Tightness

The same term may have different typing derivations. These will always provide upper bounds for the number of steps and the size of normal forms. The interesting aspect of tight type systems, however, is that the class of typing derivations that provide exact bounds for these measures is captured by tight derivations. The notion of tightness for types, typing environments, type judgments, and typing derivations is as follows:

• • A type $\tau$ is tight if $\tau$ is a tight constant (i.e., $\tau \in {{\tt tt}_{\tt v}}$ for CBV and $\tau \in {{\tt tt}_{\tt n}}$ for CBN).

• • A multi-type $\mathscr{M}$ is tight if, for all $\sigma \in \mathscr{M}$ , $\sigma$ is tight.

• • A typing environment $\Gamma$ is tight if, for all $x \in {\tt dom}(\Gamma )$ , $\Gamma (x)$ is tight.

• • A typing judgment $\Gamma \vdash ^{(b,s)} t\;:\; \tau$ is tight if $\Gamma$ and $\tau$ are tight.

• • A type derivation $\Phi$ is tight if its conclusion is tight.

Soundness and completeness

The type system for Weak Open CBN (resp. CBV) is quantitatively sound and complete with respect to ${\rightarrow }_{\texttt {n}}$ (resp. ${\rightarrow }_{\texttt {v}}$ ). Quantitative soundness means that not only a tightly typable term $t$ is terminating (qualitative) but also that exact and split measures concerning the evaluation of $t$ to normal form (quantitative) can be extracted from its tight type derivation. More precisely, if $\Phi \triangleright {\Gamma }\vdash ^{(b,s)}_{\texttt {n}} t\;:\;\tau$ (resp. $\Phi \triangleright \, \Gamma \negmedspace \overset {(b,s)}{\vdash _{{\tt v}}}\negmedspace t\;:\; \tau$ ) is tight, then there exists $u \in {\tt no}_{{\tt n}}$ (resp. ${\tt no}_{{\tt v}}$ ) such that $t \twoheadrightarrow _{\tt n}^b u$ (resp. $t \twoheadrightarrow _{\tt v}^b u$ ) with $|{u}|_{\texttt {n}} = s$ (resp. $|{u}|_{\texttt {v}} = s$ ). Dually for quantitative completeness. Most of the properties needed for quantitative soundness and completeness are shared between the two type systems. For this reason, we are going to present each statement only once and expect the reader to take the absence of any distinction between the two strategies as an indication of such between the statements. Also, we are going to present, in what follows, only the key properties and leave the auxiliary properties to Appendix A.1, along with their proofs.

To show quantitative soundness and completeness, we need to show that a term $t$ is in Weak Open CBN (resp. CBV) normal form if, and only if, the first counter, which corresponds to the number of $\beta _{\tt n}$ -steps (resp. $\beta _{\tt v}$ -steps) needed to reach a normal form, is zero.

We also need to show that, given a term $t$ , the type system infers the correct Weak Open CBN (resp. CBV) size of the normal form of $t$ .

Finally, we show that, for any term $t$ in Weak Open CBN (resp. CBV) normal form, there exists a tight derivation, which, moreover, infers the correct Weak Open CBN (resp. CBV) size of $t$ .

We now state a quantitative form of type preservation along evaluation.

Quantitative soundness follows from the correctness of the counters for normal forms (Lemma1 and Lemma2) and quantitative subject reduction (Lemma4).

The following property guarantees that typability is preserved back along evaluation.

Quantitative completeness follows from the typability of all normal forms in a tight way (Lemma3) and quantitative subject expansion (Lemma5).

The syntax for Weak Open CBN with Global State

We consider a denumerable set of locations denoted by $l, l', \ldots$ . The sets of terms, states, and configurations are presented in Figure 7.

Figure 7.

The extended syntax of the $\lambda$ -calculus for Weak Open CBN with Global State.

The set of terms of the $\lambda$ -calculus is extended with two operations: ${\tt get}_{l}(\lambda x.t)$ and ${\tt set}_{l}(t,u)$ , where $l$ is a location drawn from some set of location names. The former is used to get the term at location $l$ and bind it to $x$ in $t$ . The latter is used to set a term $t$ to location $l$ in the state and then continue with term $u$ . States are either the empty state, denoted by $\varepsilon$ , or the result of some sequence of update operations over the empty state. Given some state $s$ , an update over $s$ consists in setting some term $t$ to some location $l$ , and it is denoted by ${\tt upd}_{l}(t,s)$ . If the same location $l$ is updated twice in a row, for example, ${\tt upd}_{l}(t,{\tt upd}_{l}(u,s))$ , only $t$ (the most recent update) should be accessible at location $l$ according to our operational semantics in Figure 9. Configurations are pairs consisting of a term and a state. The sets of free and bound variables of terms, states, and configuration are as expected.

