Linguistic$\leftrightarrow$Rational Agents’ Semantics

Abstract

We define and prove a formal semantics divided into two complementary interacting components: the strictly linguistic (i.e. linguistically marked) semantics, we call linguistic agent (LA), and the strictly logical and referential semantics, we call rational agent (RA). This Linguistic $\leftrightarrow$ Rational Agents’ Semantics (LRA semantics) applies to Deep Dependency trees (DD-trees) or more generally, to discourses, i.e. sequences of DD-trees, and interprets them by functional structures we call Meaning Representation Structures (MRS), similar to the DRT, but interpreted very differently. LRA semantics incrementally interprets the discourses by minimal finite models, called proto-models, in a monotonic logic of the LA and checks the proto-models with respect to the classical models of the RA. The proto-model is considered as the linguistic sense of the discourse. We define in full detail the LA which, as we believe, must be universal. On the other hand, we don’t propose a particular RA. We only define the scheme of interaction between the two agents and the stimuli of the RA used by the LA. After all, every discourse has in LRA semantics the single meaning and the single sense for every Rational Agent used to interact with the Linguistic Agent.

Appendix

1.1 Classical Logic Programs

In this subsection we remind the notation, definitions and main facts of the classical theory of logic programs (cf. Lloyd 1984; Doets 1994) and apply them to the ${{\mathcal {L}}{\mathcal {P}}}_{O}$-Programs used above in Sect. 5.

${{\mathcal {L}}{\mathcal {P}}}_{O}$ -Programs are finite sets of ${{\mathcal {L}}{\mathcal {P}}}_{O}$ clauses. In the classical logic programming are considered finite logic programs in a fixed finite signature. Respectively, we consider here the ${{\mathcal {L}}{\mathcal {P}}}_{O}$-programs $P=\{\phi _1,\ldots ,\phi _N\}$ in the fixed finite signature $\varOmega _P$. Their proto-models are defined above (see Definition 5). We apply to the ${{\mathcal {L}}{\mathcal {P}}}_{O}$-programs the notation of the classical logic programming and cite several important facts of this theory, which also hold for the ${{\mathcal {L}}{\mathcal {P}}}_{O}$-programs and their proto-models. In particular, we use Lemma 1 and consider proto-models as Herbrand proto-models.

Definition 22

For a ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program P, $M_P = \{A\in GA(\varOmega _{O})\Vert P\,\models A\}$.

Clearly, $M_P$ is a proto-model for every ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program P.

Lemma 7

(see Lloyd 1984; Doets 1994)

• 1. $M_P\,\models P$.

• 2. If $\varSigma \,\models P$ then $M_P \subseteq \varSigma$ for every proto-model $\varSigma$.

• 3. $M_P = \bigcap \limits _{\{\varSigma \in {\mathcal {P}}{\mathcal {M}}\Vert \varSigma \,\models P\}}\ \varSigma$

This fact implies that $M_P$ is the minimal proto-model of P. It is characterized through the immediate consequence operator $T_P$ of P.

Definition 23

(Immediate consequence operator $T_P$) Let $\varSigma = (I,\Vert ..\Vert ^{\varSigma },F^{\varSigma })$ be a proto-model. In the classical Herbrand model terms it is defined as follows:

$$\begin{aligned} T_P(\varSigma )= & {} \{A\circ \sigma \in GA({{\mathcal {L}}{\mathcal {P}}}_{O}) \Vert (A\leftarrow B_1,\ldots ,B_m)\in \\&\qquad \quad P, B_1\circ \sigma ,\ldots ,B_m\circ \sigma \in \varSigma , \sigma \in \mathbf{sub}(\varOmega _\varSigma )\}. \end{aligned}$$

In terms of extensional proto-models this corresponds to the following:

• 1. Ground atoms. Let $\phi \in GA({{\mathcal {L}}{\mathcal {P}}}_{O})$.

• 1.1. If $\phi = (\mathbf{sg}~P(i_1,\ldots ,i_k) \in i)$ for some $i_1,\ldots ,i_k\in I$ and $i \in I^\mathbf{p}$ (is propositional), then

  $$\begin{aligned} \Vert j\Vert ^{T_\phi (\varSigma )} = \left\{ \begin{array}{ll} \Vert i\Vert ^\varSigma \cup \{\mathbf{sg}~P(i_1,\ldots ,i_k) \}, &{} if\ j = i,\\ \Vert j\Vert ^\varSigma , &{} \textit{otherwise}. \\ \end{array} \right. \end{aligned}$$

• 1.2. if $\phi = (i_1 \in i_2)$ for some $i_1\in I$, $i_2\in I^\mathbf{o}$, then

  $$\begin{aligned} \Vert i\Vert ^{T_\phi (\varSigma )} = \left\{ \begin{array}{ll} \Vert i_2 \Vert ^\varSigma \cup \{ i_1 \}, &{} if\ i = i_2,\\ \Vert i \Vert ^\varSigma , &{} otherwise. \\ \end{array} \right. \end{aligned}$$

• 1.3. if $\phi = (i\ f\ v)$ for some $i \in I$, $f\in F$ and $v \in V \cup I$, then

  $$\begin{aligned} F^{T_\phi (\varSigma )} = F^\varSigma \cup \{(i\ f\ v)\}. \end{aligned}$$

• 2. Clauses. Let $\phi = (A ~\leftarrow ~B_1,\ldots ,B_n)$ be a clause in which $A, B_1,\ldots ,B_n \in AF({{\mathcal {L}}{\mathcal {P}}}_{O})$. Then $T_\phi (\varSigma )$ is as follows:²⁴

  $$\begin{aligned} T_P(\varSigma ) = \begin{array}{c} \bigcup \limits _{\sigma \in \mathbf{sub}(\varOmega _\varSigma ),}\\ {\varSigma \,\models B_j\circ \sigma , 1\le j\le n} \end{array} T_{(A\circ \sigma )}(\varSigma ) \end{aligned}$$

• 3. Programs. For a ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program $P=\{\phi _1,\ldots ,\phi _N\}$:

  $$\begin{aligned} T_P(\varSigma ) = \bigcup \limits _{\phi \in P}\ T_{\phi }(\varSigma ) \end{aligned}$$

Definition 24

Operator T on a complete lattice L is monotonic if $T(X) \subseteq T(Y)$ for all subsets $X\subseteq Y \subseteq L$.

T is finitary if for every $A\in L$ and $X\subseteq L$ such that $A\in T(X)$, there is a finite $Y\subseteq X$ such that $A\in T(Y)$. T is continuous if $T(\bigcup X) = \bigcup T(X)$ for every directed subset $X\subseteq L$ (X is directed if every its finite subset $Y\subseteq X$ has an upper bound in Y).

Lemma 8

(see Lloyd 1984; Doets 1994) The immediate consequence operator of every ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program P is monotonic and finitary.

Corollary 4

(see Lloyd 1984; Doets 1994) The immediate consequence operator of every ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program P is continuous.

Definition 25

(Powers of the immediate consequence operator) Let P be an ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program. Then:

(7.1)

Definition 26

Let L be a lattice and T be an operator over L. $X\in L$ is a fixpoint of T if $t(X) = X$. X is a least fixpoint of T (denoted lfp(T)) if $X \subseteq Y$ for every fixpoint Y of T.

Evidently, if T has lfp(T), it is unique. One of the central facts of the classical logic programming is the following characterisation theorem of van Emden and Kowalski.

Theorem 6

(see Lloyd 1984; Doets 1994) For every ${{\mathcal {L}}{\mathcal {P}}}_{O}$-program P,

• 1. the powers of its immediate consequence operator form the increasing chain:

  $$\begin{aligned} T_P\uparrow ^0 \subseteq T_P\uparrow ^1 \subseteq T_P\uparrow ^2 \subseteq \ldots \subseteq T_P\uparrow ^{\omega }; \end{aligned}$$

• 2. $M_P = lfp(T_P) = T_P\uparrow ^{\omega }$.

1.2 Examples of Meaning Computation

In this subsection we show the computation of the meaning of sentences in Examples 1–9.

$\mathbf {1.}$ Translation of the DD-tree $D_1$ of the sentence

$$\begin{aligned} {\texttt {\textit{The morning star shines by reflected light}}} \end{aligned}$$

shown in Fig. 3.

• 1. $\varDelta (r_3) =_{(4.22)} \varLambda (r_3) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in {\textsc {morning}}^{\prime })]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_1), (i_1 \in {\textsc {morning}}^{\prime })]$.

