Good-for-Game QPTL: An Alternating Hodges Semantics

ELECTRONIC APPENDICES

A PROOFS OF SECTION 2

Proposition 1. \({\mathfrak{X}} \subseteq \bar{\bar{{\mathfrak{X}} }}\) and \({\mathfrak{X}} \equiv \bar{\bar{{\mathfrak{X}} }}\), for all \({\mathfrak{X}} \in {\rm HAsg}\).

Lemma 1. The following equivalences hold true, for all QPTLformulae \(\varphi\) and hyperassignments \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\sf free}{\varphi })\).

Proof.

• (\(1a \Rightarrow 1b\)) By 1a, there is \({\rm X}\in {\mathfrak{X}}\) such that \(\chi \models \varphi\) holds for every \(\chi \in {\rm X}\). By definition of \(\bar{{\mathfrak{X}} }\), for every \(\bar{{\rm X}} \in \bar{{\mathfrak{X}} }\) there is \(\Gamma _{\bar{{\rm X}}}\) such that \(\Gamma _{\bar{{\rm X}}}({\rm X}) \in {\rm X}\) and \(\bar{{\rm X}} = \lbrace \Gamma _{\bar{{\rm X}}}({\rm X}) : {\rm X}\in {\mathfrak{X}} \rbrace\); by 1a, \(\Gamma _{\bar{{\rm X}}}({\rm X}) \models \varphi\); since, in addition, \(\Gamma _{\bar{{\rm X}}}({\rm X}) \in \bar{{\rm X}}\), the thesis holds.

• (\(1b \Rightarrow 1a\)) By 1b, for every \(\bar{{\rm X}} \in \bar{{\mathfrak{X}} }\) there is \(\chi _{\bar{{\rm X}}} \in \bar{{\rm X}}\) such that \(\chi _{\bar{{\rm X}}} \models \varphi\). Consider the function \(\Gamma \in {\sf Chc}{\bar{{\mathfrak{X}} }}\) defined as: \(\Gamma (\bar{{\rm X}}) = \chi _{\bar{{\rm X}}}\), for every \(\bar{{\rm X}} \in \bar{{\mathfrak{X}} }\). By definition of \(\bar{\bar{{\mathfrak{X}} }}\), we have that \(\lbrace \Gamma (\bar{{\rm X}}) : \bar{{\rm X}} \in \bar{{\mathfrak{X}} } \rbrace \in \bar{\bar{{\mathfrak{X}} }}\). By Proposition 1, it holds \(\bar{\bar{{\mathfrak{X}} }} \sqsubseteq {\mathfrak{X}}\), which means that there is \({\rm X}\in {\mathfrak{X}}\), with \({\rm X}\subseteq \lbrace \Gamma (\bar{{\rm X}}) : \bar{{\rm X}} \in \bar{{\mathfrak{X}} } \rbrace\). Since, by construction, \(\Gamma (\bar{{\rm X}}) \models \varphi\) for every \(\bar{{\rm X}} \in \bar{{\mathfrak{X}} }\), the thesis holds.

• (\(2a \Leftrightarrow 2b\)) By statement 1 of this lemma, we have that 1a is false if and only if 1b is false (\(\mathit {not} \ 1a \Leftrightarrow \mathit {not} \ 1b\), for short). By instantiating, in this last equivalence, \(\varphi\) with \(\lnot \varphi\), we have \(1a^{\prime } \Leftrightarrow 1b^{\prime }\), where \(1a^{\prime }\) and \(1b^{\prime }\) are abbreviations for, respectively:

  –	for all sets of assignments \({\rm X}\in {\mathfrak{X}}\), there exists an assignment \(\chi \in {\rm X}\) such that \(\chi \not\models \lnot \varphi\);
  –	there exists a set of assignments \({\rm X}\in \bar{{\mathfrak{X}} }\) such that, for all assignments \(\chi \in {\rm X}\), it holds that \(\chi \not\models \lnot \varphi\).

  By applying semantics of negation, it is straightforward to see that \(1a^{\prime }\) and \(1b^{\prime }\) correspond to 2a and 2b, respectively, hence the thesis follows.\(\Box\)

Lemma 2 (Boolean Connectives). The following equivalences hold true, for all QPTLformulae \(\varphi _{1}\) and \(\varphi _{2}\) and hyperassignments \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\rm P})\), with \({\rm P}\triangleq {\sf free}{\varphi _{1}} \cup {\sf free}{\varphi _{2}}\).

Proof.

• (\(1a \Rightarrow 1b\)) Let \({\rm X}\in {\mathfrak{X}}\) be such that \(\chi \models \varphi _{1} \wedge \varphi _{2}\) holds for every \(\chi \in {\rm X}\) and consider an arbitrary pair \(({\mathfrak{X}} _1, {\mathfrak{X}} _2) \in {\sf par}({{\mathfrak{X}} })\). Since \(({\mathfrak{X}} _1, {\mathfrak{X}} _2)\) is a partition of \({\mathfrak{X}}\), either \({\rm X}\in {\mathfrak{X}} _1\) or \({\rm X}\in {\mathfrak{X}} _2\): in the former case, let \(i=1\); in the latter, let \(i = 2\). Since \({\rm X}\in {\mathfrak{X}} _i\) and \(\chi \models \varphi _{i}\) holds for every \(\chi \in {\rm X}\), the thesis holds.

• (\(1b \Rightarrow 1a\)) Consider \(\mathring{{\mathfrak{X}} } = \lbrace {\rm X}\in {\mathfrak{X}} : \forall \chi \in {\rm X}\ . \ \chi \models \varphi _1 \rbrace\) and the pair \(({\mathfrak{X}} _1 \triangleq {\mathfrak{X}} \setminus \mathring{{\mathfrak{X}} }, {\mathfrak{X}} _2 \triangleq \mathring{{\mathfrak{X}} }) \in {\sf par}({{\mathfrak{X}} })\). Observe that, by definition of \({\mathfrak{X}} _1\), there is no \({\rm X}\in {\mathfrak{X}} _1\) such that \(\chi \models \varphi _{1}\) holds for every \(\chi \in {\rm X}\). Thus, by 1b, there must exist \({\rm X}\in {\mathfrak{X}} _2\) such that \(\chi \models \varphi _{2}\) holds for every \(\chi \in {\rm X}\). By definition of \({\mathfrak{X}} _2\), it also holds that \(\chi \models \varphi _{1}\) for every \(\chi \in {\rm X}\), hence the thesis.

• By statement 1 of this lemma, we have that 1a is false if and only if 1b is false (\(\mathit {not} \ 1a \Leftrightarrow \mathit {not} \ 1b\), for short). By instantiating, in this last equivalence, \(\varphi _1\) with \(\lnot \varphi _1\) and \(\varphi _2\) with \(\lnot \varphi _2\), we have \(1a^{\prime } \Leftrightarrow 1b^{\prime }\), where \(1a^{\prime }\) and \(1b^{\prime }\) are abbreviations for, respectively:

  –	for all sets of assignments \({\rm X}\in {\mathfrak{X}}\), there exists an assignment \(\chi \in {\rm X}\) such that \(\chi \not\models \lnot \varphi _{1} \wedge \lnot \varphi _{2}\);
  –	there exists a pair of hyperassignments \(({\mathfrak{X}} _1, {\mathfrak{X}} _2) \in {\sf par}({{\mathfrak{X}} })\) such that, for all indexes \(i \in \lbrace 1, 2 \rbrace\) and sets of assignments \({\rm X}\in {\mathfrak{X}} _i\), there exists an assignment \(\chi \in {\rm X}\) for which it holds that \(\chi \not\models \lnot \varphi _{i}\).

  By applying the semantics of negation and the classical De Morgan’s laws on the semantic rules, it is straightforward to see that \(1a^{\prime }\) and \(1b^{\prime }\) correspond to 2a and 2b, respectively, hence the thesis.\(\Box\)

Lemma 3 (Hyperassignment Extensions) The following equivalences hold true, for all QPTLformulae \(\varphi\), atomic propositions \(p\in \text{AP}\), and hyperassignments \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\sf free}{\varphi } \setminus \lbrace p\rbrace)\).

Proof.

• (\(1a \Rightarrow 1b\)) Let \({\rm X}\in {\mathfrak{X}}\) be such that \(\chi \models \exists p. \varphi\) holds for every \(\chi \in {\rm X}\). By semantics (Def. 2, item 3a), for every \(\chi \in {\rm X}\), there is a temporal function \({\sf f}_{\chi } \in {\mathbb {N}}\rightarrow {\mathbb {B}}\) such that \({\chi }[p\mapsto {\sf f}_{\chi }] \models \varphi\). Let \({\sf F}\in {\rm Fnc}(\text {ap} {{\mathfrak{X}} })\) be such that \({\sf F}(\chi) = {\sf f}_{\chi }\) for every \(\chi \in {\rm X}\) and let \({\rm X}_{{\sf F}} = \lbrace {\chi }[p\mapsto {\sf F}(\chi)] : \chi \in {\rm X}\rbrace\). Since \({\rm X}_{{\sf F}} \in {\sf ext}({{\mathfrak{X}} , p})\) and \(\chi \models \varphi\) holds for every \(\chi \in {\rm X}_{{\sf F}}\), the thesis holds.

• (\(1b \Rightarrow 1a\)) Let \({\rm X}_{{\sf F}} \in {\sf ext}({{\mathfrak{X}} , p})\) be such that \(\chi \models \varphi\) holds for every \(\chi \in {\rm X}_{{\sf F}}\). By definition of \({\sf ext}({{\mathfrak{X}} , p})\), there are \({\rm X}\in {\mathfrak{X}}\) and \({\sf F}\in {\rm Fnc}(\text {ap} {{\mathfrak{X}} })\) such that \({\rm X}_{{\sf F}} = \lbrace {\chi }[p\mapsto {\sf F}(\chi)] : \chi \in {\rm X}\rbrace\). Clearly, by semantics (Def. 2, item 3a), \(\chi \models \exists p. \varphi\) holds for every \(\chi \in {\rm X}\), hence the thesis follows.

• (2a ⇔ 2b) By statement 1 of this lemma, we have that 1a is false if and only if 1b is false (\(\mathit {not} \ 1a \Leftrightarrow \mathit {not} \ 1b\), for short). By instantiating, in this last equivalence, \(\varphi\) with \(\lnot \varphi\), we have \(1a^{\prime } \Leftrightarrow 1b^{\prime }\), where \(1a^{\prime }\) and \(1b^{\prime }\) are abbreviations for, respectively:

  –	for all sets of assignments \({\rm X}\in {\mathfrak{X}}\), there exists an assignment \(\chi \in {\rm X}\) such that \(\chi \not\models \exists p. \lnot \varphi\);
  –	for all sets of assignments \({\rm X}\in {\sf ext}({{\mathfrak{X}} , p})\), there exists an assignment \(\chi \in {\rm X}\) such that \(\chi \not\models \lnot \varphi\).

  By applying semantics of negation and duality of \(\exists\) and \(\forall\), it is straightforward to see that \(1a^{\prime }\) and \(1b^{\prime }\) correspond to 2a and 2b, respectively, hence the thesis. \(\Box\)

Next, we prove Theorem 1. Here is the graph of dependency presenting the lemmata and propositions used for this proof, an edge meaning that the source is directly cited in the proof of the target.

Theorem 1 (Semantics Adequacy) For all QPTLformulae \(\varphi\) and hyperassignments \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\sf free}{\varphi })\):

Proof. Both claims 1 and 2 are proved together, by induction on the structure of the formula.

