2.1. Kleene algebra with tests

This section recalls some basic information about Kleene algebra with tests (Kozen Reference Kozen1997; Kozen and Smith Reference Kozen, Smith, van Dalen and Bezem1997). We assume that the reader is familiar with the notion of an idempotent semiring.

Definition 1. A Kleene algebra (Reference KozenKozen 1994a ) is an idempotent semiring $(K, +, \cdot, 0, 1)$ expanded with an operation $^{*} : K \to K$ such that

(1) \begin{align}1 + x x^{*} \leq x^{*}\end{align}

(2) \begin{align}1 + x^{*}x \leq x^{*}\end{align}

(3) \begin{align}y + xz \leq z \implies x^{*}y \leq z\end{align}

(4) \begin{align}y + zx \leq z \implies y x^{*} \leq z\end{align}

(We define $x \leq y$ as $x + y = y$ and we write xy instead of $x \cdot y$ .)

It follows from the definition that $x^{*}$ is the least element z such that $1 \leq z$ , $xz \leq z $ and $zx \leq z$ . A standard example of a Kleene algebra is a relational Kleene algebra where K is a set of binary relations over some set S, $\cdot$ is relational composition, $+$ is set union, $^{*}$ is reflexive transitive closure, 1 is the identity relation on S, and 0 is the empty set. Another standard example is the Kleene algebra of regular languages over a finite alphabet where $\cdot$ is concatenation of languages and $^{*}$ is finite iteration (Kleene star).

Definition 2. A Kleene algebra with tests (Kozen Reference Kozen1997) is a structure of the form:

$$(K, B, +, \cdot, \,^{*}, \,^{-}, 0, 1 ) \, ,$$

where

• • $(K, +, \cdot,\,^{*}, 0, 1)$ is a Kleene algebra,

• • $B \subseteq K$ , and $(B, +, \cdot, \, \bar{ }\, , 0, 1) $ is a Boolean algebra.

(That is, we assume that B is closed under the semiring operations; the negation operator $\,^{-}$ is a partial function defined only on B.)

Every Kleene algebra is a Kleene algebra with tests; take $B = \{ 0, 1 \}$ and define $\bar{0} = 1$ , $\bar{1} = 0$ . A standard example of a Kleene algebra with tests is a relational Kleene algebra expanded with a Boolean subalgebra of the negative cone, that is, the elements $x \leq 1$ , also called subidentities, which in the relational case are subsets of the identity relation. The class of Kleene algebras with tests is denoted as $\mathsf{KAT}$ .

Kleene algebras with tests are able to represent while programs and facilitate equational reasoning about their properties such as partial correctness and equivalence:

• • $\mathbf{skip} = 1$

• • $p\,{;}\,q = pq$

• • $\textbf{if}\: b \: \textbf{then} \: p \: \textbf{else}\: q$ $= (bp) +(\bar{b}q)$

• • $\textbf{while}\: b\: \textbf{do}\: p$ $= (bp)^{*}\bar{b}$

• • $\{ b \} \,p\, \{ c \}$ corresponds to $bp\bar{c} = 0$ (equivalently, $bp = bpc$ )

The reader is referred to Kozen (Reference Kozen1997) and references therein for more details.

2.2 Substructural logic of partial correctness

This section outlines the logic $\mathsf{S}$ (Kozen and Tiuryn Reference Kozen and Tiuryn2003). As with $\mathsf{KAT}$ , the logic $\mathsf{S}$ is many-sorted. Let $\mathsf{B} = \{ \mathtt{b}_i \mid i \in \omega \}$ be the set of test variables and let $\mathsf{P} = \{ \mathtt{p}_i \mid i \in \omega \}$ be the set of program variables. The language of $\mathsf{S}$ , $\mathcal{L}_{\mathsf{S}}$ , consists of the following sorts of syntactic objects:

We define $1 := 0 \Rightarrow 0$ , $\neg b := b \Rightarrow 0$ and $p^{*} := 1 \oplus p^{+}$ . We will sometimes write pq instead of $p \otimes q$ and $\bar{b}$ instead of $\neg b$ . Let $\mathsf{E} = \mathsf{B} \cup \mathsf{P}$ and let $\mathcal{E}_{\mathsf{S}}$ , the set of $\mathsf{S}$ -expressions, be the union of the sets of formulas, programs, and environments.