• 2. $\varDelta (r_2) =_{(4.23)} \varLambda (r_2) \circ _{a} \varDelta (r_3)=_{(4.13)} ((\lambda \, \xi \, x\, Z_1\, .\, [oi(x), (x \in {\textsc {star}}^{\prime }), {\textsc {APPEL}}(x)=oi(Z_1),$

        ${\textsc {NUM}}(x)={\textsc {SG}}, Z_1]\ \varXi ) \ \mathbf{newoi}) \circ _{a} \varDelta (r_3)$

        $= \lambda \, Z_1\, .\, [oi(i_2), (i_2 \in {\textsc {star}}^{\prime }), {\textsc {APPEL}}(i_2)=oi(Z_1), {\textsc {NUM}}(i_2)={\textsc {SG}}, Z_1] \circ _{a} \varDelta (r_3)$

        $=_{(4.16)} [oi(i_2), (i_2 \in {\textsc {star}}^{\prime }), {\textsc {APPEL}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}}, [oi(i_1), (i_1 \in {\textsc {morning}}^{\prime })]]$.

• 3. $\varDelta (r_5) =_{(4.22)} \varLambda (r_5) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in {\textsc {reflected}}^{\prime })]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_3), (i_3 \in {\textsc {reflected}}^{\prime })]$.

• 4. $\varDelta (r_4) =_{(4.23)} \varLambda (r_4) \circ _{a} \varDelta (r_5)=_{(4.5)} ((\lambda \, \xi \, x\, Z_1\, .\, [oi(x), (x \in {\textsc {light}}^{\prime }), {\textsc {NATURE}}(x)=$

  $oi(Z_1),{\textsc {COUNT}}(x)= -, Z_1]\ \varXi ) \ \mathbf{newoi}) \circ _{a} \varDelta (r_5)$

        $= \lambda \, Z_1\, .\, [oi(i_4), (i_4 \in {\textsc {light}}^{\prime }), {\textsc {NATURE}}(i_4)=oi(Z_1), {\textsc {COUNT}}(i_4)= -, Z_1] \circ _{a} \varDelta (r_5)$

  $=_{(4.16)} [oi(i_4), (i_4 \in {\textsc {light}}^{\prime }), {\textsc {NATURE}}(i_4)=i_3, {\textsc {COUNT}}(i_4)= -, [oi(i_3),$

  $(i_3 \in {\textsc {reflected}}^{\prime })]]$.

• 5. $\varDelta (D_1) = \varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2),\varDelta (r_4))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, Z_1\, .\, [oi(x), (x \in {\textsc {shine}}^{\prime }), ({\textsc {shine}}(oi(U_1)) \in x), dd(oi(U_1)) = \mathbf{D}\, ,$

  ${\textsc {MEANS}}(x) = oi(Z_1), dd(oi(Z_1)) = \mathbf{G}\,, {\textsc {A/T}}(x) = {\textsc {DURAT/PRES}}, {\textsc {M}}(x) = {\textsc {IND}}, U_1, Z_1] \varXi )\ \mathbf{newoi})$

        $\circ _{a} (\varDelta (r_2), \varDelta (r_4))$

  $=_{(4.16)} \lambda \, U_1\, Z_1\, .\, [oi(i_5), (i_5 \in {\textsc {shine}}^{\prime }), ({\textsc {shine}}(oi(U_1)) \in i_1), dd(oi(U_1)) = \mathbf{D}\, , {\textsc {MEANS}}(i_5) =$

        $oi(Z_1), dd(oi(Z_1)) = \mathbf{G}\,, {\textsc {A/T}}(i_5) = {\textsc {DURAT/PRES}}, {\textsc {M}}(i_5) = {\textsc {IND}}, U_1, Z_1] \circ _{a} (\varDelta (r_2), \varDelta (r_4))$

        $= [oi(i_5), (i_5 \in {\textsc {shine}}^{\prime }), ({\textsc {shine}}(i_2) \in i_5), dd(i_2) = \mathbf{D}\, , {\textsc {MEANS}}(i_5) = i_4, {\textsc {A/T}}(i_5) = {\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_5) = {\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {star}}^{\prime }), {\textsc {APPEL}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}},$

        $[oi(i_1), (i_1 \in {\textsc {morning}}^{\prime })]], [oi(i_4), (i_4 \in {\textsc {light}}^{\prime }), {\textsc {NATURE}}(i_4)=i_3, {\textsc {COUNT}}(i_4)= -,$

        $[oi(i_3), (i_3 \in {\textsc {reflected}}^{\prime })]] ]$.

$\mathbf {2.}$ Translation of the DD-tree $D_2$ of the sentence

$$\begin{aligned}&{\texttt {\textit{The Persians were definitely defeated by Greeks near}}}\\&{\texttt {\textit{Plataea}}} \end{aligned}$$

shown in Fig. 5.

• 1.

        $=+]\ \varXi ) \ \mathbf{newoi}) = [oi(i_1), (i_1 \in {\textsc {greek}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {PL}}, {\textsc {HUMAN}}(i_1)=+]$.

• 2.

        $=+]\ \varXi ) \ \mathbf{newoi}) = [oi(i_2), (i_2 \in {\textsc {persian}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {PL}}, {\textsc {HUMAN}}(i_2)=+]$.

• 3. $\varDelta (r_6) =_{(4.22)} \varLambda (r_6) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in {\textsc {near}}^{\prime }) ]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_3), (i_3 \in {\textsc {near}}^{\prime })]$.

• 4. $\varDelta (r_4) =_{(4.23)} \varLambda (r_4) \circ _{a} \varDelta (r_6)$

        $=_{(4.5)} ((\lambda \, \xi \, x\, Z_1\, .\, [oi(x), (x \in {\textsc {plataea}}^{\prime }), {\textsc {DIST}}(x) = oi(Z_1),{\textsc {APPEL}}(x)= {\textsc {PLATAEA}}, Z_1]$

        $\varXi ) \ \mathbf{newoi}) \circ _{a} \varDelta (r_6)$

        $= \lambda \, Z_1\, .\, [oi(i_4), (i_4 \in {\textsc {plataea}}^{\prime }), {\textsc {DIST}}(i_4)=oi(Z_1), {\textsc {APPEL}}(i_4)= {\textsc {PLATAEA}}, Z_1] \circ _{a} \varDelta (r_6)$

        $=_{(4.16)} [oi(i_4), (i_4 \in {\textsc {plataea}}^{\prime }), {\textsc {DIST}}(i_4)=i_3, {\textsc {APPEL}}(i_4)\!= {\textsc {PLATAEA}}, [oi(i_3), (i_3\! \in {\textsc {near}}^{\prime })]]$.

• 5. $\varDelta (r_5) =_{(4.22)} \varLambda (r_5) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in {\textsc {definitely}}^{\prime }) ]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_5), (i_5 \in {\textsc {definitely}}^{\prime })]$.

• 6. $\varDelta (D_2) = \varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2),\varDelta (r_3),\varDelta (r_4),\varDelta (r_5))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, Z_1\, Z_2\, .\, [oi(x), (x \in {\textsc {defeat}}^{\prime }), ({\textsc {defeat}}(oi(U_1),oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{G}\,, dd(oi(U_2)) = \mathbf{G}\,, {\textsc {LOC}}(x) = oi(Z_1)$, ${\textsc {MANNER}}(x) = oi(Z_2)$,

        ${\textsc {A/T}}(x) = {\textsc {POINT/PAST}}, {\textsc {M}}(x) = {\textsc {IND}}, {\textsc {DTH}}(x) = {\textsc {PASS}}, U_1, U_2, Z_1, Z_2]$

        $\varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_2),\varDelta (r_3),\varDelta (r_4),\varDelta (r_5))$

        $=_{(4.16)} \lambda \, U_1\, U_2\, Z_1\, Z_2\, .\, [oi(i_6), (i_6 \in {\textsc {defeat}}^{\prime }), ({\textsc {defeat}}(oi(U_1), oi(U_2)) \in i_6),$

        $dd(oi(U_1)) = \mathbf{G}\,, dd(oi(U_2)) = \mathbf{G}\,, {\textsc {LOC}}(i_6) = oi(Z_1)$, ${\textsc {MANNER}}(i_6) = oi(Z_2)$,

        ${\textsc {A/T}}(i_6) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_6) = {\textsc {IND}}, {\textsc {DTH}}(i_6) = {\textsc {PASS}}, U_1, U_2, Z_1, Z_2]$

        $\circ _{a}\ (\varDelta (r_2),\varDelta (r_3),\varDelta (r_4),\varDelta (r_5))$

        $= [oi(i_6), (i_6 \in {\textsc {defeat}}^{\prime }), ({\textsc {defeat}}(i_1,i_2) \in i_6), dd(i_1) = \mathbf{G}\,, dd(i_2) = \mathbf{G}\,,$

        ${\textsc {LOC}}(i_6) = i_4, {\textsc {MANNER}}(i_6) = i_5, {\textsc {A/T}}(i_6) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_6) = {\textsc {IND}},$

        ${\textsc {DTH}}(i_6) = {\textsc {PASS}}, [oi(i_1), (i_1 \in {\textsc {greek}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {PL}}, {\textsc {HUMAN}}(i_1)=+], [oi(i_2),$

        $(i_2 \in {\textsc {persian}}^{\prime }), {\textsc {NUM}}(i_2) ={\textsc {PL}}, {\textsc {HUMAN}}(i_2)=+], [oi(i_4), (i_4 \in {\textsc {plataea}}^{\prime }),$

        ${\textsc {DIST}}(i_4)=i_3, {\textsc {APPEL}}(i_4)= {\textsc {PLATAEA}}, [oi(i_3), (i_3 \in {\textsc {near}}^{\prime })]], [oi(i_5), (i_5 \in {\textsc {definitely}}^{\prime })] ]$.