B PROOFS OF SECTION 3

Proposition 2. Let \({\rm P}\subseteq \text{AP}\) be a set of atomic propositions, \(\chi _1, \chi _2 \in {\rm Asg}({\rm P})\) two assignments, \({\Theta} \in {\Theta}\) a quantifier specification, and \(k \in \mathbb {N}\) a time instant. Then, \(\chi _1 \approx _{{\Theta} }^{k} \chi _2\) iffthe following hold true:

Proposition 3. If \(\chi _1 \approx _{{\Theta} }^{k} \chi _2\) then \({\sf F}(\chi _1)(k) = {\sf F}(\chi _2)(k)\), for all assignments \(\chi _1, \chi _2 \in {\rm Asg}({\rm P})\), quantifier specifications \({\Theta} \in {\Theta}\), time instants \(k \in {\mathbb {N}}\), and \({\Theta}\)-functors \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\).

Proposition 4. Let \({\mathfrak{X}} _1, {\mathfrak{X}} _2 \in {\rm HAsg}\) be two hyperassignments with \({\mathfrak{X}} _1 \sqsubseteq {\mathfrak{X}} _2\). Then, the following properties hold true:

Next, we prove Theorem 2. Here is the graph of dependency presenting the only proposition used for this proof.

Theorem 2 (Hyperassignment Refinement). Let \(\varphi\) be a GFG-QPTLformula and \({\mathfrak{X}} _1, {\mathfrak{X}} _2 \in {\rm HAsg}_{\subseteq } ({\sf free}{\varphi })\) two hyperassignments with \({\mathfrak{X}} _1 \sqsubseteq {\mathfrak{X}} _2\). Then, \({\mathfrak{X}} _1 \models ^{\exists \forall } \varphi\) implies \({\mathfrak{X}} _2 \models ^{\exists \forall } \varphi\) and \({\mathfrak{X}} _2 \models ^{\forall \exists } \varphi\) implies \({\mathfrak{X}} _1 \models ^{\forall \exists } \varphi\).

Proof. Assume \({\mathfrak{X}} _1 \!\sqsubseteq \! {\mathfrak{X}} _2\). Thus, there is a function \(f : {\mathfrak{X}} _1 \rightarrow {\mathfrak{X}} _2\), such that \(f({\rm X}_1) \subseteq {\rm X}_1\) for every \({\rm X}_1\in {\mathfrak{X}} _1\). The claim is proved by induction on the structure of the formula and the alternation flags. More precisely, we consider, as a basis for the induction, a well-founded preorder \(\preceq\) over the set of pairs \(\lbrace \langle \varphi , \alpha \rangle |\varphi \text{ is a} {\sf GFG}-{\sf QPTL} \text{formula and } \alpha \in \lbrace \exists \forall , \forall \exists \rbrace \rbrace\), such that \(\langle \varphi , \alpha \rangle \preceq \langle \varphi ^{\prime }, \alpha ^{\prime } \rangle\) if and only if \(\varphi\) is a subformula of \(\varphi ^{\prime }\) or one of the following holds:

• \(\varphi = \varphi ^{\prime } = \psi _1 \wedge \psi _2\), \(\alpha = \exists \forall\), and \(\alpha ^{\prime } = \forall \exists\),

• \(\varphi = \varphi ^{\prime } = \psi _1 \vee \psi _2\), \(\alpha = \forall \exists\), and \(\alpha ^{\prime } = \exists \forall\),

• \(\varphi = \varphi ^{\prime } = \exists p:\! {\Theta} .\psi\), \(\alpha = \exists \forall\), and \(\alpha ^{\prime } = \forall \exists\),

• \(\varphi = \varphi ^{\prime } = \forall p:\! {\Theta} .\psi\), \(\alpha = \forall \exists\), and \(\alpha ^{\prime } = \exists \forall\).

Next, we prove Theorem 3. Here is the graph of dependency presenting the lemma, proposition, corollary, and theorem used for this proof. The ellipsis symbolizing dependencies is already presented in a previous graph.

Theorem 3 (Double Dualization) Let \(\varphi\) be a GFG-QPTLformula and \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\sf free}{\varphi })\) a hyperassignment. Then, \(\bar{{\mathfrak{X}} } \models ^{\bar{\alpha }} \varphi\) iff\(\bar{\bar{{\mathfrak{X}} }} \models ^{\alpha } \varphi\) iff\({\mathfrak{X}} \models ^{\alpha } \varphi\).

Proof. The fact that \(\begin{equation*} {\mathfrak{X}} \models ^{\alpha } \varphi { {iff} } \bar{\bar{{\mathfrak{X}} }} \models ^{\alpha } \varphi \text{Thm. 3a} \end{equation*}\) immediately follows from \({\mathfrak{X}} \equiv \bar{\bar{{\mathfrak{X}} }}\) (Proposition 1) and Corollary 1.

We now turn to proving that \({\mathfrak{X}} \models ^{\alpha } \varphi\) iff\(\bar{{\mathfrak{X}} } \models ^{\bar{\alpha }} \varphi\), for all \({\mathfrak{X}} \in {\rm HAsg}_{\subseteq }({\sf free}{\varphi })\). The proof is done by structural induction on the formula.

• If \(\varphi \in {\sf LTL}\), then the claim follows immediately from the semantics and Lemma 1.

• If \(\varphi = \lnot \psi\), then we have: \({\mathfrak{X}} \models ^{\alpha }\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} {\mathfrak{X}} {{⊭} ^{\bar{\alpha }}}\: \psi \mathrel {\stackrel{\text{ind.hp.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } {⊭} ^{\alpha }\: \psi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } \models ^{\bar{\alpha }}\: \varphi\).

• If \(\varphi = \varphi _1 \wedge \varphi _2\), then we have:

  –	\(\bar{{\mathfrak{X}} } \models ^{\bar{\exists \forall }}\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{\bar{{\mathfrak{X}} }} \models ^{\exists \forall }\: \varphi \mathrel {\stackrel{\text{{Thm. 3a}}}{\Leftrightarrow }} {\mathfrak{X}} \models ^{\exists \forall }\: \varphi\); and
  –	\({\mathfrak{X}} \models ^{\forall \exists }\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } \models ^{\exists \forall }\: \varphi\).

• If \(\varphi = \varphi _1 \vee \varphi _2\), then we have:

  –	\({\mathfrak{X}} \models ^{\exists \forall }\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } \models ^{\forall \exists }\: \varphi\); and
  –	\(\bar{{\mathfrak{X}} } \models ^{\bar{\forall \exists }}\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{\bar{{\mathfrak{X}} }} \models ^{\forall \exists }\: \varphi \mathrel {\stackrel{\text{{Thm. 3a}}}{\Leftrightarrow }} {\mathfrak{X}} \models ^{\forall \exists }\: \varphi\).

• If \(\varphi = \exists p:\! {\Theta} .\psi\), then we have:

  –	\(\bar{{\mathfrak{X}} } \models ^{\bar{\exists \forall }}\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{\bar{{\mathfrak{X}} }} \models ^{\exists \forall }\: \varphi \mathrel {\stackrel{\text{{Thm. 3a}}}{\Leftrightarrow }} {\mathfrak{X}} \models ^{\exists \forall }\: \varphi\); and
  –	\({\mathfrak{X}} \models ^{\forall \exists }\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } \models ^{\exists \forall }\: \varphi\).

• If \(\varphi = \forall p:\! {\Theta} .\psi\), then we have:

  –	\({\mathfrak{X}} \models ^{\exists \forall }\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{{\mathfrak{X}} } \models ^{\forall \exists }\: \varphi\);
  –	\(\bar{{\mathfrak{X}} } \models ^{\bar{\forall \exists }}\: \varphi \mathrel {\stackrel{\text{sem.}}{\Leftrightarrow }} \bar{\bar{{\mathfrak{X}} }} \models ^{\forall \exists }\: \varphi \mathrel {\stackrel{\text{{Thm. 3a}}}{\Leftrightarrow }} {\mathfrak{X}} \models ^{\forall \exists }\: \varphi\). \(\Box\)

Lemma 4 (Boolean Laws). Let \(\varphi\), \(\varphi _{1}\), \(\varphi _{2}\) be GFG-QPTLformulae:

Proof. Thanks to Corollary 2, it suffices to prove the equivalence for \(\equiv ^{\alpha }\) for some \(\alpha \in \lbrace \exists \forall , \forall \exists \rbrace\). Let \(\varphi\) be a QPTLformula and \({\mathfrak{X}} \in {\rm HAsg}({\sf free}{\varphi })\) a hyperassignment.

Proposition 5.

Let \({\mathfrak{X}} _1, {\mathfrak{X}} _2 \in {\rm HAsg}\) be two hyperassignments with \({\mathfrak{X}} _1 \sqsubseteq {\mathfrak{X}} _2\) and \({\wp} \in {\rm Qn}\). Then, the following holds true: \({\sf evl}_\alpha ({\mathfrak{X}} _1, {\wp}) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} _2, {\wp})\).

Proof. The proof proceeds by induction on the length of the quantification prefix .

Lemma 5 (Prefix Evolution). Let \({\wp} \phi\) be a GFG-QPTLformula with quantifier prefix \({\wp} \in {\rm Qn}\). Then, \({\mathfrak{X}} \models ^{\alpha } {\wp} \phi\) iff\({\sf evl}_\alpha ({\mathfrak{X}} , {\wp}) \models ^{\alpha } \phi\), for all hyperassignments \({\mathfrak{X}} \in {\rm HAsg}({\sf free}{{\wp} \phi })\).

Proof. The proof is done by induction on the quantifier prefix \({\wp}\).

C PROOFS OF SECTION 4

Now, we showcase the graph of dependency for Theorem 4, presenting the lemma, corollary, and theorem used for the proof in the main paper.

In order to provide the missing proofs of Theorems 5 and 8 and Proposition 6, in this appendix we shall also need to prove the auxiliary Propositions 7, 8, 9, 10, 11, 12, 13, and 14 and to introduce, later on, the notion of normal evolution function and a refinement of the order between hyperassignments.

Proof. We consider the two conditions separately.

• [1] If \(\alpha\) and \({\rm {Q}}\) are coherent, by definition of evolution function, we have \(\begin{equation*} {\sf evl}_\alpha ({\mathfrak{X}} , \rm{Q} ^{{\Theta} } p) = {\sf ext}_{\Theta} ({{\mathfrak{X}} , p}) = \{ { {\sf ext}({{\rm X}, {\sf F}, p} })|{ {\rm X}\in {\mathfrak{X}} , {\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {({\mathfrak{X}} })) }\}. \end{equation*}\) Thus, for every set of assignments \({\sf {W}} \subseteq {\rm Asg}({\rm P}\cup \lbrace p\rbrace)\), it holds that \({\sf {W}} \in {\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)\) iffthere exists a \({\Theta}\)-functor \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\) and a set of assignments \({\rm X}\in {\mathfrak{X}}\) such that \({\sf {W}} = {\sf ext}{{\rm X}, {\sf F}, p}\). Hence, Condition 7 immediately follows.

