when we construe A Bp e q in (A.7) as a list representation (A.2) with m “ n “ m ` n . Case 4.2: m ą n “ e ¨ e as follows: e ” e ¨ e “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ¨ e (by the induction hypothesis,using representation (A.6)) “ BBP ´´ m ÿ i “ a i ¯ ` ¯ ¨ e (since n “ “ BBP ´ m ÿ i “ a i ¨ e ¯ ` ¨ e (by axiom B4) “ BBP ´ m ÿ i “ a i ¨ e ¯ ` Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version “ BBP ` ´ m ÿ i “ a i ¨ e ¯ (by axioms (B1) and (B6))This chain of provable equalities demonstrates that e satisfies (A.1) when we construe A Bp e q in (A.7), recalling that n “
0, as a list representation (A.2) with m “ n “ m . Case 4.3: m “ n ą e ¨ e as follows: e ” e ¨ e “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ¨ e (by the induction hypothesis,using representation (A.6)) “ BBP ´ ` ´ n ÿ j “ b j ¨ e j ¯¯ ¨ e (since m “ “ BBP ¨ e ` ´ n ÿ j “ p b j ¨ e j q ¨ e ¯ (by axiom (B4)) “ BBP ` ´ n ÿ j “ p b j ¨ e j q ¨ e ¯ (by axiom (B7)) “ BBP ` ´ n ÿ j “ b j ¨ p e j ¨ e q ¯ (by axiom (B5))This chain of provable equalities demonstrates that e satisfies (A.1) when we construe A Bp e q in (A.7), recalling that m “
0, as a list representation (A.2) with m “ n “ n . Case 4.4: m “ n “ e ¨ e as follows: e ” e ¨ e “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ ¨ e (by the induction hypothesis,using representation (A.6)) “ BBP ` ` ˘ ¨ e (since m “ n “ “ BBP ¨ e (by axiom (B6)) “ BBP “ BBP ` e satisfies (A.1) when we construe A Bp e q in (A.7), recalling that m “ n “
0, as a list representation (A.2) with m “ n “ Case 5: e ” e f e .As in Case 3 we may assume that the sets of action derivatives of the constituent expressions e and e of e ` e havelist representations of the form (A.4). Then it follows from the forms of the three rules in Definition 2.4 concerningtransitions from expressions with binary iteration as their outermost symbol, that the set of action derivatives of e f e has a list representation of the form: A Bp e f e q “ ␣ x a , e f e y , . . . , x a m , e f e y , x b , e ¨ p e f e qy , . . . , x b n , e n ¨ p e f e qy , x a , ‘ y , . . . , x a m , ‘ y , x b , e y , . . . , x b n , e n y ( . By permuting the action derivatives with tick to the front, this representation can be changed into: A Bp e f e q “ ␣ x a , ‘ y , . . . , x a m , ‘ y , x a , e f e y , . . . , x a m , e f e y , x b , e ¨ p e f e qy , . . . , x b n , e n ¨ p e f e qy , x b , e y , . . . , x b n , e n y ( . ,////.////- (A.8) eport version Clemens Grabmayer and Wan Fokkink Now we argue as follows in order to reassemble e f e from its action derivatives in A Bp e q : e ” e f e (assumption in this case) “ BBP e ¨ p e f e q ` e (by axiom (BKS1)) “ BBP ´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯ ¨ p e f e q ¯ ` ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ (by the induction hypothesis,using representation (A.4)) “ BBP ´´ m ÿ i “ a i ¨ p e f e q ¯ ` ´ n ÿ j “ p b j ¨ e j q ¨ p e f e q ¯¯ ` ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ (by axiom (B4)) “ BBP ´´ m ÿ i “ a i ¨ p e f e q ¯ ` ´ n ÿ j “ b j ¨ p e j ¨ p e f e qq ¯¯ ` ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯ (by axiom (B5)) “ ACI ´ m ÿ i “ a i ¯ ` ´´ m ÿ i “ a i ¨ p e f e q ¯ ` ´´ n ÿ j “ b j ¨ p e j ¨ p e f e qq ¯ ` ´ n ÿ j “ b j ¨ e j ¯¯¯ (by axioms (B2) and (B1))This chain of provably equalities demonstrates, together with applications of the axiom (B2) that are needed to bringeach of the subexpressions of the right outermost summand into a form with association of summation subterms to theleft, that e satisfies (A.1) when we construe A Bp e q in (A.8) as a list representation of the form (A.2) with m “ m and n “ m ` n ` n .In each of these five possible cases concerning the outermost structure of e we have successfully performed the induction step.In this way we have proved the statement of the lemma. □ Proof of Proposition 2.9.
Let C p e q “ x V p e q , ‘ , e , A , T p e qy be the chart interpretation of a star expression e P StExp p A q .Let f P V p e q Ď StExp p A q be a vertex of C p e q . By Lemma A.2 every star expression in StExp p A q can be reassembled as the BBP -provable sum over products of over its action derivatives x a , ξ y , that is, over all actions a P A and a -derivatives ξ of e .In particular, (A.1) guarantees that id V p e q p f q “ f satisfies the condition for id V p e q to be a provable solution at the vertex f of C p e q , relative to a representation (A.2) of the action derivatives of f which corresponds to a representation as assumed inDefinition 2.8. Since f P V p e q was arbitrary in this argument, it follows that id V p e q is a provable solution of C p e q . □ A.2 Proofs in Section 3: Layered loop existence and eliminationProposition (= Proposition 3.7) . For every e P StExp p A q , the entry/body-labeling y C p e q of C p e q is a LLEE -witness of C p e q .Proof. To verify (W1) it suffices to show that there are no infinite body step paths from any star expression e (this is alsoa preparation for (W2)(a), part (L2)). We prove, by induction on the syntactic structure of e , the stronger statement that if e Ñ ` f , then there does not exist an infinite body step path from f . The base cases, in which e is of the form a or 0, aretrivial. Suppose e ” e ` e . Then e i Ñ ` f for some i P t , u . So by induction, f does not exhibit an infinite body step path.Suppose e ” e ¨ e . Then e Ñ ` f means either e Ñ ` f and f ” f ¨ e , or e Ñ ˚ f . In the first case, by induction, f and e do not exhibit infinite body step paths. This induces that f ¨ e does not exhibit an infinite body step path. In the secondcase, by induction, f does not exhibit an infinite body step path. Suppose e ” e f e . Then e Ñ ` f means (A) f ” e f e , or (B) e Ñ ` f and f ” f ¨ p e f e q , or (C) e Ñ ` f . In case (A), each body step path from f starts with either f ÝÑ bo e ¨ p e f e q where e ÝÑ e and e is not normed, or f ÝÑ bo e where e ÝÑ e . In the first case, by induction, e does not exhibit an infinitebody step path, so since e is not normed, e ¨ p e f e q does not exhibit an infinite body step path. In the second case, byinduction, e does not exhibit an infinite body step path. In case (B), since by induction f and by case (A) e f e do not exhibitinfinite body step paths, f ¨ p e f e q does not exhibit an infinite body step path. In case (C), by induction, f does not exhibitan infinite body step path.We verify (W2). From the TSS-rules in Definition 2.4 it follows that if e has a loop-entry transition, then e ” pp¨ ¨ ¨ pp e f e q ¨ f q ¨ ¨ ¨ q ¨ f n q for some n ě e normed. Let ˆ C denote the entry/body-labeling defined by the TSS-rules in Definition 2.4on the ‘free’ (= start-vertex free) chart of all star expressions in StExp p A q . We prove (W2) for a subchart C ˆ C p e , α q of ˆ C . We firstconsider the case n “
