针对 Milner 证明系统 Mil 的完备性开放问题，本文提出余归纳重构系统 cMil，通过允许循环推导的余归纳规则替换原不动点规则，并开发双向有效证明转换证明其与 Mil 定理等价，为完备性证明提供新立足点。
摘要：
Milner (1984) 将正则表达式作为有限状态过程的操作语义。为了在过程语义下公理化正则表达式的互模拟，他改编了 Salomaa 的证明系统，该系统在语言语义下对正则表达式的等式是完全的。除了大部分等式公理外，Milner 的系统 Mil 继承了 Salomaa 系统中用于求解单不动点方程的非代数规则。由于识别到过程语义的独特性质使得 Salomaa 的证明策略不适用，Milner 将系统 Mil 的完备性作为一个开放问题提出。作为该问题的证明论方法，我们刻画了不动点规则为 Mil 的纯等式部分 Mil$^-$ 所增加的推导能力。我们通过余归纳规则实现这一点，该规则允许循环推导，其由满足分层循环存在与消除属性 LLEE 的带空步骤的有限过程图，及其两个 Mil$^-$-可证解组成。以该规则替换 Mil 中的不动点规则，我们将余归纳重构 cMil 定义为 Mil$^-$ 的扩展。为证明 cMil 与 Mil 定理等价，我们开发了从 Mil 到 cMil 及其反向的有效证明转换。由于 cMil 位于互模拟与 Milner 系统 Mil 证明的中间位置，它可能成为 Mil 完备性证明的立足点。本文扩展了我们发表于 CALCO 2022 会议论文集的贡献。在此我们将证明转换细化为可导出与可采纳规则的消除，并将余归纳证明与过程图解的余代数表述联系起来。
英文摘要：
Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete for equality of regular expressions under the language semantics. Apart from most equational axioms, Milner's system Mil inherits from Salomaa's system a non-algebraic rule for solving single fixed-point equations. Recognizing distinctive properties of the process semantics that render Salomaa's proof strategy inapplicable, Milner posed completeness of the system Mil as an open question.
As a proof-theoretic approach to this problem we characterize the derivational power that the fixed-point rule adds to the purely equational part Mil$^-$ of Mil. We do so by means of a coinductive rule that permits cyclic derivations that consist of a finite process graph with empty steps that satisfies the layered loop existence and elimination property LLEE, and two of its Mil$^{-}$-provable solutions. With this rule as replacement for the fixed-point rule in Mil, we define the coinductive reformulation cMil as an extension of Mil$^{-}$. In order to show that cMil and Mil are theorem equivalent we develop effective proof transformations from Mil to cMil, and vice versa. Since it is located half-way in between bisimulations and proofs in Milner's system Mil, cMil may become a beachhead for a completeness proof of Mil.
This article extends our contribution to the CALCO 2022 proceedings. Here we refine the proof transformations by framing them as eliminations of derivable and admissible rules, and we link coinductive proofs to a coalgebraic formulation of solutions of process graphs.
Robin Milner (1984) gave a sound proof system for bisimilarity of regular expressions interpreted as processes: Basic Process Algebra with unary Kleene star iteration, deadlock 0, successful termination 1, and a fixed-point rule. He asked whether this system is complete. Despite intensive research over the last 35 years, the problem is still open. This paper gives a partial positive answer to Milner's problem. We prove that the adaptation of Milner's system over the subclass of regular expressions that arises by dropping the constant 1, and by changing to binary Kleene star iteration is comple…
Milner (1984) defined a process semantics for regular expressions. He formulated a sound proof system for bisimilarity of process interpretations of regular expressions, and asked whether this system is complete. We report conceptually on a proof that shows that Milner's system is complete, by motivating, illustrating, and describing all of its main steps. We substantially refine the completeness proof by Grabmayer and Fokkink (2020) for the restriction of Milner's system to `1-free' regular expressions. As a crucial complication we recognize that process graphs with empty-step transitions tha…
In type theory, coinductive types are used to represent processes, and are thus crucial for the formal verification of non-terminating reactive programs in proof assistants based on type theory, such as Coq and Agda. Currently, programming and reasoning about coinductive types is difficult for two reasons: The need for recursive definitions to be productive, and the lack of coincidence of the built-in identity types and the important notion of bisimilarity. Guarded recursion in the sense of Nakano has recently been suggested as a possible approach to dealing with the problem of productivity, a…
By adapting Salomaa's complete proof system for equality of regular expressions under the language semantics, Milner (1984) formulated a sound proof system for bisimilarity of regular expressions under the process interpretation he introduced. He asked whether this system is complete. Proof-theoretic arguments attempting to show completeness of this equational system are complicated by the presence of a non-algebraic rule for solving fixed-point equations by using star iteration. We characterize the derivational power that the fixed-point rule adds to the purely equational part $\text{Mil$^{\b…
We study bisimulation and context equivalence in a probabilistic $λ$-calculus. The contributions of this paper are threefold. Firstly we show a technique for proving congruence of probabilistic applicative bisimilarity. While the technique follows Howe's method, some of the technicalities are quite different, relying on non-trivial "disentangling" properties for sets of real numbers. Secondly we show that, while bisimilarity is in general strictly finer than context equivalence, coincidence between the two relations is attained on pure $λ$-terms. The resulting equality is that induced by Levy-…
Bisimulation up-to enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain up-to techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similar…
In a paper presented at SOS 2010, we developed a framework for big-step semantics for interactive input-output in combination with divergence, based on coinductive and mixed inductive-coinductive notions of resumptions, evaluation and termination-sensitive weak bisimilarity. In contrast to standard inductively defined big-step semantics, this framework handles divergence properly; in particular, runs that produce some observable effects and then diverge, are not "lost". Here we scale this approach for shared-variable concurrency on a simple example language. We develop the metatheory of our se…
