 Research
 Open Access
An algebra of reversible computation
 Yong Wang^{1}Email author
 Received: 13 April 2016
 Accepted: 6 September 2016
 Published: 26 September 2016
Abstract
We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules: basic reversible processes algebra, algebra of reversible communicating processes, recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.
Keywords
 Reversible computation
 Process algebra
 Algebra of communicating processes
 Axiomatization
Background
Reversible computation (Perumalla 2013) has gained more and more attention in many application areas, such as the modeling of biochemical systems, program debugging and testing, and also quantum computing. For the excellent properties reversible computing has, it will be exploited in many computing devices in the future.
There are several research works on reversible computation. Abramsky maps functional programs into reversible automata (Abramsky 2005). Danos and Krivine’s reversible RCCS (Danos and Krivine 2005) uses the concept of thread to reverse a CCS (Milner 1989; Milner et al. 1992) process. Reversible CCS (RCCS) has been proposed as a first causalconsistent reversible calculus. It introduces the idea of attaching memories to threads in order to keep the history of the computation. Boudol and Castellani (1988, 1994) compare three different noninterleaving models for CCS: proved transition systems, event structures and Petri nets. Phillips and Ulidowski’s CCSK (Phillips 2007; Ulidowski et al. 2014; Phillips and Ulidowski 2012) formulates a procedure for converting operators of standard algebraic process calculi such as CCS into reversible operators, while preserving their operational semantics. CCSK defines the socalled forward–reverse bisimulation and show that it is preserved by all reversible operators. CCSK is the extension of CCS for a general reversible process calculus. The main novelty of CCSK is that the structure of processes is not consumed, but simply annotated when they are executed. This is obtained by making all the rules defining the semantics static. Thus, no memories are needed. And other efforts on reversible computations, such as reversibility on pi (Lanese et al. 2010, 2011, 2013), reversibility and compensation (Lanese et al. 2012), reversibility and faulttolerances (Perumalla and Park 2013), and reversibility in massive concurrent systems (Cardelli and Laneve 2011). And the recently quantitative analysis of concurrent reversible computations (Marin and Rossi 2015).
In process algebra (Baeten 2005), ACP (Fokkink 2007) can be treated as a refinement of CCS (Milner 1989; Milner et al. 1992). CCSK uses the socalled communication key to mark the histories of an atomic action (called past action) and remains the structural operational semantics. We are inspired by the way of CCSK: is there an axiomatic algebra to refine CCSK, just like the relation to ACP and CCS? We do it along the way paved by CCSK and ACP, and lead to a new reversible axiomatic algebra, we called it as reversible ACP (RACP).
 1.
It has more concise structural operation semantics for forward transitions and reverse transitions, without more predicates, such as standard process predicate and freshness predicate.
 2.
It has four extendible modules, basic reversible processes algebra (BRPA), algebra of reversible communicating processes (ARCP), recursion and abstraction. While in CCSK, recursion and abstraction are not concerned.
 3.
In comparison to ACP, it is almost a brand new algebra for reversible computation which has the same advantages of ACP, such as modularity, axiomatization, etc. Firstly, in RACP, the alternative composition is replaced by choice composition, since in reversible computing, all choice branches should be retained. Secondly, the parallel operator cannot be captured by an interleaving semantics. Thirdly, more importantly to establish a full axiomatization, all the atomic actions are distinct, the same atomic action in different branches (including choice branches and parallel branches) will be deemed as the same one atomic action. Also autoconcurrency is out of scope for our work here.
The paper is organized as follows. In section “Preliminaries”, some basic concepts related to equational logic, structural operational semantics and process algebra ACP are introduced. The BRPA is introduced in section “BRPA: basic reversible process algebra”, ARCP is introduced in section “ARCP: algebra of reversible communicating processes”, recursion is introduced in section “Recursion”, and abstraction is introduced in section “Abstraction”. An application of RACP is introduced in section “Verification for business protocols with compensation support”. We discuss the extensions of RACP in section “Extensions”. Finally, we conclude this paper in section “Conclusions”.
Preliminaries
For convenience of the reader, we introduce some basic concepts about equational logic, structural operational semantics and process algebra ACP (please refer to Plotkin 1981, Fokkink 2007 for more details).
Equational logic
We introduce some basic concepts related to equational logic briefly, including signature, term, substitution, axiomatization, equality relation, model, term rewriting system, rewrite relation, normal form, termination, weak confluence and several conclusions. These concepts originate from Fokkink (2007), and are introduced briefly as follows. About the details, please see Fokkink (2007).
Definition 1
(Signature) A signature \(\varSigma \) consists of a finite set of function symbols (or operators) \(f,g,\ldots \), where each function symbol f has an arity ar(f), being its number of arguments. A function symbol a, b, c, …of arity zero is called a constant, a function symbol of arity one is called unary, and a function symbol of arity two is called binary.
Definition 2
(Term) Let \(\varSigma \) be a signature. The set \({\mathbb {T}}(\varSigma )\) of (open) terms s, t, u, …over \(\varSigma \) is defined as the least set satisfying: (1) each variable is in \({\mathbb {T}}(\varSigma )\); (2) if \(f\in \varSigma \) and \(t_1,\ldots ,t_{ar(f)}\in {\mathbb {T}}(\varSigma )\), then \(f(t_1,\ldots ,t_{ar(f)}\in {\mathbb {T}}(\varSigma ))\). A term is closed if it does not contain variables. The set of closed terms is denoted by \({\mathcal {T}}(\varSigma )\).
Definition 3
(Substitution) Let \(\varSigma \) be a signature. A substitution is a mapping \(\sigma \) from variables to the set \({\mathbb {T}}(\varSigma )\) of open terms. A substitution extends to a mapping from open terms to open terms: the term \(\sigma (t)\) is obtained by replacing occurrences of variables x in t by \(\sigma (x)\). A substitution \(\sigma \) is closed if \(\sigma (x)\in {\mathcal {T}}(\varSigma )\) for all variables x.
Definition 4
(Axiomatization) An axiomatization over a signature \(\varSigma \) is a finite set of equations, called axioms, of the form \(s=t\) with \(s,t\in {\mathbb {T}}(\varSigma )\).
Definition 5
(Equality relation) An axiomatization over a signature \(\varSigma \) induces a binary equality relation \(=\) on \({\mathbb {T}}(\varSigma )\) as follows. (1) (Substitution) If \(s=t\) is an axiom and \(\sigma \) a substitution, then \(\sigma (s)=\sigma (t)\). (2) (Equivalence) The relation = is closed under reflexivity, symmetry, and transitivity. (3) (Context) The relation = is closed under contexts: if \(t=u\) and f is a function symbol with \(ar(f)>0\), then \(f(s_1,\ldots ,s_{i1},t,s_{i+1}, \ldots ,s_{ar(f)})=f(s_1,\ldots ,s_{i1},u,s_{i+1},\ldots ,s_{ar(f)})\).
Definition 6
(Model) Assume an axiomatization \({\mathcal {E}}\) over a signature \(\varSigma \), which induces an equality relation =. A model for \({\mathcal {E}}\) consists of a set \({\mathcal {M}}\) together with a mapping \(\phi : {\mathcal {T}}(\varSigma )\rightarrow {\mathcal {M}}\). (1) \(({\mathcal {M}},\phi )\) is sound for \({\mathcal {E}}\) if \(s=t\) implies \(\phi (s)\equiv \phi (t)\) for \(s,t\in {\mathcal {T}}(\varSigma )\); (2) \(({\mathcal {M}},\phi )\) is complete for \({\mathcal {E}}\) if \(\phi (s)\equiv \phi (t)\) implies \(s=t\) for \(s,t\in {\mathcal {T}}(\varSigma )\).
Definition 7
(Term rewriting system) Assume a signature \(\varSigma \). A rewrite rule is an expression \(s\rightarrow t\) with \(s,t\in {\mathbb {T}}(\varSigma )\), where: (1) the lefthand side s is not a single variable; (2) all variables that occur at the righthand side t also occur in the lefthand side s. A term rewriting system (TRS) is a finite set of rewrite rules.
Definition 8
(Rewrite relation) A TRS over a signature \(\varSigma \) induces a onestep rewrite relation \(\rightarrow \) on \({\mathbb {T}}(\varSigma )\) as follows. (1) (Substitution) If \(s\rightarrow t\) is a rewrite rule and \(\sigma \) a substitution, then \(\sigma (s)\rightarrow \sigma (t)\). (2) (Context) The relation \(\rightarrow \) is closed under contexts: if \(t\rightarrow u\) and f is a function symbol with \(ar(f)>0\), then \(f(s_1,\ldots ,s_{i1},t,s_{i+1},\ldots ,s_{ar(f)})\rightarrow f(s_1,\ldots ,s_{i1},u,s_{i+1},\ldots ,s_{ar(f)})\). The rewrite relation \(\rightarrow ^*\) is the reflexive transitive closure of the onestep rewrite relation \(\rightarrow \): (1) if \(s\rightarrow t\), then \(s\rightarrow ^* t\); (2) \(t\rightarrow ^* t\); (3) if \(s\rightarrow ^* t\) and \(t\rightarrow ^* u\), then \(s\rightarrow ^* u\).
Definition 9
(Normal form) A term is called a normal form for a TRS if it cannot be reduced by any of the rewrite rules.
Definition 10
(Termination) A TRS is terminating if it does not induce infinite reductions \(t_0\rightarrow t_1\rightarrow t_2\rightarrow \cdots \).
Definition 11
(Weak confluence) A TRS is weakly confluent if for each pair of onestep reductions \(s\rightarrow t_1\) and \(s\rightarrow t_2\), there is a term u such that \(t_1\rightarrow ^* u\) and \(t_2\rightarrow ^* u\).
Theorem 1
(Newman’s lemma) If a TRS is terminating and weakly confluent, then it reduces each term to a unique normal form.
Definition 12
(Commutativity and associativity) Assume an axiomatization \({\mathcal {E}}\). A binary function symbol f is commutative if \({\mathcal {E}}\) contains an axiom \(f(x,y)=f(y,x)\) and associative if \({\mathcal {E}}\) contains an axiom \(f(f(x,y),z)=f(x,f(y,z))\).
Definition 13
(Convergence) A pair of terms s and t is said to be convergent if there exists a term u such that \(s\rightarrow ^* u\) and \(t\rightarrow ^* u\).
