Supplementary Material for Grammar-Based Grounded Lexicon Learning

In the supplementary material, we describe the domain specific languages used in our experiments (Section 1), demonstrate how the proposed CKY-E2 method works by a concrete example (Section 2.1), show formal properties of CKY-E2 (Section 2.2), present dataset setups and analyze model behaviors (Section 3), and list environmental details for experiments (Section??). In this section, we will present and discuss the domain-specific languages (DSLs) we use for two domains: visual reasoning and language-guided navigation. We will further introduce the neurosymbolic module we have designed for executing programs in these two domains. Overall, each DSL contains a set of types and a set of deterministic modules that have been manually designed for realizing necessary operations in these domains. However, in contrast to realizing them as we do in standard programming languages (with for-loops and if-conditions), we will be using tensor operations (e.g., tensor additions and multiplications) to realize them so that the output of each program is differentiable with respect to all of its inputs. We refer readers to the original papers for a detailed introduction to the DSL and neuro-symbolic program execution. Here we only highlight the key aspects of our language and its neuro-symbolic realization, and discuss the difference between our implementation and the ones in original papers. Our visual reasoning DSL is a subset of CLEVR, containing 6 types and 8 primitive operations. Table 1 illustrates all 6 types and how they are internally represented in neuro-symbolic execution. Table 2 further shows all operations in the DSL. There are two main differences between the DSL used by G2L2 and the original CLEVRDSL.