Kozen and Tiuryn (Reference Kozen and Tiuryn2003) introduce three kinds of semantics for their language: semantics based on guarded strings, traces, and binary relations, respectively. We will work only with binary relational semantics.

Observe that ${\texttt{[}} f {\texttt{]}}_M \subseteq \textrm{id}_W$ for all formulas f; if $(s,s) \in {\texttt{[}} f {\texttt{]}}_M$ , then we may say that formula f is true in s. Note that ${\texttt{[}} bp \Rightarrow c {\texttt{]}}_M$ is the set of (s,s) such that, for all t, if $(s,s) \in {\texttt{[}} b {\texttt{]}}_M$ and $(s,t) \in {\texttt{[}} p {\texttt{]}}_M$ , then $(t,t) \in {\texttt{[}} c {\texttt{]}}_M$ . Hence, $bp \Rightarrow c$ represents a partial correctness assertion: the formula is true in s iff b is true in s and p connects s with a state t only if c is true in t.

Fig. 1 shows the sequent proof system for $\mathsf{S}$ . A sequent $\Gamma \vdash f$ is provable in $\mathsf{S}$ iff there is a finite sequence of sequents that ends with $\Gamma \vdash f$ each of which is either of the form (Id) or (I0) or is derived from previous sequents using some of the inference rules.

Figure 1: The sequent proof system for $\mathsf{S}$ .

It is evident from the semantics that $p \Rightarrow f$ corresponds to the (test of the) modal formula ${\texttt{[}} p {\texttt{]}}f$ of Propositional Dynamic Logic, $\mathsf{PDL}$ (Fischer and Ladner Reference Fischer and Ladner1979; Harel et al. Reference Harel, Kozen and Tiuryn2000). Hence, each environment $\Gamma$ corresponds to a program of $\mathsf{PDL}$ and it can be shown that $\Gamma \vdash f$ is provable in $\mathsf{S}$ iff ${\texttt{[}} \Gamma {\texttt{]}}f$ is valid in $\mathsf{PDL}$ . Similarly, $ep \vdash f$ is provable in $\mathsf{S}$ iff $e \to {\texttt{[}} p {\texttt{]}}f$ is valid in $\mathsf{PDL}$ . The logic $\mathsf{S}$ is clearly a syntactic expansion of $\mathsf{KAT}$ (modulo the choice of primitive operators). Kozen and Tiuryn (Reference Kozen and Tiuryn2003) show that $\mathsf{KAT} \models p = q$ iff both $p \Rightarrow b \vdash q \Rightarrow b$ and $q \Rightarrow b \vdash p \Rightarrow b$ are provable in $\mathsf{S}$ where b is a Boolean variable not occurring in p or q.

$\mathsf{S}$ can be seen as a substructural logic (Galatos et al. Reference Galatos, Jipsen, Kowalski and Ono2007; Restall Reference Restall2000). However, $\mathsf{S}$ contains some rules that are unusual from the substructural logic perspective, namely, the sort-specific weakening rules (Wf) and (Wp), and the implication-formula rules (TC) and (I $\Rightarrow$ ). It is therefore interesting to inquire how $\mathsf{S}$ relates to mainstream substructural logics. Results of the next two sections will shed some light on the matter.