0, and then generalize it. Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
Let e ” e f e with e normed, and α “ | e | f `
1. Either e Ñ r α s e or e Ñ r α s e ¨ e for some normed e with e Ñ e .In the first case (L1) is clearly satisfied; we focus on the second case. It can be argued, by induction on syntactic structure,that every normed star expression has a body step path to ‘ . Then so does e . This means e ¨ e has a body step path to e . Hence (L1) holds. For the remainder of (W2) it suffices to consider loop-entry transitions e Ñ r α s e ¨ e where e Ñ e .Since we showed above there are no body step cycles, every body step path from e eventually leads to deadlock or ‘ ; inthe first case the corresponding body step path of e ¨ e also deadlocks, and in the second case it returns to e . Hence (L2)holds. Since e ¨ e cannot reach ‘ without returning to e , (L3) holds. It can be shown, by induction on derivation depth, that f ÝÑ f implies | f | f ě | f | f , and clearly f ÝÑ r β s implies β ď | f | f . So if e Ñ ˚ ¨ Ñ r β s , then β ď | e | f ď | e | f . Hence, if e ¨ e ÝÝÝÝÝÑ t p e f e q ˚ bo ¨ Ñ r β s , then β ă | e | f ` “ α . So (W2) p b q holds.Now consider e ” pp¨ ¨ ¨ pp e f e q ¨ f q ¨ ¨ ¨ q ¨ f n q for n ą
0, with e normed. Again α “ | e | f `
1. The subchart C ˆ C p e , α q basically coincides with C ˆ C p e f e , α q , except that the star expressions in the first chart are post-fixed with f , . . . , f n ; itstransitions are derived by n additional applications of the first rule for concatenation in Definition 2.4, to affix these expressions.This chart isomorphism between C ˆ C p e f e , α q and C ˆ C p e , α q preserves action labels as well as the loop-labeling, because the firstrule for concatenation preserves these labels. We showed that C ˆ C p e f e , α q satisfies (W2), so the same holds for C ˆ C p e , α q . □ We now turn to the proof of Lemma 3.9, which expresses properties of the body transition relation Ñ bo , the descends-in-loop-to relation ñ , the loops-back-to relation ü , and the directly-loops-back-to relation d ü . Lemma (= Lemma 3.9) . The relations Ñ bo , ñ , ü , d ü as defined by a LLEE -witness ˆ C on a chart C satisfy the followingproperties:(i) ˆ C does not have infinite Ñ bo paths (so no Ñ bo cycles).(ii) If scc p u q “ scc p v q , then u ñ ˚ v implies v ü ˚ u .(iii) If v ñ w and ␣p w ü q , then w is not normed.(iv) scc p u q “ scc p v q if and only if u ü ˚ w and v ü ˚ w for some vertex w .(v) ü ˚ is a partial order that has the least-upper-bound property: if a nonempty set of vertices has an upper bound with respectto ü ˚ , then it has a least upper bound.(vi) ü is a total order on ü -successor vertices: if w ü v and w ü v , then v ü v or v “ v or v ü v .(vii) If v d ü u and v d ü u for distinct v , v , then there is no vertex w such that both w ü ˚ v and w ü ˚ v . We split the proof into the arguments for the parts (i)–(vii), respectively. In doing so we repeat these statements as individuallemmas, and add a few more on the way.
Lemma A.3.
In a chart with a
LLEE -witness, if v α ñ ¨ ñ ˚ ¨ Ñ r β s , then α ą β .Proof. By induction on the number n of ñ -steps in a path v α ñ ¨ ñ n ¨ Ñ r β s . If n “
0, then from v α ñ ¨ Ñ r β s we get α ą β bymeans of the LLEE-witness condition (W2)(b). If n ą
0, then the path v α ñ ¨ ñ n ¨ Ñ r β s is of the form v α ñ ¨ ñ n ´ ¨ γ ñ ¨ Ñ r β s for some loop name γ . This path contains an initial segment v α ñ ¨ ñ n ´ ¨ Ñ r γ s . Then α ą γ follows by the inductionhypothesis. From the part γ ñ ¨ Ñ r β s of this path we get γ ą β by LLEE-witness condition (W2)(b). So we conclude that α ą β holds. □ Lemma A.4.
In a chart with a
LLEE -witness, if v ñ ` w , then w ‰ ‘ .Proof. Let ˆ C be a LLEE-witness of a chart C . It suffices to show that ñ w implies w ‰ ‘ . For this, we let v and w be verticessuch that v ñ w . Then we can pick α P N ` such that v α ñ w . Since this means v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ bo w , it follows that w P C ˆ C p v , α q . Now since C ˆ C p v , α q is a loop chart by condition (W2)(a) for the LLEE-witness ˆ C , it follows that w ‰ ‘ . □ Lemma A.5.
In a chart with a
LLEE -witness (assumed to be start-vertex connected, see Definition 2.2), every vertex is reachableby an acylic Ñ ˚ bo ¨ ñ ˚ path from the start vertex v s , that is, v s Ñ ˚ bo ¨ ñ ˚ w holds for all vertices w .Proof. Let π be a path from v s to w . By removing cycles from π we obtain an acyclic path π from v s to w that consistsof a sequence of loop-entry and body transitions. Hence π is of the form v s Ñ ˚ bo w or v s Ñ ˚ bo u ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p u q ˚ bo u ÝÝÝÑ t p u q r α s ¨ ÝÝÝÑ t p u q ˚ bo ¨ ¨ ¨ ÝÝÝÑ t p u q r α s ¨ ÝÝÝÝÝÑ t p u n ´ q ˚ bo u n ” w for some n P N , and α , . . . , α n P N ` , where the target-avoidanceparts are due to acyclicity of π . Hence π is of the form v s Ñ ˚ bo u α ñ ¨ α ñ ¨ ¨ ¨ α n ´ ñ ¨ α n ´ ñ w , for some n P N , and α , . . . , α n P N ` , and therefore of the form v s Ñ ˚ bo ¨ ñ ˚ w . □ eport version Clemens Grabmayer and Wan Fokkink Lemma A.6.
In a chart with a
LLEE -witness, for every path v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ w there is an acyclic path v α ñ ¨ ñ ˚ w .Proof. Let π be a path from v to w that starts with a loop-entry step with loop name α such that all targets of transitions in π avoid v . By removing cycles we obtain an acyclic path π from v to w that starts with an α -loop-entry step whose targetis not v . We can write π as a sequence of loop-entry and body steps of the form v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ bo u ÝÝÝÑ t p u q r α s ¨ ÝÝÝÑ t p u q ˚ bo ¨ ¨ ¨ u n ´ ÝÝÝÝÝÑ t p u n ´ q r α n ´ s ¨ ÝÝÝÝÝÑ t p u n ´ q ˚ bo u n ´ ÝÝÝÝÝÑ t p u n ´ q r α n ´ s ¨ ÝÝÝÝÝÑ t p u n ´ q ˚ bo w for some n ě
1, where the target-avoidance parts aredue to acyclicity of π . Hence π is of the form v α ñ ¨ α ñ ¨ ¨ ¨ α n ´ ñ ¨ α n ´ ñ w , and therefore of the form v α ñ ¨ ñ ˚ w . □ The following lemma was also used implicitly in the proof of Lem. 3.9, (v).
