Bisimilarity of Open Terms in Stream GSOS

Abstract

Stream GSOS is a specification format for operations and calculi on infinite sequences. The notion of bisimilarity provides a canonical proof technique for equivalence of closed terms in such specifications. In this paper, we focus on open terms, which may contain variables, and which are equivalent whenever they denote the same stream for every possible instantiation of the variables. Our main contribution is to capture equivalence of open terms as bisimilarity on certain Mealy machines, providing a concrete proof technique. Moreover, we introduce an enhancement of this technique, called bisimulation up-to substitutions, and show how to combine it with other up-to techniques to obtain a powerful method for proving equivalence of open terms.

The research leading to these results has received funding from the European Research Council (FP7/2007-2013, grant agreement nr. 320571); as well as from the LABEX MILYON (ANR-10-LABX-0070, ANR-11-IDEX-0007), the project PACE (ANR-12IS02001) and the project REPAS (ANR-16-CE25-0011).

1 Introduction

Structural operational semantics (SOS) can be considered the de facto standard to define programming languages and process calculi. The SOS framework relies on defining a specification consisting of a set of operation symbols, a set of labels or actions and a set of inference rules. The inference rules describe the behaviour of each operation, typically depending on the behaviour of the parameters. The semantics is then defined in terms of a labelled transition system over (closed) terms constructed from the operation symbols. Bisimilarity of closed terms ($\sim$) provides a canonical notion of behavioural equivalence.

It is also interesting to study equivalence of open terms, for instance to express properties of program constructors, like the commutativity of a non-deterministic choice operator. The latter can be formalised as the equation $\mathcal {X}+ \mathcal {Y}= \mathcal {Y}+ \mathcal {X}$, where the left and right hand sides are terms with variables $\mathcal {X},\mathcal {Y}$. Equivalence of open terms ($\sim _o$) is usually based on $\sim :$ for all open terms $t_1,t_2$

$$\begin{aligned} t_1\sim _ot_2 \text { iff for all closed substitutions } \phi , \; \; \phi (t_1)\sim \phi (t_2)\text {.}\end{aligned}$$

(1)

The main problem of such a definition is the quantification over all substitutions: one would like to have an alternative characterisation, possibly amenable to the coinduction proof principle. This issue has been investigated in several works, like [1, 3, 7, 11, 13, 15, 20].

Fig. 1.

A stream GSOS specification (a) is transformed first into a monadic specification (b), then in a Mealy specification (c) and finally in a specification for open terms (d). In these rules, n and m range over real numbers, b over an arbitrary set B, $\mathcal {X}$ over variables and $\varsigma$ over substitutions of variables into reals.

In this paper, we continue this line of research, focusing on the simpler setting of streams, which are infinite sequences over a fixed data type. More precisely, we consider stream languages specified in the stream GSOS format [10], a syntactic rule format enforcing several interesting properties. We show how to transform a stream specification into a Mealy machine specification that defines the operational semantics of open terms. Moreover, a notion of bisimulation – arising in a canonical way from the theory of coalgebras [16] – exactly characterises $\sim _o$ as defined in (1).

Our approach can be illustrated by taking as running example the fragment of the stream calculus [18] presented in Fig. 1(a). The first step is to transform a stream GSOS specification (Sect. 2) into a monadic one (Sect. 3). In this variant of GSOS specifications, no variable in the source of the conclusion appears in the target of the conclusion. For example, in the stream specification in Fig. 1(a), the rule associated to ${\otimes }$ is not monadic. The corresponding monadic specification is illustrated in Fig. 1(b). Notice this process requires the inclusion of a family of prefix operators (on the right of Fig. 1(b)) that satisfy the imposed restriction.

The second step – based on [8] – is to compute the pointwise extension of the obtained specification (Sect. 4). Intuitively, we transform a specification of streams with outputs in a set A into a specification of Mealy machines with inputs in an arbitrary set B and outputs in A, by replacing each transition $\xrightarrow {\, {a} \, }$ (for $a\in A$) with a transition $\xrightarrow {\, {b|a} \, }$ for each input $b\in B$. See Fig. 1(c).

In the last step (Sect. 5), we fix $B = \mathcal {V}\rightarrow A$, the set of functions assigning outputs values in A to variables in $\mathcal {V}$. To get the semantics of open terms, it only remains to specify the behaviours of variables in $\mathcal {V}$. This is done with the leftmost rule in Fig. 1(d).

As a result of this process, we obtain a notion of bisimilarity over open terms, which coincides with behavioural equivalence of all closed instances, and provides a concrete proof technique for equivalence of open terms. By relating open terms rather than all its possible instances, this novel technique often enables to use finite relations, while standard bisimulation techniques usually require relations of infinite size on closed terms. In Sect. 6 we further enhance this novel proof technique by studying bisimulation up-to [14]. We combine known up-to techniques with a novel one which we call bisimulation up-to substitutions.

2 Preliminaries

We define the two basic models that form the focus of this paper: stream systems, that generate infinite sequences (streams), and Mealy machines, that generate output streams given input streams.

Definition 2.1

A stream system with outputs in a set A is a pair $(X, \langle o,d \rangle )$ where X is a set of states and $\langle o,d \rangle :X \rightarrow A \times X$ is a function, which maps a state $x\in X$ to both an output value $o(x)\in A$ and to a next state $d(x) \in X$. We write $x \xrightarrow {\, {a} \, } y$ whenever $o(x)=a$ and $d(x) = y$.

Definition 2.2

A Mealy machine with inputs in a set B and outputs in a set A is a pair (X, m) where X is a set of states and $m :X \rightarrow (A \times X)^B$ is a function assigning to each $x\in X$ a map $m(x) = \langle o_x, d_x \rangle :B \rightarrow A \times X$. For all inputs $b \in B$, $o_x(b)\in A$ represents an output and $d_x(b) \in X$ a next state. We write $x \xrightarrow {\, {b|a} \, } y$ whenever $o_x(b)=a$ and $d_x(b) = y$.

We recall the notion of bisimulation for both models.

Definition 2.3

Let $(X,\langle o, d \rangle )$ be a stream system. A relation ${\mathrel {\mathcal {R}}} \subseteq X\times X$ is a bisimulation if for all $(x,y) \in {\mathrel {\mathcal {R}}}$, $o(x) = o(y)$ and $(d(x),d(y)) \in {\mathrel {\mathcal {R}}}$.

Definition 2.4

Let (X, m) be a Mealy machine. A relation ${\mathrel {\mathcal {R}}} \subseteq X\times X$ is a bisimulation if for all $(x,y) \in {\mathrel {\mathcal {R}}}$ and $b\in B$, $o_x(b) = o_y(b)$ and $(d_x(b), d_y(b)) \in {\mathrel {\mathcal {R}}}$.

For both kind of systems, we say that x and y are bisimilar, notation $x \sim y$, if there is a bisimulation $\mathrel {\mathcal {R}}$ s.t. $x \mathrel {\mathcal {R}}y$.

Stream systems and Mealy machines, as well as the associated notions of bisimulation, are instances of the theory of coalgebras [16]. Coalgebras provide a suitable mathematical framework to study state-based systems and their semantics at a high level of generality. In the current paper, the theory of coalgebras underlies and enables our main results.