2.3 Dynamic consequence and dynamic implication

It is clear from the previous section that $\mathsf{S}$ can be seen as an extension of $\mathsf{KAT}$ that has two specific features. First, $\mathsf{S}$ comes with a somewhat unusual notion of semantic entailment. In fact, the usual notion of entailment based on the subset relation would not make much sense in the present setting owing to the syntactic restriction on sequents allowing only formulas in the consequent. Second, the language of $\mathsf{S}$ contains an implication connective which differs semantically from the residuals of relational composition, as found in the relational semantics for Pratt’s action logic, for example. This is related to the first point since, as the reader can easily verify, the implication connective gives rise to the following semantic deduction theorem:

$$ \Gamma, p \vdash_M f \iff \Gamma \vdash_M p \Rightarrow f \, .$$

As a matter of fact, the entailment relation and the implication connective of $\mathsf{S}$ can be seen as restrictions of dynamic consequence and dynamic implication in $\mathsf{DPL}$ (Groenendijk and Stokhof Reference Groenendijk and Stokhof1991). In $\mathsf{DPL}$ , formulas of the standard first-order language are evaluated on pairs of valuations on a relational structure. Without going into details, we just mention the operations on binary relations that give rise to semantic interpretations of dynamic negation $\mathord{\sim}$ and dynamic implication $\to$ in models of $\mathsf{DPL}$ :

• • $\mathord{\sim} R = \{ (s, s) \mid \neg \exists t. (s, t) \in R \}$ ;

• • $R \to Q = \{ (s,s) \mid \forall t. ((s, t) \in R \implies \exists u. (t, u) \in Q) \}$ .

Moreover, a formula $\varphi$ entails $\psi$ in $\mathsf{DPL}$ iff, for all models and all valuations s,t, if (s,t) is in the interpretation of $\varphi$ , then there is u such that (t,u) is in the interpretation of $\psi$ . It is easily seen that $R \to Q = \mathord{\sim} (R \circ \mathord{\sim} Q)$ , and this observation will be important later on in Section 4.

It is clear that $\mathsf{S}$ uses the same semantic clause for $\Rightarrow$ as $\mathsf{DPL}$ does for $\to$ with the proviso of the syntactic restriction on implicational formulas in $\mathsf{S}$ : the consequent of an implicational formula is always a formula, that is, an expression whose semantic value is a subset of the identity relation. A similar remark applies to the comparison between the notion of entailment in $\mathsf{S}$ and the one in $\mathsf{DPL}$ . This semantic observation entails that $\mathsf{S}$ can be seen as a fragment of a combination of relational Kleene algebra with tests with a propositional fragment of $\mathsf{DPL}$ . Such a combination was studied by Bochman and Gabbay (Reference Bochman and Gabbay2012).

2.4 Sequential Dynamic Logic

Bochman and Gabbay’s Sequential Dynamic Logic $\mathsf{SDL^*}$ (Bochman and Gabbay Reference Bochman and Gabbay2012) combines features of relational Kleene algebra with tests and the propositional fragment of $\mathsf{DPL}$ . In particular, it adds the dynamic negation connective $\mathord{\sim}$ to $\mathsf{KAT}$ . Bochman and Gabbay provide a sound and weakly complete sequent system for dynamic entailment, and they observe that $\mathsf{SDL^*}$ bears strong resemblance to $\mathsf{S}$ while lifting the syntactic restrictions of $\mathsf{S}$ . They leave a more thorough investigation of relations with $\mathsf{S}$ to another occasion. In this section, we formulate the semantics of $\mathsf{SDL^*}$ and we make the straightforward observation that $\mathsf{S}$ corresponds to a syntactic fragment of $\mathsf{SDL^*}$ .

Formulas of the language of $\mathsf{SDL^*}$ are defined using the following grammar:

$$ \varphi, \psi := \mathtt{p} \mid \mathtt{b} \mid 0 \mid \varphi \oplus \psi \mid \varphi \otimes \psi \mid \varphi^{*} \mid \mathord{\sim} \varphi \,$$

where $\mathtt{p} \in \mathsf{P}$ and $\mathtt{b} \in \mathsf{B}$ . $\mathsf{SDL^*}$ -sequents are expressions of the form $\Gamma \vdash \varphi$ where $\Gamma$ is a finite sequence of $\mathsf{SDL^*}$ -formulas and $\varphi$ is a $\mathsf{SDL^*}$ -formula. (Bochman and Gabbay use $\land$ instead of $\otimes$ and $\lor$ instead of $\oplus$ . They do not use the constant 0; instead, their sequent system allows an empty conclusion.) Expressions of the language of $\mathsf{SDL^*}$ are finite sequences of $\mathsf{SDL^*}$ -formulas.