It is important to note that, if ${\tt upd}_{l}(t,{\tt upd}_{l}(u,s))$ is assumed to be equal to ${\tt upd}_{l}(t,s)$ , as in de’Liguoro and Treglia (Reference de’Liguoro, Treglia, Veltri, Benton and Ghilezan2021) and Alves et al. (Reference Alves, Kesner, Ramos, Hansen, Scedrov and de Queiroz2023), this corresponds to presenting the global state as a mapping from locations to terms, that is, the coalgebraic structure for the state monad (Plotkin and Power, Reference Plotkin, Power, Bauer and Mislove2008). We assume instead that if ${\tt upd}_{l}(t,{\tt upd}_{l}(u,s))$ is not equal to ${\tt upd}_{l}(t,s)$ , this corresponds to presenting the global state as a mapping from locations to lists of terms, that is, the coalgebraic structure for the state-logging monad (Ahman and Uustalu, Reference Ahman, Uustalu, Jacobs, Silva and Staton2014). We also consider the update operation to be commutative in the following sense:

\begin{equation*} \begin{array}{rclr} {\tt upd}_{l}(t,{\tt upd}_{l'}(u,s)) & \equiv _{\tt c} & {\tt upd}_{l'}(u,{\tt upd}_{l}(t,s)) \text{ if $l \not = l'$} \end{array} \end{equation*}

The equivalence relation generated by the algebraic equation $\equiv _{\tt c}$ is denoted by $\equiv$ . We write $l \in {\tt labels}(s)$ , if $s \equiv {\tt upd}_{l}(t,q)$ , for some term $t$ and state $q$ . Moreover, for every location $l$ , these $t$ and $q$ are unique. A consequence of the equivalence relation $\equiv$ on states is that, whenever we want to refer to the most up-to-date content of a location in a given state $s$ , we can always write $s$ so that the first update operation is the latest one over that particular location. Said differently, for every $l \in {\tt labels}(s)$ , $s \equiv {\tt upd}_{l}(u,q)$ , for some term $u$ and some state $q$ . From now on, if $s \equiv {\tt upd}_{l}(u,q)$ , we write $s(l)$ to denote the (unique) term $u$ .

Weak Open CBN size

The notion of Weak Open CBN size is extended to operations, states, and configurations according to Figure 8.

Figure 8.

Weak Open CBN size extended for global state.

Evaluation in Weak Open CBN with Global State

Weak Open CBN evaluation ${\rightarrow }_{\texttt {n}}$ is extended to global state by lifting it to configurations and taking into consideration the interaction between get and set operations and the global state. Evaluation for Weak Open CBN with Global State $\rightsquigarrow _{{\tt n}}$ is defined as the union of the reduction rules in Figure 9.

Figure 9.

Reduction rule for Weak Open CBN with Global State.

Rules ( $\beta _{\tt n}$ ) and (appL) are the same as for the Weak Open CBN, if we take only the term part of the configuration. Rules (get) and (set) describe how the get and set operations interact with the global state. In particular, notice that, while rule (set) can always be fired, rule (get) can get stuck if it tries to interact with a location on the global state that is empty.

Blocked configurations

A configuration $(t,s)$ is said to be blocked if: $t = {\tt get}_{l}(\lambda x.u)$ and $l \not \in {\tt labels}(s)$ ; or $t = up$ and $(u,s)$ is blocked. We say that $(t,s)$ is an unblocked otherwise.

Example 1. Let $l,l'$ be two locations such that $l \neq l'$ . Then, the three following configurations are all blocked:

\begin{equation*} \begin{array}{c} ({\tt get}_{l}(\lambda x.x), \varepsilon ) \quad (({\tt get}_{l}(\lambda x.x)) (\lambda y.y), {\tt upd}_{l'}(t,\varepsilon )) \quad ((({\tt get}_{l}(\lambda x.x)) (\lambda y.y)) (\lambda z.z), {\tt upd}_{l'}(t,\varepsilon ))\end{array} \end{equation*}

Normal configurations

Clearly, blocked configurations are a kind of normal form for configurations. However, they are not the desired result of a computation – even though they are in normal form – but rather a runtime error that should be caught by the type system (i.e., blocked configurations will be untyped). A configuration $(t,s)$ is said to be normal if it is a blocked configuration or $t \in {\tt no}_{{\tt n}}$ , where ${\tt no}_{{\tt n}}$ is defined in Figure 10.

Figure 10.

Normal and neutral terms for Weak Open CBN with Global State.

Example 2. Consider the configuration $c_0 = ((\lambda x.{\tt set}_{l}(z,x)) ({\tt get}_{l}(\lambda y.y)), \varepsilon )$ . Then we can reach a normal configuration as follows:

\begin{align*} ((\lambda x.{\tt set}_{l}(z,x)) ({\tt get}_{l}(\lambda y.y)), \varepsilon ) & \rightsquigarrow _{{\beta _{{\tt n}}}} ({\tt set}_{l}(z,{\tt get}_{l}(\lambda y.y)), \varepsilon ) \\ &\rightsquigarrow _{{\tt s}} ({\tt get}_{l}(\lambda y.y), {\tt upd}_{l}(z,\varepsilon ))\\ & \rightsquigarrow _{{\tt g}} (z, {\tt upd}_{l}(z,\varepsilon )) \end{align*}

The following proposition, whose proof is in Appendix A.2, shows that normal configurations are indeed irreducible.