Definition 2.5

Given a functor $F :\textit{Set}\rightarrow \textit{Set}$, an F-coalgebra is a pair (X, d), where X is a set (called the carrier) and $d :X \rightarrow FX$ is a function (called the structure). An F-coalgebra morphism from $d :X \rightarrow FX$ to $d':Y \rightarrow FY$ is a map $h :X \rightarrow Y$ such that $F h \circ d = d' \circ h$.

Stream systems and Mealy machines are F-coalgebras for the functors $FX = A \times X$ and $FX = (A \times X)^B$, respectively.

The semantics of systems modelled as coalgebras for a functor F is provided by the notion of final coalgebra. A coalgebra $\zeta :Z \rightarrow FZ$ is called final if for every F-coalgebra $d :X \rightarrow FX$ there is a unique morphism $|\![{-}]\!| :X \rightarrow Z$ such that $|\![{-}]\!|$ is a morphism from d to $\zeta$. We call $|\![{-}]\!|$ the coinductive extension of d.

Intuitively, a final coalgebra $\zeta :Z \rightarrow FZ$ defines all possible behaviours of F-coalgebras, and $|\![{-}]\!|$ assigns behaviour to all states $x,y \in X$. This motivates to define x and y to be behaviourally equivalent iff $|\![{x}]\!| = |\![{y}]\!|$. Under the condition that F preserves weak pullbacks, behavioural equivalence coincides with bisimilarity, i.e., $x \sim y$ iff $|\![{x}]\!| = |\![{y}]\!|$ (see [16]). This condition is satisfied by (the functors for) stream systems and Mealy machines. In the sequel, by $\sim$ we hence refer both to bisimilarity and behavioural equivalence.

Final coalgebras for stream systems and Mealy machines will be pivotal for our exposition. We briefly recall them, following [9, 16]. The set $A^\omega$ of streams over A carries a final coalgebra for the functor $FX=A\times X$. For every stream system $\langle o,d \rangle :X \rightarrow A \times X$, the coinductive extension $|\![{-}]\!| :X \rightarrow A^\omega$ assigns to a state $x \in X$ the stream $a_0a_1a_2 \dots$ whenever $x\xrightarrow {\, {a_0} \, }x_1\xrightarrow {\, {a_1} \, }x_2 \xrightarrow {\, {a_2} \, } \dots$

Recalling a final coalgebra for Mealy machines requires some more care. Given a stream $\beta \in B^\omega$, we write $\beta {\upharpoonright _n}$ for the prefix of $\beta$ of length n. A function $\mathfrak {c}:B^\omega \rightarrow A^\omega$ is causal if for all $n \in \mathbb {N}$ and all $\beta , \beta ' \in B^\omega$: $\beta {\upharpoonright _n} = \beta ' {\upharpoonright _n}$ entails $\mathfrak {c}(\beta ){\upharpoonright _n} = \mathfrak {c}(\beta '){\upharpoonright _n}$. The set $\varGamma (B^\omega ,A^\omega ) = \{\mathfrak {c}:B^\omega \rightarrow A^\omega \mid \mathfrak {c}\text { is causal}\}$ carries a final coalgebra for the functor $FX=(A\times X)^B$. For every Mealy machine $m :X \rightarrow (A \times X)^B$, the coinductive extension $|\![{-}]\!| :X \rightarrow \varGamma (B^\omega ,A^\omega )$ assigns to each state $x \in X$ and each input stream $b_0b_1b_2\dots \in B^\omega$ the output stream $a_0a_1a_2\dots \in A^\omega$ whenever $x\xrightarrow {\, {b_0|a_0} \, }x_1\xrightarrow {\, {b_1|a_1} \, }x_2 \xrightarrow {\, {b_2|a_2} \, } \dots$

2.1 System Specifications

Different kinds of transition systems, like stream systems or Mealy machines, can be specified by means of algebraic specification languages. The syntax is given by an algebraic signature $\varSigma$, namely a collection of operation symbols $\{f_i \mid i \in I\}$ where each operator $f_i$ has a (finite) arity $n_i \in \mathbb {N}$. For a set X, $T_\varSigma X$ denotes the set of $\varSigma$-terms with variables over X. The set of closed $\varSigma$-terms is denoted by $T_{\varSigma }\emptyset$. We omit the subscript when $\varSigma$ is clear from the context.

A standard way to define the operational semantics of these languages is by means of structural operational semantics (SOS) [12]. In this approach, the semantics of each of the operators is described by syntactic rules, and the behaviour of a composite system is given in terms of the behaviour of its components. We recall stream GSOS [10], a specification format for stream systems.

Definition 2.6

A stream GSOS rule r for a signature $\varSigma$ and a set A is a rule

$$\begin{aligned} \frac{x_1 \xrightarrow {\, {a_1} \, } x'_1 \quad \cdots \quad x_n \xrightarrow {\, {a_n} \, }x'_n}{f(x_1, \ldots , x_n) \xrightarrow {\, {a} \, } t} \end{aligned}$$

(2)

where $f \in \varSigma$ with arity n, $x_1, \ldots , x_n, x'_1, \ldots , x'_n$ are pairwise distinct variables, t is a term built over variables $\{x_1, \ldots , x_n, x'_1, \ldots , x'_n\}$ and $a,a_1, \ldots , a_n \in A$. We say that r is triggered by $(a_1, \ldots , a_n) \in A^n$.

A stream GSOS specification is a tuple $(\varSigma , A, R)$ where $\varSigma$ is a signature, A is a set of actions and R is a set of stream GSOS rules for $\varSigma$ and A s.t. for each $f\in \varSigma$ of arity n and each tuple $(a_1, \ldots , a_n)\in A^n$, there is only one rule $r \in R$ for f that is triggered by $(a_1, \ldots , a_n)$.

A stream GSOS specification allows us to extend any given stream system $\langle o,d \rangle :X \rightarrow A \times X$ to a stream system $\overline{\langle o,d \rangle } :TX \rightarrow A \times TX$, by induction: the base case is given by $\langle o,d \rangle$, and the inductive cases by the specification. This construction can be defined formally in terms of proof trees, or by coalgebraic means; we adopt the latter approach, which is recalled later in this section.

There are two important uses of the above construction: (A) applying it to the (unique) stream system carried by the empty set $\emptyset$ yields a stream system over closed terms, i.e., of the form $T\emptyset \rightarrow A \times T\emptyset$; (B) applying the construction to the final coalgebra yields a stream system of the form $TA^\omega \rightarrow A \times TA^\omega$. The coinductive extension $|\![{-}]\!| :TA^\omega \rightarrow A^\omega$ of this stream system is, intuitively, the interpretation of the operations in $\varSigma$ on streams in $A^\omega$.

Fig. 2.

The GSOS-rules of our running example

Fig. 3.

A stream system

Example 2.1

Let $(\varSigma , A, R)$ be a stream GSOS specification where the signature $\varSigma$ consists of constants $\{\textsf {a}\mid a \in A\}$ and a binary operation $\textsf {alt}$. The set R contains the rules in Fig. 2. For an instance of (A), the term $\textsf {alt}(\textsf {a}, \textsf {alt}(\textsf {b},\textsf {c})) \in T\emptyset$ defines the stream system depicted in Fig. 3. For an instance of (B), the operation $\textsf {alt}:A^\omega \times A^\omega \rightarrow A^\omega$ maps streams $a_0a_1a_2\dots$, $b_0b_1b_2\dots$ to $a_0b_1a_2b_2 \dots$.