• [2] If \(\alpha\) and \({\rm {Q}}\) are not coherent, by definition of evolution function, we have \(\begin{equation*} {\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p) = \bar{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}} = \{ { {\sf img}{\Gamma } }{ \Gamma \in {\sf Chc}{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}} }\}. \end{equation*}\) Thus, for every set of assignments \({\sf {W}} \subseteq {\rm Asg}({\rm P}\cup \lbrace p\rbrace)\), it holds that \({\sf {W}} \in {\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)\) iffthere exists a choice function \(\Gamma \in {\sf Chc}{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}}\) such that \({\sf {W}} = {\sf img}{\Gamma } = \lbrace \Gamma (Z) | Z\in {\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}\rbrace\). This means that \({\sf {W}} \subseteq \Psi\) iff\(\Gamma (Z) \in \Psi\), for all \(Z\in {\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}\). Now, it is clear that there exists a choice function \(\Gamma \in {\sf Chc}{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}}\) such that \(\Gamma (Z) \in \Psi\), for all \(Z\in {\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}\) iff, for every \(Z\in {\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p} = \lbrace {\sf ext}{{\rm Y}, {\sf F}, p} | {\rm Y}\in \bar{{\mathfrak{X}} }, {\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P}) \rbrace\), there exists \(\chi _{Z} \in Z\) such that \(\chi _{Z} \in \Psi\). The latter property, however, means that, for every \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\) and \({\rm Y}\in \bar{{\mathfrak{X}} } = \lbrace {\sf img}{\Lambda } | \Lambda \in {\sf Chc}{{\mathfrak{X}} } \rbrace\), there exists \(\chi _{{\sf F}, {\rm Y}} \in {\sf ext}{{\rm Y}, {\sf F}, p}\) such that \(\chi _{{\sf F}, {\rm Y}} \in \Psi\), which in turn can be written as, for every \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\) and \(\Lambda \in {\sf Chc}{{\mathfrak{X}} }\), there exists \(\chi _{{\sf F}, \Lambda } \in {\sf ext}{{\sf img}{\Lambda }, {\sf F}, p} = {\sf ext}{\lbrace \Lambda ({\rm X}) | {\rm X}\in {\mathfrak{X}} \rbrace , {\sf F}, p}\) such that \(\chi _{{\sf F}, \Lambda } \in \Psi\). Now, notice that \(\chi _{{\sf F}, \Lambda } \in {\sf ext}{\lbrace \Lambda ({\rm X}) | {\rm X}\in {\mathfrak{X}} \rbrace , {\sf F}, p}\) iffthere exists \({\rm X}\in {\mathfrak{X}}\) such that \(\chi _{{\sf F}, \Lambda } = {\sf ext}{\Lambda ({\rm X}), {\sf F}, p}\).

  Thus, up to this point, we have shown that the following two properties are equivalent:

  –	there exists \({\sf {W}} \in {\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)\) such that \({\sf {W}} \subseteq \Psi\);
  –	for all \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\) and \(\Lambda \in {\sf Chc}{{\mathfrak{X}} }\), there exists \({\rm X}\in {\mathfrak{X}}\) such that \({\sf ext}{\Lambda ({\rm X}), {\sf F}, p} \in \Psi\).

  Now, by deHerbrandizing³ the universal quantification of \(\Lambda\) w.r.t.the existential quantification of \({\rm X}\) in the last item and recalling that \(\Lambda ({\rm X}) \in {\rm X}\), we obtain that, for all \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\), there exists \({\rm X}\in {\mathfrak{X}}\) such that \({\sf ext}{\chi , {\sf F}, p} \in \Psi\), for all \(\chi \in {\rm X}\). But this means that, for all \({\sf F}\in {\rm Fnc}_{{\Theta} }({\rm P})\), there exists \({\rm X}\in {\mathfrak{X}}\) such that \({\sf ext}{{\rm X}, {\sf F}, p} \subseteq \Psi\), as required by Condition 7.\(\Box\)

Next, we prove Theorem 5. Here is the graph of dependency presenting the proposition used for the proof in the main paper.

Theorem 5 (Quantification Game I)

For each behavioural quantification prefix \({\wp} \in {\rm Qn}_{{\rm {B}} }\) and Borelian property \(\Psi \subseteq {\rm Asg}(\text {ap} {{\wp} })\), the game \({⅁} [{\wp} ][\Psi ]\) satisfies the following two properties:

Proof. Let \({⅁} [{\wp} ][\Psi ]\) be the game defined as prescribed in Construction 1. Obviously, this is a Borelian game, due to the hypothesis on the property \(\Psi\).

Before continuing, first observe that, thanks to the specific structure of the game, every history \({\rho} \cdot v\in {{\rm {Hst}} _\alpha }\) is bijectively correlated with the sequence of positions \({\sf obs}({\rho}) \cdot v\in {\rm Ob}^{*} \cdot {\rm {Ps}} [\alpha ]\), for any player \(\alpha \in \lbrace {\sf E}, {\sf A}\rbrace\). In other words, the functions \(J_\alpha :{{\rm {Hst}} _\alpha } \rightarrow {\rm Ob}^{*} \cdot {\rm {Ps}} [\alpha ]\) defined as \(J_\alpha ({\rho} \cdot v) \triangleq {\sf obs}({\rho}) \cdot v\) are bijective. Thanks to this observation, it is thus immediate to show that, for each strategy \({\sigma _{\sf E}} \in {\rm Str}_{\sf E}\), there is a unique function \(\widehat{{\sigma _{\sf E}} } :{\rm Ob}^{*} \cdot {\rm P}_{{\rm S}_{\sf E}}\rightarrow {\rm {Ps}}\) and, vice versa, for each function \(\widehat{{\sigma _{\sf E}} } :{\rm Ob}^{*} \cdot {\rm P}_{{\rm S}_{\sf E}}\rightarrow {\rm {Ps}}\), there is a unique strategy \({\sigma _{\sf E}} \in {\rm Str}_{\sf E}\) such that \(\begin{equation*} \widehat{{\sigma _{\sf E}} }(J_{\sf E}({\rho})) = {\sigma _{\sf E}} ({\rho}), \text{ for all histories } {\rho} \in {\rm Hst}_{\sf E}. \end{equation*}\) Similarly, for each strategy \({\sigma _{\sf A}}\in {\rm Str}_{\sf A}\), there is a unique function \(\widehat{{\sigma _{\sf A}}} :{\rm Ob}^{*} \cdot {\rm Ps}_{\sf A}\rightarrow {\rm {Ps}}\) and, vice versa, for each function \(\widehat{{\sigma _{\sf A}}} :{\rm Ob}^{*} \cdot {\rm Ps}_{\sf A}\rightarrow {\rm {Ps}}\), there is a unique strategy \({\sigma _{\sf A}}\in {\rm Str}_{\sf A}\) satisfying the equality \(\begin{equation*} \widehat{{\sigma _{\sf A}}}(J_{\sf A}({\rho})) = {\sigma _{\sf A}}({\rho}), \text{ for all histories } {\rho} \in {\rm Hst}_{\sf A}. \end{equation*}\)

We can now proceed with the proof of the two properties.

• [1] Since Eloise wins the game, she has a winning strategy, i.e., there is \({\sigma _{\sf E}} \in {\rm Str}_{\sf E}\) such that \({\sf obs}({\sf play}({\sigma _{\sf E}} , {\sigma _{\sf A}})) \in {\rm Wn}\), for all \({\sigma _{\sf A}}\in {\rm Str}_{\sf A}\). We want to prove that there exists \(E\in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp}))\) such that \(E\subseteq \Psi\).

  First, recall that \({\sf C}_{\forall \exists }({\wp}) = \forall ^{{\rm {B}} } \vec{p} .\exists ^{\vec{{\Theta} }} \vec{p}\), for some vectors of atomic propositions \(\vec{p}, \vec{p} \in \text{AP}^{*}\) and quantifier specifications \(\vec{{\Theta} } \in {\Theta} ^{|\vec{p}|}\). Moreover, thanks to Proposition 7, the following claim can be proved by induction on the number of existential variables.

  \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp}))\), iffthere exists a vector of functors \(\vec{{\sf F}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\) such that \({\sf ext}{\chi , \vec{{\sf F}}, \vec{p}} \in \Psi\), for all assignments \(\chi \in {\rm Asg}(\vec{p})\).

  As previously observed, \({\sf C}_{\forall \exists }({\wp}) = \forall ^{{\rm {B}} } \vec{p} .\exists ^{\vec{{\Theta} }} \vec{p}\), for some vectors \(\vec{p}, \vec{p} \in \text{AP}^{*}\) and \(\vec{{\Theta} } \in {\Theta} ^{|\vec{p}|}\). Thus, \({\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp})) = {\sf evl}_{\exists \forall }(\forall ^{{\rm {B}} } \vec{p} .\exists ^{\vec{{\Theta} }} \vec{p}) = {\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\forall ^{{\rm {B}} } \vec{p}), \exists ^{\vec{{\Theta} }} \vec{p}) = {\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , \exists ^{\vec{{\Theta} }} \vec{p})\). At this point, the proof proceeds by induction on the length of the vector \(\vec{p}\). If \(|\vec{p}| = 0\), there is nothing really to prove, as the thesis follows immediately from the fact that \({\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp})) = \lbrace {\rm Asg}(\vec{p}) \rbrace\). Let us now consider the case \(|\vec{p}| \gt 0\) and split both \(\vec{p}\) and \(\vec{{\Theta} }\) as follows: \(\vec{p} = \vec{p}^{\prime } \cdot p\) and \(\vec{{\Theta} } = \vec{{\Theta} }^{\prime } \cdot {\Theta}\). Obviously, \({\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp})) = {\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , \exists ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime }), \exists ^{{\Theta} } p)\). Now, by Item 7 of Proposition 7, \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp}))\), iffthere exist a functor \({\sf F}\in {\rm Fnc}_{{\Theta} }(\vec{p} \cdot \vec{p}^{\prime })\) and a set \({\rm X}\in {\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , \exists ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime })\) such that \({\sf ext}{{\rm X}, {\sf F}, p} \subseteq \Psi\). The latter inclusion can be rewritten as \({\rm X}\subseteq {\sf prj}{\Psi , {\sf F}, p}\), where \({\sf prj}{\Psi , {\sf F}, p} \triangleq \, \lbrace \chi \in {\rm Asg}(\vec{p} \cdot \vec{p}^{\prime }) | {\sf ext}{\chi , {\sf F}, p} \in \Psi \rbrace\). At this point, by the inductive hypothesis applied to the inclusion \({\rm X}\subseteq {\sf prj}{\Psi , {\sf F}, p}\), for some \({\rm X}\in {\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , \exists ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime })\), we obtain that \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp}))\), iffthere exist a functor \({\sf F}\in {\rm Fnc}_{{\Theta} }(\vec{p} \cdot \vec{p}^{\prime })\) and a vector of functors \(\vec{{\sf F}}^{\prime } \in {\rm Fnc}_{\vec{{\Theta} }^{\prime }}(\vec{p})\) such that \({\sf ext}{ {\rm Asg}(\vec{p}), \vec{{\sf F}}^{\prime }, \vec{p}^{\prime }} \subseteq {\sf prj}{\Psi , {\sf F}, p}\). The latter inclusion can now be rewritten as \({\sf ext}{{\sf ext}{ {\rm Asg}(\vec{p}), \vec{{\sf F}}^{\prime }, \vec{p}^{\prime }}, {\sf F}, p} \subseteq \Psi\). To conclude the proof, the vector of functors \(\vec{{\sf F}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\) is obtained by juxtaposing the vector \(\vec{{\sf F}}^{\prime }\) with the functor \({\sf F}^* \in {\rm Fnc}_{{\Theta} }(\vec{p})\) obtained by composing \({\sf F}\) with \(\vec{{\sf F}}^{\prime }\) as follows: \({\sf F}^*(\chi) \triangleq {\sf F}({\sf ext}{\chi , \vec{{\sf F}}^{\prime }, \vec{p}^{\prime }})\).□

  Due to the above characterisation of the existence of a set \(E\in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }({\wp}))\) such that \(E\subseteq \Psi\), the thesis can be proved by defining a suitable vector of functors \(\vec{{\sf F}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\).