Let $\delta : \mathcal{E}_{\mathsf{S}} \to \mathcal{E}_{\mathsf{SDL^*}}$ such that $\delta (\mathtt{p}) = \mathtt{p}$ , $\delta(\mathtt{b}) = \mathtt{b}$ , $\delta$ commutes with $\oplus, \otimes$ and the environment-forming comma operator, $\delta(p^{+}) = \delta (p) \otimes \delta (p)^{*}$ , and

$$ \delta (p \Rightarrow f) = \mathord{\sim} (\delta (p) \otimes \mathord{\sim} \delta (f)) \, .$$

Proof. It is sufficient to observe that every $\mathsf{S}$ -model that is a counterexample to $\Gamma \vdash f$ can be turned into a $\mathsf{SDL^*}$ -model that is a counterexample to $\delta (\Gamma) \Vdash \delta (f)$ and vice versa. We only need to note that if $F \subseteq \textrm{id}_W$ , then

$$ P \Rightarrow F = P \to F = \mathord{\sim} (P \circ \mathord{\sim} F) \,.$$

Bochman and Gabbay provide a sound and weakly complete sequent system for dynamic consequence over $\mathsf{SDL^*}$ which bears strong resemblance to the sequent system for $\mathsf{S}$ and extends sequent systems for dynamic consequence over weaker languages studied earlier (Kanazawa Reference Kanazawa, Dekker and Stokhof1993; van Benthem 1995, Reference van Benthem1996; van der Does et al. Reference van der Does, Groeneveld and Veltman1997). We omit the details.

3.1 Kleene algebra with domain

In this section, we recall KAD. Our discussion will be brief, and the reader is referred to Desharnais et al. (Reference Desharnais, Möller and Struth2006) and Desharnais and Struth (Reference Desharnais and Struth2011) for details. We note that we assume the one-sorted version of KAD where the domain operator is defined using the primitive antidomain operator (Desharnais and Struth Reference Desharnais and Struth2011). An extension of $\mathsf{KAT}$ with a primitive domain operator (also called KAD at that time) is presented in Desharnais et al. (Reference Desharnais, Möller and Struth2006).

The language of $\mathsf{KAD}$ , $\mathcal{L}_{\mathsf{KAD}}$ , is one-sorted:

$$ p, q := \mathtt{p}_i \mid 0 \mid 1 \mid p \cdot q \mid p + q \mid p^{*} \mid \mathsf{a} (p) \, .$$

We define $\mathsf{d}(p) := \mathsf{a} (\mathsf{a} (p))$ . A domain term is a term of the form $\mathsf{d} (p)$ .

Kleene algebras with domain are expansions of Kleene algebras with a unary antidomain operator. The abstract definition below generalizes the properties of the dynamic negation operator on binary relations:

$$ \mathord{\sim} R = \{ (s,s) \mid \neg \exists t . (s,t) \in R \} \, .$$

We note that, in the literature on KAD, the link between this “relational antidomain” operator and dynamic negation does not seem to have been noted yet. Relational antidomain has a natural interpretation independent of the linguistic motivations of $\mathsf{DPL}$ : if R is seen as the input–output relation determined by a program, then $\mathord{\sim} R$ is the input–output relation determined by the test whether the program diverges.

Definition 6. A KAD is a structure of the form

$$ \mathbf{K} = (K, +, \cdot, \,^{*}, \mathsf{a}, 0, 1)$$

such that $(K, +, \cdot, \,^{*}, 0, 1)$ is a Kleene algebra, $\mathsf{a} : K \to K$ and the following are satisfied for all $x,y,z \in K$ , assuming that $\mathsf{d} (x) := \mathsf{a} (\mathsf{a} (x))$ :

(5) \begin{align}\mathsf{a} (x) x & = 0\end{align}

(6) \begin{align}\mathsf{a} (xy) & \leq \mathsf{a} (x\, \mathsf{d} (y))\end{align}

(7) \begin{align}\mathsf{d} (x) + \mathsf{a} (x) & = 1\end{align}

An equation $p = q$ is valid in $\mathbf{K}$ iff $v(p) = v(q)$ for all valuations v (that is, all homomorphisms from $\mathcal{L}_{\mathsf{KAD}}$ into $\mathbf{K}$ ).