We look at the operational semantics of languages with interactive I/O through the glasses of constructive type theory. Following on from our earlier work on coinductive trace-based semantics for While, we define several big-step semantics for While with interactive I/O, based on resumptions and termination-sensitive weak bisimilarity. These require nesting inductive definitions in coinductive definitions, which is interesting both mathematically and from the point-of-view of implementation in a proof assistant. After first defining a basic semantics of statements in terms of resumptions with …
We revisit the standard bisimulation equalities in process models free of the restriction operator. As is well-known, in general the weak bisimilarity is coarser than the strong bisimilarity because it abstracts from internal actions. In absence of restriction, those internal actions become somewhat visible, so one might wonder if the weak bisimilarity is still 'weak'. We show that in both CCScore (i.e., Milner's standard CCS without $τ$-prefix, summation and relabelling) and its higher-order variant (named HOCCScore), the weak bisimilarity indeed remains weak, i.e., still strictly coarser tha…
In 1987 Ernst-Rüdiger Olderog provided an operational Petri net semantics for a subset of CCSP, the union of Milner's CCS and Hoare's CSP. It assigns to each process term in the subset a labelled, safe place/transition net. To demonstrate the correctness of the approach, Olderog established agreement (1) with the standard interleaving semantics of CCSP up to strong bisimulation equivalence, and (2) with standard denotational interpretations of CCSP operators in terms of Petri nets up to a suitable semantic equivalence that fully respects the causal structure of nets. For the latter he employed…
First, we extend Leifer-Milner RPO theory, by giving general conditions to obtain IPO labelled transition systems (and bisimilarities) with a reduced set of transitions, and possibly finitely branching. Moreover, we study the weak variant of Leifer-Milner theory, by giving general conditions under which the weak bisimilarity is a congruence. Then, we apply such extended RPO technique to the lambda-calculus, endowed with lazy and call by value reduction strategies. We show that, contrary to process calculi, one can deal directly with the lambda-calculus syntax and apply Leifer-Milner technique …
An open problem posed by Milner asks for a proof that a certain axiomatisation, which Milner showed is sound with respect to bisimilarity for regular expressions, is also complete. One of the main difficulties of the problem is the lack of a full Kleene theorem, since there are automata that can not be specified, up to bisimilarity, by an expression. Grabmayer and Fokkink (2020) characterise those automata that can be expressed by regular expressions without the constant 1, and use this characterisation to give a positive answer to Milner's question for this subset of expressions. In this pape…
Description logics (DLs) are a suitable formalism for representing knowledge about domains in which objects are described not only by attributes but also by binary relations between objects. Fuzzy extensions of DLs can be used for such domains when data and knowledge about them are vague and imprecise. One of the possible ways to specify classes of objects in such domains is to use concepts in fuzzy DLs. As DLs are variants of modal logics, indiscernibility in DLs is characterized by bisimilarity. The bisimilarity relation of an interpretation is the largest auto-bisimulation of that interpret…
We introduce the wire calculus. Its dynamic features are inspired by Milner's CCS: a unary prefix operation, binary choice and a standard recursion construct. Instead of an interleaving parallel composition operator there are operators for synchronisation along a common boundary and non-communicating parallel composition. The (operational) semantics is a labelled transition system obtained with SOS rules. Bisimilarity is a congruence with respect to the operators of the language. Quotienting terms by bisimilarity results in a compact closed category.
There exist two notions of equivalence of behavior between states of a Labelled Markov Process (LMP): state bisimilarity and event bisimilarity. The first one can be considered as an appropriate generalization to continuous spaces of Larsen and Skou's probabilistic bisimilarity, while the second one is characterized by a natural logic. C. Zhou expressed state bisimilarity as the greatest fixed point of an operator $\mathcal{O}$, and thus introduced an ordinal measure of the discrepancy between it and event bisimilarity. We call this ordinal the "Zhou ordinal" of $\mathbb{S}$, $\mathfrak{Z}(\ma…
The unique solution of contractions is a proof technique for bisimilarity that overcomes certain syntactic constraints of Milner's "unique solution of equations" technique. The paper presents an overview of a rather comprehensive formalisation of the core of the theory of CCS in the HOL theorem prover (HOL4), with a focus towards the theory of unique solutions of contractions. (The formalisation consists of about 20,000 lines of proof scripts in Standard ML.) Some refinements of the theory itself are obtained. In particular we remove the constraints on summation, which must be weakly-guarded, …
Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For `1-free' regular expressions, their process interpretations sa…
Milner (1984) introduced a process semantics for regular expressions as process graphs. Unlike for the language semantics, where every regular (that is, DFA-accepted) language is the interpretation of some regular expression, there are finite process graphs that are not bisimilar to the process interpretation of any regular expression. For reasoning about graphs that are expressible by regular expressions modulo bisimilarity it is desirable to have structural representations of process graphs in the image of the interpretation. For '1-free' regular expressions, their process interpretations sa…
State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the particular case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time -- branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in …
