SlowGuess/NAACL_2022_OCR huggingface dataset:
Rimvydas Rubavicius and Alex Lascarides
rimvydas.rubavicius@ed.ac.uk, alex@inf.ed.ac.uk
We present a procedure for learning to ground symbols from a sequence of stimuli consisting of an arbitrarily complex noun phrase (e.g. "all but one green square above both red circles.") and its designation in the visual scene. Our distinctive approach combines: a) lazy few-shot learning to relate open-class words like green and above to their visual percepts; and b) symbolic reasoning with closed-class word categories like quantifiers and negation. We use this combination to estimate new training examples for grounding symbols that occur within a noun phrase but aren't designated by that noun phase (e.g, red in the above example), thereby potentially gaining data efficiency. We evaluate the approach in a visual reference resolution task, in which the learner starts out unaware of concepts that are part of the domain model and how they relate to visual percepts.
The subfield of robotics known as Interactive Task Learning (ITL, see Laird et al. (2017) for a survey) addresses scenarios where a robot must learn to adapt its behaviour to novel and unforeseen objects, relations, and attributes that are introduced into the environment after deployment. The ITL agent learns its novel task via evidence from its own actions and reactive guidance from a teacher. This paper focuses on symbol grounding (Harnad, 1999) in the context of ITL (Matuszek, 2018): the learner must use the teacher's embodied natural language utterance and its context to learn a mapping from natural language expressions to their denotations, given the visual percepts.
There are two challenges in learning symbol grounding models (grounders) in ITL. Firstly, in contrast to many grounders (Ye et al., 2019; Datta et al., 2019), ITL requires incremental learning: knowledge is acquired piecemeal via an extended interaction, and it must influence planning as and
when it occurs. Secondly, previous work limits the teacher's language to bare nouns (e.g., square) or very short phrases (e.g., blue square, square above circle) (Hristov et al., 2018, 2019). But there's evidence from Dale and Reiter (1995) that speakers use complex referring expressions even when simpler ones would successfully refer. Such language creates the possibility that novel symbols—neologisms—are introduced in a context where their denotation is not designated by the teacher. In this work we study the natural language of complex referential expressions (REs) like "a blue square behind both red circles" which teachers can use when designating an object.
Our aim is for the learner to extract knowledge that improves their domain representation and state estimates—a necessary condition for successful planning. Contemporary grounders miss learning opportunities that complex REs afford: the RE example above not only entails that its referents are blue and square, but also that there exists two objects that are both red and circle and that they are above the designated objects, and everything else in the domain is either not red or not a circle (thanks to the meaning of both). Thus, a complex RE and its designation can be used to gather multiple (noisy) training exemplars (both positive and negative) for several symbols at once, even if they have not been designated.
In this work, we develop a method to integrate knowledge from interactively gathered evidence in the form of complex RE-designation pairs to aid data acquisition for a (neural) few-shot grounder. We explore the effect of such a method on data efficiency and the overall grounder's performance. A major novel component to our procedure is that we exploit the formal semantics of closed class word categories (e.g., quantifiers and negation) to boost the data efficiency of few-shot neural grounding models. Our experiments show these symbolic inductive biases are successful.
Human Language Technologies, pages 4863 - 4874
Symbol Grounding. Contemporary grounders extensively utilize batch learning (e.g. Shridhar and Hsu (2018)). Yet, ITL requires incremental learning because without it the teacher guidance cannot influence the learner's inferences about plans as and when the advice is given. Further, many grounders assume that the learner starts out with a complete and accurate conceptualisation of the domain using pre-defined visual features and a known vocabulary (Kennington et al., 2015; Kennington and Schlangen, 2017; Wang et al., 2017). In ITL, both of these assumptions are unrealistic; therefore in this paper we explore models for which these assumptions don't apply. Finally, in contrast to all prior grounders, we support incremental learning when the training exemplars feature REs that are linguistically complex: e.g., "two red circles that aren't to the right of both green squares".
