Appendix for: Cut-free Calculi and Relational Semantics for Temporal STIT logics

Appendix 0.A Completeness of $L d m$

We give the definitions and lemmas sufficient to prove the completeness of $L d m$ relative to $T s t i t$ frames [4, 2]. We make use of the canonical model of $L d m$ (obtained by standard means [3, 1]) to construct a $T s t i t$ model. A truth-lemma is then given relative to this model, from which, completeness follows as a corollary.

Definition 1 ( $L d m$ -Cs, $L d m$ -Mcs)

A set $Θ \subset L_{L d m}$ is a $L d m$ consistent set ( $L d m$ -CS) iff $Θ ⊬_{L d m} ⊥$ . We call a set $Θ \subset L_{L d m}$ a $L d m$ maximally consistent set ( $L d m$ -MCS) iff $Θ$ is a $L d m$ -CS and for any set $Θ^{'}$ such that $Θ \subset Θ^{'}$ , $Θ^{'} ⊢_{L d m} ⊥$ .

Lemma 1 (Lindenbaum’s Lemma [3])

Every $L d m$ -CS can be extended to a $L d m$ -MCS.

Definition 2 (Present and Future Pre-Canonical $T s t i t$ Model)

The present pre-canonical $T s t i t$ model is the tuple $M^{p r e s} = (W^{p r e s}, R_{□}^{p r e s},$ ${R_{i}^{p r e s} | i \in A g}, V^{p r e s})$ defined below left, and the future pre-canonical $T s t i t$ model is the tuple $M^{f u t} = (W^{f u t}, R_{□}^{f u t}, {R_{i}^{f u t} | i \in A g}, V^{f u t})$ defined below right:

$W^{p r e s}$ is the set of all $L d m$ -MCSs;
$R_{□}^{p r e s} w u$ iff for all $□ ϕ \in w$ , $ϕ \in u$ ;
$R_{i}^{p r e s} w u$ iff for all $[i] ϕ \in w$ , $ϕ \in u$ ;
$V^{p r e s} (p) = {w \in W | p \in w}$ .

$W^{f u t} = W^{p r e s}$ ;
$R_{□}^{f u t} (w) = ⋂_{i \in A g} R_{i}^{p r e s} (w)$ ;
$R_{i}^{f u t} (w) = ⋂_{i \in A g} R_{i}^{p r e s} (w)$ ;
$V^{f u t} (p) = V^{p r e s} (p)$ .

Definition 3 (Canonical Temporal Kripke STIT Model)

We define the canonical temporal Kripke STIT model to be the tuple $M^{L d m} = (W^{L d m}, R_{□}^{L d m},$ ${R_{i}^{L d m} | i \in A g}, R_{A g}^{L d m}, R_{G}^{L d m}, R_{H}^{L d m},$ $V^{L d m})$ such that:

$W^{L d m} = W^{p r e s} \times N$ ¹¹1Note that we choose to write each world $(w, j) \in W^{L d m}$ as $w^{j}$ to simplify notation. Moreover, we write $ϕ \in w^{j}$ to mean that the formula $ϕ$ is in the $L d m$ -MCS $w$ associated with $j$ .;
$R_{□}^{L d m} w^{j} u^{j}$ iff (i) $R_{□}^{p r e s} w u$ and $j = 0$ , or (ii) $R_{□}^{f u t} w u$ and $j > 0$ ;
$R_{i}^{L d m} w^{j} u^{j}$ iff (i) $R_{i}^{p r e s} w u$ and $j = 0$ , or (ii) $R_{i}^{f u t} w u$ and $j > 0$ ;
$R_{A g}^{L d m} (w^{j}) = ⋂_{1 \leq i \leq n} R_{i}^{L d m} (w^{j})$ ;
$R_{G}^{L d m} = {(w^{j}, w^{k}) | w^{j}, w^{k} \in W^{L d m} and j < k}$ ;
$R_{H}^{L d m} = {(u^{i}, w^{i}) | (w^{i}, u^{i}) \in R_{G}^{L d m}}$ ;
$V^{L d m} (p) = {w^{j} \in W^{L d m} | w \in V^{p r e s} (p)}$ .

