Structured Specification of Paraconsistent Transition Systems

Abstract

This paper sets the basis for a compositional and structured approach to the specification of paraconsistent transitions systems, framed as an institution. The latter and theirs logics were previously introduced in [CMB22] to deal with scenarios of inconsistency in which several requirements are on stake, either reinforcing or contradicting each other.

The present study was developed in the scope of the Project “Agenda ILLIANCE” [C644919832-00000035 | Project no. 46], financed by PRR – Plano de Recuperação e Resiliência under the Next Generation EU from the European Union. FCT, the Portuguese funding agency for Science and Technology suports with the projects UIDB/04106/2020 and PTDC/CCI-COM/4280/2021.

Appendix

Proposition 1 Let $\sigma :(\textrm{Prop},\textrm{Act})\rightarrow (\textrm{Prop}',\textrm{Act}')$ be a signature morphism, $M'$ a $(\textrm{Prop}',\textrm{Act}')$-PLTS, and $\varphi \in \textsf{Sen}(\textrm{Prop},\textrm{Act})$ a formula. Then, for any $w\in W$,

$$\begin{aligned} \big (M'|_\sigma ,w \models \varphi \big )\; = \; \big (M',w\models \textsf{Sen}(\sigma )(\varphi )\big ) \end{aligned}$$

(5)

Proof

The proof is by induction over the structure of sentences. To simplify notation we will write $\sigma (p)$ instead of $\sigma _{\textrm{Prop}}(p)$ for any $p\in \textrm{Prop}$ and $\sigma (a)$ instead of $\sigma _{\textrm{Act}}(a)$ for any $a \in \textrm{Act}$. The case of $\bot$ is trivial, by the definition of $\models$ and $\textsf{Sen}$ we have that $(M'|_\sigma ,w \models \bot ) = (0,1) =(M',w \models Sen(\sigma )(\bot ))$. For sentences $p \in \textrm{Prop}$, one observes that by defn of $\textsf{Sen}$, of $\models$ and of reducts, $(M',w\models \textsf{Sen}(\sigma )(p))=(M',w\models \sigma (p))=V'(w,\sigma (p))=V(w,p)=(M'|_\sigma ,w\models p)$. For sentences $\lnot \varphi$ we observe that, by definition of $\textsf{Sen}$ and of $\models$, we have that . By induction hypothesis and, again, by definition of $\textsf{Sen}$ and of $\models$, we have .

Let us consider now formulas composed by Boolean operators. Firstly, we can observe that, by definition of $\textsf{Sen}$ and of $\models$, . By I.H. we have that and by definition of $\models$, it is equal to $M'|_\sigma ,w\models (\varphi \wedge \varphi ')$. The proof for sentences $\varphi \vee \varphi '$ and $\varphi \rightarrow \varphi '$ is analogous.

(step $\star$) We have by reduct that $R'_{\sigma (a)}[w]=R_a[w]$. Moreover, by I.H., it is true that $(M',w \models \textsf{Sen}(\sigma )(\varphi ))=(M'|_\sigma ,w \models \varphi )$, and hence

$$\begin{aligned} \begin{aligned} \bigg ((M',w \models \textsf{Sen}(\sigma )(\varphi ))^+, (M',w \models \textsf{Sen}(\sigma )(\varphi ))^-)\bigg )= \\ \bigg ((M'|_\sigma ,w \models \varphi )^+,(M'|_\sigma ,w \models \varphi )^-\bigg )\end{aligned} \end{aligned}$$

(6)

Therefore, $M',w \models \textsf{Sen}(\sigma )(\varphi ))^+ = (M'|_\sigma ,w \models \varphi )^+$ and

$(M',w \models \textsf{Sen}(\sigma )(\varphi ))^-$ $= (M'|_\sigma ,w \models \varphi )^-$.

The proofs for sentences $\langle a\rangle \varphi$ and are analogous.

Finally, let us consider the proof for sentences $\circ \, \varphi$. By definition of $\textsf{Sen}$, $M',w \models \textsf{Sen}(\sigma )( \circ \, \varphi )=(M',w \models \circ \, \textsf{Sen}(\sigma )(\varphi ))$. By definition of $\models$, this evaluates to (1, 0), if $(M',w \models \textsf{Sen}(\sigma )(\varphi )) \in \varDelta _C$ and to (0, 1) otherwise. Hence, by I.H, it evaluates to (1, 0) when $(M'|_\sigma ,w \models \varphi ) \in \varDelta _C$ and to (0, 1), i.e., we have $(M'|_\sigma ,w \models \circ \, \varphi )$.