Figure 11.

The CBN monadic translation for types, typing environments, and judgments.

The CBN monadic translation

In order to lift the type system for Weak Open CBN into the global state monad, we are going to follow the CBN translations of the simply-typed $\lambda$ -calculus in Wadler (Reference Wadler1990). Naturally, we will have to slightly adapt the translation in order to accommodate for multi-types. The resulting translation of the Weak Open CBN types in Figure 5 is presented in Figure 11.

This figure deserves some comments. Notice that we choose to use two different arrows to denote our types: the first one $\Rightarrow$ is used for the global state monad, while the second one $\rightarrow$ is the usual functional arrow type. Also, the $T$ translation is meant to be understood as a template for building the type system.

State and monadic (multi-)types

The grammar for types is presented in Figure 12.

Figure 12.

Types for Weak Open CBN with Global State.

State types, denoted by $\mathscr{T\;}$ , can be seen as partial functions mapping all but a finite set of locations to the empty list multi-type $\star$ . Notice that $\mathscr{T\;}$ is the set of all state types and corresponds to $S$ in Figure 11. The domain of a state type $\mathscr{T\;}$ is denoted by ${\tt dom}(\mathscr{T\;})$ and defined as ${\tt dom}(\mathscr{T\;}) := \{ l \mid (l\;:\;\mathscr{L\;}) \in \mathscr{T\;} \mbox{ and } \mathscr{L\;} \neq \star \}$ . We write $\mathscr{T\;}(l)$ to denote the list multi-type associated with location $l$ in $\mathscr{T\;}$ , whenever $l \in {\tt dom}(\mathscr{T\;})$ . To simplify the notation, we will omit $\star$ whenever a list multi-type is not empty; thus, for example, $[ \mu ] \cdot \star$ will be simply written as $[ \mu ]$ . It is important to remark that the use of list multi-types is crucial in order to match the behavior exhibited by our notion of state; for example, if an empty location $l$ is updated twice, first with a term of type $\mathscr{M}_1$ and then with a term of type $\mathscr{M}_2$ , then it will be assigned type $\mathscr{M}_2 \cdot (\mathscr{M}_1 \cdot \star )$ .

We write $\mathscr{T\;};\; \mathscr{T\;}'$ to mean that ${\tt dom}(\mathscr{T\;}) \cap {\tt dom}(\mathscr{T\;}') = \emptyset$ . Notice that $\{l \;:\; \star \};\; \mathscr{T\;} = \mathscr{T\;}$ and $\{l \;:\; \mathscr{L\;}\};\; \mathscr{T\;} \neq \mathscr{T\;}$ if $\mathscr{L\;} \neq \star$ , so in particular $\{l \;:\; [ \, ]\};\; \mathscr{T\;} \neq \mathscr{T\;}$ . However, recall that $x \;:\; [ \, ];\; \Gamma = \Gamma$ .

Multi-types are now multi-sets of monadic types. And monadic types are lifted types of the form $\mathscr{T\;} \Rightarrow (\tau \times \mathscr{T\;}')$ , where the state type $\mathscr{T\;}\;$ is called an input state, and $\tau \times \mathscr{T\;}'$ is a configuration type consisting of a type $\tau$ and a state type $\mathscr{T\;}'$ called output state. When dealing with effects, it is important to distinguish between pure and monadic types. Indeed, pure types are monadic types that have the same input and output state type, for example, $\mathscr{T\;} \Rightarrow (\tau \times \mathscr{T\;})$ .

Typing system for Weak Open CBN with Global State

The type system for Weak Open CBN with Global State is defined by the typing rules in Figure 13.

Figure 13.

Typing rules for Weak Open CBN with Global State.

Like in the case for Weak Open CBN and CBV, we now present some necessary definitions:

• • The notions of typing environments are extended to global state as expected.

• • Typing rules follow the structure introduced in the previous section.

• • Typing judgments are of three different kinds:

  1. – Term typing judgments are used to type terms and are of the form $\Gamma \Vdash ^{(b,m,d)} t\;:\; \mathscr{G\,}$ , where $(b,m,d)$ is a counter, $\Gamma$ is a typing context, and $\mathscr{G\,}$ represents either a multi-type $\mathscr{M}$ or a monadic type $\mu$ .

  2. – State typing judgments are used to type states and are of the form $\Delta \Vdash ^{(b,m,d)} s\;:\; \mathscr{T\;}$ , where $(b,m,d)$ is a counter, $\Delta$ is a typing context, and $\mathscr{T\;}$ a state type.

  3. – Configuration typing judgments are used to type configurations and are of the form $\Gamma \Vdash ^{(b,m,d)} (t,s): \kappa$ , where $(b,m,d)$ is a counter, $\Gamma$ is a typing context, and $\kappa$ is a configuration type.