  Consider an arbitrary assignment \(\chi \in {\rm Asg}(\vec{p})\) and define the function \(\widehat{{\sigma _{\sf A}}}^{\chi } :{\rm Ob}^{*} \cdot {\rm Ps}_{\sf A}\rightarrow {\rm {Ps}}\) as follows, for all finite sequences of observable positions \(w\in {\rm Ob}^{*}\) and Abelard’s positions \({\chi} \in {\rm Ps}_{\sf A}\): \(\begin{equation*} \widehat{{\sigma _{\sf A}}}^{\chi }(w\cdot {\chi}) \triangleq {\left\lbrace \begin{array}{ll} {{⦰} }, & \text{if } {{\chi} \in {\rm Ob}}; \\ {{{\chi} }[{x}\mapsto \chi ({x})(|w|)]}, & \text{otherwise}; \end{array}\right.} \end{equation*}\) where \({x}\in \vec{p}\) is the atomic proposition at position \(\#({\chi})\) in the prefix \({\wp}\), i.e., \(({\wp})_{\#({\chi})} = \forall ^{{\rm {B}} } {x}\). Due to the bijective correspondence previously described, there is a unique strategy \({\sigma _{\sf A}}^{\chi } \in {\rm Str}_{\sf A}\) such that \({\sigma _{\sf A}}^{\chi }({\rho}) = \widehat{{\sigma _{\sf A}}}^{\chi }(J_{\sf A}({\rho}))\), for all histories \({\rho} \in {\rm Hst}_{\sf A}\). Obviously, the induced play \({\pi} ^{\chi } \triangleq {\sf play}({\sigma _{\sf E}} , {\sigma _{\sf A}}^{\chi })\) is won by Eloise, i.e., \(w^{\chi } \triangleq {\sf obs}({\pi} ^{\chi }) \in {\rm Wn}\).

  Thanks to all the infinite sequences \(w^{\chi }\), one for each assignment \(\chi \in {\rm Asg}(\vec{p})\), we can now define every component \((\vec{{\sf F}})_{i}\) of the vector of functors \(\vec{{\sf F}} \in ({\rm Fnc}(\vec{p}))^{|\vec{p}|}\) as follows, for all instants of time \(t \in {\mathbb {N}}\), where \(i \in {[0,|\vec{p}|)}\): \(\begin{equation*} (\vec{{\sf F}})_{i}(\chi)(t) \triangleq \, (w^{\chi })_{t}((\vec{p})_{i}). \end{equation*}\) It is not too hard to show that, by construction, this functor complies with the vector \(\vec{{\Theta} }\) of quantifier specifications.

  \(\vec{{\sf F}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\).

  At this point, for all assignments \(\chi \in {\rm Asg}(\vec{p})\), let \(\chi _{\vec{{\sf F}}} \triangleq {\sf ext}{\chi , \vec{{\sf F}}, \vec{p}}\). We can argue that \(\chi _{\vec{{\sf F}}} \in \Psi\). Indeed, by construction of the strategy \({\sigma _{\sf A}}^{\chi }\) and the vector of functors \(\vec{{\sf F}}\), it holds that \(\chi _{\vec{{\sf F}}}({x})(t) = (w^{\chi })_{t}({x})\), for all instants of time \(t \in {\mathbb {N}}\) and atomic propositions \({x}\in \vec{p} \cdot \vec{p}\). Hence, \(\texttt{wrd} (\chi _{\vec{{\sf F}}}) = w^{\chi }\), which implies \(\chi _{\vec{{\sf F}}} \in \Psi\), since \(w^{\chi } \in {\rm Wn}\).

• [2] Since Abelard wins the game, he has a winning strategy, i.e., there is \({\sigma _{\sf A}}\in {\rm Str}_{\sf A}\) such that \({\sf obs}({\sf play}({\sigma _{\sf E}} , {\sigma _{\sf A}})) \not\in {\rm Wn}\), for all \({\sigma _{\sf E}} \in {\rm Str}_{\sf E}\). We want to prove that, for all \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\), it holds that \(E\not\subseteq \Psi\).

  First, recall that \({\sf C}_\exists \forall ({\wp}) = \exists ^{{\rm {B}} } \vec{p} .\forall ^{\vec{{\Theta} }} \vec{p}\), for some vectors of atomic propositions \(\vec{p}, \vec{p} \in \text{AP}^{*}\) and quantifier specifications \(\vec{{\Theta} } \in {\Theta} ^{|\vec{p}|}\). Moreover, thanks to Proposition 7, the following claim can be proved by induction on the number of universal variables.

  \(E\not\subseteq \Psi\), for all \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\), iffthere exists a vector of functors \(\vec{{\sf G}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\) such that \({\sf ext}{\chi , \vec{{\sf G}}, \vec{p}} \not\in \Psi\), for all assignments \(\chi \in {\rm Asg}(\vec{p})\).

  For technical convenience, we prove the counter-positive version of the statement: \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\), iff, for all vectors of functors \(\vec{{\sf G}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\), it holds that \({\sf ext}{{\rm Asg}(\vec{p}), \vec{{\sf G}}, \vec{p}} \cap \Psi \ne {\emptyset}\). As previously observed, \({\sf C}_\exists \forall ({\wp}) = \exists ^{{\rm {B}} } \vec{p} .\forall ^{\vec{{\Theta} }} \vec{p}\), for some vectors \(\vec{p}, \vec{p} \in \text{AP}^{*}\) and \(\vec{{\Theta} } \in {\Theta} ^{|\vec{p}|}\). Thus, \({\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp})) = {\sf evl}_{\exists \forall }(\exists ^{{\rm {B}} } \vec{p} .\forall ^{\vec{{\Theta} }} \vec{p}) = {\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\exists ^{{\rm {B}} } \vec{p}), \forall ^{\vec{{\Theta} }} \vec{p}) = {\sf evl}_{\exists \forall }(\lbrace \lbrace \chi \rbrace | \chi \!\in \! {\rm Asg}(\vec{p}) \rbrace \!, \forall ^{\vec{{\Theta} }} \vec{p})\). At this point, the proof proceeds by induction on the length of the vector \(\vec{p}\). If \(|\vec{p}| = 0\), there is nothing really to prove, as the thesis follows immediately from the fact that \({\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp})) = \lbrace \lbrace \chi \rbrace | \chi \in {\rm Asg}(\vec{p}) \rbrace\). Let us now consider the case \(|\vec{p}| \gt 0\) and split both \(\vec{p}\) and \(\vec{{\Theta} }\) as follows: \(\vec{p} = \vec{p}^{\prime } \cdot p\) and \(\vec{{\Theta} } = \vec{{\Theta} }^{\prime } \cdot {\Theta}\). Obviously, \({\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp})) = {\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace \lbrace \chi \rbrace | \chi \in {\rm Asg}(\vec{p}) \rbrace , \forall ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime }), \forall ^{{\Theta} } p)\). Now, by Item 7 of Proposition 7, \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\), iff, for all functors \({\sf G}\in {\rm Fnc}_{{\Theta} }(\vec{p} \cdot \vec{p}^{\prime })\), there exists a set \({\rm X}\in {\sf evl}_{\exists \forall }(\lbrace \lbrace \chi \rbrace | \chi \in {\rm Asg}(\vec{p}) \rbrace , \forall ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime })\) such that \({\sf ext}{{\rm X}, {\sf G}, p} \subseteq \Psi\). The latter inclusion can be rewritten as \({\rm X}\subseteq {\sf prj}{\Psi , {\sf G}, p}\), where \({\sf prj}{\Psi , {\sf G}, p} \triangleq \, \lbrace \chi \in {\rm Asg}(\vec{p} \cdot \vec{p}^{\prime }) | {\sf ext}{\chi , {\sf G}, p} \in \Psi \rbrace\). At this point, by the inductive hypothesis applied to the inclusion \({\rm X}\subseteq {\sf prj}{\Psi , {\sf G}, p}\), for some \({\rm X}\in {\sf evl}_{\exists \forall }(\lbrace \lbrace \chi \rbrace | \chi \in {\rm Asg}(\vec{p}) \rbrace , \forall ^{\vec{{\Theta} }^{\prime }} \vec{p}^{\prime })\), we obtain that \(E\subseteq \Psi\), for some \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\), ifffor all functors \({\sf G}\in {\rm Fnc}_{{\Theta} }(\vec{p} \cdot \vec{p}^{\prime })\) and vectors of functors \(\vec{{\sf G}}^{\prime } \in {\rm Fnc}_{\vec{{\Theta} }^{\prime }}(\vec{p})\), it holds that \({\sf ext}{{\rm Asg}(\vec{p}), \vec{{\sf G}}^{\prime }, \vec{p}^{\prime }} \cap {\sf prj}{\Psi , {\sf G}, p} \ne {\emptyset}\). The latter inequality can now be rewritten as \({\sf ext}{{\sf ext}{{\rm Asg}(\vec{p}), \vec{{\sf G}}^{\prime }, \vec{p}^{\prime }}, {\sf G}, p} \cap \Psi \ne {\emptyset}\). To conclude the proof, it is enough to observe that the vectors of functors \(\vec{{\sf G}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\) can always be obtained by juxtaposing the vectors \(\vec{{\sf G}}^{\prime }\) with the functors \({\sf G}^* \in {\rm Fnc}_{{\Theta} }(\vec{p})\) obtained by composing \({\sf G}\) with \(\vec{{\sf G}}^{\prime }\) as follows: \({\sf G}^*(\chi) \triangleq {\sf G}({\sf ext}{\chi , \vec{{\sf G}}^{\prime }, \vec{p}^{\prime }})\).□

  Due to the above characterisation of non-existence of a set \(E\in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall ({\wp}))\) such that \(E\subseteq \Psi\), the thesis can be proved by defining a suitable vector of functors \(\vec{{\sf G}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\).

  Consider an arbitrary assignment \(\chi \in {\rm Asg}(\vec{p})\) and define the function \(\widehat{{\sigma _{\sf E}} }^{\chi } :{\rm Ob}^{*} \cdot {\rm P}_{{\rm S}_{\sf E}}\rightarrow {\rm {Ps}}\) as follows, for all finite sequences of observable positions \(w\in {\rm Ob}^{*}\) and Eloise’s positions \({\chi} \in {\rm P}_{{\rm S}_{\sf E}}\): \(\begin{equation*} \widehat{{\sigma _{\sf E}} }^{\chi }(w\cdot {\chi}) \triangleq {{\chi} }[{x}\mapsto \chi ({x})(|w|)], \end{equation*}\) where \({x}\in \vec{p}\) is the atomic proposition at position \(\#({\chi})\) in the prefix \({\wp}\), i.e., \(({\wp})_{\#({\chi})} = \exists ^{{\rm {B}} } {x}\). Due to the bijective correspondence previously described, there is a unique strategy \({\sigma _{\sf E}} ^{\chi } \in {\rm Str}_{\sf E}\) such that \({\sigma _{\sf E}} ^{\chi }({\rho}) = \widehat{{\sigma _{\sf E}} }^{\chi }(J_{\sf E}({\rho}))\), for all histories \({\rho} \in {\rm Hst}_{\sf E}\). Obviously, the induced play \({\pi} ^{\chi } \triangleq {\sf play}({\sigma _{\sf E}} ^{\chi }, {\sigma _{\sf A}})\) is won by Abelard, i.e., \(w^{\chi } \triangleq {\sf obs}({\pi} ^{\chi }) \not\in {\rm Wn}\).