Lemma A.7.
In a chart with a
LLEE -witness, if v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ ¨ Ñ r β s , then α ą β .Proof. This is a direct consequence of Lem. A.6 and Lem. A.3. □ Lemma A.8.
In a chart with a
LLEE -witness, if u ü ˚ v ü ˚ w , then each path u Ñ ˚ bo w visits v .Proof. Let v ‰ u , w , as else the lemma trivially holds. Since u ü ` v ü ` w , there is a path w ÝÝÝÑ t p w q r α s ¨ÝÝÝÑ t p w q ˚ v ÝÝÝÑ t p v q r β s ¨ÝÝÝÑ t p v q ˚ u .By layeredness, α ą β . A path u ÝÝÝÑ t p v q ˚ bo w would yield v ÝÝÝÑ t p v q r β s ¨ ÝÝÝÑ t p v q ˚ u ÝÝÝÑ t p v q ˚ bo w Ñ r α s . Then layeredness would require β ą α , which cannot be the case. □ Lemma (= Lemma 3.9, (i)) . In a chart with a
LLEE -witness, there are no infinite Ñ bo paths (so no Ñ bo cycles).Proof. Let C be a chart with LLEE-witness ˆ C , and with start vertex v s . Due to Lemma A.5 every vertex of v is reachableby a Ñ ˚ bo ¨ ñ ˚ path from v s . In order to show that there are no infinite Ñ bo paths in ˆ C it therefore suffices to show that if v s Ñ ˚ bo ¨ ñ n v , then there is no infinite Ñ bo path from v .For the base case, n “
0, let w be such that v s Ñ ˚ bo w . Now suppose that there is an infinite Ñ ˚ bo path from w in ˆ C . Then dueto v s Ñ ˚ bo w it follows that there is also an infinite Ñ ˚ bo path from v s in ˆ C . This, however, contradicts with the condition (W1)that the LLEE-witness ˆ C must satisfy. We conclude that there is no infinite Ñ ˚ bo path from w in ˆ C .For performing the induction step from n to n `
1, let w be such that v s Ñ ˚ bo ¨ ñ n ` w . Then we can pick w with v s Ñ ˚ bo ¨ ñ n w ñ w . It follows that w ÝÝÝÑ t p w q r α s ¨ ÝÝÝÑ t p w q ˚ bo w for some α P N ` , which we pick accordingly. Now supposethat there is an infinite Ñ ˚ bo path π from w in ˆ C . Then it cannot be the case that π avoids w forever, because otherwise itwould give rise to an infinite path w ÝÝÝÑ t p w q r α s ¨ ÝÝÝÑ t p w q bo w ÝÝÝÑ t p w q bo w ÝÝÝÑ t p w q bo w ÝÝÝÑ t p w q bo ¨ ¨ ¨ , which is not possible sincethe condition (W2)(a) for the LLEE-witness C implies that C ˆ C p u , α q is a loop chart. Therefore it follows that π must visit v .But then π also gives rise to an infinite Ñ ˚ bo path from w . This, however, contradicts the the statement that the inductionhypothesis guarantees for w due to v s Ñ ˚ bo ¨ ñ n w , namely that there is no infinite Ñ ˚ bo path from w . We have reached acontradiction. Therefore we can conclude that there is no infinite Ñ ˚ bo path π from w in ˆ C . In this way we have successfullyperformed the induction step. □ Lemma (= Lemma 3.9, (ii)) . In a chart with a
LLEE -witness, if scc p u q “ scc p v q , then u ñ ˚ v implies v ü ˚ u .Proof. We prove that u ñ n v implies v ü n u for all n ě
0, by induction on n . The base case n “ u “ v .If n ą u ñ n ´ u ñ v for some u . Clearly scc p u q “ scc p u q “ scc p v q . By induction, u ü n ´ u . Since u ñ v , there is anacyclic path u Ñ r α s ¨ Ñ ˚ bo v . And since scc p u q “ scc p v q , there is an acyclic path v Ñ ˚ bo ¨ Ñ r β s ¨ Ñ ˚ bo ¨ ¨ ¨ ¨ ¨ Ñ r β k s ¨ Ñ ˚ bo u . By(W2)(b), α ą β ą ¨ ¨ ¨ ą β k ą α . This means k “
0, so v Ñ ˚ bo u . This implies v ü u and hence v ü n u . □ Lemma (= Lemma 3.9, (iii)) . If, in a chart with a
LLEE -witness, ñ w and ␣p w ü q , then w is not normed.Proof. We argue indirectly by showing that the negation of the implication in the statement of the lemma leads to a contradiction.For this, suppose that v ñ w and ␣p w ü q hold for some vertices v and w , and that additionally w is normed. From v ñ w and ␣p w ü q we obtain by Lem. 3.9, (ii) that w R scc p v q . Since v ñ w entails v Ñ ˚ w this entails ␣p w Ñ ˚ v q . Now since that w is normed means w Ñ ˚ ‘ , we obtain v ñ ˚ w ÝÝÝÑ t p v q ˚ ‘ , which means v ÝÝÝÑ t p v q r α s ¨ ÝÝÝÑ t p v q ˚ bo w ÝÝÝÑ t p v q ˚ ‘ for some α P N ` .Then it follows from Lemma A.6 that v ñ ` ‘ . This, however, contradicts, Lemma A.4. □ Lemma (= Lemma 3.9, (iv)) . In a chart with a
LLEE -witness, scc p u q “ scc p v q if and only if u ü ˚ w and v ü ˚ w for somevertex w . Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
Proof.
The direction from right to left of the lemma trivially holds; we focus on the direction from left to right. Let scc p u q “ scc p v q . The case u “ v is trivial. Let u ‰ v . Then they are on a cycle, which, since there is no body step cycle, contains aloop-entry transition from some w . Without loss of generality, suppose w ‰ u . Then w ñ ` u , so by Lemma 3.9, (ii), u ü ` w .If w “ v we have v ü ˚ w , and if w ‰ v we can argue in the same fashion that v ü ` w . □ Lemma A.9.
In a chart with a
LLEE -witness, ü ` is irreflexive.Proof. Let ˆ C be a LLEE-witness of a LLEE-chart C . Suppose that w ü ` w holds for some vertex w of C and ˆ C . Then it followsfrom the definition of ü ` that there is a Ñ bo path of non-zero length from w to w itself. But such a Ñ bo cycle in ˆ C is notpossible, as it would give rise to an infinite Ñ bo path in ˆ C , contradicting Lemma 3.9, (i). □ Lemma A.10.