  The meaning of the counters $b$ , $m$ , and $s$ will again be made precise later.

• • The notions of type derivation and typability are extended to global state as expected.

We distinguish between consuming and persistent rules just like before. The intuitions behind the rules are the following (again, please ignore the counters $(b,m,d)$ for now):

• • Rules (ax), ( $\lambda$ ), ( $\lambda _{\tt p}$ ), (@ $_{{\tt p}}$ ), and (m). These rules are essentially the same as for Weak Open CBN, but types are now replaced by monadic types.

• • Rule (@). In this rule, we can see how CBN influences the type system. Since, according to CBN, we only reduce on the left-hand side of applications, only the term $t$ on the left can (potentially) change the global state. Therefore, CBN tells us the following: (1) $t$ is reduced until reaching an abstraction $\lambda x.p$ , going from input state type $\mathscr{T\;}$ to an output state type $\mathscr{T\;}'$ ; then (2) the $\beta _{\tt n}$ -rule will be fired, resulting in the term $p \{x \setminus u\}$ ; and then (3) reduction continues with $p \{x \setminus u\}$ , thus going from input state type $\mathscr{T\;}'$ to the output state type in the configuration type $\kappa$ . Remark that the arrow type $\mathscr{M} \rightarrow (\mathscr{T\;}' \Rightarrow \kappa )$ appearing in the monadic type of $t$ represents the fact that, after $\lambda x.p$ interacts with term $u$ of type $\mathscr{M}$ , $p \{x \setminus u\}$ continues the computation with behavior $\mathscr{T\;}' \Rightarrow \kappa$ .

• • Rule (get). This rule is used to type the get operation. Given some state $s$ , ${\tt get}_{l}(\lambda x.t)$ binds a copy of $s(l)$ to $x$ and returns $t \{x \setminus s(l)\}$ . This is reflected by this typing rule: it assigns input state type $\{l \;:\; \Gamma (x) \sqcup \mathscr{M} \cdot \mathscr{L\;}\} ;\; \mathscr{T\;}$ to ${\tt get}_{l}(\lambda x.t)$ under typing context $\Gamma \setminus x$ , so that $l \in {\tt labels}(s)$ (by Lemma 42), and assigns input state $s$ to $t$ under typing context $\Gamma$ , to indicate that evaluating $t$ consumes a copy of $s(l)$ . It is also important to note that the premise of the rule assigns input state $\{l \;:\; \mathscr{M} \cdot \mathscr{L\;}\};\; \mathscr{T\;}$ to $t$ because $t$ has access to the same input state as ${\tt get}_{l}(\lambda x.t)$ . Of course, $t$ is not required to use the contents of location $l$ , in which case $\mathscr{M} = [ \, ]$ .

• • Rule (set). This rule is used to type the set operation, which is – in a sense – dual to the get operation. Given a state $s$ and terms $t$ and $u$ , ${\tt set}_{l}(t,u)$ (possibly) extends $s$ to location $l$ by assigning $t$ to $s(l)$ and returns $u$ , which now has access to $t$ in location $l$ . This is reflected by this typing rule: it assigns input state type $\{l \;:\; \mathscr{L\;}\};\; \mathscr{T\;}$ to ${\tt set}_{l}(t,u)$ , which is the type of the state before it is changed by set, while assuming the input state type $\{l \;:\; \mathscr{M} \cdot \mathscr{L\;}\};\; \mathscr{T\;}$ for $u$ , where $\mathscr{M}$ is the type of $t$ .

• • Rule (emp). This rule is used to type the empty state. By definition, the empty state has an empty set type $\emptyset$ because no location is set.

• • Rule (upd). This rule is used to type the update operation and is very similar to rule (set). However, instead of describing the behavior of a term after a set operation, it describes the behavior of a state $s$ after it is updated to ${\tt upd}_{l}(t,s)$ , for some location $l$ and term $t$ . Of course, if location $l$ was already defined in $s$ , this information is already recorded in the type $\{l\;:\;\mathscr{L\;}\};\; \mathscr{T\;}$ of $s$ . Moreover, the new type of ${\tt upd}_{l}(t,s)$ keeps track of this old information by adding the multi-type $\mathscr{M}$ of the term $t$ to the head of list multi-type $\mathscr{L\;}$ .

• • Rule (conf). This rule is used to type configurations. In order to type a configuration of the form $(t,s)$ , both $t$ and $s$ need to be typed. Note that $s$ is the input state of $t$ ; therefore, the input state type of $t$ needs to match the type of $s$ . The type of the configuration is a configuration type, and it reflects the behavior of the normal configuration reached from $(t,s)$ .