  Thanks to all the infinite sequences \(w^{\chi }\), one for each assignment \(\chi \in {\rm Asg}(\vec{p})\), we can now define every component \((\vec{{\sf G}})_{i}\) of the vector of functors \(\vec{{\sf G}} \in ({\rm Fnc}(\vec{p}))^{|\vec{p}|}\) as follows, for all instants of time \(t \in {\mathbb {N}}\), where \(i \in {[0,|\vec{p}|)}\): \(\begin{equation*} (\vec{{\sf G}})_{i}(\chi)(t) \triangleq \, (w^{\chi })_{t}((\vec{p})_{i}). \end{equation*}\) It is not too hard to show that, by construction, this functor complies with the vector \(\vec{{\Theta} }\) of quantifier specifications.

  \(\vec{{\sf G}} \in {\rm Fnc}_{\vec{{\Theta} }}(\vec{p})\).

  At this point, for all assignments \(\chi \in {\rm Asg}(\vec{p})\), let \(\chi _{\vec{{\sf G}}} \triangleq {\sf ext}{\chi , \vec{{\sf G}}, \vec{p}}\). We can argue that \(\chi _{\vec{{\sf G}}} \not\in \Psi\). Indeed, by construction of the strategy \({\sigma _{\sf E}} ^{\chi }\) and the vector of functors \(\vec{{\sf G}}\), it holds that \(\chi _{\vec{{\sf G}}}({x})(t) = (w^{\chi })_{t}({x})\), for all instants of time \(t \in {\mathbb {N}}\) and atomic propositions \({x}\in \vec{p} \cdot \vec{p}\). Hence, \(\texttt{wrd} (\chi _{\vec{{\sf G}}}) = w^{\chi }\), which implies \(\chi _{\vec{{\sf G}}} \not\in \Psi\), since \(w^{\chi } \not\in {\rm Wn}\).\(\Box\)

The two conditions stated in Proposition 7 allow us to introduce a different, but equivalent (in terms of the equivalence relation \(\equiv\) between hyperassignments), definition of evolution function that we call normal, in symbols \({\sf nevl}\). This new notion will be useful to show important properties that would be, otherwise, much more cumbersome to prove by appealing directly to the original definition of the evolution function \({\sf evl}\). \(\begin{equation*} {{\sf nevl}}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p) \triangleq {\left\lbrace \begin{array}{ll} {{\sf ext}[{\Theta} ]{{\mathfrak{X}} , p}}, & \text{ if } {{\rm {Q}} } \text{ is } \alpha \text{-coherent}; \\ {\{ { {\sf ext}{\eth , p} }{ \eth \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} }) \rightarrow {\mathfrak{X}} }}\}, & \text{ otherwise}; \end{array}\right.} \end{equation*}\) where \({\sf ext}{\eth , p} \triangleq \bigcup \lbrace {\sf ext}{\eth ({\sf F}), {\sf F}, p} | {\sf F}\in {\sf dom}{\eth } \rbrace\). Intuitively, w.r.t.\({\sf evl}\), we just modified the non \(\alpha\)-coherent case, in order to avoid the double application of the dualization function, by replacing this with a choice of a selection map \(\eth \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} }) \rightarrow {\mathfrak{X}}\) selecting, in fact, for each \({\Theta}\)-functor \({\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} })\), a set of assignments \(\eth ({\sf F}) \in {\mathfrak{X}}\).

The new evolution operator lifts naturally to an arbitrary quantification prefix \({\wp} \in {\rm Qn}\) as follows: (1) \({{\sf nevl}}_\alpha ({\mathfrak{X}} , \epsilon) \triangleq {\mathfrak{X}}\); (2) \({{\sf nevl}}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p.{\wp}) \triangleq {{\sf nevl}}_\alpha ({{\sf nevl}}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p), {\wp})\). As we have done for \({\sf evl}\), we also set \({{\sf nevl}}_\alpha ({\wp}) \triangleq {{\sf nevl}}_\alpha (\lbrace \!\lbrace {⦰} \rbrace \!\rbrace , {\wp})\).

Proof. The proof proceeds by a case analysis on the coherence of \(\alpha\) and \({\rm {Q}}\).

• [\({\rm {Q}}\) is \(\alpha\)-coherent] By definition, \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} }\! p) \!=\! {\sf ext}[{\Theta} ]{{\mathfrak{X}} _1, p}\) and \({\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} }\! p) \!=\! {\sf ext}[{\Theta} ]{{\mathfrak{X}} _2, p}\). Since \({\mathfrak{X}} _1 \equiv {\mathfrak{X}} _2\), by Proposition 4, it holds that \({\sf ext}[{\Theta} ]{{\mathfrak{X}} _1, p} \equiv {\sf ext}[{\Theta} ]{{\mathfrak{X}} _2, p}\), which conclude this case of the proof.

• [\({\rm {Q}}\) is not \(\alpha\)-coherent] By definition, \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) = \lbrace {\sf ext}{\eth , p} | \eth \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _1}) \rightarrow {\mathfrak{X}} _1 \rbrace\) and \({\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p) = \bar{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} _2}, p}}\). We now prove the two inclusions \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\) and \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \sqsupseteq {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\) separately.

  –

  [\(\sqsubseteq\)] To prove \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\), we need to show that, for any \(\Psi \in {{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p)\) there is \({\sf {W}} [\Psi ] \in {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\) such that \({\sf {W}} [\Psi ] \subseteq \Psi\). Obviously, for any \(\Psi \in {{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p)\), it holds that \(\Psi = {\sf ext}{\eth , p} = \bigcup \lbrace {\sf ext}{\eth ({\sf F}), {\sf F}, p} | {\sf F}\in {\sf dom}{\eth } \rbrace\), for some selection function \(\eth \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _1}) \rightarrow {\mathfrak{X}} _1\). This means that, for every \({\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _1})\), there is \({\rm X}_{{\sf F}} \triangleq \eth ({\sf F}) \in {\mathfrak{X}} _1\) such that \({\sf ext}{{\rm X}_{{\sf F}}, {\sf F}, p} \subseteq \Psi\). Now, by Item 7 of Proposition 7, there exists \({\sf {W}} _1 \in {\sf evl}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p)\) such that \({\sf {W}} _1 \subseteq \Psi\). Since, thanks to Proposition 5, \({\mathfrak{X}} _1 \equiv {\mathfrak{X}} _2\) implies \({\sf evl}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \equiv {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\), we have that there is \({\sf {W}} _2 \in {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\) such that \({{\sf {W}} _2} \subseteq {\sf {W}} _1 \subseteq \Psi\). Finally, by setting \({\sf {W}} [\Psi ] \triangleq {\sf {W}} _2\), we obtain what is required.

  –

  [\(\sqsupseteq\)] To prove \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \sqsupseteq {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\), we need to show that, for any \(\Psi \in {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\) there is \({\sf {W}} [\Psi ] \in {{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p)\) such that \({\sf {W}} [\Psi ] \subseteq \Psi\). By instantiating \({\sf {W}}\) with \(\Psi\) in Proposition 7, since \({\sf {W}} = \Psi \in {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\), from Item 7 we derive that, for all \({\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _2})\), there is \({\rm X}_{{\sf F}2} \in {\mathfrak{X}} _2\) such that \({\sf ext}{{\rm X}_{{\sf F}2}, {\sf F}, p} \subseteq \Psi\). Now, since \({\mathfrak{X}} _1 \equiv {\mathfrak{X}} _2\), there is \({\rm X}_{{\sf F}1} \in {\mathfrak{X}} _1\) such that \({\rm X}_{{\sf F}1} \subseteq {\rm X}_{{\sf F}2}\), which in turn implies \({\sf ext}{{\rm X}_{{\sf F}1}, {\sf F}, p} \subseteq {\sf ext}{{\rm X}_{{\sf F}2}, {\sf F}, p} \subseteq \Psi\). At this point, define the selection map \(\eth \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _1}) \rightarrow {\mathfrak{X}} _1\) as follows: \(\eth ({\sf F}) \triangleq {\rm X}_{{\sf F}1}\), for every \({\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _1}) = {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _2})\). Clearly, by setting \({\sf {W}} [\Psi ] \triangleq {\sf ext}{\eth , p}\), both \({\sf {W}} [\Psi ] = \bigcup \lbrace {\sf ext}{\eth ({\sf F}), {\sf F}, p} | {\sf F}\in {\sf dom}{\eth } \rbrace \subseteq \Psi\) and \({\sf {W}} [\Psi ] \in {{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p)\) holds true, as required.

  This concludes the proof of the second and last case.\(\Box\)

The following examples are meant to show how the normal \(\alpha\)-evolution function for non-coherent quantifier simulates the \(\alpha\)-evolution function for the same quantifier.

Proof. The proof proceeds by simple induction on the length of the quantification prefix \({\wp}\).

• [Base case \({\wp} = \varepsilon\)] \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, \varepsilon) = {\mathfrak{X}} _1 \equiv {\mathfrak{X}} _2 = {\sf evl}_\alpha ({\mathfrak{X}} _2, \varepsilon)\).

• [Inductive case \({\wp} = {\rm {Q}} ^{{\Theta} } p.{\wp} ^{\prime }\)] By definition, we have \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p.{\wp} ^{\prime }) = {{\sf nevl}}_\alpha ({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p), {\wp} ^{\prime })\) and \({\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p.{\wp} ^{\prime }) = {\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p), {\wp} ^{\prime })\). Now, by Proposition 8, \({{\sf nevl}}_\alpha ({\mathfrak{X}} _1, {\rm {Q}} ^{{\Theta} } p) \equiv {\sf evl}_\alpha ({\mathfrak{X}} _2, {\rm {Q}} ^{{\Theta} } p)\), since \({\mathfrak{X}} _1 \equiv {\mathfrak{X}} _2\). Thus, the thesis follows by a straightforward application of the inductive hypothesis.\(\Box\)

In the following, by \({\Theta} _{{\rm {B}} }\) we denote the set of behavioural quantifier specifications, i.e., quantifier specifications of the form \({\rm {B}} \cup \left\lt {\rm {S}} : {\rm P}_{\rm {S}} \right\gt\) for some set of atomic propositions \({\rm P}_{\rm {S}} \subseteq \text{AP}\).

Towards the proof of Proposition 6, we show the following more general result.

Proof. We separately prove the two chains of inequalities forming the statement, namely \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}({\wp})) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} _1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp})\) and \({\sf evl}_\alpha ({\mathfrak{X}} , {\wp}) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} _1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))\), by using different technical expedients.