In a chart with a
LLEE -witness, ü ˚ is a partial order.Proof. By definition, ü is transitive–reflexive. Moreover, ü is anti-symmetric, because u ü ` v and v ü ` u for u ‰ v wouldimply u ü ` v and v ü ` u , in contradiction with irreflexivity of ü ` , see Lemma A.9. □ Lemma (= Lemma 3.9, (v)) . In a chart with a
LLEE -witness, ü ˚ is a partial order that has the least-upper-bound property: if anonempty set of vertices has an upper bound with respect to ü ˚ , then it has a least upper bound.Proof. Let C be a chart with a LLEE-witness C . Let the relation ü be defined on C according to ˆ C . ü ˚ is a partial order by Lemma A.10. Since C as a chart is finite, it suffices to show that for each vertex v the set of vertices x with v ü ˚ x is totally ordered with regard to ü ˚ . Let v ü ` u and v ü ` u with u ‰ u . There is a path u ÝÝÝÑ t p u q r α s ¨ ÝÝÝÑ t p u q ˚ v Ñ ` bo u ÝÝÝÑ t p u q r β s ¨ ÝÝÝÑ t p u q ˚ v Ñ ` bo u . Without loss of generality, suppose β ě α . Then layeredness implies that each path v Ñ ` bo u must visit u , so v ÝÝÝÑ t p u q ` bo u Ñ ` bo u . Hence there is a path u ÝÝÝÑ t p u q r β s ¨ ÝÝÝÑ t p u q ˚ v ÝÝÝÑ t p u q ` bo u Ñ ` bo u , whichimplies u ü ` u . □ Lemma (= Lemma 3.9, (vii)) . In a chart with a
LLEE -witness, if v d ü u and v d ü u for distinct v , v , then there is no vertex w such that both w ü ˚ v and w ü ˚ v .Proof. ␣p v ü ` v q and ␣p v ü ` v q , for else the definition of d ü would imply u ü ˚ v or u ü ˚ v , and so v ü ` v or v ü ` v , contradicting irreflexivity of ü ` , see Lemma A.9. In the proof of Lemma 3.9, (v), we furthermore saw that for each w , t x | w ü ˚ x u is totally ordered with regard to ü ˚ , which implies that any such sets cannot contain both v and v . □ A.3 Proofs in Section 5: Extraction of star expressions from, and transferral between, LLEE-chartsProposition (= Proposition 5.1, requires
BBP -axioms (B1), (B2), (B3)) . Let ϕ : V Ñ V be a functional bisimulation betweencharts C and C . Let s : V z t ‘ u Ñ StExp p A q be a provable solution of C . Then s ˝ ϕ : V z t ‘ u Ñ StExp p A q is a provablesolution of C with the same principal value as s .Proof. Let s be a provable solution of C . Let v P V zt ‘ u . Since ϕ is a functional bisimulation between C and C , the forth,back, and termination conditions for the graph of ϕ as a bisimulation hold for the pair x v , ϕ p v qy of vertices. This makes itpossible to bring the sets of transitions T p v q from v in C , and T p ϕ p v qq from ϕ p v q in C into a 1–1 correspondence such that ϕ again relates their targets: T p v q “ ␣ v a i ÝÑ ‘ ˇˇ i “ , . . . , m ( Y ␣ v b j ÝÑ v j ˇˇ j “ , . . . , n ( , (A.9) T p ϕ p v qq “ ␣ ϕ p v q a i ÝÑ ‘ ˇˇ i “ , . . . , m ( Y ␣ ϕ p v q b j ÝÑ v j ˇˇ j “ , . . . , n ( , (A.10) ϕ p v j q “ v j , for all j P t , . . . , n u , (A.11)with n , m P N , and vertices v j P V z t ‘ u , and v j P V z t ‘ u , for j P t , . . . , n u . Note that the same transition may be listedmultiple times in the set T p ϕ p v qq . On this basis we can argue as follows. p s ˝ ϕ qp v q ” s p ϕ p v qq “ BBP ´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ s p v j q ¯ (since s is a provable solution of C , using (A.10) and axioms (B1), (B2), (B3)) ” ´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ p s ˝ ϕ qp v j q ¯ (using (A.11) and p s ˝ ϕ qp v j q ” s p ϕ p v j qq ) eport version Clemens Grabmayer and Wan Fokkink This shows, in view of (A.9), that s ˝ ϕ satisfies the condition for a provable solution at v . Now as v P V z t ‘ u was arbitrary, s ˝ ϕ (with domain V z t ‘ u ) is a provable solution of C . Since furthermore the functional bisimulation ϕ must relate the startvertices of C and C , the principal value of s ˝ ϕ coincides with that of s . □ Lemma (= Lemma 5.2) . In a chart with a
LLEE -witness, for all vertices v , w :(i) v Ñ bo w ñ ∥ v ∥ bo ą ∥ w ∥ bo ,(ii) v ñ w ñ | v | en ą | w | en .Proof. For statement (i) we argue as follows. Recall that the body step norm ∥ v ∥ bo in a LLEE-witness ˆ C was defined as themaximal length of a body step path from v in ˆ C . This was well-defined due to Lemma 3.9, (i), and the finiteness of charts. Nowsuppose that v Ñ bo w . Then every body step path from w gives rise to a body step path from v that starts with the transition v Ñ bo w . Hence a longest body step path from w of length ∥ w ∥ bo gives rise to a body step path from v of length ∥ w ∥ bo `
1. Itfollows that ∥ v ∥ bo ě ∥ w ∥ bo ` ą ∥ w ∥ bo , and hence ∥ v ∥ bo ą ∥ w ∥ bo .For showing statement (ii), suppose that v ñ w . Then v α ñ w holds for some α P N ` . Then | v | en ě α . If there is noloop-entry transition that departs from w , then | w | en “ | v | en ě α ą “ | w | en . Otherwise we let β P N ` be the maximal index of a loop-entry transition from w . Then v α ñ w Ñ r β s . By Lemma A.3 it follows that α ą β .Consequently we find | v | en ě α ą β “ | w | en . In both cases we have shown | v | en ą | w | en . □ Lemma (= Lemma 5.4, uses the
BBP -axioms (B1)–(B6), (BKS2), but not the rule RSP f ) . For a