We will usually write $p \equiv_{\mathbf{K}} q$ to indicate that $p = q$ is valid in $\mathbf{K}$ , and we will write $p \equiv_{\mathsf{KAD}} q$ to indicate that the equation $p = q$ belongs to the equational theory of $\mathsf{KAD}$ (i.e. it is valid in every algebra belonging to $\mathsf{KAD}$ ); the latter will often be shortened to $p \equiv q$ if the class of algebras in question is clear from the context.

A standard example of a KAD is the Kleene algebra of binary relations over a set S extended with the dynamic negation operation $\mathord{\sim}$ . Note that the relational domain operation D defined by:

$$ D(R) := \mathord{\sim}\mathord{\sim} R = \{ (s,s) \mid \exists t. (s, t) \in R \} $$

is related to the projection operation familiar from relational databases. If R is seen as the input–output relation determined by a program, then D(R) is the input–output relation determined by the test whether the program halts. The Kleene algebra of regular languages over a finite alphabet $\Sigma$ can be extended to a KAD by adding $\mathsf{a} : 2^{\Sigma^{*}} \to 2^{\Sigma^{*}}$ such that

\begin{align*}\mathsf{a} (L) & = \begin{cases}\{ \varepsilon \} & \text{if } L = \emptyset \\\emptyset & \text{otherwise.}\end{cases}\end{align*}

The quasivariety of Kleene algebras with domain will be denoted as $\mathsf{KAD}$ . Not every Kleene algebra expands to a KAD (Desharnais and Struth Reference Desharnais and Struth2011), but the above example of a Kleene algebra of regular languages with domain shows that the equational theory of $\mathsf{KAD}$ is a conservative extension of the equational theory of $\mathsf{KA}$ . The equational theory of $\mathsf{KAD}$ is EXPTIME-complete (Sedlár Reference Sedlár, Glück, Santocanale and Winter2023).

A domain element of a KAD is an x such that $x = \mathsf{d} (y)$ for some y. In what follows, we indicate the assumption that a given x is a domain element by writing $\hat{x}$ . A similar notational convention is applied to domain terms.

In the rest of the paper, we will often use the equalities stated in the following lemma without explicit mention.

3.2 Embedding $\mathsf{\mathbf{S}}$ into Kleene algebra with domain

In this section, we define a translation function Tr from the language of $\mathsf{S}$ into the language of $\mathsf{KAD}$ and we show that it embeds the set of sequents provable in $\mathsf{S}$ into the equational theory of $\mathsf{KAD}$ .

It is easily verified that, for each formula $f \in \mathcal{E}_{\mathsf{S}}$ , the term Tr (f) is equivalent to a domain term; see Lemma 15 in the appendix. Moreover, note that $Tr (\bar{b}) = Tr (b \Rightarrow 0) = \mathsf{a} ( Tr (b) \cdot \mathsf{a} (0))$ is equivalent to $\mathsf{a} (Tr (b))$ .

Theorem. For all $\Gamma, f \in \mathcal{E}_{\mathsf{S}}$ ,

Proof. To establish the implication from right to left, let us assume that $\Gamma \vdash f$ is not provable in $\mathsf{S}$ . By Theorem 2.2, there is an $\mathsf{S}$ -model $M = (W, V)$ and states $s,t \in W$ such that $(s,t) \in {\texttt{[}} \Gamma {\texttt{]}}_M$ but $(t,t) \not\in {\texttt{[}} f {\texttt{]}}_M$ . Equivalently,

\begin{equation*}{\texttt{[}} \Gamma {\texttt{]}}_M \neq {\texttt{[}} \Gamma {\texttt{]}}_M \circ {\texttt{[}} f {\texttt{]}}_M \, .\end{equation*}