Example 2.2

We now consider the specification $(\varSigma , \mathbb {R}, R)$ which is the fragment of the stream calculus [17, 18] consisting of the constants $n \in \mathbb {R}$ and the binary operators sum ${\oplus }$ and (convolution) product ${\otimes }$. The set R is defined in Fig. 1(a). For an example of (A), consider $n \oplus m \xrightarrow {\, {n+m} \, } 0\oplus 0 \xrightarrow {\, {0} \, } 0\oplus 0 \xrightarrow {\, {0} \, } \dots$. For (B), the induced operation $\oplus :\mathbb {R}^\omega \times \mathbb {R}^\omega \rightarrow \mathbb {R}^\omega$ is the pointwise sum of streams, i.e., it maps any two streams $n_0n_1\dots$, $m_0m_1\dots$ to $(n_0+m_0)(n_1+m_1) \dots$.

Definition 2.7

We say that a stream GSOS rule r as in (2) is monadic if t is a term built over variables $\{x'_1, \ldots , x'_n\}$. A stream GSOS specification is monadic if all its rules are monadic.

The specification of Example 2.1 satisfies the monadic stream GSOS format, while the one of Example 2.2 does not since, in the rules for $\otimes$, the variable y occurs in the arriving state of the conclusion.

The notions introduced above for stream GSOS, as well as the analogous ones for standard (labeled transition systems) GSOS [5], can be reformulated in an abstract framework – the so-called abstract GSOS [10, 19] – that will be pivotal for the proof of our main result.

In this setting, signatures are represented by polynomial functors: a signature $\varSigma$ corresponds to the polynomial functor $\varSigma X = \coprod _{i \in I} X^{n_i}$. For instance, the signature $\varSigma$ in Example 2.1 corresponds to the functor $\varSigma X = A + (X \times X)$, while the signature of Example 2.2 corresponds to the functor $\varSigma X = \mathbb {R}+ (X \times X) + (X \times X)$. Models of a signature are seen as algebras for the corresponding functor.

Definition 2.8

Given a functor $F :\textit{Set}\rightarrow \textit{Set}$, an F-algebra is a pair (X, d), where X is the carrier set and $d :FX \rightarrow X$ is a function. An algebra homomorphism from an F-algebra (X, d) to an F-algebra $(Y,d')$ is a map $h :X \rightarrow Y$ such that $h \circ d = d' \circ Fh$.

Particularly interesting are initial algebras: an F-algebra is called initial if there exists a unique algebra homomorphism from it to every F-algebra. For a functor corresponding to a signature $\varSigma$, the initial algebra is $(T\emptyset ,\kappa )$ where $\kappa :\varSigma T\emptyset \rightarrow T\emptyset$ maps, for each $i\in I$, the tuple of closed terms $t_1,\dots t_{n_i}$ to the closed term $f_i(t_1,\dots t_{n_i})$. For every set X, we can define in a similar way $\kappa _X :\varSigma TX \rightarrow TX$. The free monad over $\varSigma$ consists of the endofunctor $T :\textit{Set}\rightarrow \textit{Set}$, mapping every set X to TX, together with the natural transformations $\eta :\textit{Id}\Longrightarrow T$ (interpretation of variables as terms) and $\mu :TT \Longrightarrow T$ (glueing terms built of terms). Given an algebra $\sigma :\varSigma Y \rightarrow Y$, for any function $f :X \rightarrow Y$ there is a unique algebra homomorphism $f^\dagger :TX \rightarrow Y$ from $(TX,\kappa _X)$ to $(Y,\sigma )$. In particular, the identity function $\textit{id}:X \rightarrow X$ induces a unique algebra homomorphism from TX to X, which we denote by $\sigma ^\sharp :T X \rightarrow X$; this is the interpretation of terms in $\sigma$.

Definition 2.9

An abstract GSOS specification (of $\varSigma$ over F) is a natural transformation $\lambda :\varSigma (\textit{Id}\times F) \Longrightarrow FT$. A monadic abstract GSOS specification (in short, monadic specification) is a natural transformation $\lambda :\varSigma F \Longrightarrow FT$.

By instantiating the functor F in the above definition to the functor for streams ($FX=A\times X$) one obtains all and only the stream GSOS specifications. Instead, by taking the functor for Mealy machines ($FX=(A\times X)^B$) one obtains the Mealy GSOS format [10]: for the sake of brevity, we do not report the concrete definition here but this notion will be important in Sect. 5 where, to deal with open terms, we transform stream specifications into Mealy GSOS specifications.

Example 2.3

For every set X, the rules in Example 2.1 define a function $\lambda _X : A + (A\times X) \times (A\times X) \rightarrow (A\times T_{\varSigma }X)$ as follows: each $a\in A$ is mapped to $(a, \textsf {a})$ and each pair $(a,x'),(b,y') \in (A\times X) \times (A\times X)$ is mapped to $(a, \textsf {alt}(y',x'))$ [10].

We focus on monadic distributive laws for most of the paper, and since they are slightly simpler than abstract GSOS specifications, we only recall the relevant concepts for monadic distributive laws. However, we note that the concepts below can be extended to abstract GSOS specifications; see, e.g., [4, 10] for details.

A monadic abstract GSOS specification induces a distributive law $\rho :T F \Longrightarrow FT$. This distributive law allows us to extend any F-coalgebra $d :X \rightarrow FX$ to an F-coalgebra on terms:

This construction generalises and formalises the aforementioned extension of stream systems to terms by means of a stream GSOS specification. In particular, (A) the unique coalgebra on the empty set $!:\emptyset \rightarrow F \emptyset$ yields an F-coalgebra on closed terms $T\emptyset \rightarrow FT\emptyset$. If F has a final coalgebra $(Z,\zeta )$, the unique morphism $|\![{-}]\!|_c :T\emptyset \rightarrow Z$ defines the semantics of closed terms.

Further (B), the final coalgebra $(Z,\zeta )$ yields a coalgebra on TZ. By finality, we then obtain a T-algebra over the final F-coalgebra, which we denote by $|\![{-}]\!|_a :T Z \rightarrow Z$ and we call it the abstract semantics. We define the algebra induced by $\lambda$ as the $\varSigma$-algebra $\sigma :\varSigma Z \rightarrow Z$ given by

(3)

3 Making Arbitrary Stream GSOS Specifications Monadic

The results presented in the next section are restricted to monadic specifications, but one can prove them for arbitrary GSOS specifications by exploiting some auxiliary operators, introduced in [8] with the name of buffer. Theorem 6.1 in Sect. 6 only holds for monadic GSOS specifications. This does not restrict the applicability of our approach: as we show below, arbitrary stream GSOS specifications can be turned into monadic ones.