Axiomatizations can give rise to TRSs that are not weakly confluent, which can be remedied by Knuth–Bendix completion (Knuth and Bendix 1970). It determines overlaps in left hand sides of rewrite rules, and introduces extra rewrite rules to join the resulting right hand sides, which are called critical pairs.
Theorem 2
A TRS is weakly confluent if and only if all its critical pairs are convergent.
Structural operational semantics
The concepts about structural operational semantics include labelled transition system (LTS), transition system specification (TSS), transition rule and its source, sourcedependent, conservative extension, fresh operator, panth format, congruence, bisimulation, etc. These concepts are coming from Fokkink (2007), and are introduced briefly as follows. About the details, please see Plotkin (1981). Also, to support reversible computation, we introduce a new kind of bisimulation called forward–reverse bisimulation (FR bisimulation) which occurred in De Nicola et al. (1990) and Phillips (2007).
We assume a nonempty set S of states, a finite, nonempty set of transition labels A and a finite set of predicate symbols.
Definition 14
(Labeled transition system) A transition is a triple \((s,a,s')\) with \(a\in A\), or a pair (s, P) with P a predicate, where \(s,s'\in S\). A labeled transition system (LTS) is possibly infinite set of transitions. An LTS is finitely branching if each of its states has only finitely many outgoing transitions.
Definition 15
(Transition system specification) A transition rule \(\rho \) is an expression of the form \(\frac{H}{\pi }\), with H a set of expressions \(t\xrightarrow {a}t'\) and tP with \(t,t'\in {\mathbb {T}}(\varSigma )\), called the (positive) premises of \(\rho \), and \(\pi \) an expression \(t\xrightarrow {a}t'\) or tP with \(t,t'\in {\mathbb {T}}(\varSigma )\), called the conclusion of \(\rho \). The lefthand side of \(\pi \) is called the source of \(\rho \). A transition rule is closed if it does not contain any variables. A transition system specification (TSS) is a (possible infinite) set of transition rules.
Definition 16
(Proof) A proof from a TSS T of a closed transition rule \(\frac{H}{\pi }\) consists of an upwardly branching tree in which all upward paths are finite, where the nodes of the tree are labelled by transitions such that: (1) the root has label \(\pi \); (2) if some node has label l, and K is the set of labels of nodes directly above this node, then (a) either K is the empty set and \(l\in H\), (b) or \(\frac{K}{l}\) is a closed substitution instance of a transition rule in T.
Definition 17
(Generated LTS) We define that the LTS generated by a TSS T consists of the transitions \(\pi \) such that \(\frac{\emptyset }{\pi }\) can be proved from T.
Definition 18
A set N of expressions \(t\nrightarrow ^{a}\) and \(t\lnot P\) (where t ranges over closed terms, a over A and P over predicates) hold for a set \({\mathcal {S}}\) of transitions, denoted by \({\mathcal {S}}\vDash N\), if: (1) for each \(t\nrightarrow ^{a} \in N\) we have that \(t\xrightarrow {a}t' \notin {\mathcal {S}}\) for all \(t'\in {\mathcal {T}}(\varSigma )\); (2) for each \(t\lnot P\in N\) we have that \(tP \notin {\mathcal {S}}\).
Definition 19
(Threevalued stable model) A pair \(\langle {\mathcal {C}},\mathcal {U}\rangle \) of disjoint sets of transitions is a threevalued stable model for a TSS T if it satisfies the following two requirements: (1) a transition \(\pi \) is in \({\mathcal {C}}\) if and only if T proves a closed transition rule \(\frac{N}{\pi }\) where N contains only negative premises and \({\mathcal {C}}\cup \mathcal {U}\vDash N\); (2) a transition \(\pi \) is in \({\mathcal {C}}\cup \mathcal {U}\) if and only if T proves a closed transition rule \(\frac{N}{\pi }\) where N contains only negative premises and \({\mathcal {C}}\vDash N\).
Definition 20
(Ordinal number) The ordinal numbers are defined inductively by: (1) 0 is the smallest ordinal number; (2) each ordinal number \(\alpha \) has a successor \(\alpha + 1\); (3) each sequence of ordinal number \(\alpha< \alpha + 1< \alpha + 2 < \cdots \) is capped by a limit ordinal \(\lambda \).
Definition 21
(Positive after reduction) A TSS is positive after reduction if its least threevalued stable model does not contain unknown transitions.
Definition 22
(Stratification) A stratification for a TSS is a weight function \(\phi \) which maps transitions to ordinal numbers, such that for each transition rule \(\rho \) with conclusion \(\pi \) and for each closed substitution \(\sigma \): (1) for positive premises \(t\xrightarrow {a}t'\) and tP of \(\rho , \phi (\sigma (t)\xrightarrow {a}\sigma (t'))\le \phi (\sigma (\pi ))\) and \(\phi (\sigma (t)P\le \phi (\sigma (\pi )))\), respectively; (2) for negative premise \(t\nrightarrow ^{a}\) and \(t\lnot P\) of \(\rho , \phi (\sigma (t)\xrightarrow {a}t')< \phi (\sigma (\pi ))\) for all closed terms \(t'\) and \(\phi (\sigma (t)P < \phi (\sigma (\pi )))\), respectively.
Theorem 3
If a TSS allows a stratification, then it is positive after reduction.
Definition 23
(Process graph) A process (graph) p is an LTS in which one state s is elected to be the root. If the LTS contains a transition \(s\xrightarrow {a} s'\), then \(p\xrightarrow {a} p'\) where \(p'\) has root state \(s'\). Moreover, if the LTS contains a transition sP, then pP. (1) A process \(p_0\) is finite if there are only finitely many sequences \(p_0\xrightarrow {a_1}p_1\xrightarrow {a_2}\cdots \xrightarrow {a_k} P_k\). (2) A process \(p_0\) is regular if there are only finitely many processes \(p_k\) such that \(p_0\xrightarrow {a_1}p_1\xrightarrow {a_2}\cdots \xrightarrow {a_k} P_k\).
Definition 24
(Reverse transition) There are two processes p and \(p'\), two transitions \(p \xrightarrow {a} p'\) and \(p' \mathop {\twoheadrightarrow}{a[m]} p\), the transition \(p' \mathop {\twoheadrightarrow}{a[m]} p\) is called reverse transition of \(p \xrightarrow {a} p'\), and the transition \(p \xrightarrow {a} p'\) is called forward transition. If \(p \xrightarrow {a} p'\) then \(p' \mathop {\twoheadrightarrow}{a[m]} p\), the forward transition \(p \xrightarrow {a} p'\) is reversible. Where a[m] is a kind of special action constant \(a[m]\in A \times {\mathcal {K}}, {\mathcal {K}}\subseteq {\mathbb {N}}\), called the histories of an action a, and \(m\in {\mathcal {K}}\).
Definition 25
(Bisimulation) A bisimulation relation \({\mathcal {B}}\) is a binary relation on processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then \(q\xrightarrow {a}q'\) with \(p'{\mathcal {B}}q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then \(p\xrightarrow {a}p'\) with \(p'{\mathcal {B}}q'\); (3) if \(p{\mathcal {B}}q\) and pP, then qP; (4) if \(p{\mathcal {B}}q\) and qP, then pP. Two processes p and q are bisimilar, denoted by \(p\underline{\leftrightarrow } q\), if there is a bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 26
(Forward–reverse bisimulation) A forward–reverse (FR) bisimulation relation \({\mathcal {B}}\) is a binary relation on processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then \(q\xrightarrow {a}q'\) with \(p'{\mathcal {B}}q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then \(p\xrightarrow {a}p'\) with \(p'{\mathcal {B}}q'\); (3)if \(p{\mathcal {B}}q\) and \(p\mathop {\twoheadrightarrow}{a[m]}p'\) then \(q\mathop {\twoheadrightarrow}{a[m]}q'\) with \(p'{\mathcal {B}}q'\); (4) if \(p{\mathcal {B}}q\) and \(q\mathop {\twoheadrightarrow}{a[m]}q'\) then \(p\mathop {\twoheadrightarrow}{a[m]}p'\) with \(p'{\mathcal {B}}q'\); (5) if \(p{\mathcal {B}}q\) and pP, then qP; (6) if \(p{\mathcal {B}}q\) and qP, then pP. Two processes p and q are FR bisimilar, denoted by \(p\underline{\leftrightarrow }^{fr} q\), if there is a FR bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 27
(Congruence) Let \(\varSigma \) be a signature. An equivalence relation \({\mathcal {B}}\) on \({\mathcal {T}}(\varSigma )\) is a congruence if for each \(f\in \varSigma \), if \(s_i{\mathcal {B}}t_i\) for \(i\in \{1,\ldots ,ar(f)\}\), then \(f(s_1,\ldots ,s_{ar(f)}){\mathcal {B}}f(t_1,\ldots ,t_{ar(f)})\).
Definition 28
(Panth format) A transition rule \(\rho \) is in panth format if it satisfies the following three restrictions: (1) for each positive premise \(t\xrightarrow {a} t'\) of \(\rho \), the righthand side \(t'\) is single variable; (2) the source of \(\rho \) contains no more than one function symbol; (3) there are no multiple occurrences of the same variable at the righthand sides of positive premises and in the source of \(\rho \). A TSS is said to be in panth format if it consists of panth rules only.
Theorem 4
If a TSS is positive after reduction and in panth format, then the bisimulation equivalence that it induces is a congruence.
Definition 29
(Branching bisimulation) A branching bisimulation relation \({\mathcal {B}}\) is a binary relation on the collection of processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then either \(a\equiv \tau \) and \(p'{\mathcal {B}}q\) or there is a sequence of (zero or more) \(\tau \)transitions \(q\xrightarrow {\tau }\cdots \xrightarrow {\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0\xrightarrow {a}q'\) with \(p'{\mathcal {B}}q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then either \(a\equiv \tau \) and \(p{\mathcal {B}}q'\) or there is a sequence of (zero or more) \(\tau \)transitions \(p\xrightarrow {\tau }\cdots \xrightarrow {\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0\xrightarrow {a}p'\) with \(p'{\mathcal {B}}q'\); (3) if \(p{\mathcal {B}}q\) and pP, then there is a sequence of (zero or more) \(\tau \)transitions \(q\xrightarrow {\tau }\cdots \xrightarrow {\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0P\); (4) if \(p{\mathcal {B}}q\) and qP, then there is a sequence of (zero or more) \(\tau \)transitions \(p\xrightarrow {\tau }\cdots \xrightarrow {\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0P\). Two processes p and q are branching bisimilar, denoted by \(p\underline{\leftrightarrow }_b q\), if there is a branching bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 30