Representation Learning. Models for jointly learning a representation for vision and language utilize either explicit alignment via bounding boxes or instance segmentation (Lu et al., 2019; Chen et al., 2020; Tan and Bansal, 2019; Kamath et al., 2021; Yu et al., 2021), or a large-volume of weakly labeled data in the form of image-caption pairs (Radford et al., 2021). These models rely on offline learning with large datasets. This work, on the other hand, explores how to incrementally extract knowledge from few-shot learning, using sequentially observed evidence that includes neologisms.
Visual Questions Answering (VQA). This is a task of answering free-form questions about an image (Antol et al., 2015). VQA has reached impressive performance in recent years (Fukui et al., 2016; Li et al., 2020), yet VQA models struggle with out-of distribution generalization for new types of questions, requiring multi-step reasoning, with analysis revealing that they often rely on shortcuts (Jiang and Bansal, 2019; Subramanian et al., 2019, 2020). Grounded VQA models like (Yi et al., 2018) and Bogin et al. (2021) tackle these shortcomings by grounding parts of the question and then learning to compose those parts via the question's syntax to compute the answer. They thus estimate denotations of linguistic parts that are not denoted by the answer to the question. These 'compositional' models help to achieve out-of-distribution generalization for novel questions. But they lack ITL's requirement for incremental learning: model training
relies on batch learning. Furthermore, while their performance is impressive, error analysis shows that it makes mistakes when language includes logical concepts like quantifiers and negation (e.g. Boin et al. (2021) Figure 9 shows that the determiner most incorrectly denotes an arbitrary subset of entities). Our view is that there is little benefit in trying to learn to ground logic concepts as they are domain independent and can be interpreted using formal semantics. In our experiments, we are testing the extent to which knowing and reasoning with the logical meanings of these symbols helps incremental grounding, and in particular estimating denotations of symbols within an RE that are not designated.
Grounded Language Acquisition. This task is often realized as grounded grammar induction from image-caption pairs (Shi et al., 2019; Zhao and Titov, 2020), or as learning (neural) semantic parsers from a reward signal (Williams, 1992) in VQA (Mao et al., 2019; Yi et al., 2018) or in planning (Azaria et al., 2016; Wang et al., 2016, 2017; Karamcheti et al., 2020). There, the main objective is to learn to map natural language to logical forms, which in turn get associated with visual percepts during the learning process. This paper does not aim to learn a semantic parser. Instead, we obtain logical forms from an existing broad-coverage grammar which is hard to engineer, but is robust on lexical variation (Curran et al., 2007). Our focus instead is on exploiting the logical consequences of those logical forms during symbol grounding—i.e., our focus is to utilise the interpretation of logical forms, and in particular the truth functional meanings of close-class words like quantifiers and negation, to inform the learning of mappings from (open-class) symbols like red to their denotations, given the visual percepts.
Visual Reference Resolution. In previous experiments, it is often assumed that there is a unique referent in the visual scene for the given RE in the test phase (Kazemzadeh et al., 2014; Whitney et al., 2016). We aim to cope with situations where the RE has multiple referents: identifying all the referents that satisfy an RE enables efficient planning, because it affords free choice when executing certain commands—e.g., "move a square above both red circles" when there is more than one square affords choosing a control policy so that resources are optimized.
3.1 Formal Semantics of Natural Language
Predicate logic with generalized quantifiers  $\mathfrak{L}$  (Barwise and Cooper, 1981; van Benthem, 1984) is a canonical meaning representation for natural languages.  $\mathfrak{L}$ -sentences  $\phi$  are constructed recursively from predicates  $P$ , terms  $T$  (i.e., variables  $V$  and constants  $C$ ), logical connectives  $O = \{\neg, \land, \lor, \to\}$  and quantifiers  $Q$  (see Table 1 column 1):
$$
\begin{array}{l} \phi : := \mathbf {p} (t _ {1}, \dots , t _ {n}) \equiv \mathbf {p} (t ^ {n}) \\ | (\neg \phi) | (\phi_ {1} \wedge \phi_ {2}) | (\phi_ {1} \vee \phi_ {2}) | (\phi_ {1} \rightarrow \phi_ {2}) \\ \left| \left(\mathrm {Q x} \left(\phi_ {1}, \phi_ {2}\right)\right) \right. \\ \end{array}
$$
where  $\mathfrak{p}$  is an  $n$ -place predicate,  $\mathfrak{t}_i \in T$  are terms,  $\mathbb{Q} \in Q$  is a quantifier, and  $\mathbf{x} \in V$  is a variable (in  $\mathbb{Q}\mathbf{x}(\phi_1, \phi_2)$ ,  $\phi_1$  is the restrictor and  $\phi_2$  the body). We also introduce  $\lambda$ -expressions of the form  $\lambda \mathbf{x}.\phi$ , where  $\mathbf{x} \in V$  is free or absent in  $\phi$ .