Let $(\varSigma , A, R)$ be a stream GSOS specification. The extended signature $\tilde{\varSigma }$ is given by $\{\tilde{f} \mid f \in \varSigma \} \cup \{a.\_ \mid a \in A\}$. The set of rules $\tilde{R}$ is defined as follows:

• For all $a, b \in A$, $\tilde{R}$ contains the following rule

  $$\begin{aligned} \frac{x \xrightarrow {\, {b} \, } x'}{a.x \xrightarrow {\, {a} \, } b.x'} \end{aligned}$$

  (4)

• For each rule $r=\frac{x_1 \xrightarrow {\, {a_1} \, } x'_1 \quad \cdots \quad x_n \xrightarrow {\, {a_n} \, }x'_n}{f(x_1, \ldots , x_n) \xrightarrow {\, {a} \, } t(x_1, \ldots , x_n,x'_1, \ldots , x'_n)} \in R$, the set $\tilde{R}$ contains

  $$\begin{aligned} \tilde{r}= \frac{x_1 \xrightarrow {\, {a_1} \, } x'_1 \quad \cdots \quad x_n \xrightarrow {\, {a_n} \, }x'_n}{\tilde{f}(x_1, \ldots , x_n) \xrightarrow {\, {a} \, } \tilde{t}(a_1.x'_1, \ldots , a'_n.x'_n,x'_1, \ldots , x'_n)} \end{aligned}$$

  (5)

  where $\tilde{t}$ is the term obtained from t by replacing each $g \in \varSigma$ by $\tilde{g} \in \tilde{\varSigma }$.

The specification $(\tilde{\varSigma }, A, \tilde{R})$ is now monadic and preserves the original semantics as stated by the following result.

Theorem 3.1

Let $(\varSigma , A, R)$ be a stream GSOS specification and $(\tilde{\varSigma }, A, \tilde{R})$ be the corresponding monadic one. Then, for all $t \in T_{\varSigma }\emptyset$, $t \sim \tilde{t}$.

Example 3.1

Consider the non-monadic specification in Example 2.2. The corresponding monadic specification consists of the rules in Fig. 1(b) where, to keep the notation light, we used operation symbols f rather than $\tilde{f}$.

4 Pointwise Extensions of Monadic GSOS Specifications

The first step to deal with the semantics of open terms induced by a stream GSOS specification is to transform the latter into a Mealy GSOS specification. We follow the approach in [8] which is defined for arbitrary GSOS but, as motivated in Sect. 3, we restrict our attention to monadic specifications.

Let $(\varSigma , A, R)$ be a monadic stream GSOS specification and B some input alphabet. The corresponding monadic Mealy GSOS specification is a tuple $(\varSigma , A, B, \overline{R})$, where $\overline{R}$ is the least set of Mealy rules which contains, for each stream GSOS rule $r=\frac{x_1 \xrightarrow {\, {a_1} \, } x'_1 \quad \cdots \quad x_n \xrightarrow {\, {a_n} \, }x'_n}{f(x_1, \ldots , x_n) \xrightarrow {\, {a} \, } t(x'_1, \ldots , x'_n)}\in R$ and $b\in B$, the Mealy rule $\overline{r}_b$ defined by

$$\begin{aligned} \overline{r}_b= \frac{x_1 \xrightarrow {\, {b | a_1} \, } x'_1 \quad \cdots \quad x_n \xrightarrow {\, {b | a_n} \, }x'_n}{ f(x_1, \ldots , x_n) \xrightarrow {\, {b | a} \, } t(x'_1, \ldots , x'_n)} \end{aligned}$$

(6)

An example of this construction is shown in Fig. 1(c).

Recall from Sect. 2 that any abstract GSOS specification induces a $\varSigma$-algebra on the final F-coalgebra. Let $\sigma :\varSigma A^\omega \rightarrow A^\omega$ be the algebra induced by the stream specification and $\overline{\sigma } :\varSigma \varGamma (B^\omega ,A^\omega ) \rightarrow \varGamma (B^\omega ,A^\omega )$ the one induced by the corresponding Mealy specification. Theorem 4.1, at the end of this section, informs us that $\overline{\sigma }$ is the pointwise extension of $\sigma$.

Definition 4.1

Let $g :(A^\omega )^n \rightarrow A^\omega$ and $\bar{g} :(\varGamma (B^\omega ,A^\omega ))^n \rightarrow \varGamma (B^\omega ,A^\omega )$ be two functions. We say that $\bar{g}$ is the pointwise extension of g iff for all $\mathfrak {c}_1, \ldots ,\mathfrak {c}_{n} \in \varGamma (B^\omega ,A^\omega )$ and $\beta \in B^\omega$, $\bar{g}(\mathfrak {c}_1, \ldots , \mathfrak {c}_{n})(\beta ) = g (\mathfrak {c}_1(\beta ), \ldots , \mathfrak {c}_{n}(\beta ))$. This notion is lifted in the obvious way to $\varSigma$-algebras for an arbitrary signature $\varSigma$.

Example 4.1

Recall the operation $\oplus :A^\omega \times A^\omega \rightarrow A^\omega$ from Example 2.2 that arises from the specification in Fig. 1(a) (it is easy to see that the same operation also arises from the monadic specification in Fig. 1(b)). Its pointwise extension $\bar{\oplus } :\varGamma (B^\omega ,\mathbb {R}^\omega ) \times \varGamma (B^\omega ,\mathbb {R}^\omega ) \rightarrow \varGamma (B^\omega ,\mathbb {R}^\omega )$ is defined for all $\mathfrak {c}_1,\mathfrak {c}_2 \in \varGamma (B^\omega , \mathbb {R}^\omega )$ and $\beta \in B^\omega$ as $(\mathfrak {c}_1 \bar{\oplus } \mathfrak {c}_2)(\beta ) = \mathfrak {c}_1(\beta ) \oplus \mathfrak {c}_2(\beta )$. Theorem 4.1 tells us that $\bar{\oplus }$ arises from the corresponding Mealy GSOS specification (Fig. 1(c)).

In [8], the construction in (6) is generalised from stream specifications to arbitrary abstract GSOS. The key categorical tool is the notion of costrength for an endofunctor $F:\textit{Set}\rightarrow \textit{Set}$. Given two sets B and X, we first define $\epsilon ^b :X^B \rightarrow X$ as $\epsilon ^b(f) = f(b)$ for all $b\in B$. Then, $\textit{cs}^F_{B,X} :F (X^B) \rightarrow (FX)^B$ is a natural map in B and X, given by $\textit{cs}^F_{B,X}(t)(b) = (F\epsilon ^b)(t)$.

Now, given a monadic specification $\lambda :\varSigma F \Longrightarrow FT$, we define $\bar{\lambda } :\varSigma (F^B) \Longrightarrow (FT)^B$ as the natural transformation that is defined for all sets X by

(7)

Observe that $\bar{\lambda }$ is also a monadic specification, but for the functor $F^B$ rather than the functor F. The reader can easily check that for F being the stream functor $FX=A\times X$, the resulting $\bar{\lambda }$ is indeed the Mealy specification corresponding to $\lambda$ as defined in (6).

It is worth to note that the construction of $\bar{\lambda }$ for an arbitrary abstract GSOS $\lambda :\varSigma (\textit{Id}\times F) \Longrightarrow FT$, rather than a monadic one, would not work as in (7). The solution devised in [8] consists of introducing some auxiliary operators as already discussed in Sect. 3. The following result has been proved in [8] for arbitrary abstract GSOS, with these auxiliary operators. Our formulation is restricted to monadic specifications.