Counters

Our counters are now of the form $(b,m,d)$ . This will allow us to not only keep track of the number of $\beta _{\tt n}$ -steps and the size of normal configurations, just like for Weak Open CBN, but also the number of $\tt g$ / $\tt s$ -steps. We are going to use $d$ to denote the size of normal configurations instead of $s$ to avoid confusion with notation for states. Let $\Phi \triangleright \, \Gamma \Vdash ^{(b,m,d)}_{\tt n} (t,s) \;:\; \kappa$ . Then $b$ is the number of $\beta _{\tt n}$ -steps needed to normalize $t$ , $d$ is the size of the normal configurations reached from $(t,s)$ , and $m$ is the number of $\tt g$ / $\tt s$ steps. Note that, while the first and last counters depend on the interplay between consuming and persistent typing rules, the second counter is simply counting the number of get and set operations that occur during evaluation. This is due to the fact that blocked configurations are not typable (by Proposition4), and thus, get and set never get stuck. However, this is not the same thing as the total number of get and set operations in a configuration. As an example, consider configuration $(\lambda x.{\tt get}_{l}(\lambda y.t), s)$ . This configuration has (at least) one get operation, but it is already a normal configuration. Therefore, no operation occurs during evaluation, and this should be reflected by the counter.

CBN tightness

We generalize the notion of tightness for types, typing environments, type judgments, and typing derivations as follows:

• • A type $\tau$ is tight if $\tau$ is a tight constant (i.e., $\tau \in {{\tt tt}_{\tt n}}$ ).

• • A multi-type is tight if, for all $\mu \in \mathscr{M}$ , $\mu$ is tight.

• • A list multi-type $\mathscr{L\;}$ is tight if $\mathscr{L\;} = \star$ , or $\mathscr{L\;} = \mathscr{M} \cdot \mathscr{L\;}'$ , $\mathscr{M} = [ \, ]$ , and $\mathscr{L\;}'$ is tight.

• • A monadic type $\mathscr{T\;} \Rightarrow (\tau \times \mathscr{T\;}')$ is tight if $\tau \in {{\tt tt}_{\tt n}}$ .

• • A state type $\mathscr{T\;}$ is tight if, for all $l \in {\tt dom}(\mathscr{T\;})$ , $\mathscr{T\;}(l)$ is tight. This tightness restriction on state types ensures that any information provided by the input state that is not used during evaluation is discarded by the (tight) type system.

• • A configuration type $(\tau \times \mathscr{T\;})$ is tight if $\tau \in {{\tt tt}_{\tt n}}$ and $\mathscr{T\;}$ is tight.

• • A typing environment $\Gamma$ is tight if, for all $x \in {\tt dom}(\Gamma )$ , all $\mu \in \Gamma (x)$ are of the form $\mathscr{T\;} \Rightarrow (\tau \times \mathscr{T\;})$ , and $\tau \in {{\tt tt}_{\tt n}}$ . This tightness restriction on typing environments has one crucial consequence: persistent variables cannot change the global state and thus are assigned types that reflect that fact, that is, types that have the same input and output state type (see Lemma6).

• • A term typing judgment $\Gamma \Vdash ^{(b,m,d)} t\;:\; \mathscr{G\,}$ is tight if $\Gamma$ is tight and $\mathscr{G\,}$ is tight.

• • A state typing judgment $\Delta \Vdash ^{(b,m,d)} s\;:\; \mathscr{T\;}$ is tight if $\Delta$ and $\mathscr{T\;}$ are tight.

• • A configuration typing judgment $\Gamma \Vdash ^{(b,m,d)} (t,s): \kappa$ is tight if $\Gamma$ and $\kappa$ are tight.

• • A typing derivation $\Phi$ is tight if it concludes with a tight judgment.

Soundness and completeness

We now extend the soundness and completeness results for Weak Open CBN to Weak Open CBN with Global State. The type system for Weak Open CBN with Global State is quantitatively sound and complete with respect to the extended reduction relation $\rightsquigarrow _{{\tt n}}$ to configurations. Quantitative soundness means that not only a tightly typable configuration $(t,s)$ is terminating (qualitative) but also that exact and split measures (quantitative) concerning the evaluation of $(t,s)$ to a normal configuration can be extracted from its tight type derivation. More precisely, if $\Phi \triangleright \, \Gamma \, \Vdash _{{\tt n}}^{(b,m,d)} (t,s) \;:\; \kappa$ is a tight derivation, then there exists a normal configuration $(u,q)$ such that $(t,s) \rightsquigarrow _{\tt n}^{(b,m)} (u,q)$ with $|{(u,q)}|_{\texttt {n}} = d$ . Dually for quantitative completeness. The key properties necessary to show quantitative soundness and completeness for Weak Open CBN are extended to Weak Open CBN with Global State, and additional properties strictly related to the presence of the global state are necessary. All proofs are in Appendix A.2.

Correctness of the counters

To show quantitative completeness, we need to show that if a typable configuration $(t,s)$ is normal, then the counters of its tight derivation are as follows: the first counter, which corresponds to the number of $\beta _{\tt n}$ -steps needed to reach a Weak Open CBN with Global State normal configuration, is zero; the second counter, which corresponds to the number of $\tt g$ / $\tt s$ -steps, is zero; and the last counter is equal to the Weak Open CBN with Global State size of $(t,s)$ .