Define $\mathbf{K}$ as the KAD over the set of all binary relations on W, and let ${\texttt{[}}\!{\texttt{[}} \, {\texttt{]}}\!{\texttt{]}}$ be the unique valuation on $\mathbf{K}$ such that, for all $n \in \omega$ ,

• • ${\texttt{[}}\!{\texttt{[}} \mathtt{p}_{2n} {\texttt{]}}\!{\texttt{]}} = {\texttt{[}} \mathtt{p}_n {\texttt{]}}_M$ , and

• • ${\texttt{[}}\!{\texttt{[}} \mathtt{p}_{2n+1} {\texttt{]}}\!{\texttt{]}} = {\texttt{[}} \mathtt{b}_n {\texttt{]}}_M$ .

It can be shown by induction on the complexity of expressions $\chi \in \mathcal{E}_{\mathsf{S}}$ that

\begin{equation*}{\texttt{[}}\!{\texttt{[}} Tr (\chi) {\texttt{]}}\!{\texttt{]}} = {\texttt{[}} \chi {\texttt{]}}_M \, .\end{equation*}

The base case for $\mathtt{p}_n$ holds by definition and the base case for $\mathtt{b}_n$ is established by noting that ${\texttt{[}} \mathtt{b}_n {\texttt{]}}_M = D ({\texttt{[}} \mathtt{b}_n {\texttt{]}}_M)$ since ${\texttt{[}} \mathtt{b}_n {\texttt{]}}_M \subseteq \textrm{id}_W$ , and $D ({\texttt{[}} \mathtt{b}_n {\texttt{]}}_M) = {\texttt{[}}\!{\texttt{[}} \mathsf{d} (\mathtt{p}_{2n+1}) {\texttt{]}}\!{\texttt{]}}$ . The induction step is uneventful, perhaps with the following exception:

\begin{align*}{\texttt{[}} p \Rightarrow f {\texttt{]}}_M & = \mathord{\sim} ({\texttt{[}} p {\texttt{]}}_M \circ \mathord{\sim} {\texttt{[}} f {\texttt{]}}_M)\\& = \mathord{\sim} ({\texttt{[}}\!{\texttt{[}} Tr (p) {\texttt{]}}\!{\texttt{]}} \circ \mathord{\sim} {\texttt{[}}\!{\texttt{[}} Tr (f) {\texttt{]}}\!{\texttt{]}})\\& = {\texttt{[}}\!{\texttt{[}} \mathsf{a} ( Tr (p) \cdot \mathsf{a} (Tr (f))) {\texttt{]}}\!{\texttt{]}}\\& = {\texttt{[}}\!{\texttt{[}} Tr (p \Rightarrow f) {\texttt{]}}\!{\texttt{]}} \, .\end{align*}

(For a more detailed justification, see Lemma 16 in the appendix.) It follows that

\begin{equation*}{\texttt{[}}\!{\texttt{[}} Tr (\Gamma) {\texttt{]}}\!{\texttt{]}} \neq {\texttt{[}}\!{\texttt{[}} Tr (\Gamma) {\texttt{]}}\!{\texttt{]}} \circ {\texttt{[}}\!{\texttt{[}} Tr (f) {\texttt{]}}\!{\texttt{]}} \, .\end{equation*}

Hence, $Tr (\Gamma)$ is not equivalent to $Tr(\Gamma) \cdot Tr (f) $ in $\mathsf{KAD}$ .

The converse implication is established by induction on the length of proofs. Most of the cases of the inductive proof use only Kleene algebra; here, we prove the cases of the induction step containing implication. (The proof is carried out in more detail in the appendix; see Lemma 17.)