(Branching forward–reverse bisimulation) A branching forward–reverse (FR) bisimulation relation \({\mathcal {B}}\) is a binary relation on the collection of processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then either \(a\equiv \tau \) and \(p'{\mathcal {B}}q\) or there is a sequence of (zero or more) \(\tau \)transitions \(q\xrightarrow {\tau }\cdots \xrightarrow {\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0\xrightarrow {a}q'\) with \(p'{\mathcal {B}}q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then either \(a\equiv \tau \) and \(p{\mathcal {B}}q'\) or there is a sequence of (zero or more) \(\tau \)transitions \(p\xrightarrow {\tau }\cdots \xrightarrow {\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0\xrightarrow {a}p'\) with \(p'{\mathcal {B}}q'\); (3) if \(p{\mathcal {B}}q\) and pP, then there is a sequence of (zero or more) \(\tau \)transitions \(q\xrightarrow {\tau }\cdots \xrightarrow {\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0P\); (4) if \(p{\mathcal {B}}q\) and qP, then there is a sequence of (zero or more) \(\tau \)transitions \(p\xrightarrow {\tau }\cdots \xrightarrow {\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0P\); (5) if \(p{\mathcal {B}}q\) and \(p\mathop {\twoheadrightarrow}{a[m]}p'\) then either \(a\equiv \tau \) and \(p'{\mathcal {B}}q\) or there is a sequence of (zero or more) \(\tau \)transitions \(q\mathop {\twoheadrightarrow }\limits ^{\tau }\ldots \mathop {\twoheadrightarrow }\limits ^{\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0\mathop {\twoheadrightarrow}{a[m]}q'\) with \(p'{\mathcal {B}}q'\); (6) if \(p{\mathcal {B}}q\) and \(q\mathop {\twoheadrightarrow}{a[m]}q'\) then either \(a\equiv \tau \) and \(p{\mathcal {B}}q'\) or there is a sequence of (zero or more) \(\tau \)transitions \(p\mathop {\twoheadrightarrow }\limits ^{\tau }\ldots \mathop {\twoheadrightarrow }\limits ^{\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0\mathop {\twoheadrightarrow}{a[m]}p'\) with \(p'{\mathcal {B}}q'\); (7) if \(p{\mathcal {B}}q\) and pP, then there is a sequence of (zero or more) \(\tau \)transitions \(q\mathop {\twoheadrightarrow }\limits ^{\tau }\ldots \mathop {\twoheadrightarrow }\limits ^{\tau }q_0\) such that \(p{\mathcal {B}}q_0\) and \(q_0P\); (8) if \(p{\mathcal {B}}q\) and qP, then there is a sequence of (zero or more) \(\tau \)transitions \(p\mathop {\twoheadrightarrow }\limits ^{\tau }\ldots \mathop {\twoheadrightarrow }\limits ^{\tau }p_0\) such that \(p_0{\mathcal {B}}q\) and \(p_0P\). Two processes p and q are branching FR bisimilar, denoted by \(p{\underline{\leftrightarrow }}^{fr}_b q\), if there is a branching FR bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 31
(Rooted branching bisimulation) A rooted branching bisimulation relation \({\mathcal {B}}\) is a binary relation on processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then \(q\xrightarrow {a}q'\) with \(p'\underline{\leftrightarrow }_b q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then \(p\xrightarrow {a}p'\) with \(p'\underline{\leftrightarrow }_b q'\); (3) if \(p{\mathcal {B}}q\) and pP, then qP; (4) if \(p{\mathcal {B}}q\) and qP, then pP. Two processes p and q are rooted branching bisimilar, denoted by \(p\underline{\leftrightarrow }_{rb} q\), if there is a rooted branching bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 32
(Rooted branching forward–reverse bisimulation) A rooted branching forward–reverse (FR) bisimulation relation \({\mathcal {B}}\) is a binary relation on processes such that: (1) if \(p{\mathcal {B}}q\) and \(p\xrightarrow {a}p'\) then \(q\xrightarrow {a}q'\) with \(p'{\underline{\leftrightarrow }}^{fr}_b q'\); (2) if \(p{\mathcal {B}}q\) and \(q\xrightarrow {a}q'\) then \(p\xrightarrow {a}p'\) with \(p'{\underline{\leftrightarrow }}^{fr}_b q'\); (3) if \(p{\mathcal {B}}q\) and \(p\mathop {\twoheadrightarrow}{a[m]}p'\) then \(q\mathop {\twoheadrightarrow}{a[m]}q'\) with \(p'{\underline{\leftrightarrow }}^{fr}_b q'\); (4) if \(p{\mathcal {B}}q\) and \(q\mathop {\twoheadrightarrow}{a[m]}q'\) then \(p\mathop {\twoheadrightarrow}{a[m]}p'\) with \(p'{\underline{\leftrightarrow }}^{fr}_b q'\); (5) if \(p{\mathcal {B}}q\) and pP, then qP; (6) if \(p{\mathcal {B}}q\) and qP, then pP. Two processes p and q are rooted branching FR bisimilar, denoted by \(p{\underline{\leftrightarrow }}^{fr}_{rb} q\), if there is a rooted branching FR bisimulation relation \({\mathcal {B}}\) such that \(p{\mathcal {B}}q\).
Definition 33
(Lookahead) A transition rule contains lookahead if a variable occurs at the lefthand side of a premise and at the righthand side of a premise of this rule.
Definition 34
(Patience rule) A patience rule for the ith argument of a function symbol f is a panth rule of the form
Definition 35
(RBB cool format) A TSS T is in RBB cool format if the following requirements are fulfilled. (1) T consists of panth rules that do not contain lookahead. (2) Suppose a function symbol f occurs at the righthand side the conclusion of some transition rule in T. Let \(\rho \in T\) be a nonpatience rule with source \(f(x_1,\ldots ,x_{ar(f)})\). Then for \(i\in \{1,\ldots ,ar(f)\}, x_i\) occurs in no more than one premise of \(\rho \), where this premise is of the form \(x_iP\) or \(x_i\xrightarrow {a}y\) with \(a\ne \tau \). Moreover, if there is such a premise in \(\rho \), then there is a patience rule for the ith argument of f in T.
Theorem 5
If a TSS is positive after reduction and in RBB cool format, then the rooted branching bisimulation equivalence that it induces is a congruence.
Definition 36
(Conservative extension) Let \(T_0\) and \(T_1\) be TSSs over signatures \(\varSigma _0\) and \(\varSigma _1\), respectively. The TSS \(T_0\oplus T_1\) is a conservative extension of \(T_0\) if the LTSs generated by \(T_0\) and \(T_0\oplus T_1\) contain exactly the same transitions \(t\xrightarrow {a}t'\) and tP with \(t\in {\mathcal {T}}(\varSigma _0)\).
Definition 37
(Sourcedependency) The sourcedependent variables in a transition rule of \(\rho \) are defined inductively as follows: (1) all variables in the source of \(\rho \) are sourcedependent; (2) if \(t\xrightarrow {a}t'\) is a premise of \(\rho \) and all variables in t are sourcedependent, then all variables in \(t'\) are sourcedependent. A transition rule is sourcedependent if all its variables are. A TSS is sourcedependent if all its rules are.
Definition 38
(Freshness) Let \(T_0\) and \(T_1\) be TSSs over signatures \(\varSigma _0\) and \(\varSigma _1\), respectively. A term in \({\mathbb {T}}(T_0\oplus T_1)\) is said to be fresh if it contains a function symbol from \(\varSigma _1{\setminus}\varSigma _0\). Similarly, a transition label or predicate symbol in \(T_1\) is fresh if it does not occur in \(T_0\).
Theorem 6
Let \(T_0\) and \(T_1\) be TSSs over signatures \(\varSigma _0\) and \(\varSigma _1\),respectively, where \(T_0\) and \(T_0\oplus T_1\) are positive after reduction. Under the following conditions, \(T_0\oplus T_1\) is a conservative extension of \(T_0\). (1) \(T_0\) is sourcedependent. (2) For each \(\rho \in T_1\), either the source of \(\rho \) is fresh, or \(\rho \) has a premise of the form \(t\xrightarrow {a}t'\) or tP, where \(t\in {\mathbb {T}}(\varSigma _0)\), all variables in t occur in the source of \(\rho \) and \(t', a\) or P is fresh.
Process algebra: ACP
ACP (Fokkink 2007) is a kind of process algebra which focuses on the specification and manipulation of process terms by use of a collection of operator symbols. In ACP, there are several kind of operator symbols, such as basic operators to build finite processes (called BPA), communication operators to express concurrency (called PAP), deadlock constants and encapsulation enable us to force actions into communications (called ACP), liner recursion to capture infinite behaviors (called ACP with linear recursion), the special constant silent step and abstraction operator (called \(ACP_{\tau }\) with guarded linear recursion) allows us to abstract away from internal computations.
Bisimulation or rooted branching bisimulation based structural operational semantics is used to formally provide each process term used the above operators and constants with a process graph. The axiomatization of ACP (according the above classification of ACP, the axiomatizations are \({\mathcal {E}}_{\text {BPA}}, {\mathcal {E}}_{\text {PAP}}, {\mathcal {E}}_{\text {ACP}}, {\mathcal {E}}_{\text {ACP}}\) + RDP (Recursive Definition Principle) + RSP (Recursive Specification Principle), \({\mathcal {E}}_{\text {ACP}_\tau }\) + RDP + RSP + CFAR (Cluster Fair Abstraction Rule) respectively) imposes an equation logic on process terms, so two process terms can be equated if and only if their process graphs are equivalent under the semantic model.
ACP can be used to formally reason about the behaviors, such as processes executed sequentially and concurrently by use of its basic operator, communication mechanism, and recursion, desired external behaviors by its abstraction mechanism, and so on.
ACP is organized by modules and can be extended with fresh operators to express more properties of the specification for system behaviors. These extensions are required both the equational logic and the structural operational semantics to be extended. Then the extension can use the whole outcomes of ACP, such as its concurrency, recursion, abstraction, etc.
BRPA: basic reversible process algebra
In the following, the variables \(x,x',y,y',z,z'\) range over the collection of process terms, the variables \(\upsilon ,\omega \) range over the set A of atomic actions, \(a,b\in A, s,s',t,t'\) are closed items, \(\tau \) is the special constant silent step, \(\delta \) is the special constant deadlock. We define a kind of special action constant \(a[m]\in A \times {\mathcal {K}}\) where \({\mathcal {K}}\subseteq {\mathbb {N}}\), called the histories of an action a, denoted by \(a[m],a[n],\ldots \) where \(m,n\in {\mathcal {K}}\). Let \(A=A\cup \{A\times {\mathcal {K}}\}\).
BRPA includes three kind of operators: the execution of atomic action a, the choice composition operator + and the sequential composition operator \(\cdot \). Each finite process can be represented by a closed term that is built from the set A of atomic actions or histories of an atomic action, the choice composition operator +, and the sequential composition operator \(\cdot \). The collection of all basic process terms is called Basic Reversible Process Algebra (BRPA), which is abbreviated to BRPA.
Transition rules of BRPA