Given some term $t$ in normal form, the type system will always infer a tight monadic type for $t$ with the same input and output state types. This is due to the fact that normal forms do not interact with the state (at least in an effectful way) and is formalized as follows.

A stronger crucial property can (and will) be shown for the CBV setting (Lemma 15). For the CBN setting, this weaker property is enough.

The following property shows that terms in normal form and tight states have correct size.

The fact that unblocked normal configurations also have correct size follows from Lemmas 6 and 7.

Lemma 9 (Normal configurations have correct size). Let $\Phi \triangleright \, \Gamma \Vdash ^{(b,m,d)}_{\tt n} (t,s) \;:\; \kappa$ be a tight derivation of the following form:

\begin{equation*} \frac { {\Phi _t \triangleright \,{\Gamma _t} \Vdash ^{(b_t,m_t,d_t)}_{\texttt {n}}{t}\;:\;{{\mathscr{T\;}}\Rightarrow ({\tau }\times {\mathscr{T\;}'})}}\,\,\,\,\, {\Pi _s \triangleright \, {\Delta _s} \Vdash ^{(b_t,m_t,d_t)}_{\texttt {n}}\,\,{s}\;:\;{\mathscr{T\;}}}} {{\Gamma _t + \Delta _s}\Vdash ^{(b_t+b_s,m_t+m_s,d_t+d_s)}{(t,s)}\;:\;{{\tau }\times {\mathscr{T\;}'}}} (\texttt {conf}) \end{equation*}

where $\Gamma = \Gamma _t + \Delta _s$ , $\kappa = (\tau \times \mathscr{T\;}')$ , $b = b_t + b_s$ , $m = m_t + m_s$ , and $d = d_t + d_s$ . If $b_t = m_t = 0$ , then (i) $d = |{(t,s)}|_{\texttt {n}}$ and $b = m = 0$ .

The fact that this is also the case for normal configurations follows from Lemma 8.

Another important property that needs to be shown is that the type system only types unblocked configurations.

This property allows us to conclude that a typable configuration $(t,s)$ that takes zero steps is such that $t \in {\tt no}_{{\tt n}}$ .

The following property is crucial in order to show that any state and normal form can be typed in a tight way.

And the fact that any unblocked normal configuration $(t,s)$ can be typed in a tight way follows easily from Lemma 11.

We now show a quantitative form of type preservation along evaluation for configurations.

Quantitative soundness follows from the correctness of the counters for configurations that are normal but unblocked (Lemma 7 and Lemma 10) and quantitative subject reduction (Lemma 13)

The following property guarantees that typability is also preserved back along evaluation.

Quantitative completeness follows from the typability of all configurations normal but unblocked in a tight way (Lemma 12) and quantitative subject expansion (Lemma 14).

The syntax for Weak Open CBV with Global State

The sets of values, terms, states, and configurations are presented in Figure 14.

Figure 14.

The extended syntax of the $\lambda$ -calculus for Weak Open CBV with Global State.

All notions regarding free and bound variables of terms, states, and configurations, as well as the notion of the domain of a state, are similar to the case of Weak Open CBN with Global State (c.f. Section 3.1).

Weak Open CBV size

The notion of Weak Open CBV size is extended to operations, states, and configurations according to Figure 15.

Figure 15.

Weak Open CBV size extended for global state.

Evaluation and blocked configurations in Weak Open CBV with Global State

Weak Open CBV evaluation is extended to global state in the same way as we did for Weak Open CBN with Global State. The extension is denoted by $\rightsquigarrow _{{\tt v}}$ , and due to the particular nature of CBV, this notion is mutually defined with that of blocked configurations in Figure 16. Indeed, evaluation for Weak Open CBV with Global State is defined as the union of the reduction rule in Figure 16a, where the notion of blocked configurations is in Figure 16b.

Figure 16.

Reduction rules and blocked configurations for Weak Open CBV with Global State.

In particular, note that reduction rule (appR) only evaluates terms on the right of applications if the term on the left is not blocked. Rule ( $\beta _{\tt v}$ ) and (appL) are the same as for Weak Open CBV, and rules (get) and (set) are the same as for Weak Open CBN with Global State, with the only difference being the fact that the global state only holds values.

Example 5. All configurations in Example 1 are also blocked configurations in this setting, and so are the following ones, when assuming $l \neq l'$ :

\begin{equation*} \begin{array}{c} ((\lambda y.y) ({\tt get}_{l}(\lambda x.x)), {\tt upd}_{l'}(t,\varepsilon )) \quad (((\lambda y.y) ({\tt get}_{l}(\lambda x.x))) (\lambda z.z), {\tt upd}_{l'}(t,\varepsilon )) \end{array} \end{equation*}

Normal configurations

The notion of normal configuration is also the same as for Weak Open CBN with Global State: a configuration $(t,s)$ is said to be normal if it is a blocked configuration or $t \in {\tt no}_{{\tt v}}$ , where ${\tt no}_{{\tt v}}$ is defined in Figure 17.