• [\({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}({\wp})) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} _1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp})\)] To prove that the first chain of inequalities holds, let us fix a well-founded preorder \(\preceq\) over the set of triples of quantifier prefixes \({\mathcal {T}}= \lbrace {\langle {{\wp} _1}, {{\wp} _2}, {{\wp} _3}\rangle } \in {\rm Qn}_{{\rm {B}} } \times {\rm Qn}_{{\rm {B}} } \times {\rm Qn}_{{\rm {B}} } | {\wp} = {{\wp} ^1} .{{\wp} _2} .{\wp} _3\rbrace\) defined as follows: \({\langle {{\wp} ^1} ,{{\wp} _2} ,{{\wp} _3}\rangle } \preceq {\langle {{\wp} {^{1\prime}}}, {{\wp} _{2}^{\prime }}, {{\wp} _{3}^{\prime }}\rangle }\) iff\({\wp} _2^{\prime } = {{\wp} _l} \cdot {\wp} _2 \cdot {\wp} [r]\), for some \({{\wp} _l}, {\wp} [r] \in {\rm Qn}_{{\rm {B}} }\), i.e., \({\wp} _2\) is a (not necessarily proper) infix of \({\wp} _2^{\prime }\). Notice that, given the definition of the set \({\mathcal {T}}\), the relation \({\langle {{\wp} ^1}, {{\wp} _2}, {{\wp} _3}\rangle } \preceq {\langle {{\wp} ^{1\prime }}, {{\wp} _2^{\prime }}, {{{\wp} _3^\prime }}\rangle }\) also implies \({{\wp} ^1} = {\wp} ^{1\prime } \cdot {{\wp} _l}\) and \({\wp} _3 = {\wp} [r] \cdot {{\wp} _3^\prime }\). In addition, let us introduce \({\sf C}_{\bar{\alpha }}(T)\) as an abbreviation for \({\wp} ^1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3\), given an arbitrary triple \(T= {\langle {{\wp} ^1}, {{\wp} _2}, {{\wp} _3}\rangle } \in {\mathcal {T}}\). Now, to show that the chain of inequalities holds true, it is enough to prove that \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime })) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))\), for all \(T, T^{\prime } \in {\mathcal {T}}\) with \(T\preceq T^{\prime }\). The proof shall proceed by structural induction on the preorder \(\preceq\).

  –

  [Base case \(T= T^{\prime }\)] Obviously \({\sf C}_{\bar{\alpha }}(T) = {\sf C}_{\bar{\alpha }}(T^{\prime })\). Thus, the property trivially holds, as \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime })) = {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))\).

  –

  [Inductive case \(T\prec T^{\prime }\)] Since \(T\prec T^{\prime }\), there necessarily exists a triple \(T^{\prime \prime } = {\langle {{{\wp} _1^{\prime \prime }}}, {{\wp} _2^{\prime \prime }}, {{\wp} _3^{\prime \prime }}\rangle } \in {\mathcal {T}}\) such that \(T\prec T^{\prime \prime } \preceq T^{\prime }\) and either (a) \({\wp} ^1 = {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } p\), \({\wp} _2^{\prime \prime } = {\rm {Q}} ^{{\rm {B}} } p.{\wp} _2\), and \({\wp} _3 = {\wp} _3^{\prime \prime }\), or (b) \({\wp} ^1 = {{\wp} _1^{\prime \prime }}\), \({\wp} _2^{\prime \prime } = {\wp} _2 .{\rm {Q}} ^{{\rm {B}} } p\), and \({\wp} _3 = {\rm {Q}} ^{{\rm {B}} } p.{\wp} _3^{\prime \prime }\), for some quantifier symbol \({\rm {Q}} \in \lbrace \exists , \forall \rbrace\) and atomic proposition \(p\in \text{AP}\). By inductive hypothesis, it holds that \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime })) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime \prime }))\). Thus, to conclude, we need to show that \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime \prime })) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))\). If \({\sf C}_{\bar{\alpha }}(T) = {\sf C}_{\bar{\alpha }}(T^{\prime \prime })\), there is nothing really to prove, as \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime \prime })) = {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))\). Hence, let us assume \({\sf C}_{\bar{\alpha }}(T) \ne {\sf C}_{\bar{\alpha }}(T^{\prime \prime })\). The proof now proceeds with the following case analysis. (a) [\({\wp} ^1 = {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } p\), \({\wp} _2^{\prime \prime } = {\rm {Q}} ^{{\rm {B}} } p.{\wp} _2\), and \({\wp} _3 = {\wp} _3^{\prime \prime }\)] First observe that \({\rm {Q}}\) is \(\alpha\)-coherent. If this were not the case, indeed, we would have had \({\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) = {\sf C}_{\bar{\alpha }}({\rm {Q}} ^{{\rm {B}} } p.{\wp} _2) = {\rm {Q}} ^{{\rm {B}} } p.{\sf C}_{\bar{\alpha }}({\wp} _2)\), which in turn would have implied \({\sf C}_{\bar{\alpha }}(T^{\prime \prime }) = {{\wp} _1^{\prime \prime }} .{\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) .{\wp} _3^{\prime \prime } = {{\wp} _1^{\prime \prime }} .{\sf C}_{\bar{\alpha }}({\rm {Q}} ^{{\rm {B}} } p.{\wp} _2) .{\wp} _3^{\prime \prime } = {{\wp} _1^{\prime \prime }} .{{\rm {Q}} ^{{\rm {B}} }} p.{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3^{\prime \prime } = {\wp} ^1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3, = {\sf C}_{\bar{\alpha }}(T)\), contradicting the previous assumption \({\sf C}_{\bar{\alpha }}(T) \ne {\sf C}_{\bar{\alpha }}(T^{\prime \prime })\). Both \({\sf C}_{\bar{\alpha }}({\wp} _2)\) and \({\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime })\) are prefix canonicalisation, featuring at most one quantifier alternation starting with a \(\bar{\alpha }\)-coherent quantifier \(\bar{{\rm {Q}} }\). Specifically, these can be written as \({\sf C}_{\bar{\alpha }}({\wp} _2) = \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{\vec{{\Theta} }} \vec{r}\) and \({\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) = {\sf C}_{\bar{\alpha }}({\rm {Q}} ^{{\rm {B}} } p.{\wp} _2) = \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} \cup \left\lt {\rm {S}} : \vec{p} \right\gt } p.{\rm {Q}} ^{\vec{{\Theta} }} \vec{r}\), for some vectors of atomic propositions \(\vec{p}\) and \(\vec{r}\), and a vector of quantifiers specifications \(\vec{{\Theta} } \in {\Theta} _{{\rm {B}} }^*\) with \(|\vec{{\Theta} }| = |\vec{r}|\). At this point, the induction proof terminates by checking the following chain of equalities/inequalities: \(\begin{align} {{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime \prime }))} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) .{\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime })), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} \cup \left\lt {\rm {S}} : \vec{p} \right\gt } p.{\rm {Q}} ^{\vec{{\Theta} }} \vec{r}), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} \cup \left\lt {\rm {S}} : \vec{p} \right\gt } p), {\rm {Q}} ^{\vec{{\Theta} }} \vec{r} .{\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{{\rm {B}} } \vec{p}), {\rm {Q}} ^{\vec{{\Theta} }} \vec{r} .{\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{\vec{{\Theta} }} \vec{r}), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } p), \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{\vec{{\Theta} }} \vec{r}), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1), {\sf C}_{\bar{\alpha }}({\wp} _2)), {\wp} _3)} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3)} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))}. \end{align}\) Step (3e) is due to Proposition 11 applied to \({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), \bar{{\rm {Q}} }^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} \cup \left\lt {\rm {S}} : \vec{p} \right\gt } p)\) combined with Proposition 5. All the other steps are just immediate consequences of the definition of evolution function and the structure of both the quantifier prefixes \({{\wp} _1^{\prime \prime }}\), \({\wp} _2^{\prime \prime }\), and \({\wp} _3^{\prime \prime }\), and the canonical forms \({\sf C}_{\bar{\alpha }}(T)\) and \({\sf C}_{\bar{\alpha }}(T^{\prime \prime })\). (b) [\({\wp} ^1 = {{\wp} _1^{\prime \prime }}\), \({\wp} _2^{\prime \prime } = {\wp} _2 .{\rm {Q}} ^{{\rm {B}} } p\), and \({\wp} _3 = {\rm {Q}} ^{{\rm {B}} } p.{\wp} _3^{\prime \prime }\)] Similarly to the previous case, from \({\sf C}_{\bar{\alpha }}(T) \ne {\sf C}_{\bar{\alpha }}(T^{\prime \prime })\), one can derive that \({\rm {Q}}\) is \(\bar{\alpha }\)-coherent. Consequently, \({\sf C}_{\bar{\alpha }}({\wp} _2)\) and \({\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime })\) can be written as \({\sf C}_{\bar{\alpha }}({\wp} _2) = {\rm {Q}} ^{{\rm {B}} } \vec{p} .\bar{{\rm {Q}} }^{\vec{{\Theta} }} \vec{r}\) and \({\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) = {\sf C}_{\bar{\alpha }}({\wp} _2 .{\rm {Q}} ^{{\rm {B}} } p) = {\rm {Q}} ^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{\vec{{\Theta} }^{\prime }} \vec{r}\), for some vectors of atomic propositions \(\vec{p}\) and \(\vec{r}\), and vectors of quantifiers specifications \(\vec{{\Theta} }, \vec{{\Theta} }^{\prime } \in {\Theta} _{{\rm {B}} }^*\) with \(|\vec{{\Theta} }| = |\vec{{\Theta} }^{\prime }|= |\vec{r}|\) and \(\vec{{\Theta} ^{\prime }} = \vec{{\Theta} } \cup \left\lt {\rm {S}} : p \right\gt\). At this point, the induction proof terminates by checking the following chain of equalities/inequalities: \(\begin{align} {{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T^{\prime \prime }))} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime }) .{\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\sf C}_{\bar{\alpha }}({\wp} _2^{\prime \prime })), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\rm {Q}} ^{{\rm {B}} } \vec{p} .{\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{\vec{{\Theta} }^{\prime }} \vec{r}), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } \vec{p}), {\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{\vec{{\Theta} }^{\prime }} \vec{r}), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } \vec{p}), \bar{{\rm {Q}} }^{\vec{{\Theta} }} \vec{r} .{\rm {Q}} ^{{\rm {B}} } p), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\rm {Q}} ^{{\rm {B}} } \vec{p} .\bar{{\rm {Q}} }^{\vec{{\Theta} }} \vec{r} .{\rm {Q}} ^{{\rm {B}} } p), {\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }}), {\rm {Q}} ^{{\rm {B}} } \vec{p} .\bar{{\rm {Q}} }^{\vec{{\Theta} }} \vec{r}), {\rm {Q}} ^{{\rm {B}} } p.{\wp} _3^{\prime \prime })} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1), {\sf C}_{\bar{\alpha }}({\wp} _2)), {\wp} _3)} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3)} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}(T))}. \end{align}\) Step (4e) is due to Proposition 11 applied to \({\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {{\wp} _1^{\prime \prime }} .{\rm {Q}} ^{{\rm {B}} } \vec{p}), {\rm {Q}} ^{{\rm {B}} } p.\bar{{\rm {Q}} }^{\vec{{\Theta} }^{\prime }} \vec{r})\) combined with Proposition 5. All the other steps are just immediate consequences of the definition of evolution function and the structure of both the quantifier prefixes \({{\wp} _1^{\prime \prime }}\), \({\wp} _2^{\prime \prime }\), and \({\wp} _3^{\prime \prime }\), and the canonical forms \({\sf C}_{\bar{\alpha }}(T)\) and \({\sf C}_{\bar{\alpha }}(T^{\prime \prime })\).

• [\({\sf evl}_\alpha ({\mathfrak{X}} , {\wp}) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))\)] In order to show that the second chain of inequalities holds as well, we first state the following two simple auxiliary results, one regarding a duality property between the two syntactic canonicalisations of a quantifier prefix and the other concerning the dualization of the evolution function.

  \(\bar{{\sf C}_{\alpha }({\wp})} = {\sf C}_{\bar{\alpha }}(\bar{{\wp} })\), for all quantifier prefixes \({\wp} \in {\rm Qn}\).

  \(\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})} \equiv {\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} })\), for all hyperassignments \({\mathfrak{X}} \in {\rm HAsg}\) and quantifier prefixes \({\wp} \in {\rm Qn}\).

  Proof. The proof proceeds by induction on the length of \({\wp}\).

  –

  [Base step \({\wp} = \varepsilon\)] \(\bar{{\sf evl}_\alpha ({\mathfrak{X}} , \varepsilon)} = \bar{{\mathfrak{X}} } = {\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \varepsilon)\).