The first transition rule says that each atomic action \(\upsilon \) can execute successfully, and leads to a history \(\upsilon [m]\). The forward transition rule \(\frac{}{\upsilon \xrightarrow {\upsilon }\upsilon [m]}\) implies a successful forward execution.

The next four transition rules say that \(s+t\) can execute only one branch, that is, it can execute either s or t, but the other branch remains.

The next four transition rules say that \(s+t\) can execute both branches, only by executing the same atomic actions. When one branch s or t is forward executed successfully, we define \(s+t\) is forward executed successfully.

The last four transition rules say that \(s\cdot t\) can execute sequentially, that is, it executes s in the first and leads to a successful history, after successful execution of s, then execution of t follows. When both s and t are forward executed successfully, we define \(s\cdot t\) is forward executed successfully.

The first transition rule says that each history of an atomic action \(\upsilon [m]\) can reverse successfully, and leads to an atomic action \(\upsilon \). Similarly, the reverse transition rule \(\frac{}{\upsilon [m]\mathop {\twoheadrightarrow }\limits ^{\upsilon [m]}\upsilon }\) implies a successful reverse.

The next four transition rules say that \(s+t\) can reverse only one branch, that is, it can reverse either s or t, but the other branch remains.

The next four transition rules say that \(s+t\) can reverse both branches, only by executing the same histories of atomic actions. When one branch s or t is reversed successfully, we define \(s+t\) is reversed successfully.

The last four transition rules say that \(s\cdot t\) can reverse sequentially, that is, it reverses t in the first and leads to a successful atomic action, after successful reverse of t, then reverse of s follows. When both s and t are reversed successfully, we define \(s\cdot t\) is reversed successfully.
Axiomatization for BRPA
Axioms for BRPA
No.  Axiom 

RA1  \(x + y = y + x\) 
RA2  \(x + x = x\) 
RA3  \((x + y) + z = x + (y + z)\) 
RA4  \(x \cdot (y + z) = x\cdot y + x\cdot z\) 
RA5  \((x\cdot y)\cdot z = x\cdot (y\cdot z)\) 
The following conclusions can be obtained.
Theorem 7
FR bisimulation equivalence is a congruence with respect to BRPA.
Proof
The forward and reverse TSSs are all in panth format, so FR bisimulation equivalence that they induce is a congruence. \(\square \)
Theorem 8
\({\mathcal {E}}_{\text {BRPA}}\) is sound for BRPA modulo FR bisimulation equivalence.
Proof
Since FR bisimulation is both an equivalence and a congruence for BRPA, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {BRPA}}\) and \(\sigma \) a closed substitution that maps the variable in s and t to basic reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}\sigma (t)\).

RA1 (commutativity of +) says that \(s+t\) and \(t+s\) are all execution branches and are equal modulo FR bisimulation.

RA2 (idempotency of +) is used to eliminate redundant branches.

RA3 (associativity of +) says that \((s+t)+u\) and \(s+(t+u)\) are all execution branches of s, t, u.

RA4 (left distributivity of \(\cdot \)) says that both \(s \cdot (t+u)\) and \(s\cdot t + s\cdot u\) represent the same execution branches. It must be pointed out that the right distributivity of \(\cdot \) does not hold modulo FR bisimulation. For example, \((a+b)\cdot c\xrightarrow {a}(a[m]+b)\cdot c\xrightarrow {c}(a[m]+b)\cdot c[n]\mathop {\twoheadrightarrow }\limits ^{c[n]}(a[m]+b)\cdot c\mathop {\twoheadrightarrow}{a[m]}(a+b)\cdot c;\) while \(a\cdot c + b\cdot c\xrightarrow {a}a[m]\cdot c+b\cdot c\mathop {\nrightarrow }\limits ^{c}\).

RA5 (associativity of \(\cdot \)) says that both \((s\cdot t)\cdot u\) and \(s\cdot (t\cdot u)\) represent forward execution of s followed by t followed by u, or, reverse execution of u followed by t followed by s.
These intuitions can be made rigorous by means of explicit FR bisimulation relations between the left and righthand sides of closed instantiations of the axioms in Table 1. Hence, all such instantiations are sound modulo FR bisimulation equivalence. \(\square \)
Theorem 9
\({\mathcal {E}}_{\text {BRPA}}\) is complete for BRPA modulo FR bisimulation equivalence.
Proof
We refer to Fokkink (2007) for the completeness proof of \({\mathcal {E}}_{\text {BPA}}\).
To prove that \({\mathcal {E}}_{\text {BRPA}}\) is complete for BRPA modulo FR bisilumation equivalence, it means that \(s\underline{\leftrightarrow }^{fr} t\) implies \(s=t\).
We consider basic reversible process terms modulo associativity and commutativity (AC) of the + (RA1,RA2), and this equivalence relation is denoted by \(=_{AC}\). A basic reversible process term s then represents the collection of basic reversible process term t such that \(s=_{AC} t\). Each equivalence class s modulo AC of the + can be represented in the form \(s_1+\cdots +s_k\) with each \(s_i\) either an atomic action or of the form \(t_1\cdot t_2\). We refer to the subterms \(s_1,\ldots ,s_k\) as the summands of s.
Then these rewrite rules are applied to basic reversible process terms modulo AC of the +.
We can see that the TRS is terminating modulo AC of the +.

Consider a summand a of n. Then \(n\xrightarrow {a}a[m]+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\xrightarrow {a}a[m]+u\), meaning that \(n'\) also contains the summand a.

Consider a summand a[m] of n. Then \(n\mathop {\twoheadrightarrow}{a[m]}a+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\mathop {\twoheadrightarrow}{a[m]}a+u\), meaning that \(n'\) also contains the summand a[m].

Consider a summand \(a_1\ldots a_i\ldots a_k\) of n. Then \(n\xrightarrow {a_1}\cdots \xrightarrow {a_i}\cdots \xrightarrow {a_k} a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\xrightarrow {a_1}\cdots \xrightarrow {a_i}\cdots \xrightarrow {a_k} a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]+u\), meaning that \(n'\) also contains the summand \(a_1\ldots a_i\ldots a_k\).