Figure 17.

Normal and neutral terms for Weak Open CBV with Global State.

Example 6. Consider configuration $c_1 = ((\lambda x.{\tt get}_{l}(\lambda y.yx)) ({\tt set}_{l}((\lambda w.w),z)), \varepsilon )$ . Then we can reach a normal configuration as follows:

\begin{align*} ((\lambda x.{\tt get}_{l}(\lambda y.yx)) ({\tt set}_{l}((\lambda w.w),z)), \varepsilon ) & \rightsquigarrow _{{\tt s}} ((\lambda x.{\tt get}_{l}(\lambda y.yx)) z, {\tt upd}_{l}((\lambda w.w),\varepsilon )) \\&\rightsquigarrow _{{\beta _{{\tt v}}}} ({\tt get}_{l}(\lambda y.yz), {\tt upd}_{l}((\lambda w.w),\varepsilon ))\\ & \rightsquigarrow _{{\tt g}} ((\lambda w.w) z, {\tt upd}_{l}((\lambda w.w),\varepsilon )) \\&\rightsquigarrow _{{\beta _{{\tt v}}}} (z, {\tt upd}_{l}((\lambda w.w),\varepsilon )) \end{align*}

The following proposition, whose proof is in Appendix A.3, shows that normal configurations are indeed irreducible.

The CBV monadic translation

In order to lift the type system for Weak Open CBV into the global state monad, we are going to follow the CBV translations of the simply-typed $\lambda$ -calculus in Wadler (Reference Wadler1990). Again, we will have to slightly adapt the translation in order to accommodate for tight constants and multi-types. The resulting translation is presented in Figure 18.

Figure 18.

The CBV monadic translation for types, typing environments, and judgments.

State and monadic (multi-)types

The grammar for types is presented in Figure 19.

Figure 19.

Types for Weak Open CBV with Global State.

State Types for Weak Open CBV with Global State, and all related notions, are similar to those for Weak Open CBN with Global State.

Multi-types are multi-sets of value types. Monadic types and configuration types are exactly the same as for Weak Open CBN with Global State. Moreover, since some typing rules will apply both to value types and multi-types, we will use $\pi$ as a new auxiliary symbol ranging over both these kinds of types. All other notions related to types are common to Weak Open CBN with Global State. Note that the set of tight constants now consists of three different symbols, as in Weak Open CBV.

Typing system for Weak Open CBV with Global State

The type system for Weak Open CBV with Global State is defined by the typing rules in Figure 20.

Figure 20.

Typing rules for Weak Open CBV with Global State.

The notions of typing environments, typing judgments, typing rules, and typability are the same as those for Weak Open CBN with Global State.

Like in Weak Open CBN with Global State, we distinguish between consuming and persistent rules. The intuition behind the rules is the following (again, ignore the counters $(b,m,d)$ ):

• • Rules (ax), ( $\lambda$ ), (@ $^1_{{\tt p}}$ ), (@ $^2_{{\tt p}}$ ), and (m). These rules are essentially the same as for Weak Open CBV, but with types replaced with monadic types.

• • Rule ( $\lambda _{\tt p}$ ) is the same as for Weak Open CBV.

• • Rule (@). In this rule, we see how CBV, which is more intricate than the CBN, influences the type system. Note that, just like in the case of Weak Open CBN with Global State, this rule is used to type consuming applications. Therefore, CBV tells us the following: (1) $t$ is reduced until reaching an abstraction $\lambda x.p$ , going from input state type $\mathscr{T\;}$ to an output state type $\mathscr{T\;}''$ ; then (2) $u$ is reduced until reaching a value $v$ , going from the input state type $\mathscr{T\;}''$ to an output state type $\mathscr{T\;}'$ ; then (3) the $\beta _{\tt v}$ -rule will be fired, resulting in the term $p \{x \setminus v\}$ ; and then (4) reduction continues with $p \{x \setminus v\}$ , thus going from input state type $\mathscr{T\;}'$ to the output state type in the configuration type $\kappa$ . Remark that the arrow type $\mathscr{M} \rightarrow (\mathscr{T\;}' \Rightarrow \kappa )$ appearing in the monadic type of $t$ represents the fact that, after $\lambda x.p$ interacts with value $v$ of type $\mathscr{M}$ , $p \{x \setminus v\}$ continues the computation with behavior $\mathscr{T\;}' \Rightarrow \kappa$ .

• • Rule ( $\uparrow$ ). In CBV, variables and abstractions are values. Rule ( $\uparrow$ ) lifts value types and multi-types to pure types, that is, monadic types with the same input and output state types.

• • Rules (get), (set), (emp), (upd), and (conf). These rules are essentially the same as for Weak Open CBN with Global State, with the only difference being that rules (set) and (upd) place values into locations instead of arbitrary terms.