To establish the case corresponding to (TC), it is sufficient to show that

\begin{equation*}\dfrac{x \hat{u} y \trianglelefteq \hat{v} \qquad x \mathsf{a}(\hat{u})y \trianglelefteq \hat{v}}{xy \trianglelefteq \hat{v}} \, .\end{equation*}

This is established using Lemma 9(2). The assumptions entail that $x \hat{u} y + x \mathsf{a}(\hat{u}) y \trianglelefteq \hat{z}$ , but

$$x \hat{u} y + x \mathsf{a}(\hat{u}) y = x (\hat{u} + \mathsf{a} (\hat{u}))y = xy \, .$$

To establish the case corresponding to (R $\Rightarrow$ ), it is sufficient to show that

\begin{equation*}\dfrac{xy = xy \hat{u}}{x = x \mathsf{a} (y \mathsf{a} (\hat{u}))} \, .\end{equation*}

If $xy = xy \hat{u}$ , then the following holds. First,

\begin{align*}\mathsf{d} (x \mathsf{d} (y \mathsf{a} (\hat{u}))) & = \mathsf{d} (xy \mathsf{a} (\hat{u}))\\& = \mathsf{d} (xy\hat{u} \mathsf{a} (\hat{u}))\\& = 0 \, ,\end{align*}

and so $x \mathsf{d} (y \mathsf{a} (\hat{u})) = 0$ . Second,

\begin{align*}x &= x \mathsf{d} (y \mathsf{a} (\hat{u})) + x \mathsf{a} (y \mathsf{a} (\hat{u}))\\& = 0 + x \mathsf{a} (y \mathsf{a} (\hat{u}))\\& = x \mathsf{a} (y \mathsf{a} (\hat{u})) \, .\end{align*}

To establish (I $\Rightarrow$ ), it is sufficient to show that

\begin{equation*}\mathsf{a} (x \cdot \mathsf{a} (\hat{z})) x \leq x \hat{z}\, .\end{equation*}

(The desired result is then obtained using Lemma 9(1).) We reason as follows:

\begin{align*}\mathsf{a} (x \mathsf{a} (\hat{z})) x & = \mathsf{a} (x \mathsf{a} (\hat{z})) (x \hat{z} + x \mathsf{a} (\hat{z}))\\& = \mathsf{a} (x \mathsf{a} (\hat{z})) x \hat{z} + \mathsf{a} (x \mathsf{a} (\hat{z})) x \mathsf{a} (\hat{z})\\& = \mathsf{a} (x \mathsf{a} (\hat{z})) x \hat{z} + 0\\& = \mathsf{a} (x \mathsf{a} (\hat{z})) x \hat{z}\\& \leq x \hat{z}\end{align*}

This (together with the other cases discussed in the appendix) concludes the proof of Theorem 3.2.

We note that essentially the same technique can be used to show that $\mathsf{SDL^{*}}$ embeds into $\mathsf{KAD}$ in the sense that $\Gamma \vdash \varphi$ is valid in all $\mathsf{SDL^{*}}$ -models M iff $\lambda (\Gamma) \trianglelefteq \mathsf{d} (\lambda(\varphi))$ is valid in $\mathsf{KAD}$ for a translation $\lambda$ that sends $\mathtt{p}_n$ to $\mathtt{p}_{2n}$ and $\mathtt{b}_n$ to $\mathsf{d} (\mathtt{p}_{2n+1})$ .

4.1 Kleene algebra with dynamic implication

We observed in the previous section that $\mathsf{a} (x \cdot \mathsf{a} (\hat{z}))$ behaves like Kozen and Tiuryn’s implication $x \Rightarrow \hat{z}$ . From this point of view, $\mathsf{a} (x)$ is equivalent to “x implies 0”, that is, to $\mathsf{a} (x \cdot \mathsf{a} (0))$ . This motivates the following question: Can we capture antidomain in terms of a primitive implication operator?

Definition 11. An iKAD is an algebra of the form:

$$ \mathbf{K} = (K, +, \cdot, \to, \,^{*}, 0, 1)$$

where $(K, +, \cdot, \,^{*}, 0, 1)$ is a Kleene algebra and $\to$ is a binary operation satisfying the following axioms:

(8) \begin{align} (x \to 0)x & = 0\end{align}

(9) \begin{align} (x \to y) + ((x \to y) \to 0) & = 1 \end{align}

(10) \begin{align} (xy \to z) & = (x \to (y \to z)) \end{align}

(11) \begin{align} (x \to y) & = (x \to ((y \to 0) \to 0))\end{align}

We define $\mathord{\sim} x := x \to 0$ .