Consider a summand \(a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]\) of n. Then \(n\mathop {\twoheadrightarrow }\limits ^{a_k[m_k]}\cdots \mathop {\twoheadrightarrow }\limits ^{a_i[m_i]}\cdots \mathop {\twoheadrightarrow }\limits ^{a_1[m_1]}a_1\ldots a_i\ldots a_k+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\mathop {\twoheadrightarrow }\limits ^{a_k[m_k]}\cdots \mathop {\twoheadrightarrow }\limits ^{a_i[m_i]}\cdots \mathop {\twoheadrightarrow }\limits ^{a_1[m_1]}a_1\ldots a_i\ldots a_k+u\), meaning that \(n'\) also contains the summand \(a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]\).
Hence, each summand of n is also a summand of \(n'\). Vice versa, each summand of \(n'\) is also a summand of n. In other words, \(n=_{AC} n'\).
Finally, let the basic reversible process terms s and t be FR bisimilar. The TRS is terminating modulo AC of the +, so it reduces s and t to normal forms n and \(n'\), respectively. Since the rewrite rules and equivalence modulo AC of the + can be derived from the axioms, \(s=n\) and \(t=n'\). Soundness of the axioms then yields \(s\underline{\leftrightarrow }^{fr} n\) and \(t\underline{\leftrightarrow }^{fr} n'\), so \(n\underline{\leftrightarrow }^{fr} s\underline{\leftrightarrow }^{fr} t\underline{\leftrightarrow }^{fr} n'\). We showed that \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n=_{AC}n'\). Hence, \(s=n=_{AC} n'=t\). \(\square \)
ARCP: algebra of reversible communicating processes
It is well known that process algebra captures parallelism and concurrency by means of the socalled interleaving pattern in contrast to the socalled true concurrency. ACP uses left merge and communication merge to bridge the gap between the parallel semantics, and sequential semantics. But in reversible computation, Milner’s expansion law modeled by left merge does not hold any more, as pointed out in Phillips (2007). \(a\parallel b \ne a\cdot b + b\cdot a\), because \(a\parallel b\xrightarrow {a}a[m]\parallel b\xrightarrow {b}a[m]\parallel b[n]\) and \(a\cdot b + b\cdot a\mathop {\nrightarrow }\limits ^{a}\). That is, the left merge to capture the asynchronous concurrency in an interleaving fashion will be instead by a real static parallel fashion and the parallel branches cannot be merged. But, the communication merge used to capture synchrony will be retained.
Static parallelism and communication merge
We use a parallel operator \(\parallel \) to represent the whole parallelism semantics, a static parallel \({\text{operator}}\,\) to represent the real parallelism semantics, and a communication merge \(\between \) to represent the synchronisation. We call BRPA extended with the whole parallel operator \(\parallel \), the static parallel \({\text{operator}}\,\) and the communication merge operator \(\between \) Reversible Process Algebra with Parallelism, which is abbreviated to RPAP.
Transition rules of RPAP
The above eight transition rules are reverse transition rules for the static parallel \({\text{operator}}\,\) and say that \(s\mid t\) can reverse in a real parallel pattern. When both s and t are reversed successfully, we define \(s\mid t\) is reversed successfully.
Theorem 10
RPAP is a conservative extension of BRPA.
Proof
Since the TSS of BRPA is sourcedependent, and the transition rules for the static parallel \({\text{operator}}\, \), communication merge \(\between \) contain only a fresh operator in their source, so the TSS of RPAP is a conservative extension of that of BRPA. That means that RPAP is a conservative extension of BRPA. \(\square \)
Theorem 11
FR bisimulation equivalence is a congruence with respect to RPAP.
Proof
The TSSs for RPAP and BRPA are all in panth format, so FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axiomatization for RPAP
Axioms for RPAP
No.  Axiom 

RP1  \(x\parallel y = x\mid y + x\between y\) 
RP2  \(x\mid x = x\) 
RP3  \((x\mid y)\mid z = x\mid (y\mid z)\) 
RP4  \(x \mid (y + z) = x\mid y + x\mid z\) 
RP5  \((x + y) \mid z = x\mid z + y\mid z\) 
RP6  \(x \cdot (y \mid z) = x\cdot y \mid x\cdot z\) 
RP7  \((x \mid y) \cdot z = x\cdot z \mid y\cdot z\) 
RC8  \(\upsilon \between \omega =\gamma (\upsilon ,\omega )\) 
RC9  \(\upsilon [m]\between \omega [m]=\gamma (\upsilon ,\omega )[m]\) 
RC10  \(\upsilon \between (\omega \cdot y) = \gamma (\upsilon ,\omega )\cdot y\) 
RC11  \(\upsilon [m]\between (\omega [m]\cdot y) = \gamma (\upsilon ,\omega )[m]\cdot y\) 
RC12  \((\upsilon \cdot x)\between \omega = \gamma (\upsilon ,\omega )\cdot x\) 
RC13  \((\upsilon [m]\cdot x)\between \omega [m] = \gamma (\upsilon ,\omega )[m]\cdot x\) 
RC14  \((\upsilon \cdot x)\between (\omega \cdot y) = \gamma (\upsilon ,\omega )\cdot (x\parallel y)\) 
RC15  \((\upsilon [m]\cdot x)\between (\omega [m]\cdot y) = \gamma (\upsilon ,\omega )[m]\cdot (x\parallel y)\) 
RC16  \((x + y)\between z = x\between z + y\between z\) 
RC17  \(x\between (y+z) = x\between y + x\between z\) 
Then, we can obtain the soundness and completeness theorems as follows.
Theorem 12
\({\mathcal {E}}_{\text {RPAP}}\) is sound for RPAP modulo FR bisimulation equivalence.
Proof
Since FR bisimulation is both an equivalence and a congruence for RPAP, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {RPAP}}\) and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}\sigma (t)\).

RP1 says that \(s\parallel t\) is a real static parallel or is a communication of initial transitions from s and t.

RP2 says that \(s\mid s\) can eliminate redundant parallel branches to s.

RP3RP7 say that the static parallel operator satisfies associativity, left distributivity and right distributivity to + and \(\cdot \).

RC8RC15 are the defining axioms for the communication merge, which say that \(s\between t\) makes as initial transition a communication of initial transitions from s and t.

RC16RC17 say that the communication merge \(\between \) satisfies both left distributivity and right distributivity.
These intuitions can be made rigorous by means of explicit FR bisimulation relations between the left and righthand sides of closed instantiations of the axioms in Table 2. Hence, all such instantiations are sound modulo FR bisimulation equivalence. \(\square \)
Theorem 13
\({\mathcal {E}}_{\text {RPAP}}\) is complete for RPAP modulo FR bisimulation equivalence.
Proof
To prove that \({\mathcal {E}}_{\text {RPAP}}\) is complete for RPAP modulo FR bisilumation equivalence, it means that \(s\underline{\leftrightarrow }^{fr} t\) implies \(s=t\).
(1) We consider the introduction to the static \({\text{parallel}}\, \).
We consider reversible process terms contains +, \(\cdot , \mid \) modulo associativity and commutativity (AC) of the + (RA1,RA2), and this equivalence relation is denoted by \(=_{AC}\). A reversible process term s then represents the collection of reversible process term t contains \(+, \cdot \), \({\text{and}}\,\) such that \(s =_{AC} t\). Each equivalence class s modulo AC of the + can be represented in the form \(s_{11}\mid \ldots \mid s_{1l}+\cdots +s_{k1}\mid \ldots \mid s_{km}\) with each \(s_{ij}\) either an atomic action or of the form \(t_1\cdot t_2\). We refer to the subterms \(s_{ij}\) and \(s_{ij}\mid s_{i,j+1}\) are the summands of s.
Then these rewrite rules are applied to the above reversible process terms modulo AC of the +.
We can see that the TRS is terminating modulo AC of the +.

Consider a summand a of n. Then \(n\xrightarrow {a}a[m]+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\xrightarrow {a}a[m]+u\), meaning that \(n'\) also contains the summand a.

Consider a summand a[m] of n. Then \(n\mathop {\twoheadrightarrow}{a[m]}a+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\mathop {\twoheadrightarrow}{a[m]}a+u\), meaning that \(n'\) also contains the summand a[m].

Consider a summand \(a_1\ldots a_i\ldots a_k\) of n. Then \(n\xrightarrow {a_1}\cdots \xrightarrow {a_i}\cdots \xrightarrow {a_k} a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\xrightarrow {a_1}\cdots \xrightarrow {a_i}\cdots \xrightarrow {a_k} a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]+u\), meaning that \(n'\) also contains the summand \(a_1\ldots a_i\ldots a_k\).

Consider a summand \(a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]\) of n. Then \(n\mathop {\twoheadrightarrow }\limits ^{a_k[m_k]}\ldots \mathop {\twoheadrightarrow }\limits ^{a_i[m_i]}\ldots \mathop {\twoheadrightarrow }\limits ^{a_1[m_1]}a_1\ldots a_i\ldots a_k+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\mathop {\twoheadrightarrow }\limits ^{a_k[m_k]}\ldots \mathop {\twoheadrightarrow }\limits ^{a_i[m_i]}\ldots \mathop {\twoheadrightarrow }\limits ^{a_1[m_1]}a_1\ldots a_i\ldots a_k+u\), meaning that \(n'\) also contains the summand \(a_1[m_1]\ldots a_i[m_i]\ldots a_k[m_k]\).

Consider a summand \(a\mid b\) of n. Then \(n\xrightarrow {a}a[m]\mid b+u\xrightarrow {b}a[m]\mid b[k]+u\), or \(n\xrightarrow {b}a\mid b[k]+u\xrightarrow {a}a[m]\mid b[k]+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\xrightarrow {a}a[m]\mid b+u\xrightarrow {b}a[m]\mid b[k]+u\), or \(n'\xrightarrow {b}a\mid b[k]+u\xrightarrow {a}a[m]\mid b[k]+u\), meaning that \(n'\) also contains the summand \(a\mid b\).

Consider a summand \(a[m]\mid b[k]\) of n. Then \(n\mathop {\twoheadrightarrow}{a[m]}a\mid b[k]+u\mathop {\twoheadrightarrow }\limits ^{b[k]}a\mid b+u\), or \(n\mathop {\twoheadrightarrow }\limits ^{b[k]}a[m]\mid b+u\mathop {\twoheadrightarrow}{a[m]}a\mid b+u\), so \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n'\mathop {\twoheadrightarrow}{a[m]}a\mid b[k]+u\mathop {\twoheadrightarrow }\limits ^{b[k]}a\mid b+u\), or \(n'\mathop {\twoheadrightarrow }\limits ^{b[k]}a[m]\mid b+u\mathop {\twoheadrightarrow}{a[m]}a\mid b+u\), meaning that \(n'\) also contains the summand \(a[m]\mid b[k]\).

