Quantification and the language of thought

Neural Information Processing Systems

Many researchers have suggested that the psychological complexity of a concept is related to the length of its representation in a language of thought. As yet, however, there are few concrete proposals about the nature of this language. This paper makes one such proposal: the language of thought allows first order quantification (quantificationover objects) more readily than second-order quantification (quantification over features). To support this proposal we present behavioral results froma concept learning study inspired by the work of Shepard, Hovland and Jenkins. Humans can learn and think about many kinds of concepts, including natural kinds such as elephant and water and nominal kinds such as grandmother and prime number.


AAAI Conferences

Most logic-based machine learning algorithms rely on an Occamist bias where textual complexity of hypotheses is minimised. Within Inductive Logic Programming (ILP), this approach fails to distinguish between the efficiencies of hypothesised programs, such as quick sort (O(n log n)) and bubble sort (O(n2)).

On the Semantics and Complexity of Probabilistic Logic Programs

Journal of Artificial Intelligence Research

We examine the meaning and the complexity of probabilistic logic programs that consist of a set of rules and a set of independent probabilistic facts (that is, programs based on Sato's distribution semantics). We focus on two semantics, respectively based on stable and on well-founded models. We show that the semantics based on stable models (referred to as the "credal semantics") produces sets of probability measures that dominate infinitely monotone Choquet capacities; we describe several useful consequences of this result. We then examine the complexity of inference with probabilistic logic programs. We distinguish between the complexity of inference when a probabilistic program and a query are given (the inferential complexity), and the complexity of inference when the probabilistic program is fixed and the query is given (the query complexity, akin to data complexity as used in database theory). We obtain results on the inferential and query complexity for acyclic, stratified, and normal propositional and relational programs; complexity reaches various levels of the counting hierarchy and even exponential levels.


AAAI Conferences

In the last few years, there has been a large effort for analyzing the computational properties of reasoning in fuzzy description logics. This has led to a number of papers studying the complexity of these logics, depending on the chosen semantics. Surprisingly, despite being arguably the simplest form of fuzzy semantics, not much is known about the complexity of reasoning in fuzzy description logics w.r.t.

Learning to Reason

arXiv.org Artificial Intelligence

Automated theorem proving has long been a key task of artificial intelligence. Proofs form the bedrock of rigorous scientific inquiry. Many tools for both partially and fully automating their derivations have been developed over the last half a century. Some examples of state-of-the-art provers are E (Schulz, 2013), VAMPIRE (Kov\'acs & Voronkov, 2013), and Prover9 (McCune, 2005-2010). Newer theorem provers, such as E, use superposition calculus in place of more traditional resolution and tableau based methods. There have also been a number of past attempts to apply machine learning methods to guiding proof search. Suttner & Ertel proposed a multilayer-perceptron based method using hand-engineered features as far back as 1990; Urban et al (2011) apply machine learning to tableau calculus; and Loos et al (2017) recently proposed a method for guiding the E theorem prover using deep nerual networks. All of this prior work, however, has one common limitation: they all rely on the axioms of classical first-order logic. Very little attention has been paid to automated theorem proving for non-classical logics. One of the only recent examples is McLaughlin & Pfenning (2008) who applied the polarized inverse method to intuitionistic propositional logic. The literature is otherwise mostly silent. This is truly unfortunate, as there are many reasons to desire non-classical proofs over classical. Constructive/intuitionistic proofs should be of particular interest to computer scientists thanks to the well-known Curry-Howard correspondence (Howard, 1980) which tells us that all terminating programs correspond to a proof in intuitionistic logic and vice versa. This work explores using Q-learning (Watkins, 1989) to inform proof search for a specific system called non-classical logic called Core Logic (Tennant, 2017).