Theorem 4.1

Let F be a functor with a final coalgebra $(Z, \zeta )$, and let $(\bar{Z}, \bar{\zeta })$ be a final $F^B$-coalgebra. Let $\lambda :\varSigma F \Longrightarrow FT$ be a monadic distributive law, and $\sigma :\varSigma Z \rightarrow Z$ the algebra induced by it. The algebra $\bar{\sigma } :\varSigma \bar{Z} \rightarrow \bar{Z}$ induced by $\bar{\lambda }$ is a pointwise extension of $\sigma$.

In the theorem above, the notion of pointwise extension should be understood as a generalisation of Definition 4.1 to arbitrary final F and $F^B$-coalgebras. This generalised notion, that has been introduced in [8], will not play a role for our paper where F is fixed to be the stream functor $FX=A\times X$.

5 Mealy Machines over Open Terms

We now consider the problem of defining a semantics for the set of open terms $T\mathcal {V}$ for a fixed set of variables $\mathcal {V}$. Our approach is based on the results in the previous sections: we transform a monadic GSOS specification for streams with outputs in A into a Mealy machine with inputs in $A^\mathcal {V}$ and outputs in A, i.e., a coalgebra for the functor $FX=(A \times X)^{A^\mathcal {V}}$. The coinductive extension of this Mealy machine provides the open semantics: for each open term $t\in T\mathcal {V}$ and variable assignment $\psi :\mathcal {V}\rightarrow A^\omega$, it gives an appropriate output stream in $A^{\omega }$. This is computed in a stepwise manner: for an input $\varsigma :\mathcal {V}\rightarrow A$, representing “one step” of a variable assignment $\psi$, we obtain one step of the output stream.

We start by defining a Mealy machine $c :\mathcal {V}\rightarrow (A \times \mathcal {V})^{A^\mathcal {V}}$ on the set of variables $\mathcal {V}$ as on the left below, for all $\mathcal {X}\in \mathcal {V}$ and $\varsigma \in A^\mathcal {V}$:

(8)

Concretely, this machine has variables as states and for each $\varsigma :\mathcal {V}\rightarrow A$ a self-loop, as depicted on the right. Now, let $\lambda :\varSigma (A \times {-}) \Rightarrow A \times T$ be a monadic stream specification and $\bar{\lambda } :\varSigma ((A \times {-})^{A^\mathcal {V}}) \Rightarrow (A \times T({-}))^{A^\mathcal {V}}$ be the induced Mealy specification, as defined in (7). As mentioned in Sect. 2, $\bar{\lambda }$ defines a distributive law $\rho :T((A \times {-})^{A^\mathcal {V}}) \Rightarrow (A \times T({-}))^{A^\mathcal {V}}$, which allows to extend c (see (8)) to a coalgebra $m_\lambda :T\mathcal {V}\rightarrow (A \times T\mathcal {V})^{A^\mathcal {V}}$, given by

(9)

This is the Mealy machine of interest.

Example 5.1

Consider the stream specification $\lambda$ of the operation $\textsf {alt}$, given in Example 2.1. The states of the Mealy machine $m_\lambda$ are the open terms $T\mathcal {V}$. The transitions of terms are defined by the set of rules

$$ \frac{}{\textsf {a}\xrightarrow {\, {\varsigma |a} \, } \textsf {a}} \qquad \frac{x \xrightarrow {\, {\varsigma |a} \, } x' \quad y \xrightarrow {\, {\varsigma |b} \, } y'}{\textsf {alt}(x,y) \xrightarrow {\, {\varsigma |a} \, } \textsf {alt}(y',x')} \qquad \text { for all } \varsigma :\mathcal {V}\rightarrow A \text { and } a,b \in A $$

together with the transitions for the variables as in (8). For instance, for each $\mathcal {X}, \mathcal {Y}, \mathcal {Z}\in \mathcal {V}$ and all $\varsigma ,\varsigma ' :\mathcal {V}\rightarrow A$, we have the following transitions in $m_\lambda$:

Example 5.2

For the fragment of the stream calculus introduced in Example 2.2, the Mealy machine over open terms is defined by the rules in Fig. 1(d). Below we draw the Mealy machines of some open terms that will be useful later.

We define the open semantics below by the coinductive extension of $m_{\lambda }$. Let $\tilde{\varGamma }= \varGamma ((A^\mathcal {V})^\omega , A^\omega )$ be the set of causal functions $\mathfrak {c}:(A^\mathcal {V})^\omega \rightarrow A^\omega$, which is the carrier of the final coalgebra for the functor $FX= (A\times X)^{A^\mathcal {V}}$. Notice that a function $\mathfrak {c}:(A^\mathcal {V})^\omega \rightarrow A^\omega$ can equivalently be presented as a function $\tilde{\mathfrak {c}} :(A^\omega )^\mathcal {V}\rightarrow A^\omega$ (swapping the arguments in the domain). Given such a function $\mathfrak {c}:(A^\mathcal {V})^\omega \rightarrow A^\omega$ and a function $\psi :\mathcal {V}\rightarrow A^\omega$, in the sequel, we sometimes abuse of notation by writing $\mathfrak {c}(\psi )$ where we formally mean $\tilde{\mathfrak {c}}(\psi )$.

Definition 5.1

Let $\lambda :\varSigma (A \times {-}) \Rightarrow A \times T$ be a monadic stream GSOS specification. The open semantics of $\lambda$ is the coinductive extension $|\![{-}]\!|_o :T\mathcal {V}\rightarrow \tilde{\varGamma }$ of the Mealy machine $m_\lambda :T\mathcal {V}\rightarrow (A\times T\mathcal {V})^{A^\mathcal {V}}$ defined in (9).

Behavioural equivalence of open terms can now be checked by means of bisimulations on Mealy machines (Definition 2.4). We define open bisimilarity, denoted by $\sim _o$, as the greatest bisimulation on $m_\lambda$. Obviously, for all open terms $t_1,t_2\in T\mathcal {V}$ it holds that $t_1 \sim _ot_2$ iff $|\![{t_1}]\!|_o=|\![{t_2}]\!|_o$. The following result provides another useful characterisation of $|\![{-}]\!|_o$.

Lemma 5.1

Let $\lambda$ be a monadic stream GSOS specification, with induced algebra $\sigma :\varSigma A^\omega \rightarrow A^\omega$. Let $\bar{\lambda }$ be the corresponding Mealy specification, with induced algebra $\bar{\sigma } :\varSigma \tilde{\varGamma }\rightarrow \tilde{\varGamma }$. Then the open semantics $|\![{-}]\!|_o$ is the unique homomorphism making the diagram below commute:

where $\eta$ and $\kappa$ are defined by initiality (Sect. 2), and for each $\mathcal {X}\in \mathcal {V}$ and $\psi :\mathcal {V}\rightarrow A^\omega$, $\textit{proj}(\mathcal {X})(\psi ) = \psi (\mathcal {X})$.

Observe that, by virtue of Theorem 4.1, the algebra $\bar{\sigma }$ is the pointwise extension of $\sigma$. This fact will be useful in the next section to relate $\sim _o$ with bisimilarity on the original stream system.