The summands \(as\mid bt\) and \(a[m]s\mid b[k]t\) are integrated cases of the above summands.
Hence, each summand of n is also a summand of \(n'\). Vice versa, each summand of \(n'\) is also a summand of n. In other words, \(n=_{AC} n'\).
Finally, let the reversible process terms s and t contains +, \(\cdot \), \({\text{and}}\,\) be FR bisimilar. The TRS is terminating modulo AC of the +, so it reduces s and t to normal forms n and \(n'\), respectively. Since the rewrite rules and equivalence modulo AC of the + can be derived from the axioms, \(s=n\) and \(t=n'\). Soundness of the axioms then yields \(s\underline{\leftrightarrow }^{fr} n\) and \(t\underline{\leftrightarrow }^{fr} n'\), so \(n\underline{\leftrightarrow }^{fr} s\underline{\leftrightarrow }^{fr} t\underline{\leftrightarrow }^{fr} n'\). We showed that \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n=_{AC}n'\). Hence, \(s=n=_{AC} n'=t\).
(2) We prove the completeness of the axioms involve the parallel operator \(\parallel \) and the communication merge \(\between \).
Then these rewrite rules are applied to the above reversible process terms modulo AC of the +.
We can see that the TRS is terminating modulo AC of the +.

If n is an atomic action, then it does not contain any parallel operators.

Suppose \(n =_{AC} s + t\) or \(n =_{AC} s\cdot t\) or \(n=_{AC}s\mid t\). Then by induction the normal forms s and t do not contain \(\parallel \) and \(\between \), so that n does not contain \(\parallel \) and \(\between \) either.

n cannot be of the form \(s\parallel t\), because in that case the directed version of RP1 would apply to it, contradicting the fact that n is a normal form.

Suppose \(n =_{AC} s\between t\). By induction the normal forms s and t do not contain \(\parallel \) and \(\between \). We can distinguish the possible forms of s and t, which all lead to the conclusion that one of the directed versions of RC8RC17 can be applied to n. We conclude that n cannot be of the form \(s\between t\).
Hence, normal forms do not contain occurrences of parallel operators \(\parallel \) and \(\between \). In other words, normal forms only contains \(+, \cdot \) \(\text{and }\, \).
Finally, let the reversible process terms s and t be FR bisimilar. The TRS is terminating modulo AC of the +, so it reduces s and t to normal forms n and \(n'\), respectively. Since the rewrite rules and equivalence modulo AC of the + can be derived from the axioms, \(s=n\) and \(t=n'\). Soundness of the axioms then yields \(s\underline{\leftrightarrow }^{fr} n\) and \(t\underline{\leftrightarrow }^{fr} n'\), so \(n\underline{\leftrightarrow }^{fr} s\underline{\leftrightarrow }^{fr} t\underline{\leftrightarrow }^{fr} n'\). We showed that \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n=_{AC}n'\). Hence, \(s=n=_{AC} n'=t\). \(\square \)
Deadlock and encapsulation
A mismatch in communication of two actions \(\upsilon \) and \(\omega \) can cause a deadlock (nothing to do), we introduce the deadlock constant \(\delta \) and extend the communication function \(\gamma \) to \(\gamma :C\times C\rightarrow C\cup \{\delta \}\). So, the introduction about communication merge \(\between \) in the above section should be with \(\gamma (\nu ,\mu )\ne \delta \). We also introduce a unary encapsulation operator \(\partial _H\) for sets H of atomic communicating actions and their histories, which renames all actions in H into \(\delta \). RPAP extended with deadlock constant \(\delta \) and encapsulation operator \(\partial _H\) is called the Algebra of Reversible Communicating Processes, which is abbreviated to ARCP.
Transition rules of ARCP
Theorem 14
ARCP is a conservative extension of RPAP.
Proof
Since the TSS of RPAP is sourcedependent, and the transition rules for encapsulation operator \(\partial _H\) contain only a fresh operator in their source, so the TSS of ARCP is a conservative extension of that of RPAP. That means that ARCP is a conservative extension of RPAP. \(\square \)
Theorem 15
FR bisimulation equivalence is a congruence with respect to ARCP.
Proof
The TSSs for ARCP and RPAP are all in panth format, so FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axiomatization for ARCP
Axioms for ARCP
No.  Axiom 

RA6  \(x+\delta = x\) 
RA7  \(\delta \cdot x = \delta \) 
RA8  \(x \cdot \delta = \delta \) 
RD1  \(\upsilon \notin H\quad \partial _H(\upsilon ) = \upsilon \) 
RD2  \(\upsilon [m]\notin H\quad \partial _H(\upsilon [m]) = \upsilon [m]\) 
RD3  \(\upsilon \in H\quad \partial _H(\upsilon ) = \delta \) 
RD4  \(\upsilon [m]\in H\quad \partial _H(\upsilon [m]) = \delta \) 
RD5  \(\partial _H(\delta ) = \delta \) 
RD6  \(\partial _H(x+y)=\partial _H(x)+\partial _H(y)\) 
RD7  \(\partial _H(x\cdot y)=\partial _H(x)\cdot \partial _H(y)\) 
RD8  \(\partial _H(x\mid y)=\partial _H(x)\mid \partial _H(y)\) 
RP8  \(\delta \mid x=\delta \) 
RP9  \(x\mid \delta =\delta \) 
RC18  \(\delta \between x = \delta \) 
RC19  \(x\between \delta = \delta \) 
The soundness and completeness theorems are following.
Theorem 16
\({\mathcal {E}}_{\text {ARCP}}\) is sound for ARCP modulo FR bisimulation equivalence.
Proof
Since FR bisimulation is both an equivalence and a congruence for ARCP, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {ARCP}}\) and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}\sigma (t)\).

RA6 says that the deadlock \(\delta \) displays no behaviour, so that in a process term \(s + \delta \) the summand \(\delta \) is redundant.

RA7RA8, RP8RP9, RC18RC19 say that the deadlock \(\delta \) blocks all behaviour.

RD1RD5 are the defining axioms for the encapsulation operator \(\partial _H\).

RD6RD8 say that in \(\partial _H(t)\), all transitions of t labelled with atomic actions from H are blocked.
These intuitions can be made rigorous by means of explicit FR bisimulation relations between the left and righthand sides of closed instantiations of the axioms in Table 3. Hence, all such instantiations are sound modulo FR bisimulation equivalence. \(\square \)
Theorem 17
\({\mathcal {E}}_{\text {ARCP}}\) is complete for ARCP modulo FR bisimulation equivalence.
Proof
To prove that \({\mathcal {E}}_{\text {ARCP}}\) is complete for ARCP modulo FR bisilumation equivalence, it means that \(s\underline{\leftrightarrow }^{fr} t\) implies \(s=t\).
The axioms RA6RA8, RD1RD8, RP8RP9, RC18RC19 are turned into rewrite rules, by directing them from left to right. The resulting TRS is applied to process terms in RPAP modulo AC of the +.
Then these rewrite rules are applied to the above reversible process terms modulo AC of the +.
We can see that the TRS is terminating modulo AC of the +.

If \(s\equiv a\), then the directed version of RA6RA8 applies to \(\partial _H(s)\).

If \(s\equiv \delta \), then the directed version of RD5 applies to \(\partial _H(s)\).

If \(s=_{AC} t + t'\), then the directed version of RD6 applies to \(\partial _H(s)\).

If \(s=_{AC} t \cdot t'\), then the directed version of RD7 applies to \(\partial _H(s)\).

If \(s=_{AC} t \mid t'\), then the directed version of RD8 applies to \(\partial _H(s)\).
Hence, normal forms do not contain occurrences of \(\partial _H\). In other words, normal forms only contains \(+, \cdot \) \(\text{and }\,\).
Finally, let the reversible process terms s and t be FR bisimilar. The TRS is terminating modulo AC of the +, so it reduces s and t to normal forms n and \(n'\), respectively. Since the rewrite rules and equivalence modulo AC of the + can be derived from the axioms, \(s=n\) and \(t=n'\). Soundness of the axioms then yields \(s\underline{\leftrightarrow }^{fr} n\) and \(t\underline{\leftrightarrow }^{fr} n'\), so \(n\underline{\leftrightarrow }^{fr} s\underline{\leftrightarrow }^{fr} t\underline{\leftrightarrow }^{fr} n'\). We showed that \(n\underline{\leftrightarrow }^{fr} n'\) implies \(n=_{AC}n'\). Hence, \(s=n=_{AC} n'=t\). \(\square \)
Recursion
To capture infinite computing, recursion is introduced in this section. In ARCP, because parallel branches cannot be merged, the static parallel \(\text{operator }\,\) is a fundamental operator like + and \(\cdot \) and cannot be replaced by + and \(\cdot \). To what extent the existence \(\text{of} \,\) will influence the recursion theory, is a topic for our future research. In this section, we discuss recursion in reversible computation based on ARCP without the static parallel \({\text{operator}}\,\) denoted as ARCPRP, the corresponding axiomatization is denoted as \({\mathcal {E}}_{\text {ARCP}}\)RP2–RP9. For recursion and abstraction, it is reasonable to do extensions based on ARCPRP (ARCP without static parallel \({\text{operator}}\,\)). Because in reversible computation, all choice branches are retained and can execute simultaneously. The choice operator + and the static parallel \({\text{operator}}\, \) have the similar behaviors, so the static parallel operator can be naturally removed from ARCP.
In the following, E, F, G are guarded linear recursion specifications, X, Y, Z are recursive variables. We first introduce several important concepts, which come from Fokkink (2007).
Definition 39
Definition 40
(Solution) Processes \(p_1,\ldots ,p_n\) are a solution for a recursive specification \(\{X_i=t_i(X_1,\ldots ,X_n)i\in \{1,\ldots ,n\}\}\) (with respect to FR bisimulation equivalence) if \(p_i\underline{\leftrightarrow }^{fr}t_i(p_1,\ldots ,p_n)\) for \(i\in \{1,\ldots ,n\}\).
Definition 41
Definition 42
where \(a_1,\ldots ,a_k,b_1,\ldots ,b_l\in A\), and the sum above is allowed to be empty, in which case it represents the deadlock \(\delta \).
Transition rules of guarded recursion
Theorem 18
ARCPRP with guarded recursion is a conservative extension of ARCPRP.
Proof
Since the TSS of ARCPRP is sourcedependent, and the transition rules for guarded recursion contain only a fresh constant in their source, so the TSS of ARCPRP with guarded recursion is a conservative extension of that of ARCPRP. \(\square \)
Theorem 19
FR bisimulation equivalence is a congruence with respect to ARCPRP with guarded recursion.
Proof
The TSSs for guarded recursion and ARCPRP are all in panth format, so FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axiomatization for guarded recursion
Recursive definition principle and recursive specification principle
No.  Axiom 