3.2 Model-theoretic Interpretation
$\mathfrak{L}$ -sentences are interpreted using a domain model  $\mathcal{M} = (E, I)$  consisting of a set of entities  $E$  (universe of discourse), and an extension function  $I$  that maps non-logical symbols  $P \cup C$  to denotations (tuples of entities). For convenience, we assume  $I: C \mapsto E$  is one-to-one. Variables are interpreted via an assignment function  $g: V \mapsto E$ .
The interpretation function  $\llbracket \cdot \rrbracket^{\mathcal{M},g}$  specifies the semantic value of well-formed expressions of  $\mathfrak{L}$ :
$$
\llbracket \mathbf {a} \rrbracket^ {\mathcal {M}, g} = \left\{ \begin{array}{l l} I (\mathbf {a}) & \text {i f} \mathbf {a} \in P \cup C \\ g (\mathbf {a}) & \text {i f} \mathbf {a} \in V \end{array} \right.
$$
$$
\llbracket \mathbf {p} (\mathsf {t} ^ {n}) \rrbracket^ {\mathcal {M}, g} = 1 \mathrm {i f f}
$$
$$
\left(\llbracket \mathbf {t} _ {1} \rrbracket^ {\mathcal {M}, g}, \dots , \llbracket \mathbf {t} _ {n} \rrbracket^ {\mathcal {M}, g}\right) \in \llbracket \mathbf {p} \rrbracket^ {\mathcal {M}, g}
$$
$$
\llbracket \neg \phi \rrbracket^ {\mathcal {M}, g} = 1 \mathrm {i f f} \llbracket \phi \rrbracket^ {\mathcal {M}, g} = 0
$$
$$
\llbracket \phi \wedge \psi \rrbracket^ {\mathcal {M}, g} = 1 \text {i f f} \llbracket \phi \rrbracket^ {\mathcal {M}, g} = 1 \text {a n d} \llbracket \psi \rrbracket^ {\mathcal {M}, g} = 1
$$
$$
\llbracket \phi \vee \psi \rrbracket^ {\mathcal {M}, g} = 1 \text {i f f} \llbracket \phi \rrbracket^ {\mathcal {M}, g} = 1 \text {o r} \llbracket \psi \rrbracket^ {\mathcal {M}, g} = 1
$$
$$
\llbracket \phi \rightarrow \psi \rrbracket^ {\mathcal {M}, g} = 1 \text {i f f} \llbracket \phi \rrbracket^ {\mathcal {M}, g} = 0 \text {o r} \llbracket \psi \rrbracket^ {\mathcal {M}, g} = 1
$$
$$
\llbracket \lambda \mathrm {x}. \phi \rrbracket^ {\mathcal {M}, g} = \left\{e \in E: \llbracket \phi \rrbracket^ {\mathcal {M}, g [ \mathrm {x} / e ]} = 1 \right\}
$$
$$
\llbracket \mathbb {Q} \mathbf {x} (\phi_ {1}, \phi_ {2}) \rrbracket^ {\mathcal {M}, g} = \mathrm {Q} (\llbracket \lambda . \mathbf {x} \phi_ {1} \rrbracket^ {\mathcal {M}, g}, \llbracket \lambda . \mathbf {x} \phi_ {2} \rrbracket^ {\mathcal {M}, g})
$$
where  $g[\mathbf{x} / e]$  is just like  $g$ , except  $g[\mathbf{x} / e](\mathbf{x}) = e$  and  $\mathrm{Q}$  is a specific relation between the restrictor  $[\lambda \mathbf{x}. \phi_1]^{M, g}$  and body  $[\lambda \mathbf{x}. \phi_2]^{M, g}$ , as defined in Table 1 column 3.  $[\cdot]^{M, g}$  is directly related to satisfiability for  $\mathfrak{L}$ -sentences:
$$
\mathcal {M}, g \vDash \phi \text {i f f} [   [ \phi ]   ] ^ {\mathcal {M}, g} = 1
$$
$$
\mathcal {M} \models \phi \mathrm {i f f} [ [ \phi ] ] ^ {\mathcal {M}} = 1
$$
where  $\mathbb{[}\phi \mathbb{]}^{\mathcal{M}} = 1$  iff  $\mathbb{[}\phi \mathbb{]}^{\mathcal{M},g} = 1$  for all  $g$ . Further, if  $\mathbf{x}$  is the only free variable in  $\phi$ , then  $\mathbb{[}\lambda \mathbf{x}. \phi \mathbb{]}^{\mathcal{M}, g} = \mathbb{[}\lambda \mathbf{x}. \phi \mathbb{]}^{\mathcal{M}, g'}$  for all  $g, g'$ ; so without a loss of generality, this is expressed as  $\mathbb{[}\lambda \mathbf{x}. \phi \mathbb{]}^{\mathcal{M}}$ . If all variables in  $\mathsf{Qx}(\phi, \psi)$  are bound by quantifiers, then this  $\mathfrak{L}$ -sentence is true iff  $\mathrm{Q}$  is true for all  $g$ .
Some quantifiers, like "both",<sup>2</sup> are presupposition triggers: "exactly two blocks are blue" is different from "both blocks are blue" in that the latter is true only if there are exactly two individuals that are blocks. We've adopted a Russellian interpretation (Russell, 1917) of these in Table 1.
3.3 Logical Forms of Referential Expressions
We now define the logical forms of REs and their interpretations with respect to a domain model  $\mathcal{M}$ . Noun phrases like "a block" are represented as  $\langle \underline{\mathbf{a}}_{\mathbf{q}\mathbf{x}}.\mathbf{block}(\mathbf{x})\rangle$ . More generally, let  $\langle \mathbb{Q}\mathbf{x}. \phi \rangle$  be the logical form of an RE, where  $\mathbb{Q} \in Q$  and  $\phi$  is an  $\mathfrak{L}$ -sentence with  $\mathbf{x} \in V$  being the only free variable in  $\phi$ . The referents  $\langle \mathbb{Q}\mathbf{x}. \phi \rangle^{\mathcal{M}}$  of this logical form with respect to  $\mathcal{M}$  are computed as follows:
$$
\langle \mathbb {Q} _ {\mathbf {x}}. \phi \rangle^ {\mathcal {M}} = \langle \mathbb {Q} \rangle^ {\pi (\mathcal {M}, \phi , \mathbf {x})} \tag {1}
$$
where  $\pi (\mathcal{M},\phi ,\mathbf{x})$  is an  $\mathcal{M}$  -projection,giving a new domain model  $\mathcal{M}'$  with entities  $E^{\prime} = E\cap$ $[\lambda .\mathbf{x}\phi ]]^{\mathcal{M}}$  and  $\langle \mathbb{Q}\rangle^{\mathcal{M}}$  is a quantifier referent-a quantifier-specific subset of the power set of  $E$  Table 1 column 4 gives the list of quantifier referents.