  –

  [Inductive step \({\wp} = {\rm {Q}} ^{{\Theta} } p.{\wp} ^{\prime }\)] First notice that \(\bar{{\wp} } = \bar{{\rm {Q}} }^{{\Theta} }\! p.\bar{{\wp} ^{\prime }}\) and observe that, thanks to the definition of evolution function and the inductive hypothesis, the following holds true: \(\begin{equation*} \bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})} = \bar{{\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p), {\wp} ^{\prime })} \equiv {\sf evl}_\alpha (\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)}, \bar{{\wp} ^{\prime }}). \end{equation*}\) Let us now distinguish two cases based on the coherence of \(\alpha\) and \({\rm {Q}}\). [\({\rm {Q}}\) is \(\alpha\)-coherent] \(\begin{align} {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})}} & \equiv {{\sf evl}_\alpha (\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)}, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\sf ext}[{\Theta} ]{{\mathfrak{X}} , p}}, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & \equiv {{\sf evl}_\alpha (\bar{{\sf ext}[{\Theta} ]{ \bar{\bar{{\mathfrak{X}} }}, p} }, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\rm {Q}} }^{{\Theta} }\! p), \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\rm {Q}} }^{{\Theta} }\! p.\bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} })} \end{align}\) Step (5b) and (5d) are due to the definition of evolution function over a single quantifier, for the cases when \(\alpha\) and \({\rm {Q}}\) are coherent and non-coherent, respectively. Step (5c) is just a simple consequence of Propositions 1, 4, and 5. Finally, Step (5e) is given by the definition of evolution function for quantifier prefixes. [\({\rm {Q}}\) is not \(\alpha\)-coherent] \(\begin{align} {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})}} & \equiv {{\sf evl}_\alpha (\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\rm {Q}} ^{{\Theta} } p)}, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{\bar{{\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}}}, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & \equiv {{\sf evl}_\alpha ({\sf ext}[{\Theta} ]{\bar{{\mathfrak{X}} }, p}, \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha ({\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\rm {Q}} }^{{\Theta} }\! p), \bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\rm {Q}} }^{{\Theta} }\! p.\bar{{\wp} ^{\prime }})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} })} \end{align}\) Step (6b) and (6d) are due to the definition of evolution function over a single quantifier, for the cases when \(\alpha\) and \({\rm {Q}}\) are non-coherent and coherent, respectively. Step (6c) is just a simple consequence of Propositions 1 and 5. Finally, Step (6e) is given by the definition of evolution function for quantifier prefixes.\(\Box\)

  In the first item of this proof, we have proved that \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}({\wp})) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\bar{\alpha }}({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp})\) holds true for every \({\mathfrak{X}} \in {\rm HAsg}\) and \({\wp} , {\wp} ^1, {\wp} _2, {\wp} _3 \in {\rm Qn}_{{\rm {B}} }\), with \({\wp} = {\wp} ^1 .{\wp} _2 .{\wp} _3\). By instantiating \({\mathfrak{X}}\) and \({\wp}\) with \(\bar{{\mathfrak{X}} }\) and \(\bar{{\wp} }\), and observing that \(\bar{{\wp} } = \bar{{\wp} ^1} .\bar{{\wp} _2} .\bar{{\wp} _3}\), we obtain \({\sf evl}_\alpha (\bar{{\mathfrak{X}} }, {\sf C}_{\bar{\alpha }}(\bar{{\wp} })) \sqsubseteq {\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1} .{\sf C}_{\bar{\alpha }}(\bar{{\wp} _2}) .\bar{{\wp} _3}) \sqsubseteq {\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} })\). Now, thanks to Claims 9 and 10 above, we obtain \(\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))} \sqsubseteq \bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3)} \sqsubseteq \bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})}\), as shown in the following two chains of equivalences/inequalities: \(\begin{align} {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))}} & \equiv {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\sf C}_{\alpha }({\wp})})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, {\sf C}_{\bar{\alpha }}(\bar{{\wp} }))} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1} .{\sf C}_{\bar{\alpha }}(\bar{{\wp} _2}) .\bar{{\wp} _3})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1} .\bar{{\sf C}_{\alpha }({\wp} _2)} .\bar{{\wp} _3})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3})} \end{align}\) \(\begin{align} & \equiv {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3)}}. \end{align}\) \(\begin{align} {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3)}} & \equiv {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1} .\bar{{\sf C}_{\alpha }({\wp} _2)} .\bar{{\wp} _3})} \end{align}\) \(\begin{align} & = {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} ^1} .{\sf C}_{\bar{\alpha }}(\bar{{\wp} _2}) .\bar{{\wp} _3})} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_\alpha (\bar{{\mathfrak{X}} }, \bar{{\wp} })} \end{align}\) \(\begin{align} & \equiv {\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})}}. \end{align}\)

  At this point, thanks to Propositions 1 and 4, we derive \({\sf evl}_\alpha ({\mathfrak{X}} , {\wp}) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))\) from \(\bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))} \sqsubseteq \bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp} ^1 .{\sf C}_{\alpha }({\wp} _2) .{\wp} _3)} \sqsubseteq \bar{{\sf evl}_\alpha ({\mathfrak{X}} , {\wp})}\).\(\Box\)

The following proposition is now an immediate consequence of the above result.

Proposition 6. \({\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\bar{\alpha }}({\wp})) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\wp}) \sqsubseteq {\sf evl}_\alpha ({\mathfrak{X}} , {\sf C}_{\alpha }({\wp}))\), for all hyperassignments \({\mathfrak{X}} \in {\rm HAsg}\) and behavioral quantifier prefixes \({\wp} \in {\rm Qn}_{{\rm {B}} }\), with \(\text {ap} {{\wp} } \cap \text {ap} {{\mathfrak{X}} } = {\emptyset}\).

At this point, we have proven everything used in the proof of Theorem 6 from the main paper. Here is the graph of dependency presenting the propositions used for this proof.

For the next proposition, given a set of assignments \({\rm Y}\) and a set of atomic propositions \({\rm P}\subseteq \text{AP}\), we introduce the notation \({\rm Y}\setminus _{{\rm P}} \triangleq \lbrace \chi \in {\rm Asg}(\text {ap} {{\rm Y}} \setminus {\rm P}) | \exists \chi ^{\prime } \in {\rm Y}.\chi \subseteq \chi ^{\prime }\rbrace\). We also use the notation \({\rm Y}\setminus _{p}\), with \(p\in \text {ap} {{\rm Y}}\), as a shortcut for \({\rm Y}\setminus _{\lbrace p\rbrace }\).

Proof. The proof proceeds by induction on the length of the quantification prefix \({\wp}\).

We now define a refinement of the order \(\sqsubseteq\) between two hyperassignments \({\mathfrak{X}} _1, {\mathfrak{X}} _2 \in {\rm HAsg}\), with \(\text {ap} {{\mathfrak{X}} _1} = \text {ap} {{\mathfrak{X}} _2}\), w.r.t.a set of assignments \({\rm X}\subseteq {\rm Asg}({\rm P})\) over some \({\rm P}\subseteq \text{AP}\) as follows: \({\mathfrak{X}} _1 \sqsubseteq _{{\rm X}} {{\mathfrak{X}} _2}\) if, for every \({{\rm X}_1} \in {\mathfrak{X}} _1\), there is \({{\rm X}_2} \in {\mathfrak{X}} _2\) such that \({\rm X}_2 \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright _{{\rm P}} \in {\rm X}\rbrace \subseteq {{\rm X}_1}\).

Proof. The proof proceeds by induction on the length of \({\wp}\).

• [Base step \({\wp} = \varepsilon\)] \({\sf evl}_\alpha ({{\mathfrak{X}} _1}, \varepsilon) = {{\mathfrak{X}} _1} \sqsubseteq _{{\rm X}} {{\mathfrak{X}} _2} = {\sf evl}_\alpha ({{\mathfrak{X}} _2}, \varepsilon)\).

• [Inductive step \({\wp} = {\rm {Q}} ^{{\Theta} } p.{\wp} ^{\prime }\)] Let us distinguish two cases based on whether \({\rm {Q}}\) is or not \(\alpha\)-coherent.

  –

  [\({\rm {Q}}\) is \(\alpha\)-coherent] Since \(\alpha\) and \({\rm {Q}}\) are coherent, \({\sf evl}_\alpha ({\mathfrak{X}} _i, {\wp}) = {\sf evl}_\alpha ({\sf ext}[{\Theta} ]{{\mathfrak{X}} _i, p}, {\wp} ^{\prime })\), for all \(i \in \lbrace 1, 2 \rbrace\). We can now focus on showing that \({\sf ext}[{\Theta} ]{{\mathfrak{X}} _1, p} \sqsubseteq _{{\rm X}} {\sf ext}[{\Theta} ]{{{\mathfrak{X}} _2}, p}\) holds true, as the thesis follows by applying the inductive hypothesis. Since \({\sf ext}[{\Theta} ]{{\mathfrak{X}} _i, p} = \lbrace {\sf ext}{{\rm X}_i, {\sf F}_i, p} | {\rm X}_i \in {\mathfrak{X}} _i, {\sf F}_i \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _i}) \rbrace\), we have to prove that, for every \({{\rm X}_1} \in {{\mathfrak{X}} _1}\) and \({\sf F}_1 \in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}})\), there exist \({\rm X}_2 \in {{\mathfrak{X}} _2}\) and \({\sf F}_2 \in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _2}})\) such that \({\sf ext}{{\rm X}_2, {\sf F}_2, p} \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [{\rm P}] \in {\rm X}\rbrace \subseteq {\sf ext}{{{\rm X}_1}, {\sf F}_1, p}\). Now, it is easy to see that such a property can be satisfied by choosing \({\sf F}_2 \triangleq {\sf F}_1\), since \({\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}}) = {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _2}})\), and \({{\mathfrak{X}} _2} \triangleq {\sf f}({{\mathfrak{X}} _1})\), where \({\sf f}:{{\mathfrak{X}} _1} \rightarrow {{\mathfrak{X}} _2}\) is a witness for the inclusion \({{\mathfrak{X}} _1} \sqsubseteq _{{\rm X}} {{\mathfrak{X}} _2}\).