$\mathbf {3.}$ Translation of the DD-tree $D_3$ of the sentence

$$\begin{aligned} {\texttt {\textit{Their son is a smart guy}}} \end{aligned}$$

shown in Fig. 7.

• 1. $\varDelta (r_4) =_{(4.22)} \varLambda (\mathbf{coref}~l_{they}) =_{(4.4)} ((\lambda \, \xi \, x\, .\, [oi( x ),\ (x \equiv \xi ( l_{they} ))]\ \varXi ) \mathbf{newoi})$

        $= [oi( i_1),\ (i_1 \equiv i_{they})]$, where $i_{they} = \varXi ( l_{they} )$.

• 2. $\varDelta (r_2) =_{(4.23)} \varLambda (r_2) \circ _{a} \varDelta (r_4)$

        $=_{(4.13)} ((\lambda \, \xi \, x\, Z\, .\, [oi(x), (x \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(x)=oi(Z), {\textsc {NUM}}(x)={\textsc {SG}}, {\textsc {HUMAN}}(x)=+, Z]$

        $\varXi ) \ \mathbf{newoi}) \circ _{a} \varDelta (r_4)$

        $= \lambda \, Z\, .\, [oi(i_1), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=oi(Z), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+, Z]$

        $\circ _{a}\, \varDelta (r_4)$

        $=_{(4.16)} [oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+, [oi( i_1),$

        $(i_1 \equiv i_{they})]]$.

• 3. $\varDelta (r_6) =_{(4.22)} \varLambda (r_6) =_{(4.5)} ((\lambda \ \xi \ x\, .\ [oi(x),\ (x \in \xi ({\textsc {neutr}}))]\ \varXi )\ \mathbf{newoi})$

        $= [oi(i_3),\ (i_3 \in {\textsc {neutr}}^{\prime })]$.

• 4. $\varDelta (r_5) =_{(4.23)} \varLambda (r_5) \circ _{a} \varDelta (r_6)$

        $=_{(4.14)} ((\lambda \ \xi \ x\ Z\ y\, . [oi( x ),\ (x \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(y, oi(Z)) \in x), {\textsc {MAGN}}(x) = oi(Z), Z]$

        $\varXi )\ \mathbf{newoi}) \circ _{a} \varDelta (r_6)$

        $= \lambda \ Z\ y\, . [oi( i_4 ),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(y, oi(Z)) \in i_4), {\textsc {MAGN}}(i_4) = oi(Z), Z]\ \varXi )$

        $\mathbf{newoi}) \circ _{a} \varDelta (r_6)$

        $= \lambda \ y\, . [oi(i_4),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(y, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3, [oi(i_3),$

        $(i_3 \in {\textsc {neutr}}^{\prime })]]$.

• 5. $\varDelta (r_3) =_{(4.23)} \varLambda (r_3) \circ _{a} \varDelta (r_5)$

        $=_{(4.13)} ((\lambda \ \xi \ x\ Y\, . [oi( x ),\ (x \in {\textsc {guy}}^{\prime }), (Y\ x), {\textsc {NUM}}(x)={\textsc {SG}}, {\textsc {HUMAN}}(x)=+ ]\ \varXi )\ \mathbf{newoi})$

        $\circ _{a} \varDelta (r_5)$

        $= \lambda \ Y\, .\, [oi( i_5 ),\ (i_5 \in {\textsc {guy}}^{\prime }), (Y\ i_5), {\textsc {NUM}}(i_5)={\textsc {SG}}, {\textsc {HUMAN}}(i_5)=+ ] \circ _{a} \varDelta (r_5)$

        $= [oi( i_5 ),\ (i_5 \in {\textsc {guy}}^{\prime }), [oi(i_4),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(i_5, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3,$

        $[oi(i_3), (i_3 \in {\textsc {neutr}}^{\prime })]], {\textsc {NUM}}(i_5)={\textsc {SG}}, {\textsc {HUMAN}}(i_5)=+ ]$.

• 6. $\varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2), \varDelta (r_3))$

        $=_{(4.10)} ((\lambda \ \xi \ x\ U_1\ U_2\, . [oi( x ),\ (x \in {\textsc {be-a}}^{\prime }),\, ({\textsc {be-a}}(oi(U_1), oi(U_2)) \in x),$

        $(oi(U_1) \in oi(U_2)), dd(oi(U_1)) = \mathbf{D}\, ,\ dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(x)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(x)={\textsc {IND}}, U_1,U_2]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_2), \varDelta (r_3))$

        $= \lambda \ U_1\ U_2\, . [oi( i_6 ),\ (i_6 \in {\textsc {be-a}}^{\prime }),\, ({\textsc {be-a}}(oi(U_1), oi(U_2)) \in i_6), (oi(U_1) \in oi(U_2)),$

        $dd(oi(U_1)) = \mathbf{D}\, ,\ dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(i_6)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_6)={\textsc {IND}}, U_1,U_2]$

        $\circ _{a} (\varDelta (r_2), \varDelta (r_3))$

        $= [oi( i_6 ),\ (i_6 \in {\textsc {be-a}}^{\prime }),\, ({\textsc {be-a}}(i_2, i_5) \in i_6), (i_2 \in i_5), dd(i_2) = \mathbf{D}\, ,\ dd(i_5) = \mathbf{E}\,,$

        ${\textsc {A/T}}(i_6)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_6)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1,$

        ${\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+, [oi( i_1),\ (i_1 \equiv i_{they})]], [oi( i_5 ),\ (i_5 \in {\textsc {guy}}^{\prime }), [oi(i_4),$

        $(i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(i_5, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3, [oi(i_3), (i_3 \in {\textsc {neutr}}^{\prime })]],$

        ${\textsc {NUM}}(i_5)={\textsc {SG}}, {\textsc {HUMAN}}(i_5)=+ ]]$.

$\mathbf {4.}$ Translation of the DD-tree $D_4$ of the sentence

$$\begin{aligned} {\texttt {\textit{Their son is very smart}}} \end{aligned}$$

shown in Fig. 9.

• 1. $\varDelta (r_4) =_{Example~3} [oi( i_1),\ (i_1 \equiv i_{they})]$, where $i_{they} = \varXi ( l_{they} )$.

• 2. $\varDelta (r_2) =_{Example~3} [oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+, [oi( i_1), (i_1 \equiv i_{they})]]$.

• 3. $\varDelta (r_5) =_{(4.22)} \varLambda (r_5) =_{(4.5)} ((\lambda \ \xi \ x\, .\ [oi(x),\ (x \in (\xi \ {\textsc {very}}))]\ \varXi )\ \mathbf{newoi})$

        $= [oi(i_3),\ (i_3 \in {\textsc {very}}^{\prime })]$.

• 4. $\varDelta (r_3) =_{Example~3} \lambda \ y\, . [oi(i_4),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(y, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3,$

        $[oi(i_3),\ (i_3 \in {\textsc {very}}^{\prime })]]$.

• 5. $\varDelta (r_1) =_{(4.24)} \varLambda (r_1) \circ _{pa} (\varDelta (r_2), \varDelta (r_3)) =_{(4.17)} (\varLambda (r_1)\, \varDelta (r_2)\, (\varDelta (r_3)\, oi(\varDelta (r_2))) )$

        $=_{(4.11)} (((\lambda \ \xi \ x\ U_1\ U_2\, . [oi( x ),\ (x \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(x)={\textsc {DURAT/PRES}}, {\textsc {M}}(x)={\textsc {IND}}, U_1, U_2 ]\ \varXi )\ \mathbf{newoi})\, \varDelta (r_2)$

        $(\varDelta (r_3)\, \varDelta (r_2)))$

        $= ( \lambda \ U_1\ U_2\, . [oi( i_5 ),\ (i_5 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in i_5),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(i_5)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_5)={\textsc {IND}}, U_1, U_2 ],\ \varDelta (r_2)\, (\varDelta (r_3)\, oi(\varDelta (r_2))) )$

        $= [oi( i_5 ),\ (i_5 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_4) \in i_5), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_5)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_5)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+, [oi( i_1),$

        $(i_1 \equiv i_{they})] ], (\lambda \ y\, . [oi(i_4),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(y, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3, [oi(i_3),$

        $(i_3 \in {\textsc {very}}^{\prime })]]\ oi([oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+, [oi( i_1), (i_1 \equiv i_{they})] ]) )]$

        $= [oi( i_5 ),\ (i_5 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_4) \in i_5), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_5)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_5)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {son}}^{\prime }), {\textsc {APPRT}}(i_2)=i_1, {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+, [oi( i_1),$

        $(i_1 \equiv i_{they})] ], [oi(i_4),\ (i_4 \in {\textsc {smart}}^{\prime }), ({\textsc {smart}}(i_2, i_3) \in i_4), {\textsc {MAGN}}(i_4) = i_3, [oi(i_3),$

        $(i_3 \in {\textsc {very}}^{\prime })]] ]$.