Lemma 2

For all $α \in {□, A g} \cup A g$ , if $R_{α}^{L d m} w^{j} u^{k}$ for $j, k \in N$ , then $j = k$ .

Proof

Follows by definition of the canonical $T s t i t$ model.

Lemma 3

For all $j \in N$ with $k \geq 1$ , $(w^{j}, u^{j}) \in R_{A g}^{L d m}$ iff $(w^{j + k}, u^{j + k}) \in R_{A g}^{L d m}$ .

Proof

This follows from the fact that $u^{0} \in R_{A g}^{L d m} (w^{0})$ iff $u \in ⋂_{i \in A g} R_{i}^{p r e s} (w)$ iff $u \in R_{i}^{f u t} (w)$ for each $i \in A g$ iff $u \in ⋂_{i \in A g} R_{i}^{f u t} (w)$ iff $u^{k} \in ⋂_{i \in A g} R_{i}^{L d m} (w^{k})$ for any $k > 0$ .

Lemma 4 ([3])

(i) For all $x \in {p r e s, f u t, L d m}$ , $R_{□}^{x} w u$ iff for all $ϕ$ , if $ϕ \in u$ , then $◊ ϕ \in w$ . (ii) For all $x \in {p r e s, f u t, L d m}$ , $R_{i}^{x} w u$ iff for all $ϕ$ , if $ϕ \in u$ , then $⟨ i ⟩ ϕ \in w$ .

Lemma 5 (Existence Lemma [3])

(i) For any world $w^{j} \in W^{L d m}$ , if $◊ ϕ \in w^{j}$ , then there exists a world $u^{j} \in W^{L d m}$ such that $R_{□}^{L d m} w^{j} u^{j}$ and $ϕ \in u^{j}$ . (ii) For any world $w^{j} \in W^{L d m}$ , if $⟨ i ⟩ ϕ \in w^{j}$ , then there exists a world $u^{j} \in W^{L d m}$ such that $R_{i}^{L d m} w^{j} u^{j}$ and $ϕ \in u^{j}$ .

Lemma 6

The Canonical Model is a temporal Kripke STIT model.

Proof

We prove that $M^{L d m}$ has all the properties of a $T s t i t$ model:

By lemma 1, the $L d m$ consistent set ${p}$ can be extended to a $L d m$ -MCS, and therefore $W^{p r e s}$ is non-empty. Since $N$ is non-empty as well, $W^{p r e s} \times N = W^{L d m}$ is a non-empty set of worlds.
We argue that $R_{□}^{L d m}$ is an equivalence relation between worlds of $W^{L d m}$ , and omit the arguments for $R_{i}^{L d m}$ and $R_{A g}^{L d m}$ , which are similar. Suppose that $w^{j} \in W^{L d m}$ . We have two cases to consider: (i) $j = 0$ , and (ii) $j > 0$ . (i) Standard canonical model arguments apply and $R_{□}^{L d m}$ is an equivalence relation between all worlds of the form $w^{0} \in W^{L d m}$ (See [3] for details). (ii) If we fix a $j > 0$ , then $R_{□}^{L d m}$ will be an equivalence relation for all worlds of the form $w^{j} \in W^{L d m}$ since the intersection of equivalence relations produces another equivalence relation. Last, since $R_{□}^{L d m}$ is an equivalence relation for each fixed $j \in N$ , and because each $W^{p r e s} \times {j} \subset W^{L d m}$ is disjoint from each $W^{p r e s} \times {j^{'}} \subset W^{L d m}$ for $j \neq j^{'}$ , we know that the union all such equivalence relations will be an equivalence relation.
Let $i$ be in $A g$ and assume that $(w^{j}, u^{j}) \in R_{i}^{L d m}$ . We split the proof into two cases: (i) $j = 0$ , or (ii) $j > 0$ . (i) Assume that $□ ϕ \in w^{0}$ . Since $w$ is a $L d m$ -MCS, it contains the axiom $□ ϕ \to [i] ϕ$ , and so, $[i] ϕ \in w$ as well. Since $(w, u) \in R_{i}^{p r e s}$ (because $j = 0$ ), we know that $ϕ \in u$ by the definition of the relation; therefore, $(w, u) \in R_{□}^{p r e s}$ , which implies that $(w^{0}, u^{0}) \in R_{□}^{L d m}$ by definition. (ii) The assumption that $j > 0$ implies that $u \in R_{i}^{f u t} (w)$ $= ⋂_{i \in A g} R_{i}^{p r e s} (w) = R_{□}^{f u t} (w)$ by definition, which implies that $(w^{j}, u^{j}) \in R_{□}^{L d m}$ .
Let $u_{1}^{j}, . . ., u_{n}^{j} \in W^{L d m}$ and assume that $R_{□}^{L d m} u_{i}^{j} u_{k}^{j}$ for all $i, k \in {1, . . ., n}$ . We split the proof into two cases: (i) $j = 0$ , or (ii) $j > 0$ . (i) We want to show that there exists a world $w^{j} \in W^{L d m}$ such that $w^{j} \in ⋂_{1 \leq i \leq n} R_{i}^{L d m} (u_{i}^{j})$ . Let $^wj=⋃1≤i≤n{ϕ|[i]ϕ∈uji}$ . Suppose that ${^w}^{j}$ is inconsistent to derive a contradiction. Then, there are $ψ_{1}$ ,…, $ψ_{k}$ such that $⊢_{L d m} ⋀_{1 \leq l \leq k} ψ_{i} \to ⊥$ . For each $i \in A g$ , we define $Φ_{i} = {ψ_{l} | [i] ψ_{l} \in u_{i}^{j}} \subseteq {ψ_{1}, . . ., ψ_{k}}$ . Observe that for each $i \in A g$ , $[i] ⋀ Φ_{i} \in u_{i}^{j}$ because $⋀ [i] Φ_{i} \in u_{i}^{j}$ and $⊢_{L d m} ⋀ [i] Φ_{i} \to [i] ⋀ Φ_{i}$ . Since by assumption $R_{□}^{L d m} u_{i}^{j} u_{k}^{j}$ for all $i, k \in {1, . . ., n}$ , this means that for any $u_{m}^{j}$ we pick (with $1 \leq m \leq n$ ), $◊ [i] ⋀ Φ_{i} \in u_{m}^{j}$ for each $i \in A g$ by lemma 4; hence, $⋀_{i \in A g} ◊ [i] ⋀ Φ_{i} \in u_{m}^{j}$ . By the $(I O A)$ axiom, this implies that $◊ ⋀_{i \in A g} [i] (⋀ Φ_{i}) \in u_{m}^{j}$ . By lemma 5, there must exist a world $v^{j}$ such that $R_{□}^{L d m} u_{m}^{j} v^{j}$ and $⋀_{i \in A g} [i] (⋀ Φ_{i}) \in v^{j}$ . But then, since $⊢_{L d m} [i] (⋀ Φ_{i}) \to ⋀ Φ_{i}$ by reflexivity, $⊢_{L d m} ⋀_{i \in A g} (⋀ Φ_{i}) \leftrightarrow ⋀_{1 \leq i \leq k} ψ_{i}$ , and $⊢_{L d m} ⋀_{1 \leq i \leq k} ψ_{i} \to ⊥$ , it follows that $⊥ \in v^{j}$ , which is a contradiction since $v^{j}$ is a $L d m$ -MCS. Therefore, ${^w}^{j}$ must be consistent and by lemma 1, it may be extended to a $L d m$ -MCS $w^{j}$ . Since for each $[i] ϕ \in u_{i}^{j}$ , $ϕ \in w^{j}$ , we have that $w \in R_{i}^{p r e s} (u_{i})$ for each $i \in A g$ . Hence, $w \in ⋂_{1 \leq i \leq n} R_{i}^{p r e s} (u_{i})$ , and so, $w^{j} \in ⋂_{1 \leq i \leq n} R_{i}^{L d m} (u_{i}^{j})$ . (ii) Suppose that $j > 0$ , so that $t^{j} \in R_{□}^{L d m} (s^{j})$ iff $t \in R_{□}^{f u t} (s)$ = $⋂_{i \in A g} R_{i}^{p r e s} (s)$ . By assumption then, $u_{m}^{j} \in ⋂_{i \in A g} R_{i}^{p r e s} (u_{k}^{j}) = R_{i}^{f u t} (u_{k}^{j})$ for all $k, m \in {1, . . ., n}$ and each $i \in A g$ . Hence, $u_{m}^{j} \in ⋂_{i \in A g} R_{i}^{f u t} (u_{k}^{j})$ for all $k, m \in {1, . . ., n}$ . If we therefore pick any $u_{k}^{j}$ , it follows that $u_{k}^{j} \in ⋂_{i \in A g} R_{i}^{f u t} (u_{i}^{j})$ , meaning that the intersection $⋂_{1 \leq i \leq n} R_{i}^{L d m} (u_{i}^{j})$ is non-empty.
Follows by definition.
$R_{G}^{L d m}$ is a transitive and serial by definition, and $R_{H}^{L d m}$ is the converse of $R_{G}^{L d m}$ by definition as well.
For all $u^{j}, u^{k}, u^{l} \in W^{L d m}$ , suppose that $R_{G}^{L d m} u^{j} u^{k}$ and $R_{G}^{L d m} u^{j} u^{l}$ . Then, $j < k$ and $j < l$ , and since $N$ is linearly ordered, we have that $k < l$ , $k = l$ , or $k > l$ , implying that $R_{G}^{L d m} u^{k} u^{l}$ , $u^{k} = u^{l}$ , or $R_{G}^{L d m} u^{l} u^{k}$ .
Similar to previous case.
Suppose that $(u^{j}, v^{j + k}) \in R_{G}^{L d m} \circ R_{□}^{L d m}$ with $k \geq 1$ . By definition of $R_{G}^{L d m}$ , $u^{j + k}$ is the only element in $R_{G}^{L d m} (u^{j})$ associated with $j + k$ , and so, $(u^{j + k}, v^{j + k}) \in R_{□}^{L d m}$ (By lemma 2 no other $u^{j + k^{'}}$ with $k^{'} \neq k$ can relate to $v^{j + k}$ in $R_{□}^{L d m}$ .). Since $k \geq 1$ , $v^{j + k} \in R_{□}^{L d m} (u^{j + k})$ iff iff $v^{0} \in R_{A g}^{L d m} (u^{0})$ . By lemma 3, $(u^{j}, v^{j}) \in R_{A g}^{L d m}$ . This implies that, and since $(v^{j}, v^{j + k}) \in R_{G}^{L d m}$ by definition, we have that $(u^{j}, v^{j + k}) \in R_{A g}^{L d m} \circ R_{G}^{L d m}$ .
Follows from the definition of the $R_{G}^{L d m}$ relation.
Last, it is easy to see that the valuation function $V^{L d m}$ is indeed a valuation function.

