Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTowards a Formal Distributional Semantics: Simulating Logical Calculi with Tensors
246
0
0.0
(
0
)
Added by
Edward Grefenstette
Publication date
2013
fields
Informatics Engineering
and research's language is
English
Authors
Edward Grefenstette
Ask ChatGPT about the research
No Arabic abstract
The development of compositional distributional models of semantics reconciling the empirical aspects of distributional semantics with the compositional aspects of formal semantics is a popular topic in the contemporary literature. This paper seeks to bring this reconciliation one step further by showing how the mathematical constructs commonly used in compositional distributional models, such as tensors and matrices, can be used to simulate different aspects of predicate logic. This paper discusses how the canonical isomorphism between tensors and multilinear maps can be exploited to simulate a full-blown quantifier-free predicate calculus using tensors. It provides tensor interpretations of the set of logical connectives required to model propositional calculi. It suggests a variant of these tensor calculi capable of modelling quantifiers, using few non-linear operations. It finally discusses the relation between these variants, and how this relation should constitute the subject of future work.
rate research
Read More
Classical scope-assignment strategies for multi-quantifier sentences involve quantifier phrase (QP)-movement. More recent continuation-based approaches provide a compelling alternative, for they interpret QPs in situ - without resorting to Logical Forms or any structures beyond the overt syntax. The continuation-based strategies can be divided into two groups: those that locate the source of scope-ambiguity in the rules of semantic composition and those that attribute it to the lexical entries for the quantifier words. In this paper, we focus on the former operation-based approaches and the nature of the semantic operations involved. More specifically, we discuss three such possible operation-based strategies for multi-quantifier sentences, together with their relative merits and costs.
For formulas F of propositional calculus I introduce a metavariable MF and show how it can be used to define an algorithm for testing satisfiability. MF is a formula which is true/false under all possible truth assignments iff F is satisfiable/unsatisfiable. In this sense MF is a metavariable with the meaning F is SAT. For constructing MF a group of transformations of the basic variables ai is used which corresponds to flipping literals to their negation. The whole procedure corresponds to branching algorithms where a formula is split with respect to the truth values of its variables, one by one. Each branching step corresponds to an approximation to the metatheorem which doubles the chance to find a satisfying truth assignment but also doubles the length of the formulas to be tested, in principle. Simplifications arise by additional length reductions. I also discuss the notion of logical primes and show that each formula can be written as a uniquely defined product of such prime factors. Satisfying truth assignments can be found by determining the missing primes in the factorization of a formula.
Constructions in type-driven compositional distributional semantics associate large collections of matrices of size $D$ to linguistic corpora. We develop the proposal of analysing the statistical characteristics of this data in the framework of permutation invariant matrix models. The observables in this framework are permutation invariant polynomial functions of the matrix entries, which correspond to directed graphs. Using the general 13-parameter permutation invariant Gaussian matrix models recently solved, we find, using a dataset of matrices constructed via standard techniques in distributional semantics, that the expectation values of a large class of cubic and quartic observables show high gaussianity at levels between 90 to 99 percent. Beyond expectation values, which are averages over words, the dataset allows the computation of standard deviations for each observable, which can be viewed as a measure of typicality for each observable. There is a wide range of magnitudes in the measures of typicality. The permutation invariant matrix models, considered as functions of random couplings, give a very good prediction of the magnitude of the typicality for different observables. We find evidence that observables with similar matrix model characteristics of Gaussianity and typicality also have high degrees of correlation between the ranked lists of words associated to these observables.
Categorical compositional distributional semantics provide a method to derive the meaning of a sentence from the meaning of its individual words: the grammatical reduction of a sentence automatically induces a linear map for composing the word vectors obtained from distributional semantics. In this paper, we extend this passage from word-to-sentence to sentence-to-discourse composition. To achieve this we introduce a notion of basic anaphoric discourses as a mid-level representation between natural language discourse formalised in terms of basic discourse representation structures (DRS); and knowledge base queries over the Semantic Web as described by basic graph patterns in the Resource Description Framework (RDF). This provides a high-level specification for compositional algorithms for question answering and anaphora resolution, and allows us to give a picture of natural language understanding as a process involving both statistical and logical resources.
We introduce a new application for inductive logic programming: learning the semantics of programming languages from example evaluations. In this short paper, we explored a simplified task in this domain using the Metagol meta-interpretive learning system. We highlighted the challenging aspects of this scenario, including abstracting over function symbols, nonterminating examples, and learning non-observed predicates, and proposed extensions to Metagol helpful for overcoming these challenges, which may prove useful in other domains.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