LLEE -chart C with LLEE -witness ˆ C the following connection holds between the extracted solution s ˆ C and the relative extracted solution t ˆ C , for all vertices v , w : v ñ w ùñ s ˆ C p w q “ BBP t ˆ C p w , v q ¨ s ˆ C p v q . (A.12) Note that if v ñ w , then v ‰ ‘ , and also w ‰ ‘ , because w is in the body of a loop at v , and therefore cannot be ‘ (see Lem. A.4).Proof. In order to show (A.12) we proceed by complete induction (without explicit treatment of the base case) on the length ∥ w ∥ bo of a longest body step path from w . For performing the induction step, we consider arbitrary v , w ‰ ‘ with v ñ w .We assume a representation of the set ˆ T p w q of transitions from w in ˆ C :ˆ T p w q “ ␣ w a i ÝÑ r α i s w ˇˇ i “ , . . . , m ( Y ␣ w b j ÝÑ r β j s w j ˇˇ w j ‰ w , j “ , . . . , n ( Y ␣ w c i ÝÑ bo v ˇˇ i “ , . . . , p ( Y ␣ w d j ÝÑ bo u j ˇˇ u j ‰ v , j “ , . . . , q ( (A.13)that partitions ˆ T p w q into loop-entry transitions to w and to other targets w , . . . , w n , and body transitions to v and to othertargets u , . . . , u q . Since w is contained in a loop at v , none of these targets can be ‘ . In order to show provable equality at theright-hand side of (A.12), we argue as follows: s ˆ C p w q ” ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´ ` ´´ p ÿ i “ c i ¨ s ˆ C p v q ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯¯ (by the definition of s ˆ C p w q , based on the representation (A.13),using that none of the target vertices is ‘ ) “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¨ s ˆ C p v q ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (using axiom (B6)) “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¨ s ˆ C p v q ¯ ` ´ q ÿ j “ d j ¨ ` t ˆ C p u j , v q ¨ s ˆ C p v q ˘¯¯ (by the induction hypothesis, using that v ñ u j and (cid:13)(cid:13) u j (cid:13)(cid:13) bo ă ∥ w ∥ bo because w Ñ bo u j for j “ , . . . , q , see (A.13)) “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯ ¨ s ˆ C p v q ¯ (using axioms (B5), (B4)) Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯ ¨ s ˆ C p v q (using axiom (BKS2)) ” t ˆ C p w , v q ¨ s ˆ C p v q (by the definition of t ˆ C p w , v q , based on the representation (A.13))This chain of provable equalities demonstrates (A.12). □ Proposition (= Proposition 5.5, uses the
BBP -axioms (B1)–(B6), (BKS1), (BKS2), but not the rule RSP f ) . In a chart C with aLLEE-witness ˆ C , s ˆ C is a provable solution of C .Proof. We prove that s ˆ C is a provable solution of the chart C . Let w ‰ ‘ . We show that s ˆ C p w q satisfies the defining equationof s ˆ C to be a provable solution of C at w .We consider a representation of the set ˆ T p w q of transitions from w in ˆ C as follows:ˆ T p w q “ ␣ w a i ÝÑ r α i s w ˇˇ i “ , . . . , m ( Y ␣ w b j ÝÑ r β j s w j ˇˇ w j ‰ w , j “ , . . . , n ( Y ␣ w c i ÝÑ bo ‘ ˇˇ i “ , . . . , p ( Y ␣ w d j ÝÑ bo u j ˇˇ u j ‰ ‘ , j “ , . . . , q ( (A.14)that partitions ˆ T p w q into loop-entry transitions to w and to other targets w , . . . , w n , and body transitions to ‘ and to othertargets u , . . . , u q . We argue as follows: s ˆ C p w q ” ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (by the definition of s ˆ C , in view of (A.14)) “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ ¨ s ˆ C p w q ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (using axiom (BKS1) and the defining equality in the first step) “ BBP ´´ m ÿ i “ a i ¨ s ˆ C p w q ¯ ` ´ n ÿ j “ b j ¨ p t ˆ C p w j , w q ¨ s ˆ C p w qq ¯¯ ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (using axioms (B5), (B4)) “ BBP ´´ m ÿ i “ a i ¨ s ˆ C p w q ¯ ` ´ n ÿ j “ b j ¨ s ˆ C p w j q ¯¯ ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (using (A.12) of Lemma 5.4, in view of w ñ w j for j “ , . . . , n ) “ BBP ´ p ÿ i “ c i ¯ ` ´´´ m ÿ i “ a i ¨ s ˆ C p w q ¯ ` ´ n ÿ j “ b j ¨ s ˆ C p w j q ¯¯ ` ´ q ÿ j “ d j ¨ s ˆ C p u j q ¯¯ (using axioms (B1), (B2))This chain of provable equalities demonstrates that s ˆ C p w q is a provable solution of C at w , in view of (A.14). As w ‰ ‘ isarbitrary, s ˆ C is indeed a provable solution of C . □ Lemma (= Lemma 5.7, uses the
BBP -axioms (B1)–(B6), and the rule RSP f ) . For every provable solution s of a chart C with LLEE -witness ˆ C , the following connection holds with the relative extraction function t ˆ C holds, for all vertices v , w : v ñ w ùñ s p w q “ BBP t ˆ C p w , v q ¨ s p v q (A.15) Note that if v ñ w , then v ‰ ‘ , and also w ‰ ‘ , because w is in the body of a loop at v , and therefore cannot be ‘ .Proof. In order to prove (A.15) we proceed by complete induction on the same measure as used in the definition of the relativeextraction function t ˆ C , namely, induction on the maximal loop level of a loop at v , with a subinduction on ∥ w ∥ bo . For performingthe induction step, consider vertices v , w with v ñ w . As in the proof of Prop. 5.5 we assume the representation (A.13) of theset ˆ T p w q of transitions from w in ˆ C , which partitions ˆ T p w q into loop-entry transitions to w and to other targets w , . . . , w n , eport version Clemens Grabmayer and Wan Fokkink and body transitions to v and to other targets u , . . . , u q . Since w is contained in a loop at v , none of these targets can be ‘ .We now argue as follows: s p w q “ BBP ` ´´ m ÿ i “ a i ¨ s p w q ¯ ` ´´ n ÿ j “ b j ¨ s p w j q ¯ ` ´ p ÿ i “ c i ¨ s p v q ¯ ` ´ q ÿ j “ d j ¨ s p u j q ¯¯¯ (since s is a provable solution of C at w , using (A.13)) “ BBP ´´ m ÿ i “ a i ¨ s p w q ¯ ` ´ n ÿ j “ b j ¨ s p w j q ¯¯ ` ´´ p ÿ i “ c i ¨ s p v q ¯ ` ´ q ÿ j “ d j ¨ s p u j q ¯¯ (using axioms (B6), (B2)) “ BBP ´´ m ÿ i “ a i ¨ s p w q ¯ ` ´ n ÿ j “ b j ¨ ` t ˆ C p w j , w q ¨ s p w q ˘¯¯ ` ´´ p ÿ i “ c i ¨ s p v q ¯ ` ´ q ÿ j “ d j ¨ ` t ˆ C p u j , v q ¨ s p v q ˘¯¯ (using the induction hypothesis, which is applicable becausethe maximal loop level at w is smaller than that at v due to v ñ w , and v ñ u i and (cid:13)(cid:13) u j (cid:13)(cid:13) bo ă ∥ w ∥ bo due to w Ñ bo u j for j “ , . . . , q , see (A.13)) “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ ¨ s p w q ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯ ¨ s p v q (using axioms (B5), (B4))This chain of provable equalities justifies: s p w q “ BBP ´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b j ¨ t ˆ C p w j , w q ¯¯ ¨ s p w q ` ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯ ¨ s p v q To this equality we can apply the rule RSP f : s p w q “ BBP ´´´ m ÿ i “ a i ¯ ` ´ n ÿ j “ b i ¨ t ˆ C p w j , w q ¯¯ f ´´ p ÿ i “ c i ¯ ` ´ q ÿ j “ d j ¨ t ˆ C p u j , v q ¯¯¯ ¨ s p v q (by applying rule RSP f ) ” t ˆ C p w , v q ¨ s p v q , The last step uses the definition of t ˆ C p w , v q , based on representation (A.13) of p T p w q . In this way we have carried out theinduction step. We conclude that (A.15) holds for all vertices v and w of C . □ A.4 Proofs in Section 6: Preservation of LLEE under collapseLemma (= Lemma 6.2) . If w Ø w in C , then C p w q w Ø C .Proof. Let C “ x V , ‘ , v s , , T y and C p w q w “ x V , ‘ , v s , , T y . Let B Ď V ˆ V be the largest bisimulation relation on C . Inparticular, x w , w y P B . We argue that B “ B X p V ˆ V q is a bisimulation relation between C and C p w q w . Take any x u , v y P B Ď B . ‚ ( forth ): Let u a ÝÑ u P T . Then x u , v y P B implies there is a v a ÝÑ v P T with x u , v y P B . If v a ÝÑ v P T , then v P V ,so x u , v y P B and we are done. If v a ÝÑ v R T , then v “ w and v a ÝÑ w P T . Since x u , w y P B and x w , w y P B ,also x u , w y P B . Since w P V , it follows that x u , w y P B . ‚ ( back ): Let v a ÝÑ v P T . If v a ÝÑ v P T , then x u , v y P B implies there is a u a ÝÑ u P T with x u , v y P B . Since v P V ,also x u , v y P B and we are done. If v a ÝÑ v R T , then v “ w and v a ÝÑ w P T . So x u , v y P B implies there is a u a ÝÑ u P T with x u , w y P B . Since x u , w y P B and x w , w y P B , also x u , w y P B . Since w P V , it follows that x u , w y P B . ‚ ( termination ): Since B Ď B clearly u “ ‘ if and only if v “ ‘ .Finally, concerning ( start ): If v s , “ v s , , then trivially x v s , , v s , y P B . If v s , ‰ v s , , then v s , “ w and v s , “ w . Since x w , w y P B and w P V , we have x w , w y P B . □ Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
Proposition (= Proposition 6.4) . If a
LLEE -chart C is not a bisimulation collapse, then it contains a pair of bisimilar vertices w , w that satisfy, for a LLEE -witness of C , one of the following conditions:(C1) ␣p w Ñ ˚ w q ^ p ñ w ùñ w is not normed q ,(C2) w ü ` w ,(C3) D v P V ` w d ü v ^ w ü ` v ˘ ^ ␣p w Ñ ˚ bo w q .More supplementary illustrations for the proof of Prop. 6.4 on pages 9–10. The proof started from a pair u , u of distinct bisimi-lar vertices. In the case scc p u q “ scc p u q , we had the following situation: u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q . (A.16)For pairs of vertices u and u such that (A.16) holds, for some v , v , and v , we used induction on ∥ u ∥ min lb in order show that u and u progress, via pairs of distinct bisimilar vertices, to bisimilar vertices w and w such that one of the conditions (C1) ,(C2) , or (C3) holds. Note that each of (C1) , (C2) , and (C3) implies that w and w are distinct.In order to carry out the induction step we used a case distinction. Below we repeat the arguments, and supplement themwith illustrations.Case 1: u ü ` v .Since u Ñ u , either u “ v or v ñ ` u . Moreover, scc p u q “ scc p u q “ scc p v q , so by Lem. 3.9, (ii), u ü ˚ v .Hence, u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q , and (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb . We apply the induction hypothesis toobtain a bisimilar pair w , w for which (C1) , (C2) , or (C3) holds. In the illustration below, we drew both of the twocases in which the transition u Ñ u is a loop-entry transition, or a body transition, from u . vv v { bo u u lb u u u r α s bo u s e i n d . h yp . u s e i n d . h y p . Case 2: u “ v .Case 2.1: u Ñ r α s u .Then either u “ u or u ñ u . Moreover, scc p u q “ scc p u q , so by Lem. 3.9, (ii), u ü ˚ u , and hence u ü ˚ v .Thus we have obtained u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q . Due to (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb , we can apply theinduction hypothesis again. vv v “ u “ u r α s { bo u u lb u s e i n d . h y p . vv v “ u r α s { bo u u lb u u s e i n d . h y p . eport version Clemens Grabmayer and Wan Fokkink Case 2.2: u Ñ bo u .Then ␣p v Ñ ˚ bo v q together with v “ u Ñ bo u and u Ñ ˚ bo v (because u ü ˚ v ) imply u ‰ u . We distinguishtwo cases.Case 2.2.1: u “ v .Then u ü ˚ v d ü v “ u , i.e., u ü ` u , so we are done, because (C2) holds for w “ u and w “ u . v “ u v v “ u { bo u u lb u s e i n d . h y p . (C2) Case 2.2.2: u ‰ v .By Lem. 3.9, (ii), u ü ` v . Hence, u ü ˚ v d ü v for some v . Since v “ u Ñ bo u ü ˚ v and ␣p v Ñ ˚ bo v q , itfollows that ␣p v Ñ ˚ bo v q . So u ü ˚ v d ü v d ý v ý ˚ u ^ ␣p v Ñ ˚ bo v q . Due to (cid:13)(cid:13) u (cid:13)(cid:13) min lb ă ∥ u ∥ min lb , we canapply the induction hypothesis again. vv v “ u { bo bo u u lb v { bo u u s e i n d . h y p . □ Proposition (= Proposition 6.8) . Let C be a LLEE -chart. If a pair x w , w y of vertices satisfies (C1), (C2), or (C3) with respect to a LLEE -witness of C , then C p w q w is a LLEE -chart.
As background for the proof of this proposition, we first give examples why conditions (C1) , (C2) , and (C3) cannot bereadily relaxed or changed. These examples showcase that, far from being artificial, the conditions (C1) , (C2) , and (C3) marksharp borders between whether, on a given LLEE-witness, a connect-through operation is possible while preserving LLEE, ornot. Thus these examples demonstrate that a further simplification of the case analysis provided by Proposition 7.3 is notreadily possible, with an eye towards LLEE-structure preserving connect-through operations. Therefore a substantial furtherimprovement of our stepwise collapse procedure appears unlikely.For convenience, the pictures in these examples neglect action labels on transitions.