Lemma 7 (Truth-Lemma)

For any formula $ϕ$ , $M^{L d m}, w^{0} ⊨ ϕ$ iff $ϕ \in w^{0}$ .

Proof

Shown by induction on the complexity of $ϕ$ (See [3]).

Appendix 0.B $G 3 X s t i t$ Derivation of IOA $^{x}$ Axiom

We make use of the system of rules $(I O A_{X})$ , to derive the $X s t i t$ IOA axiom in $G 3 X s t i t$ .

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{A} w_{4} w_{5}, R_{A} w_{2} w_{5}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{5} : ϕ, w_{5} : ¯ ¯ ¯ ϕ

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{A} w_{4} w_{5}, R_{A} w_{2} w_{5}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{5} : ϕ

(

I O A - U_{1})

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{A} w_{4} w_{5}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{5} : ϕ

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{4} : [A]^{x} ϕ

D_{1}

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{B} w_{4} w_{6}, R_{B} w_{3} w_{6}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . . .

w_{6} : ψ, w_{6} : ¯ ¯¯ ¯ ψ

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{B} w_{4} w_{6}, R_{B} w_{3} w_{6}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{6} : ψ

(

I O A - U_{2})

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, R_{B} w_{4} w_{6}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{6} : ψ

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, . . .

w_{4} : [B]^{x} ψ

D_{2}

D_{1}

D_{2}

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, w_{1} : ◊ ([A]^{x} ϕ \land [B]^{x} ψ), w_{4} : [A]^{x} ϕ \land [B]^{x} ψ