$\mathbf {5.}$ Translation of the DD-tree $D_5$ of the sentence

$$\begin{aligned} {\texttt {\textit{I met Nick and Jeff, both from a neighbouring village}}} \end{aligned}$$

shown in Fig. 11.

• 1. $\varDelta (r_2) =_{Examples~3,4} [oi( i_0),\ (i_0 \equiv i_{speaker}), {\textsc {HUMAN}}(i_0)=+]$, where $i_{speaker} = \varXi ( l_{speaker} )$.

• 2. $\varDelta (r_8) =_{(4.22)} \varLambda (r_8) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in ({\textsc {jeff}}^{\prime })), {\textsc {NUM}}(x)={\textsc {SG}}, {\textsc {HUMAN}}(x)=+]$

        $\varXi )\ \mathbf{newoi})$

        $= [oi(i_1), (i_1 \in {\textsc {jeff}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]$.

  Side effect: $\varXi (l_{jeff}) = i_1$ (see Remark 4).

• 3. $\varDelta (r_5) =_{(4.23)} \varLambda (r_5) \circ _{a} \varDelta (r_8)$

        

        $\varXi )\ \mathbf{newoi}) \circ _{a} \varDelta (r_8)$

        

        $[oi(i_1), (i_1 \in {\textsc {jeff}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]$

        

        ${\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]]$.

  Side effect: $\varXi (l_{nick}) = i_2$.

• 4. $\varDelta (r_3) =_{(4.23)} \varLambda (r_3) \circ _{a} \varDelta (r_5)$

        $=_{(4.13)} ((\lambda \ \xi \ x\ Z\, .\, [oi( x ),\ {\eqcirc }(x) = oi(Z), Z]\ \varXi )\ \mathbf{newoi}) \circ _{a} \varDelta (r_8)$

        

        ${\textsc {HUMAN}}(i_2)=+, [oi(i_1), (i_1 \in {\textsc {jeff}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]] ]$

  Side effect: $\varXi (l_{both}) = i_3$.

• 5. $\varDelta (r_6) =_{Examples~3,4} [oi( i_4),\ (i_4 \equiv i_3)]$.

• 6. $\varDelta (r_9) =_{(4.5)} [oi(i_5),\ (i_5 \in {\textsc {neighbouring}}^{\prime })]$.

• 7. $\varDelta (r_7) =_{Example~2} [oi(i_6), (i_6 \in {\textsc {village}}^{\prime }), {\textsc {DIST}}(i_6)=i_5, [oi(i_5), (i_5 \in {\textsc {neighbouring}}^{\prime })]]$.

  At the same time, $\varXi (l_v) = \varXi (r_7) = i_6$ due to the side effect defined by (4.7).

• 8. $\varDelta (r_4) =_{(4.23)} \varLambda (r_4) \circ _{a} (\varDelta (r_6), \varDelta (r_7))$

        $=_{(4.12)} ((\lambda \ \xi \ x\ U\ Z\, . [oi( x ),\ (x \in {\textsc {be-attr}}^{\prime }),\, ({\textsc {be-attr}}(oi(U),oi(Z)) \in x),$

        ${\textsc {LOC}}(oi(U)) = oi(Z), dd(oi(U)) = \mathbf{G}\,, dd(oi(Z)) = \mathbf{E}\,, {\textsc {A/T}}(x)={\textsc {DURAT/PAST}},$

        ${\textsc {M}}(x)={\textsc {IND}}, U, Z]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_6), \varDelta (r_7))$

        $=\lambda \ U\ Z\, . [oi( i_7 ),\ (i_7 \in {\textsc {be-attr}}^{\prime }),\, ({\textsc {be-attr}}(oi(U),oi(Z)) \in i_7), {\textsc {LOC}}(oi(U)) = oi(Z),$

        $dd(oi(U)) = \mathbf{G}\,, dd(oi(Z)) = \mathbf{E}\,, {\textsc {A/T}}(i_7)={\textsc {DURAT/PAST}}, {\textsc {M}}(i_7)={\textsc {IND}}, U, Z]$

        $\circ _{a} (\varDelta (r_6), \varDelta (r_7))$

        $= [oi( i_7 ),\ (i_7 \in {\textsc {be-attr}}^{\prime }),\, ({\textsc {be-attr}}(i_4,i_6) \in i_7), {\textsc {LOC}}(i_4) = i_6, dd(i_4) = \mathbf{G}\,,$

        $dd(i_6) = \mathbf{E}\,, {\textsc {A/T}}(i_7)={\textsc {DURAT/PAST}}, {\textsc {M}}(i_7)={\textsc {IND}}, [oi( i_4),\ (i_4 \equiv i_3)],$

        $[oi(i_6), (i_6 \in {\textsc {village}}^{\prime }), {\textsc {DIST}}(i_6)=i_5, [oi(i_5), (i_5 \in {\textsc {neighbouring}}^{\prime })]] ]$.

• 9. $\varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2), \varDelta (r_3), \varDelta (r_4))$

        $=_{(4.6)} ((\lambda \ \xi \ x\ U_1\ U_2\ Z\, .\, [oi( x ),\ (x \in {\textsc {meet}}^{\prime }), ({\textsc {meet}}(oi(U_1), oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , dd(oi(U_2)) = \mathbf{E}\,, \mathbf{and}(x) = oi(Z), {\textsc {A/T}}(x) = {\textsc {POINT/PAST}},$

        ${\textsc {M}}(x)={\textsc {IND}}, U_1, U_2, Z]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_2), \varDelta (r_3), \varDelta (r_4))$

        $= ((\lambda \, U_1\ U_2\ Z\, .\, [oi( i_8 ),\ (i_8 \in {\textsc {meet}}^{\prime }), ({\textsc {meet}}(oi(U_1), oi(U_2)) \in i_8), dd(oi(U_1)) = \mathbf{D}\, ,$

        $dd(oi(U_2)) = \mathbf{E}\,, \mathbf{and}(i_8) = oi(Z), {\textsc {A/T}}(i_8) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_8)={\textsc {IND}}, U_1, U_2, Z]$

        $\circ _{a} (\varDelta (r_2), \varDelta (r_3), \varDelta (r_4))$

        $= [oi( i_8 ),\ (i_8 \in {\textsc {meet}}^{\prime }), ({\textsc {meet}}(i_0, i_3) \in i_8), dd(i_0) = \mathbf{D}\, , dd(i_3) = \mathbf{E}\,, \mathbf{and}(i_8) = i_7,$

        ${\textsc {A/T}}(i_8) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_8)={\textsc {IND}}, [oi( i_0),\ (i_0 \equiv i_{speaker}), {\textsc {HUMAN}}(i_0)=+],$

        

        ${\textsc {HUMAN}}(i_2)=+, [oi(i_1), (i_1 \in {\textsc {jeff}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]] ], [oi( i_7 ),$

        $(i_7 \in {\textsc {be-attr}}^{\prime }),\, ({\textsc {be-attr}}(i_4,i_6) \in i_7), {\textsc {LOC}}(i_4) = i_6, dd(i_4) = \mathbf{G}\,, dd(i_6) = \mathbf{E}\,,$

        ${\textsc {A/T}}(i_7)={\textsc {DURAT/PAST}}, {\textsc {M}}(i_7)={\textsc {IND}}, [oi( i_4), (i_4 \equiv i_3)], [oi(i_6), (i_6 \in {\textsc {village}}^{\prime }),$

        ${\textsc {DIST}}(i_6)=i_5, [oi(i_5), (i_5 \in {\textsc {neighbouring}}^{\prime })]]] ]$.

$\mathbf {6.}$ Translation of the DD-tree $D_6$ of the sentence

$$\begin{aligned}&{\texttt {\textit{Every farmer in the village, having tractors, shares}}}\\&{\texttt {\textit{ them with neighbours}}} \end{aligned}$$

shown in Fig. 13.