Example A.11 (= Example 6.3) . To show that in (C1) it is crucial that w does not loop back, we refer back to the LLEE-witnessˆ C in Ex. 6.3. There ␣p w Ñ ˚ w q , but (C1) is not satisfied by the pair w , w because w ü p w . Since in ˆ C the levels ofloop-entry transitions that descend to w are higher than the loop levels that descend from w , the preprocessing step oftransformation I is void. We observed that the connect- w -through-to- w chart C p w q w on the left in Ex. 6.3 has no LLEE-witness.The bisimilar pair w , w in ˆ C progresses to the bisimilar pair p w , p w , for which (C1) holds. Since ˆ C p p w q p w on the right of Ex. 6.3 Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version is obtained by applying transformation I to this pair, it is guaranteed to be a LEE-witness; this will be argued in the proof ofProp. 6.8. C p w q w p w p w ‘ w ˆ C ‘ r s p w w r s p w w ‘ ˆ C p p w q p w r s p w w C p w q w ÐSS C p I q p p w q p w To avoid the creation of body step cycles in transformation II, it would seem expedient to connect transitions to w throughto w , since (C2) , w ü ` w , rules out the existence of a path w Ñ ` bo w in ˆ C . (Instead, transitions to w are connected throughto w , and resulting body step cycles are eliminated by turning the body transitions at p w into loop-entry transitions.) However,connecting transitions to w through to w may produce a chart for which no LLEE-witness exists. Example A.12 (= Example 6.6) . For the LLEE-chart C with LLEE-witness ˆ C below in the middle, the connect- w -through-to- w chart C p w q w on the left does not have a LLEE-witness: it has no loop subchart, because from each of its three vertices an infinitepath starts that does not return to this vertex. From p w this path, drawn in red, cycles between u and w . Transformation IIapplied to the pair w , w (instead of w , w ) in ˆ C yields the entry/body-labeling ˆ C p w q w for the connect- w -through-to- w chartwith additionally p w Ñ bo w turned into p w Ñ r s w . Since the pair w , w satisfies (C2) , the proof of Prop. 6.8 guarantees thatthis entry/body-labeling, drawn on the right, is a LLEE-witness. C p w q w ÐSS C p II q p w q w w p w u C p w q w w r s r s p w r s uw ˆ C p w r s r s uw ˆ C p w q w The following example shows that for transformation III it is essential to select a bisimilar pair w , w where w directly loops back to v . Example A.13 (= Example 6.7) . In the LLEE-witness ˆ C below in the middle, w , w ü ` v , and there is no body step pathfrom w to w , but (C3) does not hold for the pair w , w because ␣p w d ü v q . All loop-entry transitions from v have thesame loop label, so the preprocessing step of transformation III is void. The connect- w -through-to- w chart C p w q w on the leftdoes not have a LLEE-witness. Namely, the transition from p w can be declared a loop-entry transition, and after its removalalso two transitions from v can be declared loop-entry transitions, leading to the removal of the five transitions that aredepicted as dotted arrows. The remaining chart (of solid arrows) however has no further loop subchart, because from each ofits vertices an infinite path starts that does not return to this vertex. The bisimilar pair w , w progresses to the bisimilar pair p w , p w in ˆ C , for which (C3) holds because p w d ü v ý p w and ␣p p w Ñ ˚ bo p w q . Transformation III applied to this pair yields theentry/body-labeling ˆ C p p w q p w on the right. In the proof of Prop. 6.8 it is argued that this is guaranteed to be a LLEE-witness. Theremaining two bisimilar pairs can be eliminated by one or two further applications of transformation III. eport version Clemens Grabmayer and Wan Fokkink v C p w q w p w p w w v ˆ C p w r s r s r sr sr s w p w r s w v ˆ C p p w q p w r sr sr s w p w r s w C p w q w ÐSS C p III q p p w q p w The following example shows (C3) cannot be weakened by dropping ␣p w Ñ ˚ bo w q . Example A.14.
For the LLEE-witness ˆ C below in the middle, w d ü v ý ` w , but there is a body step path from w to w .The connect- w -through-to- w chart C p w q w on the left does not have a LLEE-witness, because from each of its vertices aninfinite path starts that does not return to it. The bisimilar pair w , w in ˆ C progresses to the bisimilar pair v , p w , to whichtransformation II is applicable because (C2) holds: p w ü v . In the resulting LLEE-witness ˆ C p v q p w , second to the right, (C3)holds for the pair w , w because w d ü p w ý w and ␣p w Ñ ˚ bo w q . Applying transformation III to this pair results in theLLEE-witness on the right. C p w q w v p w w ˆ C v r sr s r s p w r s w w ˆ C p v q p w p w r s r sr s w w p w r s r sr s w C p w q w ÐSS C p II q p v q p w p III q p w q w Supplement for the proof of Proposition 6.8.
Let ˆ C be a LLEE-chart. For vertices w , w such that (C1) , (C2) , or (C3) holds,transformation I, II, or III, respectively, produces an entry/body-labeling ˆ C p w q w . In the article submission we have proved fortransformation I that it is a LLEE-witness. Here we do the same for transformations II and III.We recall that in the proof in the article submission we have shown that it suffices to show that each of the transformationsproduces, before the final clean-up step, an entry/body-labeling that satisfies the LLEE-conditions with the exception ofpossible violations of the loop property (L1) in (W2)(a). Transformation II:
We argue the correctness of transformation II. Consider vertices w , w such that (C2) holds, that is, w ü ` w . Let p w be the d ü -predecessor of w in the d ü -chain from w to w , i.e., w ü ˚ p w d ü w .As for the transformations I and III it suffices to show, in view of the alleviation of the proof obligation at the start of theproof on page 12, that the intermediate result ˆ C of transformation II before the clean-up step satisfies the LLEE-witnessproperties, except for possible violations of (L1). By the definition of transformation II, ˆ C results by performing theadaptation step L II to the chart ˆ C : “ ˆ C p w q w that arises from ˆ C by connecting w through to w .To prove that (W1), and the part concerning (L2) for (W2)(a) is satisfied for ˆ C , it suffices to show that the transformedchart does not contain a cycle of body transitions. At first, the step of connecting w through to w in ˆ C may introduce abody step cycle in ˆ C “ ˆ C p w q w . But every such cycle is removed in the subsequent level adaptation step L II . Namely, eachbody step cycle introduced in ˆ C must stem from a transition u Ñ bo w (which is redirected to w in ˆ C ) and a path w Ñ ˚ bo u in ˆ C , for some u ‰ w . Since w ü ˚ p w d ü w , by Lem. A.8, the path w Ñ ˚ bo u Ñ bo w in ˆ C must visit p w .Since all body transitions from p w are turned into loop-entry transitions in step L II , the body step cycle w Ñ ˚ bo u Ñ bo w in ˆ C that was introduced in the connect-through step, is after step L II no longer a body step cycle in ˆ C . Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity Report version