RDP  \(\langle X_iE\rangle = t_i(\langle X_1E,\ldots ,X_nE\rangle )\quad \quad (i\in \{1,\ldots ,n\})\) 
RSP  if \(y_i=t_i(y_1,\ldots ,y_n)\) for \(i\in \{1,\ldots ,n\}\), then \(y_i=\langle X_iE\rangle \quad \quad (i\in \{1,\ldots ,n\})\) 
Theorem 20
\({\mathcal {E}}_{\text {ARCP}}\) RP2–RP9 + RDP + RSP is sound for ARCPRP with guarded recursion modulo FR bisimulation equivalence.
Proof
Since FR bisimulation is both an equivalence and a congruence for ARCPRP with guarded recursion, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {ARCP}}\)RP2–RP9 + RDP + RSP and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}\sigma (t)\).

Soundness of RDP follows immediately from the two transition rules for guarded recursion, which express that \(\langle X_iE\rangle \) and \(t_i(\langle X_1E\rangle ,\ldots ,\langle X_nE\rangle )\) have the same initial transitions for \(i\in \{1,\ldots ,n\}\).

Soundness of RSP follows from the fact that guarded recursive specifications have only one solution modulo FR bisimulation equivalence.
These intuitions can be made rigorous by means of explicit FR bisimulation relations between the left and righthand sides of RDP and closed instantiations of RSP in Table 4. \(\square \)
Theorem 21
\({\mathcal {E}}_{\text {ARCP}}\) RP2–RP9 + RDP + RSP is complete for ARCPRP with linear recursion modulo FR bisimulation equivalence.
Proof
The proof is similar to the proof of “\({\mathcal {E}}_{\text {ACP}}\) + RDP + RSP is complete for ACP with linear recursion modulo bisimulation equivalence”, see reference Fokkink (2007). \(\square \)
Then, if \(\langle X_1E_1\rangle \underline{\leftrightarrow }^{fr}\langle Y_1E_2\rangle \) for linear recursive specifications \(E_1\) and \(E_2\), then \(\langle X_1E_1\rangle =\langle Y_1E_2\rangle \) can be proved similarly.
Abstraction
A program has internal implementations and external behaviors. Abstraction technology abstracts away from the internal steps to check if the internal implementations really display the desired external behaviors. This makes the introduction of special silent step constant \(\tau \) and the abstraction operator \(\tau _I\).
Firstly, we introduce the concept of guarded linear recursive specification, which comes from Fokkink (2007).
Definition 43
A linear recursive specification E is guarded if there does not exist an infinite sequence of \(\tau \)transitions \(\langle XE\rangle \xrightarrow {\tau }\langle X'E\rangle \xrightarrow {\tau }\langle X''E\rangle \xrightarrow {\tau }\cdots \).
Silent step
A \(\tau \)transition is silent, which means that it can be eliminated from a process graph. \(\tau \) is an internal step and kept silent from an external observer.
Now, the set A is extended to \(A\cup \{\tau \}\), and \(\gamma \) to \(\gamma :A\cup \{\tau \}\times A\cup \{\tau \}\rightarrow A\cup \{\delta \}\), the predicate \(\xrightarrow {\tau }\surd \) means a successful termination after execution of \(\tau \).
Transition rules of silent step
Transition rules for choice composition, sequential composition and guarded linear recursion that involves \(\tau \)transitions are omitted.
Theorem 22
ARCPRP with silent step and guarded linear recursion is a conservative extension of ARCPRP with guarded linear recursion.
Proof
Since (1) the TSS of ARCPRP with guarded linear recursion is sourcedependent; (2) and the transition rules for the silent step \(\tau \) contain only a fresh constant in their source, (3) each transition rule for choice composition, sequential composition, or guarded linear recursion that involves \(\tau \)transitions, includes a premise containing the fresh relation symbol \(\xrightarrow {\tau }\) or predicate \(\xrightarrow {\tau }\surd \), and a lefthand side of which all variables occur in the source of the transition rule, the TSS of ARCPRP with silent step and guarded recursion is a conservative extension of that of ARCPRP with guarded linear recursion. \(\square \)
Theorem 23
Rooted branching FR bisimulation equivalence is a congruence with respect to ARCPRP with silent step and guarded linear recursion.
Proof
The TSSs for ARCPRP with silent step and guarded linear recursion are all in RBB cool format, by incorporating the successful termination predicate \(\downarrow \) in the transition rules, so rooted branching FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axioms for silent step
Axioms for silent step
No.  Axiom 

RB1  \(x + \tau = x\) 
RB2  \(\tau + x = x\) 
RB3  \(\tau \cdot x = x\) 
RB4  \(x \cdot \tau = x\) 
Theorem 24
\({\mathcal {E}}_{\text {ARCP}}\) RP2–RP9 + RB1–RB4 + RDP + RSP is sound for ARCPRP with silent step and guarded linear recursion, modulo rooted branching FR bisimulation equivalence.
Proof
Since rooted branching FR bisimulation is both an equivalence and a congruence for ARCPRP with silent step and guarded recursion, only the soundness of the first clause in the definition of the relation \(=\) is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {ARCP}}\)RP2–RP9 + RB1–RB4 + RDP + RSP and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}_{rb}\sigma (t)\).
We only provide some intuition for the soundness of axioms in Table 5.
The axioms in Table 5 says that the silent step \(\tau \) keep real silent in reversible processes, since all choice branches are retained in reversible computation.
This intuition can be made rigorous by means of explicit rooted branching FR bisimulation relations between the left and righthand sides of closed instantiations of RB1–RB4. \(\square \)
Theorem 25
\({\mathcal {E}}_{\text {ARCP}} \) RP2–RP9 + RB1–RB4 + RDP + RSP is complete for ARCPRP with silent step and guarded linear recursion, modulo rooted branching FR bisimulation equivalence.
Proof
The proof is similar to the proof of “\({\mathcal {E}}_{\text {ACP}}\,+\,\)B1–B2 + RDP + RSP is complete for ACP with silent step and guarded linear recursion modulo rooted branching bisimulation equivalence”, see reference Fokkink (2007).
Then, if \(\langle X_1E_1\rangle \underline{\leftrightarrow }^{fr}_{rb}\langle Y_1E_2\rangle \) for guarded linear recursive specifications \(E_1\) and \(E_2\), then \(\langle X_1E_1\rangle =\langle Y_1E_2\rangle \) can be proved similarly. \(\square \)
Abstraction
Abstraction operator \(\tau _I\) is used to abstract away the internal implementations. ARCPRP extended with silent step \(\tau \) and abstraction operator \(\tau _I\) is denoted by \(\text {ARCPRP}_{\tau }\).
Transition rules of abstraction operator
Theorem 26
\(\text {ARCPRP}_{\tau }\) with guarded linear recursion is a conservative extension of ARCPRP with silent step and guarded linear recursion.
Proof
Since (1) the TSS of ARCPRP with silent step and guarded linear recursion is sourcedependent; (2) and the transition rules for the abstraction operator contain only a fresh \(\tau _I\) in their source, the TSS of \(\text {ARCPRP}_{\tau }\) with guarded linear recursion is a conservative extension of that of ARCPRP with silent step and guarded linear recursion. \(\square \)
Theorem 27
Rooted branching FR bisimulation equivalence is a congruence with respect to \(\text {ARCPRP}_{\tau }\) with guarded linear recursion.
Proof
The TSSs for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion are all in RBB cool format, by incorporating the successful termination predicate \(\downarrow \) in the transition rules, so rooted branching FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axiomatization for abstraction operator
Axioms for abstraction operator
No.  Axiom 

RTI1  \(\upsilon \notin I \quad \tau _I(\upsilon )=\upsilon \) 
RTI2  \(\upsilon \in I \quad \tau _I(\upsilon )=\tau \) 
RTI3  \(\upsilon [m]\notin I \quad \tau _I(\upsilon [m])=\upsilon [m]\) 
RTI4  \(\upsilon [m]\in I \quad \tau _I(\upsilon [m])=\tau \) 
RTI5  \(\tau _I(\delta )=\delta \) 
RTI6  \(\tau _I(x+y)=\tau _I(x)+\tau _I(y)\) 
RTI7  \(\tau _I(x\cdot y)=\tau _I(x)\cdot \tau _I(y)\) 
Before we introduce the cluster fair abstraction rule, the concept of cluster is recaptured from Fokkink (2007).
Definition 44
(Cluster) Let E be a guarded linear recursive specification, and \(I\subseteq A\). Two recursion variable X and Y in E are in the same cluster for I if and only if there exist sequences of transitions \(\langle XE\rangle \xrightarrow {b_1}\cdots \xrightarrow {b_m}\langle YE\rangle \) and \(\langle YE\rangle \xrightarrow {c_1}\cdots \xrightarrow {c_n}\langle XE\rangle \), where \(b_1,\ldots ,b_m,c_1,\ldots ,c_n\in I\cup \{\tau \}\).
Cluster fair abstraction rule
No.  Axiom 

CFAR  If X is in a cluster for I with exits \(\{\upsilon _1Y_1,\ldots ,\upsilon _mY_m,\omega _1,\ldots ,\omega _n\}\), 
then \(\tau \cdot \tau _I(\langle XE\rangle )=\tau \cdot \tau _I(\upsilon _1\langle Y_1E\rangle ,\ldots ,\upsilon _m\langle Y_mE\rangle ,\omega _1,\ldots ,\omega _n)\) 
Theorem 28
\({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR is sound for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion, modulo rooted branching FR bisimulation equivalence.
Proof
Since rooted branching FR bisimulation is both an equivalence and a congruence for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}_{rb}\sigma (t)\).