To illustrate, consider the domain model where:
$$
\begin{array}{l} E = \{a, b, c, d, f \} \\ I (\operatorname {c a t}) = \{a, b \} \quad I (\operatorname {d o g}) = \{c, d, f \} \\ I (\mathsf {b i t}) = \{(c, a), (c, b), (d, b), (f, a), (f, b) \} \\ \end{array}
$$
The RE "a dog that bit both cats" has logical form  $\langle\_ a\_ q\ x.\\_ both\_ q\ y(\text{cat}(\mathbf{y}), \text{dog}(\mathbf{x}) \land \text{bit}(\mathbf{x}, \mathbf{y}))\rangle$ . By Equation 1, its referent is:
$$
\langle \_ a \_ q \rangle^ {\pi (\mathcal {M}, \_ \text {b o t h} \_ q y (c a t (y), d o g (x) \wedge b i t (x, y)), x)}
$$
The semantic value of the  $\lambda$ -expression formed from this RE is a set of entities  $e \in E$  for which the following quantifier condition is true: both_q(R, B) where  $R = \llbracket \lambda y.\mathrm{cat}(\mathbf{y})\rrbracket^{\mathcal{M},g[\mathbf{x}/e]}$  and  $B = \llbracket \lambda y.\mathrm{dog}(\mathbf{x}) \wedge \mathrm{bit}(\mathbf{x},\mathbf{y})\rrbracket^{\mathcal{M},g[\mathbf{x}/e]}$ . Only  $c, f \in E$  satisfy this condition, defining a new model  $\mathcal{M}'$  with  $E_{\mathcal{M}'} = \{c,f\}$ ; this leads to the set of possible referents as  $\{\{c\}, \{f\}\}$ , given the quantifier referent  $\langle\_a\_q\rangle^{\mathcal{M}}$ .
This fact will be used when defining referents.
2This is not an English specific phenomena: Finnish molempi has the same condition as both.
quantifiers Q	surface form	condition Q(R,B)	referent ⟨Q⟩M
Exactly_n_q	exactly n	|R∩B|=n	{A⊆E: |A|=n}
-at_most_n_q	at most n	|R∩B|≤n	{A⊆E: |A|≤n}
-at_least_n_q	at least n	|R∩B|≥n	{A⊆E: |A|≥n}
_a_q	a/an	|R∩B|≠n	{A⊆E: |A|≤1}
_every_q	all/every	|R∩B|=|R|	{A⊆E: |A|=|E|}
_the_n_q	the n	|R∩B|=n)∧|R|=n	{A⊆E: |A|=|E|∧(|E|=n)}
both_q	both	|R∩B|=2)∧|R|=2	{A⊆E: |A|=|E|∧|E|=2}
_all但不限_n_q	all but n	|R∩B|=|R|-n	{A⊆E: |A|=|E|-n∧|E|≥n}
_n_of_the_m_q	n of the m	|R∩B|=n∧|R|=m	{A⊆E: |A|=n∧|E|=m}
Table 1: Quantifiers (column 1), their surface forms (column 2), condition Q between the restrictor  $R$  and body denotations  $B$ , used to compute a semantic value for  $\mathfrak{L}$ -sentences of the form  $\mathbb{Q}\mathbf{x}(\phi, \psi)$  (column 3); and quantifier referents  $\langle \mathbb{Q}\rangle^{\mathcal{M}}$  used to compute references of the logical form of REs (column 4).
Figure 1: IGRE overview. In interaction, the learner observes an RE, which is parsed to logical form (\$5.3.2) and interpreted with respect to the extracted feature vectors for denotations (\$5.3.1) to perform reference resolution (§3.3) with respect to the estimated domain model  $\hat{\mathcal{M}}$ . In case of teacher feedback, RE and its designation is observed. This is used to build the  $\mathfrak{L}$ -sentence that is added to  $\Delta$  to update beliefs about the underlying concept vectors (\$4.3.2), which in turn are used to update the support set (\$3.3), used as parameters for the grounder (\$4.1). Elements in blue are pre-defined elements of IGRE while elements in red are learned through interaction.
Below we present the procedure of interactive grounding with referential expressions (IGRE). The overall framework is given in Figure 1.