• 1. $\varDelta (r_{17}) =_{(4.2)} \lambda \ X .\ [oi(X)]$.

• 2. $\varDelta (r_{18}) =_{(4.22)} \varLambda (r_{18}) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in {\textsc {tractor}}^{\prime }), {\textsc {NUM}}(x) = {\textsc {PL}},$

        ${\textsc {ANIM}}(x) = -]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_9), (i_9 \in {\textsc {tractor}}^{\prime }), {\textsc {NUM}}(i_9) = {\textsc {PL}}, {\textsc {ANIM}}(i_9) = -]$.

  Side effect: $\varXi (l_t) = i_9$ (see Example (5)).

• 3. $\varDelta (r_{14}) =_{(4.25)} \varLambda (r_{14}) \circ _{r 1} (\varDelta (r_{17}),\varDelta (r_{18}))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi(x), (x \in {\textsc {have}}^{\prime }), ({\textsc {have}}(oi(U_1),oi(U_2)) \in x), dd(oi(U_2)) = \mathbf{E}\,,$

        ${\textsc {A/T}}(x) = {\textsc {PROGR/PRES}}, {\textsc {M}}(x) = {\textsc {IND}}, U_1, U_2]\ \varXi )\ \mathbf{newoi})$ $\circ _{r 1} (\varDelta (r_{17}),\varDelta (r_{18}))$

        $= \lambda \, U_1\, U_2\, .\, [oi(i_{10}), (i_{10} \in {\textsc {have}}^{\prime }), ({\textsc {have}}(oi(U_1),oi(U_2)) \in i_{10}), dd(oi(U_2)) = \mathbf{E}\,,$

        ${\textsc {A/T}}(i_{10}) = {\textsc {PROGR/PRES}}, {\textsc {M}}(i_{10}) = {\textsc {IND}}, U_1, U_2] \circ _{r 1} (\varDelta (r_{17}),\varDelta (r_{18}))$

        $=_{(4.18)} ( \lambda \, U_1\, U_2\, y\, .\, [oi(i_{10}), (i_{10} \in {\textsc {have}}^{\prime }), ({\textsc {have}}(oi(U_1),oi(U_2)) \in i_{10}),$

        $dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(i_{10}) = {\textsc {PROGR/PRES}}, {\textsc {M}}(i_{10}) = {\textsc {IND}}, U_1, U_2]\ (\varDelta (r_{17})\ y)\ \varDelta (r_{18}) )$

        $= \lambda \, y\, .\, [oi(i_{10}), (i_{10} \in {\textsc {have}}^{\prime }), ({\textsc {have}}(y,i_9) \in i_{10}), dd(i_9) = \mathbf{E}\,, {\textsc {A/T}}(i_{10}) = {\textsc {PROGR/PRES}},$

        ${\textsc {M}}(i_{10}) = {\textsc {IND}}, [oi(y)], [oi(i_9), (i_9 \in {\textsc {tractor}}^{\prime }), {\textsc {NUM}}(i_9) = {\textsc {PL}}, {\textsc {ANIM}}(i_9) = -] ]$

• 4. $\varDelta (r_{15}) =_{(4.22)} \varLambda (r_{15}) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \equiv i_6), (x \in {\textsc {village}}^{\prime }), {\textsc {NUM}}(x) = {\textsc {SG}},$

        ${\textsc {ANIM}}(x) = -]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_{11}), (i_{11} \equiv i_6), (i_{11}\in {\textsc {village}}^{\prime }), {\textsc {NUM}}(i_{11}) = {\textsc {SG}}, {\textsc {ANIM}}(i_{11}) = -]$.

• 5. $\varDelta (r_{11}) =_{(4.23)} \varLambda (r_{11}) \circ _{a} (\varDelta (r_{14}), \varDelta (r_{15}))$

        $=_{(4.13)} ((\lambda \ \xi \ x\ R\ Z . [oi( x ),\ (x \in {\textsc {farmer}}^{\prime }), (R\ x), {\textsc {LOC}}(x) = oi(Z), {\textsc {NUM}}(x) = {\textsc {PL}},$

        ${\textsc {HUMAN}}(x) = +, Z]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_{14}), \varDelta (r_{15}))$

        $= \lambda \ R\ Z . [oi( i_{12} ),\ (i_{12} \in {\textsc {farmer}}^{\prime }), (R\ i_{12}), {\textsc {LOC}}(x) = oi(Z), {\textsc {NUM}}(i_{12}) = {\textsc {PL}},$

        ${\textsc {HUMAN}}(i_{12}) = +, Z]\ \circ _{a} (\varDelta (r_{14}), \varDelta (r_{15}))$

        $= [oi( i_{12} ),\ (i_{12} \in {\textsc {farmer}}^{\prime }), [oi(i_{10}), (i_{10} \in {\textsc {have}}^{\prime }), ({\textsc {have}}(i_{12},i_9) \in i_{10}),$

        $dd(i_9) = \mathbf{E}\,, {\textsc {A/T}}(i_{10}) = {\textsc {PROGR/PRES}}, {\textsc {M}}(i_{10}) = {\textsc {IND}}, [oi(i_{12})], [oi(i_9),$

        $(i_9 \in {\textsc {tractor}}^{\prime }), {\textsc {NUM}}(i_9) = {\textsc {PL}}, {\textsc {ANIM}}(i_9) = -] ], {\textsc {LOC}}(i_{12}) = i_{11}, {\textsc {NUM}}(i_{12}) = {\textsc {PL}},$

        ${\textsc {HUMAN}}(i_{12}) = +, [oi(i_{11}), (i_{11} \equiv i_6), (i_{11} \in {\textsc {village}}^{\prime }), {\textsc {NUM}}(i_{11}) = {\textsc {SG}}, {\textsc {ANIM}}(i_{11}) = -]]$.

  Side effect: $\varXi (l_f) = i_{12}$.

• 6. $\varDelta (r_{12}) =_{Example~3} = [oi(i_{13}), (i_{13} \equiv i_9)]$.

• 7. $\varDelta (r_{16}) = [oi(i_{14}), (i_{14} \equiv i_{12})]$.

• 8. $\varDelta (r_{13}) =_{(4.23)} \varLambda (r_{13}) \circ _{a} \varDelta (r_{16}) =_{(4.13)}$

        $((\lambda \, \xi \, x\, Z\, .\, [oi(x), (x \in {\textsc {neighbour}}^{\prime }), {\textsc {APPRT}}(x)=oi(Z), {\textsc {NUM}}(x)={\textsc {PL}},{\textsc {HUMAN}}(x)=+, Z]$

        $\varXi )\ \mathbf{newoi})\ \circ _{a} \varDelta (r_{16})$

        $= \lambda \, Z\, .\, [oi(i_{15}), (i_{15} \in {\textsc {neighbour}}^{\prime }), {\textsc {APPRT}}(i_{15})=oi(Z), {\textsc {NUM}}(i_{15})={\textsc {PL}},$

        ${\textsc {HUMAN}}(i_{15})=+, Z] \circ _{a}\ \varDelta (r_{16})$

        $= [oi(i_{15}), (i_{15} \in {\textsc {neighbour}}^{\prime }), {\textsc {APPRT}}(i_{15})=i_{14}, {\textsc {NUM}}(i_{15})={\textsc {PL}},{\textsc {HUMAN}}(i_{15})=+, [oi(i_{14}),$

        $(i_{14} \equiv i_{12})]]$.