5.1 Abstract, Open and Closed Semantics

Recall from Sect. 2 the abstract semantics $|\![{-}]\!|_a :T A^\omega \rightarrow A^\omega$ arising as in (B) from a monadic stream specification $\lambda$. The following proposition is the key to prove Theorem 5.1 relating open bisimilarity and abstract semantics.

Proposition 5.1

Let $|\![{-}]\!|_a$ and $|\![{-}]\!|_o$ be the abstract and open semantics respectively of a monadic stream GSOS specification $\lambda$. For any $t \in T\mathcal {V}$, $\psi :\mathcal {V}\rightarrow A^\omega$:

$$|\![{t}]\!|_o(\psi ) = |\![{(T\psi )(t)}]\!|_a \, .$$

As a simple consequence, we obtain the following characterization of $\sim _o$.

Theorem 5.1

For all $t_1, t_2 \in T\mathcal {V}$, $|\![{t_1}]\!|_o = |\![{t_2}]\!|_o$ iff for all $\psi :\mathcal {V}\rightarrow A^\omega$: $|\![{T\psi (t_1)}]\!|_a = |\![{T\psi (t_2)}]\!|_a$.

This is one of the main results of this paper: $T\psi (t_1)$ and $T\psi (t_2)$ are expressions in $TA^{\omega }$ built from symbols of the signature $\varSigma$ and streams $\alpha _1, \dots \alpha _n \in A^\omega$. By checking $t_1 \sim _ot_2$ one can prove that the two expressions are equivalent for all possible streams $\alpha _1, \dots \alpha _n \in A^\omega$.

Example 5.3

By using the Mealy machine $m_\lambda$ in Example 5.1, the relation

$$\begin{aligned} R =&\{(\textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {Y},\mathcal {Z})),\textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {W},\mathcal {Z}))), (\textsf {alt}(\textsf {alt}(\mathcal {Z},\mathcal {Y}),\mathcal {X}),\textsf {alt}(\textsf {alt}(\mathcal {Z},\mathcal {W}),\mathcal {X}))\} \end{aligned}$$

is easily verified to be a bisimulation (Definition 2.4). In particular this shows that $|\![{\textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {Y},\mathcal {Z}))}]\!|_o = |\![{ \textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {W},\mathcal {Z}))}]\!|_o$. By Theorem 5.1, we have that $|\![{T\psi (\textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {Y},\mathcal {Z})))}]\!|_a = |\![{T\psi (\textsf {alt}(\mathcal {X}, \textsf {alt}(\mathcal {W},\mathcal {Z})))}]\!|_a$ for all $\psi :\mathcal {V}\rightarrow A^\omega$, i.e.,

$$ \textsf {alt}(\alpha _1, \textsf {alt}(\alpha _2,\alpha _3)) \sim \textsf {alt}(\alpha _1, \textsf {alt}(\alpha _4,\alpha _3)) \text { for all } \alpha _1,\alpha _2,\alpha _3, \alpha _4 \in A^\omega \text {.}$$

The above law can be understood as an equivalence of program schemes stating that one can always replace the stream $\alpha _2$ by an arbitrary stream $\alpha _4$, without changing the result.

Example 5.4

By using the Mealy machines in Example 5.2, it is easy to check that both $\{ ((\mathcal {X}\oplus \mathcal {Y}) \oplus \mathcal {Z}, \mathcal {X}\oplus ( \mathcal {Y}\oplus \mathcal {Z})) \}$ and $\{ ( \mathcal {X}\oplus \mathcal {Y}, \mathcal {Y}\oplus \mathcal {X}) \}$ are bisimulations. This means that $|\![{(\mathcal {X}\oplus \mathcal {Y}) \oplus \mathcal {Z}}]\!|_o = |\![{ \mathcal {X}\oplus ( \mathcal {Y}\oplus \mathcal {Z})}]\!|_o$ and $|\![{\mathcal {X}\oplus \mathcal {Y}}]\!|_o= |\![{ \mathcal {Y}\oplus \mathcal {X}}]\!|_o$. By Theorem 5.1 we obtain associativity and commutativity of $\oplus$:

$$(\alpha _1 \oplus \alpha _2) \oplus \alpha _3 \sim \alpha _1 \oplus ( \alpha _2 \oplus \alpha _3 ) \text { and } \alpha _1 \oplus \alpha _2 \sim \alpha _2 \oplus \alpha _1 \text { for all } \alpha _1,\alpha _2,\alpha _3 \in A^\omega \text {.}$$

Example 5.5

In a similar way, one can check that $\{((a+b).(\mathcal {X}\oplus \mathcal {Y}), a.\mathcal {X}\oplus b.\mathcal {Y}) \mid a,b \in \mathbb {R}\}$ is a bisimulation. This means that $|\![{(a+b).(\mathcal {X}\oplus \mathcal {Y})}]\!|_o = |\![{ a.\mathcal {X}\oplus b.\mathcal {Y}}]\!|_o$ for all $a,b \in \mathbb {R}$ and, using again Theorem 5.1, we conclude that $(a+b).(\alpha _1 \oplus \alpha _2) \sim a.\alpha _1 \oplus b.\alpha _2 \text { for all } \alpha _1,\alpha _2 \in A^\omega \text {.}$

Often, equivalence of open terms is defined by relying on the equivalence of closed terms: two open terms are equivalent iff under all possible closed substitutions, the resulting closed terms are equivalent. For $\sim _o$, this property does not follow immediately by Theorem 5.1, where variables range over streams, i.e., elements of the final coalgebra. One could assume that all the behaviours of the final coalgebra are denoted by some term, however this restriction would rule out most of the languages we are aware of: in particular, the stream calculus that can express only the so-called rational streams [18].

The following theorem, that is the second main result of this paper, only requires that the stream GSOS specification is sufficiently expressive to describe arbitrary finite prefixes. We use that any closed substitution $\phi :\mathcal {V}\rightarrow T\emptyset$ defines $\phi ^\dagger :T\mathcal {V}\rightarrow T\emptyset$ (see Sect. 2.1).

Theorem 5.2

Suppose $\lambda :\varSigma (A \times {-}) \Rightarrow A \times T_{\varSigma }$ is a monadic stream GSOS specification which contains, for each $a \in A$, the prefix operator $a.-$ as specified in (4) in Sect. 3. Further, assume $T\emptyset$ is non-empty.

Let $|\![{-}]\!|_c$ and $|\![{-}]\!|_o$ be the closed and open semantics respectively of $\lambda$. Then for all $t_1, t_2 \in T\mathcal {V}$: $|\![{t_1}]\!|_o = |\![{t_2}]\!|_o$ iff $|\![{\phi ^\dagger (t_1)}]\!|_c = |\![{\phi ^\dagger (t_2)}]\!|_c$ for all $\phi :\mathcal {V}\rightarrow T\emptyset$.