Now we prove that (W2)(b) is preserved by the two steps from ˆ C via C “ ˆ C p w q w to ˆ C . Every path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C with u ‰ w , w arises by a, possibly empty, combination of the following three kinds of modifications in the firsttwo transformation steps:(i) A transition to w was redirected to w in the connect-through step.(ii) The loop-entry transition at the beginning of the path is from p w and was a body transition before step L II , meaningthat u “ p w and α “ γ . (Recall that γ is a loop name of maximum loop level among the loop-entries at w in ˆ C .)(iii) The loop-entry transition at the end of the path is from p w and was a body transition before step L II , meaning that β “ γ .This gives 2 “ C , and so α ą β is guaranteed; (ii) and (iii) together cannot hold, because then the path wouldreturn to u “ p w , which it cannot, because all of its steps avoid u as target. We now show that in the remaining fivecases always α ą β . Since w ü ` w , there is a path w ÝÝÝÑ t p w q r δ s ¨ ÝÝÝÑ t p w q ˚ bo w in ˆ C . By definition of γ , γ ě δ .A Let only (i) hold: there are paths u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w and w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C (which do not visit p w ). Then thereis a path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w Ñ r γ s in ˆ C , so α ą γ . We distinguish two cases.Case 1: The path w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s visits w . Then there is a path w ÝÝÑ t p u q ˚ ¨ Ñ r β s in ˆ C . So u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C . So by (W2)(b), α ą β .Case 2: The path w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s does not visit w . Then there is a path w ÝÝÝÑ t p w q r δ s ¨ ÝÝÝÑ t p w q ˚ bo w ÝÝÝÑ t p w q ˚ bo ¨ Ñ r β s inˆ C , so δ ą β . Hence α ą γ ě δ ą β .B Let only (ii) hold. Then u “ p w , α “ γ , and there is a path p w ÝÝÝÝÝÑ t p p w , w q ` bo ¨ Ñ r β s in ˆ C . As p w d ü w , there is a path w ÝÝÝÑ t p w q r δ s ¨ ÝÝÝÑ t p w q ˚ bo p w in ˆ C . Hence w ÝÝÝÑ t p w q r δ s ¨ ÝÝÝÑ t p w q ˚ bo p w ÝÝÝÑ t p w q ` bo ¨ Ñ r β s in ˆ C , so δ ą β . Hence α “ γ ě δ ą β .C Let only (iii) hold. Then β “ γ , and u ÝÝÝÝÝÑ t p u , w q r α s ¨ ÝÝÝÝÝÑ t p u , w q ˚ bo p w with u ‰ w is a path in in ˆ C . Since p w d ü w and u ‰ w , it follows that ␣p p w d ü u q . So in view of the path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo p w , there is no path p w Ñ ˚ bo u in ˆ C .Since p w d ü w , there is a path p w Ñ ˚ bo w in ˆ C , which by the previous observation is of the form p w ÝÝÑ t p u q ˚ bo w .Hence there is a path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo p w ÝÝÑ t p u q ˚ bo w Ñ r γ s in ˆ C , so α ą γ “ β .D Let only (i) and (ii) hold, meaning u “ p w , α “ γ , and there are paths p w ÝÝÝÑ t p p w q ` w and w ÝÝÝÑ t p p w q ˚ bo ¨ Ñ r β s in ˆ C . Since w ü ˚ p w d ü ` w , and u “ p w implies w ‰ p w , by Lem. A.8, the path w ÝÝÝÑ t p p w q ˚ bo ¨ Ñ r β s cannot visit w . Hence w ÝÝÝÑ t p w q r δ s ¨ ÝÝÝÑ t p w q ˚ bo w ÝÝÝÑ t p w q ˚ bo ¨ Ñ r β s in ˆ C . So δ ą β . Hence α “ γ ě δ ą β .E Let only (i) and (iii) hold. Then β “ γ , and u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w and w ÝÝÑ t p u q ˚ bo p w are paths in ˆ C . Since u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w Ñ r γ s in ˆ C , α ą γ “ β .We conclude that in all five cases, ˆ C satisfies (W2)(b).Finally we argue that part (L3) of (W2)(a) holds for ˆ C , i.e., there are no descends-in-loop-to paths of the form u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo ‘ in ˆ C . We can use part of the argumentation employed for demonstrating (W2)(b) above. Itwas demonstrated in particular that for every descends-in-loop-to path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo x in ˆ C , there is a descends-in-loop-to path ˜ u ÝÝÑ t p ˜ u q r γ s ¨ ÝÝÑ t p ˜ u q ˚ bo x with the same target x in ˆ C . From this it follows that if a descends-in-loop-to pathin ˆ C had ‘ as target, then there were a descends-in-loop-to path already in ˆ C that had ‘ as target, violating (L3) forthe LLEE-chart ˆ C . Hence ˆ C must satisfy (L3).We conclude that the result of transformation II is a LLEE-chart. eport version Clemens Grabmayer and Wan Fokkink Transformation III:
To show the correctness of transformation III, consider vertices w and w such that (C3) holds. Let v besuch that w d ü v ý ` w . We show that its intermediate result ˆ C p w q w before the clean-up step satisfies the LLEE-witnessproperties, except for possible violations of (L1).First we show that (W2)(b) is preserved by both the level adaptation and the connect-through step. A violation arisingby the first step, i.e., in ˆ C , would involve a path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo v Ñ r β s in ˆ C where β is increased to a loop label γ of maximum level among all loop-entries at v . But in this way no violation can arise, since there was already a path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo v Ñ r γ s in ˆ C , so α ą γ ě β .Now we exclude violations of (W2)(b) in the connect-through step, by showing that in ˆ C p w q w , α ą β for all newly createdpaths u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo ¨ Ñ r β s with u ‰ w that stem from paths u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w and w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C .As w ü ` v , there is a path v ÝÝÝÑ t p v q r γ s ¨ ÝÝÝÑ t p v q ˚ bo w in ˆ C . We distinguish two cases.Case 1: u “ v . Then, by the level adaptation step, α “ γ . Since u “ v , there is a path v ÝÝÝÑ t p v q r γ s ¨ ÝÝÝÑ t p v q ˚ bo w ÝÝÝÑ t p v q ˚ bo ¨ Ñ r β s in ˆ C . By (W2)(b) for ˆ C , γ ą β .Case 2: u ‰ v . Since w d ü v , there is a path w Ñ ` bo v in ˆ C and thus in ˆ C . Suppose, toward a contradiction, thatthis path visits u . Then u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w Ñ ` bo u , so w ü u in ˆ C and thus in ˆ C . Then w d ü v and u ‰ v imply v ü u , which together with u Ñ ` bo v yields a body step cycle between u and v in ˆ C . This contradicts that (W1) holds inˆ C . Therefore w ÝÝÑ t p u q ` bo v in ˆ C . We consider two cases.Case 2.1: w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s visits v , so v ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C . Then u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w ÝÝÑ t p u q ` bo v ÝÝÑ t p u q ˚ bo ¨ Ñ r β s in ˆ C .By (W2)(b) for ˆ C , α ą β .Case 2.2: w ÝÝÑ t p u q ˚ bo ¨ Ñ r β s does not visit v . Then since w ü ` v implies v ñ ` w , there is a path v ÝÝÝÑ t p v q r γ s ¨ ÝÝÝÑ t p v q ˚ bo x k ÝÝÝÑ t p x k q r δ k s ¨ ÝÝÝÑ t p x k q ˚ bo ¨ ¨ ¨ x ÝÝÝÑ t p x q r δ s ¨ ÝÝÝÑ t p x q ˚ bo w ÝÝÝÑ t p v q ˚ ¨ Ñ r β s in ˆ C , for some k ě
0. Since also u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo w ÝÝÑ t p u q ` bo v Ñ r γ s in ˆ C , by (W2)(b), α ą γ ą δ k ą ¨ ¨ ¨ ą δ ą β . So α ą β .To verify (W1) together with part (L2) of (W2)(a) for ˆ C p w q w , it suffices to show that ˆ C p w q w does not contain body stepcycles. This can be verified analogously as for transformation I. That is, under the assumption of a body step cycle wecan construct a path w Ñ ` bo w in ˆ C , which contradicts (C3) (as it contradicted (C1) ).To show part (L3) of (W2)(a) for ˆ C p w q w , we can use part of the argumentation employed above for proving (W2)(b). It wasdemonstrated in particular that for every descends-in-loop-to path u ÝÝÑ t p u q r α s ¨ ÝÝÑ t p u q ˚ bo x in ˆ C there is a descends-in-loop-to path ˜ u ÝÝÑ t p ˜ u q r γ s ¨ ÝÝÑ t p ˜ u q ˚ bo x with the same target x in ˆ C . This entails that if a descends-in-loop-to path in ˆ C had ‘ as target, then there were a descends-in-loop-to path in ˆ C with ‘ as target, contradicting (L3) for the LLEE-witness ˆ C .Hence ˆ C must satisfy part (L3) of (W2)(a).We conclude that the result of transformation III is again a LLEE-witness. □□