4.1 Grounder
Matching networks (Vinyals et al., 2016) are an extension of the  $k$  nearest-neighbour algorithm (Fix and Hodges, 1989) and has been used as a fast few-shot grounder in the ITL setting (Cano Santín et al., 2020). For predicates  $P^n \subseteq P$  of the same arity  $n$ , a grounder  $\Theta^n$  is parameterized by a support set  $\mathcal{S}^n = \{(\mathbf{x}_i^n, \mathbf{y}_i^n)\}_{i=1}^{K^n}$ , consisting of  $K^n$  pairs of feature vectors  $\mathbf{x}_i^n \in \mathbb{R}^{d_n}$  for denotations  $e^n \in E^n$
and concept vectors  $\mathbf{y}_i^n\in [0,1]^{|P^n|}$ . In  $\mathbf{y}^n$ , the dimension  $z$  corresponds the predicate  $\mathfrak{p}_z\in P^n$  and its value is the probability that  $\llbracket \mathfrak{p}_z(e^n)\rrbracket^{\mathcal{M},g} = 1$ . Concept vectors have a one-to-one correspondence with the domain model  $\mathcal{M}$ .
Given a feature vector  $\mathbf{x}^n$  for a denotation  $e^n\in E^n$ ,  $\Theta^n$  predicts the concept vector  $\hat{\mathbf{y}}^n$ , using the following inference rule:
$$
\Theta^ {n} (\mathbf {x} ^ {n}, S ^ {n}) = \sum_ {i = 1} ^ {k} \alpha^ {n} (\mathbf {x} _ {i} ^ {n}, \mathbf {x} ^ {n}; S ^ {n}) \mathbf {y} _ {i} ^ {n}
$$
where  $\alpha^n$  is an attention kernel:
$$
\alpha^ {n} (\mathbf {x} _ {i} ^ {n}, \mathbf {x} ^ {n}; S ^ {n}) = \frac {\exp \left(f ^ {n} (\mathbf {x} _ {i} ^ {n}) \cdot h ^ {n} (\mathbf {x} ^ {n})\right)}{\sum_ {j = 1} ^ {k} \exp \left(f ^ {n} (\mathbf {x} _ {j} ^ {n}) \cdot h ^ {n} (\mathbf {x} ^ {n})\right)}
$$
$$
f ^ {n} (\mathbf {x} ^ {n}) = \frac {\mathrm {R e L U} (\mathbf {w} ^ {n} \cdot \mathbf {x} ^ {n} + b ^ {n})}{| | \mathrm {R e L U} (\mathbf {w} ^ {n} \cdot \mathbf {x} ^ {n} + b ^ {n}) | | _ {2}}
$$
$$
h ^ {n} (\mathbf {x} ^ {n}) = \frac {\mathrm {R e L U} (\mathbf {v} ^ {n} \cdot \mathbf {x} ^ {n} + c ^ {n})}{| | \mathrm {R e L U} (\mathbf {v} ^ {n} \cdot \mathbf {x} ^ {n} + c ^ {n}) | | _ {2}}
$$
$$
\operatorname {R e L U} (a) = \max  (0, a)
$$
with learnable parameters  $\theta^n = \{\mathbf{w}^n,\mathbf{v}^n,b^n,c^n\}$  and  $S^n$  is  $k = 3$  nearest exemplars to  $\mathbf{x}^n$  from  $S^n$ :
$$
S ^ {n} = \left\{\left(\mathbf {x} _ {i} ^ {n}, \mathbf {y} _ {i} ^ {n}\right) \in S ^ {n}: \mathbf {x} _ {i} ^ {n} \in \mathcal {V} (k, \mathbf {x} ^ {n}, S ^ {n}) \right\}
$$
where  $\mathcal{V}(k,\mathbf{x}^n,S^n)$  is a set of  $k$  nearest feature vectors.