Example 5.6

The specification in Fig. 2 does not include the prefix operator, therefore it does not meet the assumptions of Theorem 5.2. Instead, the monadic GSOS specification in Fig. 1(b) contains the prefix. Recall from Example 5.5 that $(a+b).(\mathcal {X}\oplus \mathcal {Y}) \sim _oa.\mathcal {X}\oplus b.\mathcal {Y}$. Using Theorem 5.2, we can conclude that $(a+b).(t_1 \oplus t_2) \sim a.t_1 \oplus b.t_2 \text { for all } t_1,t_2\in T\emptyset \text {.}$

6 Bisimulation Up-To Substitutions

In the previous section, we have shown that bisimulations on Mealy machines can be used to prove equivalences of open terms specified in the stream GSOS format. In this section we introduce up-to substitutions, an enhancement of the bisimulation proof method that allows to deal with smaller, often finite, relations. We also show that up-to substitutions can be effectively combined with other well-known up-to techniques such as up-to bisimilarity and up-to context.

Intuitively, in a bisimulation up-to substitutions ${\mathrel {\mathcal {R}}}$, the states reached by a pair of states do not need to be related by ${\mathrel {\mathcal {R}}}$, but rather by ${\theta (\mathrel {\mathcal {R}}})$, for some substitution $\theta :\mathcal {V}\rightarrow T\mathcal {V}$. We give a concrete example. Suppose we extend the stream calculus of Example 2.2 with the operators f and g defined by the rules in Fig. 4. In Fig. 5, we have the pointwise extensions of these new operators. It should be clear that $f(\mathcal {X}) \sim g(\mathcal {X})$. To try to formally prove $f(\mathcal {X}) \sim g(\mathcal {X})$, consider the relation ${\mathrel {\mathcal {R}}} = \{(f(\mathcal {X}), g(\mathcal {X}))\}$. For all $\varsigma :\mathcal {V}\rightarrow A$, there are transitions $f(\mathcal {X}) \xrightarrow {\, {\varsigma | \varsigma (\mathcal {X})} \, } f(\mathcal {X}\oplus \mathcal {X})$ and $g(\mathcal {X}) \xrightarrow {\, {\varsigma | \varsigma (\mathcal {X})} \, } g(\mathcal {X}\oplus \mathcal {X})$. The outputs of both transitions coincide but the reached states are not in ${\mathrel {\mathcal {R}}}$, hence ${\mathrel {\mathcal {R}}}$ is not a bisimulation. However it is a bisimulation up-to substitutions, since the arriving states are related by ${\theta (\mathrel {\mathcal {R}}})$, for some substitution $\theta$ mapping $\mathcal {X}$ to $\mathcal {X}\oplus \mathcal {X}$. In fact, without this technique, any bisimulation relating $f(\mathcal {X})$ and $g(\mathcal {X})$ should contain infinitely many pairs.

Fig. 4.

f and g, operators over streams

Fig. 5.

Pointwise extensions of f and g.

In order to prove the soundness of this technique, as well as the fact that it can be safely combined with other known up-to techniques, we need to recall some notions of the theory of up-to techniques in lattices from [14]. Given a Mealy machine (X, m), we consider the lattice $(\mathcal {P}(X\times X), \subseteq )$ of relations over X, ordered by inclusion, and the monotone map $\mathbf {b}:\mathcal {P}(X\times X) \rightarrow \mathcal {P}(X\times X)$ defined for all ${\mathrel {\mathcal {R}}} \subseteq X\times X$ as

$$\begin{aligned} \mathbf {b}(\mathrel {\mathcal {R}}) = \{(s,t) \in X\times X \mid \forall b \in B, o_s(b)=o_t(b) \text { and } d_s(b) \mathrel {\mathcal {R}}d_t(b)\}\text {.} \end{aligned}$$

(11)

It is easy to see that post fixed points of $\mathbf {b}$, i.e., relations $\mathrel {\mathcal {R}}$ such that ${\mathrel {\mathcal {R}}} \subseteq \mathbf {b}(\mathrel {\mathcal {R}})$, are exactly bisimulations for Mealy machines (Definition 2.4) and that its greatest fixed point is $\sim$.

For a monotone map $\mathbf {f}:\mathcal {P}(X\times X) \rightarrow \mathcal {P}(X\times X)$, a bisimulation up-to $\mathbf {f}$ is a relation ${\mathrel {\mathcal {R}}}$ such that ${\mathrel {\mathcal {R}}} \subseteq \mathbf {b}\mathbf {f}(\mathrel {\mathcal {R}})$. We say that $\mathbf {f}$ is compatible with $\mathbf {b}$ if ${\mathbf {f}\mathbf {b}(\mathrel {\mathcal {R}})} \subseteq \mathbf {b}\mathbf {f}(\mathrel {\mathcal {R}})$ for all relations ${\mathrel {\mathcal {R}}}$. Two results in [14] are pivotal for us: first, if $\mathbf {f}$ is compatible and ${\mathrel {\mathcal {R}}} \subseteq \mathbf {b}\mathbf {f}(\mathrel {\mathcal {R}})$ then ${\mathrel {\mathcal {R}}} \subseteq {\sim }$; second if $\mathbf {f}_1$ and $\mathbf {f}_2$ are compatible with $\mathbf {b}$ then $\mathbf {f}_1\circ \mathbf {f}_2$ is compatible with $\mathbf {b}$. The first result informs us that bisimilarity can be proved by means of bisimulations up-to $\mathbf {f}$, whenever $\mathbf {f}$ is compatible. The second result states that compatible up-to techniques can be composed.

We now consider up-to techniques for the Mealy machine over open terms $(T\mathcal {V},m_\lambda )$ as defined in Sect. 5. Recall that bisimilarity over this machine is called open bisimilarity, denoted by $\sim _o$. Up-to substitutions is the monotone function $(-)_{\forall \theta } :\mathcal {P}( T\mathcal {V}\times T\mathcal {V}) \rightarrow \mathcal {P}( T\mathcal {V}\times T\mathcal {V})$ mapping every ${\mathrel {\mathcal {R}}}\subseteq T\mathcal {V}\times T\mathcal {V}$ to

$$\begin{aligned} (\mathrel {\mathcal {R}})_{\forall \theta } = \{(\theta (t_1),\theta (t_2)) \mid \theta :\mathcal {V}\rightarrow T\mathcal {V}\text { and } t_1 \mathrel {\mathcal {R}}t_2 \}\text {.} \end{aligned}$$

Similarly, we define up-to context as the monotone function mapping every relation ${\mathrel {\mathcal {R}}} \subseteq T\mathcal {V}\times T\mathcal {V}$ to its contextual closure $\mathcal C(\mathrel {\mathcal {R}})$ and up-to (open) bisimilarity as the function mapping $\mathrel {\mathcal {R}}$ to ${\sim _o} \mathrel {\mathcal {R}}{\sim _o} = \{(t_1,t_2) \mid \exists t_1',t_2' \text { s.t. } t_1 \sim _ot_1' \mathrel {\mathcal {R}}t_2' \sim _ot_2 \}$.

Compatibility with $\mathbf {b}$ of up-to context and up-to bisimilarity hold immediately by the results in [6]. For the novel technique, up-to substitutions, we have:

Theorem 6.1

The function $(-)_{\forall \theta }$ is compatible with $\mathbf {b}$.