RTI1–RTI5 are the defining equations for the abstraction operator \(\tau _I\): RTI2 and RTI4 says that it renames atomic actions from I into \(\tau \), while RTI1, RTI3, RTI5 say that it leaves atomic actions outside I and the deadlock \(\delta \) unchanged.

RTI6–RTI7 say that in \(\tau _I(t)\), all transitions of t labelled with atomic actions from I are renamed into \(\tau \).
This intuition can be made rigorous by means of explicit rooted branching FR bisimulation relations between the left and righthand sides of closed instantiations of RTI1–RTI7. \(\square \)
Theorem 29
\({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR is complete for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion, modulo rooted branching FR bisimulation equivalence.
Proof
The proof is similar to the proof of “\({\mathcal {E}}_{\text {ACP}_{\tau }}\) RDP + RSP +CFAR is complete for \(\text {ACP}_{\tau }\) with guarded linear recursion modulo rooted branching bisimulation equivalence”, see reference Fokkink (2007).
Firstly, each process term \(t_1\) in \(\text {ARCPRP}_{\tau }\) with guarded linear recursion is provably equal to a process term \(\langle X_1E\rangle \) with E a guarded linear recursive specification.
Then, if \(\langle X_1E_1\rangle \underline{\leftrightarrow }^{fr}_{rb}\langle Y_1E_2\rangle \) for guarded linear recursive specifications \(E_1\) and \(E_2\), then \(\langle X_1E_1\rangle =\langle Y_1E_2\rangle \) can be proved similarly. \(\square \)
Verification for business protocols with compensation support
 1.
The user plans a travel on UserAgent.
 2.
He/she submits the travel plan to the travel corporation TravelCorp via UserAgent.
 3.
TravelCorp receives the travel plan.
 4.
It books traffic tools and hotels according to the travel plan.
 5.
It sends the pay order to UserAgent.
 6.
UserAgent receives the pay order.
 7.
UserAgent sends the pay information to TravelCorp.
 8.
TravelAgent receives the pay information.
 9.
TravelAgent sends the business order to the Bank.
 10.
The Bank receives the business order and does paying.
Generating the reverse (compensation) graph
The above business protocol as Fig. 1 shows can be expressed by the following reversible process term.
\(\begin{aligned} & PlanATravel \cdot SubmitTravelPlan \cdot ReceivePayOrder \cdot PayForTravelCorp \between \\ &ReceiveTravelPlan \cdot BookTrafficTools \cdot BookHotels \cdot SendPayOrder \cdot ReceivePayInformation \cdot\\ &PayForBank \between ReceiveBusinessOrder \cdot DoPaying\end{aligned}\).
After the successful forward execution of the above process term, the following reversible process term can be obtained.
\(PlanATravel[m_1] \cdot c_{TravelPlan}[m_2] \cdot BookTrafficTools[m_3] \cdot BookHotels[m_4] \cdot c_{PayOrder}[m_5] \cdot c_{PayInformation}[m_6] \cdot c_{BusinessOrder}[m_7] \cdot DoPaying[m_8]\).
After the successful reverse execution (Compensation) the above process term, the original process term can be obtained.
Verification for business protocols with compensation support
RACP can be used in correctness verification under the framework of reversible computation for business protocols with compensation support.
Then we get the following conclusion.
Theorem 30
The business protocol as Fig. 2 shows \(\tau _I(\partial _H(U\parallel T \parallel B))\) exhibits desired external behaviors under the framework of reversible computation.
Proof
We get \(\tau _I(\langle X_1E\rangle )=\sum _{D_i\in \varDelta _i}\sum _{D_o\in \varDelta _o}receive_A(D_i)\cdot send_D(D_o)\cdot \tau _I(\langle X_1E\rangle )\), that is, \(\tau _I(\partial _H(U\parallel T \parallel B))=\sum _{D_i\in \varDelta _i}\sum _{D_o\in \varDelta _o}receive_A(D_i)\cdot send_D(D_o)\cdot \tau _I(\partial _H(U\parallel T \parallel B))\). So, the business protocol as Fig.2 shows \(\tau _I(\partial _H(U\parallel T \parallel B))\) exhibits desired external behaviors. \(\square \)
Extensions
One of the most fascinating characteristics is the modularity of RACP, that is, RACP can be extended easily. Through out this paper, we can see that RACP also inherents the modularity characteristics of ACP. By introducing new operators or new constants, RACP can have more properties. It provides RACP an elegant fashion to express a new property.
In this section, we take an example of renaming operators which are used to rename the atomic actions.
Transition rules of renaming operators
Theorem 31
\(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators is a conservative extension of \(\text {ARCPRP}_{\tau }\) with guarded linear recursion.
Proof
Since (1) the TSS of \(\text {ARCPRP}_{\tau }\) with guarded linear recursion is sourcedependent; (2) and the transition rules for the renaming operators contain only a fresh \(\rho _f\) in their source, the TSS of \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators is a conservative extension of that of \(\text {ARCPRP}_{\tau }\) with guarded linear recursion. \(\square \)
Theorem 32
Rooted branching FR bisimulation equivalence is a congruence with respect to \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators.
Proof
The TSSs for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators are all in RBB cool format, by incorporating the successful termination predicate \(\downarrow \) in the transition rules, so rooted branching FR bisimulation equivalence that they induce is a congruence. \(\square \)
Axioms for renaming operators
Axioms for renaming
No.  Axiom 

RRN1  \(\rho _f(\upsilon )=f(\upsilon )\) 
RRN2  \(\rho _f(\upsilon [m])=f(\upsilon )[m]\) 
RRN3  \(\rho _f(\delta )=\delta \) 
RRN4  \(\rho _f(x+y)=\rho _f(x)+\rho _f(y)\) 
RRN5  \(\rho _f(x\cdot y)=\rho _f(x)\cdot \rho _f(y)\) 
Theorem 33
\({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR + RRN1–RRN5 is sound for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators, modulo rooted branching FR bisimulation equivalence.
Proof
Since rooted branching FR bisimulation is both an equivalence and a congruence for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators, only the soundness of the first clause in the definition of the relation = is needed to be checked. That is, if \(s=t\) is an axiom in \({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR + RRN1RRN5 and \(\sigma \) a closed substitution that maps the variable in s and t to reversible process terms, then we need to check that \(\sigma (s)\underline{\leftrightarrow }^{fr}_{rb}\sigma (t)\).

RRN1–RRN3 are the defining equations for the renaming operator \(\rho _f\).

RRN4–RRN5 say that in \(\rho _f(t)\), the labels of all transitions of t are renamed by means of the mapping f.
This intuition can be made rigorous by means of explicit rooted branching FR bisimulation relations between the left and righthand sides of closed instantiations of RRN1RRN5. \(\square \)
Theorem 34
\({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR + RRN1RRN5 is complete for \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators, modulo rooted branching FR bisimulation equivalence.
Proof
It suffices to prove that each process term t in \(\text {ARCPRP}_{\tau }\) with guarded linear recursion and renaming operators is provably equal to a process term \(\langle XE\rangle \) with E a guarded linear recursive specification. Namely, then the desired completeness result follows from the fact that if \(\langle X_1E_1\rangle \underline{\leftrightarrow }^{fr}_{rb}\langle Y_1E_2\rangle \) for guarded linear recursive specifications \(E_1\) and \(E_2\), then \(\langle X_1E_1\rangle =\langle Y_1E_2\rangle \) can be derived from \({\mathcal {E}}_{\text {ARCPRP}_{\tau }}\) + RSP + RDP + CFAR.
Replacing \(Y_i\) by \(\rho _f(\langle X_iE\rangle )\) for \(i\in \{1,\ldots ,n\}\) is a solution for F. So by RSP, \(\rho _f(\langle X_1E\rangle )=\langle Y_1F\rangle \). \(\square \)
Conclusions
In this paper, we give reversible computation an axiomatic foundation called RACP. RACP can be widely used in verification of applications in reversible computation.
For recursion and abstraction, it is reasonable to do extensions based on ARCPRP (ARCP without static parallel \({\text{operator}}\,\)). Because in reversible computation, all choice branches are retained and can execute simultaneously. The choice operator + and the static parallel \({\text{operator}}\, \) have the similar behaviors, so the static parallel operator can be naturally removed from ARCP.
Any computable process can be represented by a process term in ACP (exactly \(\text {ACP}_\tau \) with guarded linear recursion) Baeten et al. (1987). That is, ACP may have the same expressive power as Turing machine. And RACP may have the same expressive power as ACP.
Same as ACP, RACP has good modularity and can be extended easily. Although the extensions can not improve the expressive power of RACP, it still provides an elegant and convenient way to model other properties in reversible computation.
Declarations
Competing interests
The author declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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