4.2 Batch Learning
Given  $S^n$ , one can estimate  $\Theta^n$  either via batch learning performed offline, or—when  $S^n$  is small—in real time, as outlined by Cano Santín et al. (2020). In our scenario, we learn in real time by minimizing binary cross-entropy between the ground-truth  $\mathbf{y}^n$  and predicted  $\hat{\mathbf{y}}^n$  concept vectors:
$$
\mathcal {L} (\mathbf {y} ^ {n}, \hat {\mathbf {y}} ^ {n}) = - \sum_ {z = 1} ^ {| P ^ {n} |} l \left(y _ {z} ^ {n}, \hat {y} _ {z} ^ {n}\right)
$$
$$
l (y _ {i} ^ {n}, \hat {y} _ {i} ^ {n}) = y _ {i} ^ {n} \log (\hat {y} _ {i} ^ {n}) + (1 - y _ {i} ^ {n}) \log (1 - \hat {y} _ {i} ^ {n})
$$
4.3 Incremental Learning
$S^n$  gets augmented whenever the teacher provides an RE-designation pair. This speech act provides two types of information: certain information  $\mathcal{C}^n$  in the form of denotation-symbol-semantic value triples  $(e^n,\mathfrak{p}_z,y_z^n)$ , corresponding to symbols and entities designated by the RE; and noisy information  $\mathcal{N}^n$ , corresponding to denotation-symbol-semantic value estimate triples  $(e^n,\mathfrak{p}_z,\tilde{y}_z^n)$ , which are acquired from the symbols that are part of the RE and its referent inferred via (uncertain) reasoning. E.g., the RE "a circle below a square," entails that its designation  $e\in E$  is a circle and so  $(e,\mathrm{circle},1)$  is added to  $\mathcal{C}^n$ . But it also entails there exists an entity which is a square that is not designated by the RE, but rather this entity is in the below relation with the designated entity. If the grounded is sufficiently confident about the referent for square, then the corresponding triple is added to  $\mathcal{N}^n$ .
4.3.1 Acquiring Observations and Symbols
When the learner first observes its visual scene—and so the teacher has not expressed any concepts, and so the learner is currently unaware of all concepts—the noisy support set  $\mathcal{N}^n$  is populated with  $(e^n, \mathfrak{p}_z, 0.5)$  (0.5 is a default semantic value) for all  $e^n$  in the scene and for all known  $n$ -place predicates. Whenever the teacher's REDesignation pair features a neologism  $\mathfrak{p}^*$ , then this expansion to the learner's vocabulary prompts adding  $(e^n, \mathfrak{p}^*, 0.5)$  to  $\mathcal{N}^n$  for all  $e^n$ . During interaction, each RE-designation pair uttered by the teacher adds elements to  $\mathcal{C}^n$  (for designated symbols) and triggers updates to the  $\mathcal{N}^n$  elements for all entities in the current visual scene, as we'll now describe.
4.3.2 Integrating the Teacher's Feedback
$\mathcal{N}^n$  elements are interactively updated using an incrementally built domain-level theory  $\Delta$ , which is the conjunction of  $\mathfrak{L}$ -sentences that are built from the logical forms of the REs that the teacher has uttered so far and their designations. To compute the beliefs about semantic values, given  $\Delta$ , we model the semantic value of  $\mathfrak{L}$ -sentences of the form  $\mathfrak{p}(\mathfrak{t}^n)$ , in which  $t^n$  are all constants (ground atom), as a random variable with Bernoulli's distribution  $\mathcal{B}$ . Thus a distribution over the possible domain models can be estimated using (propositional) model counting MC (Valiant, 1979), which maps each  $\mathfrak{L}$ -sentence to the number of domain models satisfying it. In this way, the semantic value of any proposition can be estimated as follows:
$$
\tilde {y} _ {z} ^ {n} = \left\{ \begin{array}{l l} \frac {\operatorname {M C} (\mathrm {p} _ {z} (e ^ {n}) \wedge \Delta)}{\operatorname {M C} (\Delta)} & \text {i f} \operatorname {M C} (\Delta) \neq 0 \\ 0. 5 & \text {o t h e r w i s e} \end{array} \right.
$$
MC can be computed exactly or approximately (Samer and Szeider, 2010). In our experiments we use the ADDMC (Dudek et al., 2020) weighted model counter, with weights set to 0.5.
4.3.3 Building the Support Set