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, R_{□} w_{1} w_{4}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, w_{1} : ◊ ([A]^{x} ϕ \land [B]^{x} ψ)

(

I O A - E

)

R_{□} w_{1} w_{2}, R_{□} w_{1} w_{3}, w_{2} : ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{3} : ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, w_{1} : ◊ ([A]^{x} ϕ \land [B]^{x} ψ)

w_{1} : □ ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ, w_{1} : □ ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ, w_{1} : ◊ ([A]^{x} ϕ \land [B]^{x} ψ)

w_{1} : □ ⟨ A ⟩^{x} ¯ ¯ ¯ ϕ \lor □ ⟨ B ⟩^{x} ¯ ¯¯ ¯ ψ \lor ◊ ([A]^{x} ϕ \land [B]^{x} ψ)

References

[1] Balbiani, P., Herzig, A., Troquard, N.: Alternative axiomatics and complexity of deliberative STIT theories. Journal of Philosophical Logic, 37(4), pp.387–406. Springer (2008)
[2]
Berkel, K., Lyon, T.: Cut-free Calculi and Relational Semantics for Temporal STIT logics. In Logics in Artificial Intelligence - 16th European Conference, JELIA 2019, Rende, Italy, May 7-10, 2019, Proceedings, To appear.
[3] Blackburn, P., de Rijke, M., Venema, Y.: Modal logic. Cambridge University Press, Cambridge (2001)
[4] Lorini, E.: Temporal STIT logic and its application to normative reasoning. Journal of Applied Non-Classical Logics 23 (4), pp. 372–399 (2013)

Appendix for: Cut-free Calculi and Relational Semantics for Temporal STIT logics

Authors

Cut-free Calculi and Relational Semantics for Temporal STIT Logics

Poset products as relational models

Relational Hypersequent S4 and B are Cut-Free Hypersequent Incomplete

Relational Hypersequents for Modal Logics

A Neutral Temporal Deontic STIT Logic

Syntactic completeness of proper display calculi

A Presheaf Semantics for Quantified Temporal Logics

Appendix 0.A Completeness of $L d m$

Definition 1 ( $L d m$ -Cs, $L d m$ -Mcs)

Lemma 1 (Lindenbaum’s Lemma [3])

Definition 2 (Present and Future Pre-Canonical $T s t i t$ Model)

Definition 3 (Canonical Temporal Kripke STIT Model)

Lemma 2

Proof

Lemma 3

Proof

Lemma 4 ([3])

Lemma 5 (Existence Lemma [3])

Lemma 6

Proof

Lemma 7 (Truth-Lemma)

Proof

Appendix 0.B $G 3 X s t i t$ Derivation of IOA $^{x}$ Axiom

References

Appendix for: Cut-free Calculi and Relational Semantics for Temporal STIT logics

Authors

Related Research

Cut-free Calculi and Relational Semantics for Temporal STIT Logics

Poset products as relational models

Relational Hypersequent S4 and B are Cut-Free Hypersequent Incomplete

Relational Hypersequents for Modal Logics

A Neutral Temporal Deontic STIT Logic

Syntactic completeness of proper display calculi

A Presheaf Semantics for Quantified Temporal Logics

This week in AI

Appendix 0.A Completeness of Ldm

Definition 1 (Ldm-Cs, Ldm-Mcs)

Lemma 1 (Lindenbaum’s Lemma [3])

Definition 2 (Present and Future Pre-Canonical Tstit Model)

Definition 3 (Canonical Temporal Kripke STIT Model)

Lemma 2

Proof

Lemma 3

Proof

Lemma 4 ([3])

Lemma 5 (Existence Lemma [3])

Lemma 6

Proof

Lemma 7 (Truth-Lemma)

Proof

Appendix 0.B G3Xstit Derivation of IOAx Axiom

References

Appendix 0.A Completeness of $L d m$

Definition 1 ( $L d m$ -Cs, $L d m$ -Mcs)

Definition 2 (Present and Future Pre-Canonical $T s t i t$ Model)

Appendix 0.B $G 3 X s t i t$ Derivation of IOA $^{x}$ Axiom