Counters

The counters work as they did for Weak Open CBN with Global State.

CBV tightness

The notions of tightness for types, typing environments, type judgments, and typing derivations for CBV are as follows:

• • A type $\tau$ is tight if $\tau$ is a tight constant (i.e., $\tau \in {{\tt tt}_{\tt v}}$ ).

• • The notions of tightness for multi-types, list multi-types, and monadic, state, and configuration types are defined just like for Weak Open CBN with Global State (Section 3.1.2).

• • A typing environment $\Gamma$ is tight if, for all $x \in {\tt dom}(\Gamma )$ , $\Gamma (x)$ is tight.

• • The notions of tightness for typing judgments and typing derivations are defined in the same way as for Weak Open CBN with Global State (Section 3.1.2).

Soundness and completeness

We now extend the soundness and completeness result for Weak Open CBN to Weak Open CBV with Global State. The key properties necessary to show quantitative soundness and completeness for Weak Open CBV with Global State are the same as for Weak Open CBN with Global State. However, a stronger pureness over normal forms is necessary in order to show quantitative completeness, due to the fact that reduction happens both on the left and on the right of applications. All proofs are in Appendix A.3.

Correctness of the counters

To show quantitative completeness, we need to show that if a configuration $(t,s)$ is normal, then the counters of its tight derivation are as follows: the first counter, which corresponds to the number of $\beta _{\tt v}$ -steps needed to reach a Weak Open CBV with Global State normal configuration, is zero; the second counter, which corresponds to the number of $\tt g$ / $\tt s$ -steps, is zero; and the last counter is equal to the Weak Open CBN with Global State size of $(t,s)$ .

Given some term $t \in {\tt no}_{{\tt v}}$ , in addition to the type system always inferring a tight monadic type for $t$ with the same input and output state types, such a type derivation exists for any state type.

The previous property is a stronger version of Lemma6. It is a crucial property in order to show the quantitative versions of the usual subject reduction (Lemma 19) and subject expansion (Lemma 20) properties for CBV. We now provide an intuitive explanation of why this is the case, and point the reader to the full proofs in Appendix A.3 for a better understanding. Consider the following configuration $(tu,s)$ , such that $(t,s) \not \rightsquigarrow _{{\tt v}}$ and $(u,s) \rightsquigarrow _{{\tt v}} (u',q)$ . Moreover, let us assume that $s \neq q$ . Then, $(tu,s) \rightsquigarrow _{{\tt v}} (tu',q)$ , and it is easy to see that $(t,q) \not \rightsquigarrow _{{\tt v}}$ (by Proposition5). Clearly, the typability of $(tu,s)$ depends on the typability of $(t,s)$ , and the typability of $(tu',q)$ depends on the typability of $(t,q)$ . However, the typability of $t$ in $(t,s)$ tells us that $t$ takes $s$ as its input state, and the typability of $t$ in $(t,q)$ tells us that $t$ should also be able to take $q$ as its input state. Hopefully, this example also clarifies why this property is not necessary in the CBN setting, where evaluation only occurs on the left of applications.

The following lemma tells us that unblocked normal configurations have correct size.

Lemma 16 (Tight configurations have correct size). Let $\Phi \triangleright \, \Gamma \Vdash ^{(b,m,d)}_{\tt v} (t,s):\kappa$ be a tight derivation of the following form:

\begin{equation*} \frac {\Phi _t \triangleright \, \Gamma _t\Vdash ^{(b_t,m_t,d_t)}_{\tt v} t\;:\;\mathscr{T\;} \Rightarrow (\tau \times \mathscr{T\;}') \quad \Pi _s \triangleright \Delta _s \Vdash ^{(b_s,m_s,d_s)}_{\tt v} s\;:\;\mathscr{T\;}}{\Gamma \Vdash ^{(b,m,d)} (t,s): \kappa }({\tt conf}) \end{equation*}

where $\Gamma = \Gamma _t + \Delta _s$ , $\kappa = (\tau \times \mathscr{T\;}')$ , $b = b_t + b_s$ , $m = m_t + m_s$ , and $d = d_t + d_s$ . If $b_t = m_t = 0$ , then (i) $b = m = 0$ and (ii) $d = |{(t,s)}|_{\texttt {v}}$ .

Like in Weak Open CBN with Global State, it is also necessary to show that typed configurations that take zero steps are not blocked.

Moreover, we also need to show that all unblocked normal configurations are tightly typable.

We now show a quantitative form of type preservation along evaluation for configurations, which relies on Lemma 15.

Quantitative soundness follows from the correctness of the counters for unblocked normal configurations (Lemma 17 and Lemma 16) and quantitative subject reduction (Lemma 19).

The following property also relies on Lemma 15 and guarantees that typability is preserved back along evaluation.

Quantitative completeness follows from the typability of all unblocked normal configurations in a tight way (Lemma 18) and quantitative subject expansion (Lemma 20).