As a consequence of the above theorem and the results in [14], up-to substitutions can be used in combination with up-to bisimilarity and up-to context (as well as any another compatible up-to technique) to prove open bisimilarity. We will show this in the next, concluding example, for which a last remark is useful: the theory in [14] also ensures that if $\mathbf {f}$ is compatible with $\mathbf {b}$, then $\mathbf {f}(\sim ) \subseteq {\sim }$. By Theorem 6.1, this means that $(\sim _o)_{\forall \theta } \subseteq {\sim _o}$. The same obviously holds for the contextual closure: $\mathcal C(\sim _o)\subseteq {\sim _o}$.

Example 6.1

We prove that the convolution product $\otimes$ distributes over the sum $\oplus$, i.e., $\alpha _1 \otimes (\alpha _2 \oplus \alpha _3) \sim (\alpha _1 \otimes \alpha _2) \oplus (\alpha _1 \otimes \alpha _3) \text { for all streams } \alpha _1,\alpha _2,\alpha _3 \in \mathbb {R}^\omega \text {.}$ By Theorems 5.1 and 6.1, to prove our statement it is enough to show that ${\mathrel {\mathcal {R}}} = \{(\mathcal {X}\otimes (\mathcal {Y}\oplus \mathcal {Z}) , (\mathcal {X}\otimes \mathcal {Y}) \oplus (\mathcal {X}\otimes \mathcal {Z}))\}$ is a bisimulation up-to ${\sim _o} \mathrel {\mathcal {C}(\sim _o(-)_{\forall \theta }\sim _o)}{\sim _o}$.

By rules in Fig. 1(d), for all $\varsigma :\mathcal {V}\rightarrow \mathbb {R}$, the transitions of the open terms are

• $\mathcal {X}\otimes (\mathcal {Y}\oplus \mathcal {Z}) \xrightarrow {\, {\varsigma | \varsigma (\mathcal {X}) \times (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z}))} \, } ( \varsigma (\mathcal {X}) \otimes (\mathcal {Y}\oplus \mathcal {Z}) ) \oplus ( \mathcal {X}\otimes (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})).(\mathcal {Y}\oplus \mathcal {Z}) )$

• $(\mathcal {X}\otimes \mathcal {Y}) \oplus (\mathcal {X}\otimes \mathcal {Z}) \xrightarrow {\, {\varsigma | \varsigma (\mathcal {X}) \times \varsigma (\mathcal {Y}) + \varsigma (\mathcal {X}) \times \varsigma (\mathcal {Z})} \, }$ $((\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y})) \oplus ( (\varsigma (\mathcal {X}) \otimes \mathcal {Z}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z}) )$

For the outputs, it is evident that $\varsigma (\mathcal {X}) \times (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})) = \varsigma (\mathcal {X}) \times \varsigma (\mathcal {Y}) + \varsigma (\mathcal {X}) \times \varsigma (\mathcal {Z})$. For the arriving states we need a few steps, where for all $\varsigma :\mathcal {V}\rightarrow \mathbb {R}$ and $\mathcal {X}\in \mathcal {V}$, $\varsigma (\mathcal {X})$ denotes either a real number (used as a prefix) or a constant of the syntax (Example 2.2).

• (a) $\mathcal {X}\otimes (\varsigma (\mathcal {Y}).\mathcal {Y}\oplus \varsigma (\mathcal {Z}).\mathcal {Z}) \mathrel {{\mathrel {\mathcal {R}}}_{\forall \theta }} (\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z})$.

• (b) By Example 5.5 and $\mathcal C(\sim _o)\subseteq \sim _o$, we have that:

  $\mathcal {X}\otimes (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})).(\mathcal {Y}\oplus \mathcal {Z}) \sim _o\mathcal {X}\otimes (\varsigma (\mathcal {Y}).\mathcal {Y}\oplus \varsigma (\mathcal {Z}).\mathcal {Z})$.

• (c) By (b) and (a):

  $\mathcal {X}\otimes (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})).(\mathcal {Y}\oplus \mathcal {Z}) \mathrel {{\sim _o}{\mathrel {\mathcal {R}}}_{\forall \theta }{\sim _o}}\ \! (\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z})$.

• (d) $\varsigma (\mathcal {X}) \otimes (\mathcal {Y}\oplus \mathcal {Z}) \mathrel {\mathrel {\mathcal {R}}_{\forall \theta }} (\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\varsigma (\mathcal {X}) \otimes \mathcal {Z})$.

• (e) Using (d) and (c) with context $\mathcal {C} = \_ \oplus \_$:

  $(\varsigma (\mathcal {X}) \otimes (\mathcal {Y}\oplus \mathcal {Z})) \oplus (\mathcal {X}\otimes (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})).(\mathcal {Y}\oplus \mathcal {Z}))$

  $\mathrel {\mathcal {C}(\sim _o\mathrel {\mathcal {R}}_{\forall \theta }\sim _o)}~ ((\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\varsigma (\mathcal {X}) \oplus \mathcal {Z})) \oplus ((\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z}))$.

• (f) By Example 5.4 (associativity and commutativity of $\oplus$) and $(\sim _o)_{\forall \rho } \subseteq {\sim _o}$:

  $((\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\varsigma (\mathcal {X}) \oplus \mathcal {Z})) \oplus ((\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z})).\mathcal {Z}))$

  $\sim _o((\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y})) \oplus ((\varsigma (\mathcal {X}) \otimes \mathcal {Z}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z}))$.

• (g) By (e) and (f): $(\varsigma (\mathcal {X}) \otimes (\mathcal {Y}\oplus \mathcal {Z}) ) \oplus ( \mathcal {X}\times (\varsigma (\mathcal {Y}) + \varsigma (\mathcal {Z})).(\mathcal {Y}\oplus \mathcal {Z}) )$

  $\mathrel {{\sim _o}\mathcal {C}(\sim _o\mathrel {\mathcal {R}}_{\forall \theta }\sim _o){\sim _o}} ((\varsigma (\mathcal {X}) \otimes \mathcal {Y}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Y}).\mathcal {Y}) ) \oplus ( (\varsigma (\mathcal {X}) \otimes \mathcal {Z}) \oplus (\mathcal {X}\otimes \varsigma (\mathcal {Z}).\mathcal {Z}) )$.

7 Final Remarks

In this paper we have studied the semantics of open terms specified in the stream GSOS format. Our recipe consists in translating the stream specification into a Mealy specification giving semantics to all open terms. Remarkably, this semantics equates two open terms if and only if they are equivalent under all possible interpretations of variables as streams (Theorem 5.1) or under the interpretation of variables as closed terms (Theorem 5.2). Furthermore, semantic equivalence can be checked by means of the bisimulation proof method enhanced with a technique called up-to substitutions (Theorem 6.1).

Our work can be considered as a first step toward a (co)algebraic understanding of the semantics of open terms in the general setting of abstract GSOS [10, 19]. While our approach exploits several peculiarities of the final coalgebra for stream systems, several intermediate results hold in the general setting: for instance, the construction in Sect. 3 transforming arbitrary stream GSOS specifications into monadic ones, seems to hold for arbitrary abstract GSOS. Another promising clue in this direction comes from the way we specified the semantics of variables in Sect. 5: it is reminiscent of the technique adopted in [2] for dealing with open terms of process calculi denoting labeled transition systems.