  –

  [\({\rm {Q}}\) is not \(\alpha\)-coherent] Since \(\alpha\) and \({\rm {Q}}\) are not coherent, by Propositions 5 and 9, it holds that \({\sf evl}_\alpha ({\mathfrak{X}} _i, {\wp}) = {\sf evl}_\alpha ({\sf evl}_\alpha ({\mathfrak{X}} _i, {\rm {Q}} ^{{\Theta} } p), {\wp} ^{\prime }) \equiv {\sf evl}_\alpha ({{\sf nevl}}_\alpha ({\mathfrak{X}} _i, {\rm {Q}} ^{{\Theta} } p), {\wp} ^{\prime })\), for all \(i \in \lbrace 1, 2 \rbrace\). As done in the previous case, we can now focus on showing that \({{\sf nevl}}_\alpha ({{\mathfrak{X}} _1}, {\rm {Q}} ^{{\Theta} } p) \sqsubseteq _{{\rm X}} {{\sf nevl}}_\alpha ({{\mathfrak{X}} _2}, {\rm {Q}} ^{{\Theta} } p)\) holds true, as the thesis follows by applying the inductive hypothesis. Since \({{\sf nevl}}_\alpha ({\mathfrak{X}} _i, {\rm {Q}} ^{{\Theta} } p) = \lbrace {\sf ext}{\eth _{i}, p} | \eth _{i} \in {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _i}) \rightarrow {\mathfrak{X}} _i \rbrace\), we have to prove that, for every \(\eth _{1} \in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}}) \rightarrow {{\mathfrak{X}} _1}\), there exists \(\eth _{2} \in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _2}}) \rightarrow {{\mathfrak{X}} _2}\) such that \({\sf ext}{\eth _{2}, p} \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [{\rm P}] \in {\rm X}\rbrace \subseteq {\sf ext}{\eth _{1}, p}\). To this end, let us define a function \({\rm g}:({\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}}) \rightarrow {{\mathfrak{X}} _1}) \rightarrow ({\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _2}) \rightarrow {{\mathfrak{X}} _2})\) as follows: \({\rm g}(\eth _{1})({\sf F}) \triangleq {\sf f}(\eth _{1}({\sf F}))\), for every \(\eth _{1} \in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}}) \rightarrow {{\mathfrak{X}} _1}\) and \({\sf F}\in {\rm Fnc}_{{\Theta} }(\text {ap} {{{\mathfrak{X}} _1}}) = {\rm Fnc}_{{\Theta} }(\text {ap} {{\mathfrak{X}} _2})\), where \({\sf f}:{{\mathfrak{X}} _1} \rightarrow {{\mathfrak{X}} _2}\) is a witness for the inclusion \({{\mathfrak{X}} _1} \sqsubseteq _{{\rm X}} {{\mathfrak{X}} _2}\). Clearly, it holds that \({\rm g}(\eth _{1})({\sf F}) \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [{\rm P}] \in {\rm X}\rbrace \subseteq \eth _{1}({\sf F})\). Thus, the required property can be satisfied by choosing \(\eth _{2} \triangleq {\rm g}(\eth _{1})\), since \({\sf ext}{{\rm g}(\eth _{1}), p} \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [{\rm P}] \in {\rm X}\rbrace \subseteq {\sf ext}{\eth _{1}, p}\) holds true.\(\Box\)

Next, we prove Theorem 8. Here is the graph of dependency presenting the theorem and the propositions used for this proof.

Theorem (Quantification Game II). Every \({\mathfrak{Q}}\)-game \({⅁}\), for some quantification-game schema \({\mathfrak{Q}} \triangleq {\langle {{\mathfrak{X}} }, {{\wp} }, {\Psi }\rangle }\), satisfies the following two properties:

Proof. First of all, recall that the game \({⅁} [{\mathfrak{Q}} ]\) of Construction 2 is obtained directly from the game \({⅁} [\widehat{{\wp} }][\widehat{\Psi }]\) of Construction 1, by defining the set of assignments \(\widehat{{\wp} }\) and the quantifier prefix \(\widehat{\Psi }\) as follows:

• \(\widehat{{\wp} } \triangleq \forall \vec{p} .{\sim} {{\wp} } .{\wp}\) and

• \(\widehat{\Psi } \triangleq \Psi \cup \lbrace \chi \in {\rm Asg}({\rm P}) | \chi \upharpoonright [\vec{p}] \,\not\in {\rm X}\rbrace\),

with \(\vec{p} \triangleq \text {ap} {{\mathfrak{X}} } \setminus \text {ap} {{\sim} {{\wp} }}\) and \({\rm P}\triangleq \text {ap} {{\wp} } \cup \text {ap} {{\mathfrak{X}} }\).

We can now proceed with the proof of the two properties.

• [1] If Eloise wins the game, by Item 5 of Theorem 5, there exists a set of assignments \(\widehat{E} \in {\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }(\widehat{{\wp} }))\) such that \(\widehat{E} \subseteq \widehat{\Psi }\). Thanks to Propositions 5 and 12, we can show that \({\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }(\widehat{{\wp} })) \sqsubseteq {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_{\forall \exists }({\wp}))\). Indeed, \(\begin{align} {{\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }(\widehat{{\wp} }))} & = {{\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }(\forall \vec{p} .{\sim} {{\wp} } .{\wp}))} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_{\exists \forall }(\forall \vec{p} .{\sim} {{\wp} } .{\sf C}_{\forall \exists }({\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\forall \vec{p}), {\sim} {{\wp} }), {\sf C}_{\forall \exists }({\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , {\sim} {{\wp} }), {\sf C}_{\forall \exists }({\wp}))} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace {\rm X}\rbrace , {\sim} {{\wp} }), {\sf C}_{\forall \exists }({\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_{\forall \exists }({\wp}))}, \end{align}\)

  where Step (8b) is due to Proposition 12, Step (8d) to the equality \({\sf evl}_{\exists \forall }(\forall \vec{p}) = \lbrace {\rm Asg}(\vec{p}) \rbrace\), and Step (8e) is derived from Proposition 5, thanks to the fact that \(\lbrace {\rm Asg}(\vec{p}) \rbrace \sqsubseteq \lbrace {\rm X}\rbrace\). Now, due to the definition of the ordering \(\sqsubseteq\) between hyperassignments, it follows that \({\sf evl}_{\exists \forall }({\sf C}_{\forall \exists }(\widehat{{\wp} })) \sqsubseteq {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_{\forall \exists }({\wp}))\) necessarily implies the existence of a set of assignments \(E\!\in \! {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_{\forall \exists }({\wp}))\) such that \(E\subseteq \widehat{E}\). Therefore, \(E\subseteq \widehat{\Psi }\). At this point, we can prove that \(E\subseteq \Psi\), since \(\widehat{\Psi } = \Psi \cup \lbrace \chi \in {\rm Asg}({\rm P}) | \chi \upharpoonright [\vec{p}] \,\not\in {\rm X}\rbrace\) and \(E\cap \lbrace \chi \in {\rm Asg}({\rm P}) | \chi \upharpoonright [\vec{p}] \,\not\in {\rm X}\rbrace = {\emptyset}\). Indeed, \(E\in {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_{\forall \exists }({\wp})) = {\sf evl}_{\exists \forall }(\lbrace {\rm X}\rbrace , {\sim} {{\wp} } .{\sf C}_{\forall \exists }({\wp}))\) and, by Proposition 13, it follows that \(E\setminus _{\vec{p}} \subseteq {\rm X}\).

• [2] If Abelard wins the game, by Item 5 of Theorem 5, it holds that \(\widehat{E} \not\subseteq \widehat{\Psi }\), for all sets of assignments \(\widehat{E} \in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\widehat{{\wp} }))\). It is easy to observe that \(\lbrace {\rm X}\rbrace \sqsubseteq _{\bar{{\rm X}}} \lbrace {\rm Asg}(\vec{p}) \rbrace\), since \({\rm Asg}(\vec{p}) \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright _{\vec{p}} \in \bar{{\rm X}} \rbrace = {\rm Asg}(\vec{p}) \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [\vec{p}] \not\in {\rm X}\rbrace = {\rm X}\). Thus, thanks to Propositions 12 and 14, we can show that \({\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp})) \sqsubseteq _{\bar{{\rm X}}} {\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\widehat{{\wp} }))\). Indeed, \(\begin{align} {{\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp}))} & = {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace {\rm X}\rbrace , {\sim} {{\wp} }), {\sf C}_\exists \forall ({\wp}))} \end{align}\) \(\begin{align} & \sqsubseteq _{\bar{{\rm X}}} {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\lbrace {\rm Asg}(\vec{p}) \rbrace , {\sim} {{\wp} }), {\sf C}_\exists \forall ({\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }({\sf evl}_{\exists \forall }(\forall \vec{p}), {\sim} {{\wp} }), {\sf C}_\exists \forall ({\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }(\forall \vec{p} .{\sim} {{\wp} } .{\sf C}_\exists \forall ({\wp}))} \end{align}\) \(\begin{align} & \sqsubseteq {{\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\forall \vec{p} .{\sim} {{\wp} } .{\wp}))} \end{align}\) \(\begin{align} & = {{\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\widehat{{\wp} }))}, \end{align}\)

  where Step (9b) is due to Proposition 14, thanks to the fact that \(\lbrace {\rm X}\rbrace \sqsubseteq _{\bar{{\rm X}}} \lbrace {\rm Asg}(\vec{p}) \rbrace\), Step (9c) to the equality \({\sf evl}_{\exists \forall }(\forall \vec{p}) = \lbrace {\rm Asg}(\vec{p}) \rbrace\), and Step (9e) is derived from Proposition 12. Now, due to the definition of the ordering \(\sqsubseteq _{\bar{{\rm X}}}\) between hyperassignments, it follows that \({\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp})) \sqsubseteq _{\bar{{\rm X}}} {\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\widehat{{\wp} }))\) necessarily implies the non existence of a set of assignments \(E\in {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp}))\) such that \(E\subseteq \widehat{\Psi }\). Indeed, assume towards a contradiction that there is \(E\in {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp}))\) such that \(E\subseteq \widehat{\Psi }\). By the above inclusion, there is \(\widehat{E} \in {\sf evl}_{\exists \forall }({\sf C}_\exists \forall (\widehat{{\wp} }))\) such that \(\widehat{E} \setminus \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [\vec{p}] \in \bar{{\rm X}} \rbrace \subseteq E\subseteq \widehat{\Psi }\). Since \(\widehat{E} \cap \lbrace \chi \in {\rm Asg}| \chi \upharpoonright [\vec{p}] \in \bar{{\rm X}} \rbrace \subseteq \lbrace \chi \in {\rm Asg}({\rm P}) | \chi \upharpoonright [\vec{p}] \not\in {\rm X}\rbrace \subseteq \widehat{\Psi }\), we have that \(\widehat{E} \subseteq \widehat{\Psi }\), which contradicts the fact that Abelard wins the game. Hence, \(E\not\subseteq \widehat{\Psi }\) holds, for all \(E\in {\sf evl}_{\exists \forall }({\mathfrak{X}} , {\sf C}_\exists \forall ({\wp}))\), which implies that \(E\not\subseteq \Psi\), being \(\Psi \subseteq \widehat{\Psi }\).\(\Box\)

Now, we have proven everything that is used for the proof of Theorem 9 from the main paper. Here is the graph of dependency presenting the lemma, the propositions, the corollaries and theorems used for this proof.

And finally, we have proven everything that is used for the proof of Theorem 7 from the main paper. Here is the graph of dependency presenting the corollary and the theorem used for this proof.

D PROOFS OF SECTION 5

The following theorem relies on both the notions of parity game and parity automaton [14, 64, 65] (see also [26, 45]). Parity games are perfect-information two-player turn-based games of infinite duration, usually played on finite directed graphs. Their vertices, called positions, are labelled by natural numbers, called priorities, and are assigned to one of two players, namely 0 and 1. The game starts at a given position and, during its evolution, players can take a move (an outgoing edge) only at their own positions. The moves selected by the players induce an infinite sequence of vertices, called play. If the maximal priority of the vertices occurring infinitely often in the play is even, then the play is winning for player 0, otherwise, player 1 takes it all. Similarly, the states of a (non-deterministic) parity automaton are labelled with natural numbers (priorities) and an infinite word given in input is accepted by the automaton iffthere exists a run induced by such a word, for which the maximal priority seen infinitely often along it has even parity.

Theorem 10 (Satisfiability Game). For every behavioral GFG-QPTLsentence \(\varphi\) there is a parity game, with \(2^{| {2^{| {O{|\varphi |}}{\cdot }}}{\cdot }}\) positions and \(2^{| {O{|\varphi |}}{\cdot }}\) priorities, won by Eloise iff\(\varphi\) is satisfiable.

ACKNOWLEDGMENTS

The authors would like to thank the anonymous reviewers for the many suggestions which helped to considerably improve the earlier version of the manuscript.