A Complete Proof System for 1-Free Regular Expressions Modulo Bisimilarity

Abstract

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Structure-Constrained Process Graphs for the Process Semantics of Regular Expressions

“…2.6 satisfies the 'loop existence and elimination' property LEE that we will explain in this section. For this purpose we summarize Section 3 in [6], and in doing so we adapt the concepts defined there from 1-free star expressions to the full class of star expressions as defined in Section 2. The property LEE is defined by a dynamic elimination procedure that analyses the structure of a chart by peeling off 'loop subcharts'.…”

“…Statements (5) and (6) can be established by rule inspection, statements (7) and (8) can be shown by induction on the structure of E. Then statement (1) of the lemma can be shown from (5) and (7), and statement (2) from (6) and (8). Statements (3) and (4) of the lemma can be shown by means of a proof by induction on the structure of E that makes crucial use of the admissible rules of T (StExp(A)) from Def.…”

“…The result of Baeten, Corradini, and myself [2] that the problem (R) is decidable in principle was based on the concept of 'well-behaved (recursive) specifications' that links process graphs with star expressions. Recently in [6,7], Wan Fokkink and I obtained a partial solution for (A) in the form of a complete proof system for '1-free' star expressions, which do not contain 1, but are formed with binary Kleene star iteration (•) ⊛ (•) instead of unary iteration. For this, we defined the efficiently decidable 'loop existence and elimination property (LEE)' of process graphs that holds for all process graph interpretations of 1-free star expressions, and for their bisimulation collapses.…”

“…In Section 3 we explain the loop existence and elimination property LEE for process graphs, and we define the concept of a 'layered LEE-witness', for short a 'LLEE-witness' for process graphs. Hereby Section 3 is an adaptation for star expressions that may contain 1 of the motivation of LLEE-witnesses in Section 3 in [6], which concerned the process semantics of '1-free star expressions'. LLEE-witnesses arise by adding natural-number labels to transitions that are subject to suitable constraints.…”

“…Also, we sketch the crucial steps of the proof of (P1), but refer to the report version [5] for the details. Finally in Section 7 we briefly report on which of the steps that we developed in [6] for obtaining solutions of the problems (A) and (R) for 1-free star expressions can be extended or adapted in the direction of solutions of these problems for all star expressions, and where the remaining difficulty resides.…”