• 9. $\varDelta (r_{10}) =_{(4.23)} \varLambda (r_{10}) \circ _{a} (\varDelta (r_{11}),\varDelta (r_{12}),\varDelta (r_{13}))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, U_3\, .\, [oi(x), (x \in {\textsc {share}}^{\prime }), ({\textsc {share}}(oi(U_1),oi(U_2),oi(U_3)) \in x),$

        $dd(oi(U_1)) = \mathbf{U}\, , dd(oi(U_2)) = \mathbf{E}\,, dd(oi(U_3)) = \mathbf{E}\,, {\textsc {A/T}}(x) = {\textsc {DURAT/PRES}},$

        ${\textsc {M}}(x) = {\textsc {IND}}, U_1, U_2, U_3]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_{11}),\varDelta (r_{12}),\varDelta (r_{13}))$

        $= \lambda \, U_1\, U_2\, U_3\, .\, [oi(i_{16}), (i_{16} \in {\textsc {share}}^{\prime }), ({\textsc {share}}(oi(U_1),oi(U_2),oi(U_3)) \in i_{16}),$

        $dd(oi(U_1)) = \mathbf{U}\, , dd(oi(U_2)) = \mathbf{E}\,, dd(oi(U_3)) = \mathbf{E}\,, {\textsc {A/T}}(i_{16}) = {\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_{16}) = {\textsc {IND}}, U_1, U_2, U_3]\ \circ _{a} (\varDelta (r_{11}),\varDelta (r_{12}),\varDelta (r_{13}))$

        $= [oi(i_{16}), (i_{16} \in {\textsc {share}}^{\prime }), ({\textsc {share}}(i_{12},i_{13},i_{15}) \in i_{16}), dd(i_{12}) = \mathbf{U}\, , dd(i_{13}) = \mathbf{E}\,,$

        $dd(i_{15}) = \mathbf{E}\,, {\textsc {A/T}}(i_{16}) = {\textsc {DURAT/PRES}}, {\textsc {M}}(i_{16}) = {\textsc {IND}}, [oi( i_{12} ),\ (i_{12} \in {\textsc {farmer}}^{\prime }), [oi(i_{10}),$

        $(i_{10} \in {\textsc {have}}^{\prime }), ({\textsc {have}}(i_{12},i_9) \in i_{10}), dd(i_9) = \mathbf{E}\,, {\textsc {A/T}}(i_{10}) = {\textsc {PROGR/PRES}}, {\textsc {M}}(i_{10}) = {\textsc {IND}},$

        $[oi(i_{12})], [oi(i_9), (i_9 \in {\textsc {tractor}}^{\prime }), {\textsc {NUM}}(i_9) = {\textsc {PL}}, {\textsc {ANIM}}(i_9) = -] ], {\textsc {LOC}}(i_{12}) = i_{11},$

        ${\textsc {NUM}}(i_{12}) = {\textsc {PL}}, {\textsc {HUMAN}}(i_{12}) = +, [oi(i_{11}), (i_{11} \equiv i_6), (i_{11} \in {\textsc {village}}^{\prime }), {\textsc {NUM}}(i_{11}) = {\textsc {SG}},$

        ${\textsc {ANIM}}(i_{11}) = -]], [oi(i_{13}), (i_{13} \equiv i_9)], [oi(i_{15}), (i_{15} \in {\textsc {neighbour}}^{\prime }), {\textsc {APPRT}}(i_{15})=i_{14},$

        ${\textsc {NUM}}(i_{14})={\textsc {PL}}, {\textsc {HUMAN}}(i_{14})=+, [oi(i_{14}), (i_{14} \equiv i_{12})]] ]$.

$\mathbf {7.}$ Translation of the DD-tree $D_7$ of the sentence

shown in Fig. 15.

• 1. $\varDelta (r_{20}) =_{(4.22)} \varLambda (r_{20}) =_{(4.5)} [oi(i_{17}), (i_{17} \equiv i_{nick})]$.

• 2. $\varDelta (r_{23}) =_{Examples~3,4} [oi( i_{18}),\ (i_{18} \equiv i_{nick})]$.

• 3. $\varDelta (r_{21}) =_{(4.23)} \varLambda (r_{21}) \circ _{a} \varDelta (r_{23}) =_{(4.13)} ((\lambda \, \xi \, x\, Z\, .\, [oi(x), (x \in {\textsc {caterpillar}}^{\prime }),$

        ${\textsc {APPRT}}(x)=oi(Z), {\textsc {NUM}}(x)={\textsc {SG}},{\textsc {HUMAN}}(x)=-, Z]\ \varXi )\ \mathbf{newoi})\ \circ _{a} \varDelta (r_{23})$

        $= \lambda \, Z\, .\, [oi(i_{19}), (i_{19} \in {\textsc {caterpillar}}^{\prime }), {\textsc {APPRT}}(i_{19})=oi(Z), {\textsc {NUM}}(i_{19})={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_{19})=-, Z] \circ _{a}\ \varDelta (r_{23})$

        $= [oi(i_{19}), (i_{19} \in {\textsc {caterpillar}}^{\prime }), {\textsc {APPRT}}(i_{19})=i_{18}, {\textsc {NUM}}(i_{19})={\textsc {SG}},{\textsc {HUMAN}}(i_{19})=-,$

        $[oi( i_{18}),\ (i_{18} \equiv i_{nick})]]$.

• 4. $\varDelta (r_{22}) =_{(4.22)} \varLambda (r_{22}) =_{(4.5)} [oi(i_{20}), (i_{20} \equiv i_{jeff})]$.

• 5. $\varDelta (r_{19}) =_{(4.23)} \varLambda (r_{19}) \circ _{a} (\varDelta (r_{20}),\varDelta (r_{21}),\varDelta (r_{22}))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, U_3\, .\, [oi(x), (x \in {\textsc {share}}^{\prime }), (\sim {\textsc {share}}(oi(U_1),oi(U_2),oi(U_3)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , dd(oi(U_2)) = \mathbf{E}\,, dd(oi(U_3)) = \mathbf{D}\, , {\textsc {A/T}}(x) = {\textsc {DURAT/PRES}},$

        ${\textsc {M}}(x) = {\textsc {IND}}, U_1, U_2, U_3]\ \varXi )\ \mathbf{newoi}) \circ _{a} (\varDelta (r_{20}),\varDelta (r_{21}),\varDelta (r_{22}))$

        $= \lambda \, U_1\, U_2\, U_3\, .\, [oi(i_{21}), (i_{21} \in {\textsc {share}}^{\prime }), (\sim {\textsc {share}}(oi(U_1),oi(U_2),oi(U_3)) \in i_{21}),$

        $dd(oi(U_1)) = \mathbf{D}\, , dd(oi(U_2)) = \mathbf{E}\,, dd(oi(U_3)) = \mathbf{D}\, , {\textsc {A/T}}(i_{21}) = {\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_{21}) = {\textsc {IND}}, U_1, U_2, U_3]\ \circ _{a} (\varDelta (r_{20}),\varDelta (r_{21}),\varDelta (r_{22}))$

        $= [oi(i_{21}), (i_{21} \in {\textsc {share}}^{\prime }), (\sim {\textsc {share}}(i_{17},i_{19},i_{20}) \in i_{21}), dd(i_{17}) = \mathbf{D}\, , dd(i_{19}) = \mathbf{E}\,,$

        $dd(i_{20}) = \mathbf{D}\, , {\textsc {A/T}}(i_{21}) = {\textsc {DURAT/PRES}}, {\textsc {M}}(i_{21}) = {\textsc {IND}}, [oi(i_{17}), (i_{17} \equiv i_{nick})], [oi(i_{19}),$

        $(i_{19} \in {\textsc {caterpillar}}^{\prime }), {\textsc {APPRT}}(i_{19})=i_{18}, {\textsc {NUM}}(i_{19})={\textsc {SG}},{\textsc {HUMAN}}(i_{19})=-, [oi( i_{18}),$

        $(i_{18} \equiv i_{nick})]], [oi(i_{20}), (i_{20} \equiv i_{jeff})] ].$

$\mathbf {8.}$ Translation of the DD-tree $D_8$ of the sentence

$$\begin{aligned} {\texttt {\textit{Jackie married a guy rich as Croesus}}} \end{aligned}$$

shown in Fig. 17.

• 1. $\varDelta (r_2) =_{(4.22)} \varLambda (r_2) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in ({\textsc {jackie}}^{\prime })), {\textsc {NUM}}(x)={\textsc {SG}},$

        ${\textsc {HUMAN}}(x)=+]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_1), (i_1 \in {\textsc {jackie}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]$.

• 2. $\varDelta (r_7) =_{(4.22)} \varLambda (r_7) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in ({\textsc {croesus}}^{\prime })), {\textsc {NUM}}(x)={\textsc {SG}},$

        ${\textsc {HUMAN}}(x)=+]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+]$.

• 3. $\varDelta (r_9) = \lambda \, X\, .\, [oi(X)]$.

• 4. $\varDelta (r_8) =_{(4.25)} \varLambda (r_8) \circ _{r 1} \varDelta (r_9)$

        $=_{(4.14)} ((\lambda \ \xi \, x\, Z\, y\, .\, [oi( x ),\ (x \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in x), {\textsc {MAGN}}(x) = oi(Z)]$

        $\varXi )\ \mathbf{newoi})\ \circ _{r 1} \varDelta (r_9)$

        $= \lambda \, Z\, y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_3), {\textsc {MAGN}}(i_3) = oi(Z)]\ \circ _{r 1} \varDelta (r_9)$

        $=_{(4.18)} (\lambda \, Z\, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_2), {\textsc {MAGN}}(i_3) = oi(Z)]$

        $(\lambda \, X\, .\, [oi(X)]\ Y) )$

        $= \lambda \, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, Y) \in i_3), {\textsc {MAGN}}(i_3) = Y]$.