The guiding example of an iKAD is a relational Kleene algebra with the dynamic implication operation of $\mathsf{DPL}$ , namely,

$$ R \to Q = \{ (s,s) \mid \forall t: (s,t) \in R \implies \exists u (t, u) \in Q \} \, .$$

Dynamic implication expresses a test of the following liveness property: the program represented by Q has a terminating execution starting in any final state of a terminating execution of the program represented by R (we can always continue with Q after R, as put by Hollenberg Reference Hollenberg1997). In the particular instance where Q is a test, this means that Q holds after every terminating execution of R (partial correctness). Note that $R \to \emptyset$ boils down to dynamic negation of R.

A domain element (term) is an element of the form $\mathord{\sim} \mathord{\sim} x$ (p instead of x). As before, we use the notation $\hat{x}$ ( $\hat{p}$ ) to indicate that the element x (term p) is a domain element (term).

Pratt’s action logic (Pratt Reference Pratt1991) is a well-known implicational extension of Kleene algebra which was studied intensively in the recent decades (Buszkowski Reference Buszkowski2006; Jipsen Reference Jipsen2004; Reference KozenKozen 1994b ; Kuznetsov Reference Kuznetsov2021; Palka Reference Palka2007). Action logic is based on residuated Kleene algebras, adding to Kleene algebras a pair of binary operations $\multimap$ and such that

(12)

As axiom (10) shows, an “axiom version” of residuation holds for $\to$ in $\mathsf{iKAD}$ , bringing $\to$ close to the operator of action logic. However, it can be shown that $\to$ does not residuate with $\cdot$ . In particular, a counterexample to

(13) \begin{equation}xy \leq z \implies x \leq y \to z\end{equation}

is a three-element Kleene algebra consisting of the linearly ordered set of elements $0 < 1 < 2$ where $\cdot$ is commutative and $x \cdot 2 = 2 $ in case $x \neq 0$ , $2^{*} = 2$ and where the following table characterizes $\to$ :

It is clear that $2 \cdot 0 \leq 0$ , but not $2 \leq 0 \to 0$ . A similar counterexample to the converse of (13) exists. In general, it is clear that (13) should fail since $y \to z \leq 1$ for all y,z. We leave a more thorough comparison of iKAD and residuated Kleene algebras to another occasion.

4.2 $\mathsf{KAD}$ and $\mathsf{iKAD}$

In this section, we show that $\mathsf{KAD}$ and $\mathsf{iKAD}$ are equivalent in the sense that the equational theory of one embeds into the equational theory of the other. A corollary of this result is that the equational theory of $\mathsf{iKAD}$ is decidable (EXPTIME-complete by Sedlár Reference Sedlár, Glück, Santocanale and Winter2023). This is an interesting contrast to residuated Kleene algebras (Pratt’s action logic).

Proof. To prove the first item, it is sufficient to show that the translations of the antidomain axioms of $\mathsf{KAD}$ are valid in $\mathsf{iKAD}$ . Validity of the translations of (5) and (7) follows easily from (8) and (9), respectively. To deal with (6), it is sufficient to use the following equivalences which are established using Lemma 18 in the appendix:

\begin{align*}\mathord{\sim} (x \cdot \mathord{\sim}\mathord{\sim} y) & = x \to \mathord{\sim} \mathord{\sim} \mathord{\sim} y\\& = x \to \mathord{\sim} y\\& = \mathord{\sim} (xy) \, .\end{align*}

To prove the second item, it is sufficient to show that the translations of the implicational axioms of $\mathsf{iKAD}$ are valid in $\mathsf{KAD}$ . This is easy; see Lemma 14 in the appendix. To prove the third item, it is sufficient to observe that $\mathsf{a} (x \cdot \mathsf{a} (0)) = \mathsf{a} (x)$ holds in $\mathsf{KAD}$ . The final item is established easily using axiom (11).

Theorem 4.2 shows that $\mathsf{S}$ embeds into a specific implicational expansion of Kleene algebra, and that this expansion is decidable (EXPTIME-complete by Sedlár Reference Sedlár, Glück, Santocanale and Winter2023).