• 5. $\varDelta (r_6) =_{(4.28)} \varLambda (r_6) \circ _{par_3} (\varDelta (r_7), \varDelta (r_8))$

        $=_{(4.11)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi( x ),\ (x \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(x)={\textsc {DURAT/PRES}}, {\textsc {M}}(x)={\textsc {IND}}, U_1, U_2 ]\ \varXi )\ \mathbf{newoi})\, \circ _{par_3} (\varDelta (r_7), \varDelta (r_8))$

        $= \lambda \, U_1\, U_2\, .\, [oi( i_4),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in i_4),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, U_1, U_2 ]\, \circ _{par_3} (\varDelta (r_7), \varDelta (r_8))$

        $=_{(4.21)} (\lambda \, U_1\, U_2\, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in i_4),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, U_1, U_2]\ \varDelta (r_7) (\varDelta (r_8)\ oi(\varDelta (r_7))\ X) )$

        $= \lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], (\lambda \, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, Y) \in i_3),$

        ${\textsc {MAGN}}(i_3) = Y]\ oi([oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+])\ X) ]$

        $= \lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, X) \in i_3), {\textsc {MAGN}}(i_3) = X] ]$

• 6. $\varDelta (r_5) =_{(4.23)} \varLambda (r_5) \circ _{a} \varDelta (r_6)$

        $=_{4.13} ((\lambda \, \xi \, x\, R\, .\, [oi( x ),\ (R\ x) ]\ \varXi )\ \mathbf{newoi})\ \circ _{a} \varDelta (r_6)$

        $= (\lambda \ R\, .\, [oi( i_5 ),\ (R\ i_5) ]\ \varDelta (r_6) )$

        $= [oi( i_5 ),\ (\lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, X) \in i_3), {\textsc {MAGN}}(i_3) = X] ]\ i_5) ]$

        $= [oi( i_5 ),\ [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ]$

• 7. $\varDelta (r_4) =_{(4.23)} \varLambda (r_4)\ \circ _{a}\ \varDelta (r_5)$

        $=_{(4.14)} ((\lambda \ \xi \, x\, Z\, y\, .\, [oi( x ),\ (x \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in x), {\textsc {MAGN}}(x) = oi(Z), Z]$

        $\varXi )\ \mathbf{newoi})\ \circ _{a}\ \varDelta (r_5)$

        $= (\lambda \ Z\, y\, .\, [oi( i_6),\ (i_6 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_6), {\textsc {MAGN}}(i_6) = oi(Z), Z] \varDelta (r_5) )$

        $= \lambda \ y\, .\, [oi( i_6 ),\ (i_6 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, i_5) \in i_6), {\textsc {MAGN}}(i_6) = i_5, [oi( i_5 ), [oi( i_4 ),\$

        $(i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),$

        $(i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ]$.

• 8. $\varDelta (r_3) =_{(4.23)} \varLambda (r_3)\ \circ _{a}\ \varDelta (r_4)$

        $=_{(4.13)} ((\lambda \ \xi \ x\ Y\, . [oi( x ),\ (x \in {\textsc {guy}}^{\prime }), (Y\ x), {\textsc {NUM}}(x)={\textsc {SG}}, {\textsc {HUMAN}}(x)=+ ]\ \varXi )\ \mathbf{newoi})$

        $\circ _{a}\ \varDelta (r_4)$

        $= ( \lambda \ Y\, . [oi( i_7 ),\ (i_7 \in {\textsc {guy}}^{\prime }), (Y\ i_7), {\textsc {NUM}}(i_7)={\textsc {SG}}, {\textsc {HUMAN}}(i_7)=+ ]\ \varDelta (r_4) )$

        $= [oi( i_7 ),\ (i_7 \in {\textsc {guy}}^{\prime }), [oi( i_6 ),\ (i_6 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_7, i_5) \in i_6), {\textsc {MAGN}}(i_6) = i_5, [oi( i_5 ),$

        $[oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),$

        $(i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ], {\textsc {NUM}}(i_7)={\textsc {SG}}, {\textsc {HUMAN}}(i_7)=+ ]$.

• 9. $\varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi(x), (x \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(oi(U_1),oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(x) = {\textsc {POINT/PAST}}, {\textsc {M}}(x) = {\textsc {IND}}, U_1, U_2] \varXi )\ \mathbf{newoi})$

        $\circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $= \lambda \, U_1\, U_2\, .\, [oi(i_8), (i_8 \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(oi(U_1),oi(U_2)) \in i_8), dd(oi(U_1)) = \mathbf{D}\, ,$

        $dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(i_8) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_8) = {\textsc {IND}}, U_1, U_2]\ \circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $= [oi(i_8), (i_8 \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(i_1,i_7) \in i_8), dd(i_1) = \mathbf{D}\, , dd(i_7) = \mathbf{E}\,,$

        ${\textsc {A/T}}(i_8) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_8) = {\textsc {IND}}, [oi(i_1), (i_1 \in {\textsc {jackie}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_1)=+], [oi( i_7 ),\ (i_7 \in {\textsc {guy}}^{\prime }), [oi( i_6 ),\ (i_6 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_7, i_5) \in i_6),$

        ${\textsc {MAGN}}(i_6) = i_5, [oi( i_5 ),\ [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ],$

        ${\textsc {NUM}}(i_7)={\textsc {SG}}, {\textsc {HUMAN}}(i_7)=+ ] ]$.

$\mathbf {9.}$ Translation of the DD-tree $D_9$ of the sentence

$$\begin{aligned} {\texttt {\textit{Jackie married a guy more rich than Croesus}}} \end{aligned}$$

shown in Fig. 19.

• 1. $\varDelta (r_2) =_{(4.22)} \varLambda (r_2) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in ({\textsc {jackie}}^{\prime })), {\textsc {NUM}}(x)={\textsc {SG}},$

        ${\textsc {HUMAN}}(x)=+]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_1), (i_1 \in {\textsc {jackie}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}}, {\textsc {HUMAN}}(i_1)=+]$.

• 2. $\varDelta (r_7) =_{(4.2)} \lambda \, X\, .\, [oi(X)]$.

• 3. $\varDelta (r_{10}) =_{(4.22)} \varLambda (r_{10}) =_{(4.5)} ((\lambda \, \xi \, x\, .\, [oi(x), (x \in ({\textsc {croesus}}^{\prime })), {\textsc {NUM}}(x)={\textsc {SG}},$

        ${\textsc {HUMAN}}(x)=+]\ \varXi ) \ \mathbf{newoi})$

        $= [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+]$.

• 4. $\varDelta (r_{12}) = \lambda \, X\, .\, [oi(X)]$.

• 5. $\varDelta (r_{11}) =_{(4.25)} \varLambda (r_{11}) \circ _{r 1} \varDelta (r_{12})$

        $=_{(4.14)} ((\lambda \ \xi \, x\, Z\, y\, .\, [oi( x ),\ (x \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in x), {\textsc {MAGN}}(x) = oi(Z)]$

        $\varXi )\ \mathbf{newoi})\ \circ _{r 1} \varDelta (r_{12})$

        $= \lambda \, Z\, y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_3), {\textsc {MAGN}}(i_3) = oi(Z)]\ \circ _{r 1} \varDelta (r_{12})$

        $=_{(4.18)} (\lambda \, Z\, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_2), {\textsc {MAGN}}(i_3) = oi(Z)]$

        $(\lambda \, X\, .\, [oi(X)]\ Y) )$

        $= \lambda \, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, Y) \in i_3), {\textsc {MAGN}}(i_3) = Y]$.

• 6. $\varDelta (r_9) =_{(4.28)} \varLambda (r_9) \circ _{par_3} (\varDelta (r_{10}), \varDelta (r_{11}))$

        $=_{(4.11)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi( x ),\ (x \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(x)={\textsc {DURAT/PRES}}, {\textsc {M}}(x)={\textsc {IND}}, U_1, U_2 ]\ \varXi )\ \mathbf{newoi})\, \circ _{par_3} (\varDelta (r_{10}), \varDelta (r_{11}))$

        $= \lambda \, U_1\, U_2\, .\, [oi( i_ 4),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in i_4),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, U_1, U_2 ]\, \circ _{par_3} (\varDelta (r_{10}), \varDelta (r_{11}))$

  $=_{(4.21)} (\lambda \, U_1\, U_2\, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(oi(U_1), oi(U_2)) \in i_4),$

        $dd(oi(U_1)) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, U_1, U_2]\ \varDelta (r_{10})\ (\varDelta (r_{11})\ oi(\varDelta (r_{10}))\ X) )$

        $= \lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], (\lambda \, y\, Y\, .\, [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, Y) \in i_3),$

        ${\textsc {MAGN}}(i_3) = Y]\ oi([oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+])\ X) ]$

        $= \lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, X) \in i_3), {\textsc {MAGN}}(i_3) = X] ]$

• 7. $\varDelta (r_8) =_{(4.23)} \varLambda (r_8) \circ _{a} \varDelta (r_9)$

        $=_{4.13} ((\lambda \, \xi \, x\, R\, .\, [oi( x ),\ (R\ x) ]\ \varXi )\ \mathbf{newoi})\ \circ _{a} \varDelta (r_9)$

        $= (\lambda \ R\, .\, [oi( i_5 ),\ (R\ i_5) ]\ \varDelta (r_9) )$

        $= [oi( i_5 ),\ (\lambda \, X\, .\, [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, X) \in i_3), {\textsc {MAGN}}(i_3) = X] ]\ i_5) ]$

        $= [oi( i_5 ),\ [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),\, ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ]$

• 8. $\varDelta (r_6) =_{(4.25)} \varLambda (r_6) \circ _{r 1} (\varDelta (r_7),\varDelta (r_8))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi(x), (x \in ({>})^{\prime }), (({>})(oi(U_1), oi(U_2)) \in x), U_1, U_2]\ \varXi ) \mathbf{newoi})$

        $\circ _{r 1} (\varDelta (r_7),\varDelta (r_8))$

        $= \lambda \, U_1\, U_2\, .\, [oi(i_6), (i_6 \in ({>})^{\prime }), (({>})(oi(U_1), oi(U_2)) \in i_6), U_1, U_2]\ \circ _{r 1} (\varDelta (r_7),\varDelta (r_8))$

        $=_{(4.18)} (\lambda \, U_1\, U_2\, Y\, .\, [oi(i_6), (i_6 \in ({>})^{\prime }), (({>})(oi(U_1), oi(U_2)) \in i_6), U_1, U_2] (\varDelta (r_7)\ Y)$

        $[oi( i_5 ),\ [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }), ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, ,$

        ${\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),\ (i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ])$

        $= \lambda \, Y\, .\, [oi(i_6), (i_6 \in {>}^{\prime }), (({>})(Y, i_5) \in i_6), [oi(Y)], [oi( i_5 ), [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),$

        $({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2),$

        $(i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ), (i_3 \in {\textsc {rich}}^{\prime }),$

        $({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ]$.

• 9. $\varDelta (r_5) =_{(4.23)} \varLambda (r_5) \circ _{a} \varDelta (r_6)$

        $=_{(4.13)} ((\lambda \ \xi \ x\ R\, .\, [oi( x ),\ (R\ x) ]\ \varXi )\ \mathbf{newoi})\ \circ _{a} \varDelta (r_6)$

        $= (\lambda \ R\, .\, [oi( i_7 ),\ (R\ i_7) ]\ \varDelta (r_6) )$

        $= [oi( i_7 ),\ [oi(i_6), (i_6 \in ({>})^{\prime }), (({>})(i_7, i_5) \in i_6), [oi(i_7)], [oi( i_5 ), [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),$

        $({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2),$

        $(i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ), (i_3 \in {\textsc {rich}}^{\prime }),$

        $({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ] ]$.

• 10. $\varDelta (r_4) =_{(4.23)} \varLambda (r_4)\ \circ _{a}\ \varDelta (r_5)$

        $=_{(4.14)} ((\lambda \ \xi \, x\, Z\, y\, .\, [oi( x ),\ (x \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in x), {\textsc {MAGN}}(x) = oi(Z), Z]$

        $\varXi )\ \mathbf{newoi})\ \circ _{a}\ \varDelta (r_5)$

        $= (\lambda \ Z\, y\, .\, [oi( i_8 ),\ (i_8 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, oi(Z)) \in i_8), {\textsc {MAGN}}(i_8) = oi(Z), Z] \varDelta (r_5) )$

        $= \lambda \ y\, .\, [oi( i_8 ),\ (i_8 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(y, i_7) \in i_8), {\textsc {MAGN}}(i_8) = i_7, [oi( i_7 ),\ [oi(i_6),$

        $(i_6 \in ({>})^{\prime }), (({>})(i_7, i_5) \in i_6), [oi(i_7)], [oi( i_5 ), [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),$

        $({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}},$

        $[oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ), (i_3 \in {\textsc {rich}}^{\prime }),$

        $({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ] ] ]$.

• 11. $\varDelta (r_3) =_{(4.23)} \varLambda (r_3)\ \circ _{a}\ \varDelta (r_4)$

        $=_{(4.13)} ((\lambda \ \xi \ x\ Y\, . [oi( x ),\ (x \in {\textsc {guy}}^{\prime }), (Y\ x), {\textsc {NUM}}(x)={\textsc {SG}}, {\textsc {HUMAN}}(x)=+ ]\ \varXi )\ \mathbf{newoi})$

        $\circ _{a}\ \varDelta (r_4)$

        $= ( \lambda \ Y\, . [oi( i_9 ),\ (i_9 \in {\textsc {guy}}^{\prime }), (Y\ i_9), {\textsc {NUM}}(i_9)={\textsc {SG}}, {\textsc {HUMAN}}(i_9)=+ ]\ \varDelta (r_4) )$

        $= [oi( i_9 ),\ (i_9 \in {\textsc {guy}}^{\prime }), [oi( i_8 ),\ (i_8 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_9, i_7) \in i_8), {\textsc {MAGN}}(i_8) = i_7, [oi( i_7 ),$

        $[oi(i_6), (i_6 \in ({>})^{\prime }), (({>})(i_7, i_5) \in i_6), [oi(i_7)], [oi( i_5 ), [oi( i_4 ),\ (i_4 \in {\textsc {be-prop}}^{\prime }),$

  $({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}}, {\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2),$

  $(i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ), (i_3 \in {\textsc {rich}}^{\prime }),$

        $({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ] ] ], {\textsc {NUM}}(i_9)={\textsc {SG}}, {\textsc {HUMAN}}(i_9)=+ ]$.

• 12. $\varDelta (r_1) =_{(4.23)} \varLambda (r_1) \circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $=_{(4.6)} ((\lambda \, \xi \, x\, U_1\, U_2\, .\, [oi(x), (x \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(oi(U_1),oi(U_2)) \in x),$

        $dd(oi(U_1)) = \mathbf{D}\, , dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(x) = {\textsc {POINT/PAST}}, {\textsc {M}}(x) = {\textsc {IND}}, U_1, U_2]\ \varXi ) \mathbf{newoi})$

        $\circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $= \lambda \, U_1\, U_2\, .\, [oi(i_{10}), (i_{10} \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(oi(U_1),oi(U_2)) \in i_{10}), dd(oi(U_1)) = \mathbf{D}\, ,$

        $dd(oi(U_2)) = \mathbf{E}\,, {\textsc {A/T}}(i_{10}) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_{10}) = {\textsc {IND}}, U_1, U_2]\ \circ _{a} (\varDelta (r_2),\varDelta (r_3))$

        $= [oi(i_{10}), (i_{10} \in {\textsc {marry}}^{\prime }), ({\textsc {marry}}(i_1,i_9) \in i_{10}), dd(i_1) = \mathbf{D}\, , dd(i_9) = \mathbf{E}\,,$

        ${\textsc {A/T}}(i_{10}) = {\textsc {POINT/PAST}}, {\textsc {M}}(i_{10}) = {\textsc {IND}}, [oi(i_1), (i_1 \in {\textsc {jackie}}^{\prime }), {\textsc {NUM}}(i_1)={\textsc {SG}},$

        ${\textsc {HUMAN}}(i_1)=+], [oi( i_9 ),\ (i_9 \in {\textsc {guy}}^{\prime }), [oi( i_8 ),\ (i_8 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_9, i_7) \in i_8),$

        ${\textsc {MAGN}}(i_8) = i_7, [oi( i_7 ), [oi(i_6), (i_6 \in ({>})^{\prime }), (({>})(i_7, i_5) \in i_6), [oi(i_7)], [oi( i_5 ), [oi( i_4 ),$

        $(i_4 \in {\textsc {be-prop}}^{\prime }), ({\textsc {be-prop}}(i_2, i_3) \in i_4), dd(i_2) = \mathbf{D}\, , {\textsc {A/T}}(i_4)={\textsc {DURAT/PRES}},$

        ${\textsc {M}}(i_4)={\textsc {IND}}, [oi(i_2), (i_2 \in {\textsc {croesus}}^{\prime }), {\textsc {NUM}}(i_2)={\textsc {SG}}, {\textsc {HUMAN}}(i_2)=+], [oi( i_3 ),$

        $(i_3 \in {\textsc {rich}}^{\prime }), ({\textsc {rich}}(i_2, i_5) \in i_3), {\textsc {MAGN}}(i_3) = i_5] ] ] ] ] ], {\textsc {NUM}}(i_9)={\textsc {SG}}, {\textsc {HUMAN}}(i_9)=+ ] ]$.