A first-order axiomatization of the theory of finite trees

We provide first-order axioms for the theories of finite trees with bounded branching and finite trees with arbitrary (finite) branching. The signature is chosen to express, in a natural way, those properties of trees most relevant to linguistic theories. These axioms provide a foundation for results in linguistics that are based on reasoning formally about such properties. We include some observations on the expressive power of these theories relative to traditional language complexity classes

What Does a Grammar Formalism Say About a Language?

Over the last ten or fifteen years there has been a shift in generative linguistics away from formalisms based on a procedural interpretation of grammars towards constraint-based formalisms—formalisms that define languages by specifying a set of constraints that characterize the set of well-formed structures analyzing the strings in the language. A natural extension of this trend is to define this set of structures model-theoretically—to define it as the set of mathematical structures that satisfy some set of logical axioms. This approach raises a number of questions about the nature of linguistic theories and the role of grammar formalisms in expressing them. We argue here that the crux of what theories of syntax have to say about language lies in the abstract properties of the sets of structures they license. This is the level that is most directly connected to the empirical basis of these theories and it is the level at which it is possible to make meaningful comparisons between the approaches. From this point of view, grammar formalisms, or (formal frameworks) are primarily means of presenting these properties. Many of the apparent distinctions between formalisms, then, may well be artifacts of their presentation rather than substantive distinctions between the properties of the structures they license. The model-theoretic approach offers a way in which to abstract away from the idiosyncrasies of these presentations. Having said that, we must distinguish between the class of sets of structures licensed by a linguistic theory and the set of structures licensed by a specific instance of the theory—by a grammar expressing that theory. Theories of syntax are not simply accounts of the structure of individual languages in isolation, but rather include assertions about the organization of the structure of human languages in general. These universal aspects of the theories present two challenges for the model-theoretic approach. First, they frequently are not properties of individual structures, but are rather properties of sets of structures. Thus, in capturing these model-theoretically one is not defining sets of structures but is rather defining classes of sets of structures; these are not first order properties. Secondly, the universal aspects of linguistic theories are frequently not explicit, but are consequences of the nature of the formalism that embodies the theory. In capturing these one must develop an explicit axiomatic treatment of the formalism. This is both a challenge and a powerful beneft of the approach. Such re-interpretations tend to raise a variety of issues that are often overlooked in the original formalization. In this report we examine these issues within the context of a model-theoretic reinterpretation of Generalized Phrase-Structure Grammar. While there is little current active research on GPSG, it provides an ideal laboratory for exploring these issues. First, the formalism of GPSG is expressly intended to embody a great deal of the accompanying linguistic theory. Thus it provides a variety of opportunities for examining principles expressed as restrictions on the formalism from a model-theoretic point of view. At the same time, the fact that these restrictions embody universal grammar principles provides us with a variety of opportunities to explore the way in which the linguistic theory expressed by a grammar can transcend the mathematical theory of the structures it licenses. Finally, GPSG, although defined declaratively, is a formalism with restricted generative capacity, a characteristic more typical of the earlier procedural formalisms. As such, one component of the theory it embodies is a claim about the language-theoretic complexity of natural languages. Such claims are difficult to establish for any of the constraint-based approaches to grammar. We can show, however, that the class of sets of trees that are definable within the logical language we employ in reformalizing GPSG is nearly exactly the class of sets of trees definable within the basic GPSG formalism. Thus we are able to capture the language-theoretic consequences of GPSGs restricted formalism by employing a restricted logical language

Mixed-initiative control of intelligent systems

Mixed-initiative user interfaces provide a means by which a human operator and an intelligent system may collectively share the task of deciding what to do next. Such interfaces are important to the effective utilization of real-time expert systems as assistants in the execution of critical tasks. Presented here is the Incremental Inference algorithm, a symbolic reasoning mechanism based on propositional logic and suited to the construction of mixed-initiative interfaces. The algorithm is similar in some respects to the Truth Maintenance System, but replaces the notion of 'justifications' with a notion of recency, allowing newer values to override older values yet permitting various interested parties to refresh these values as they become older and thus more vulnerable to change. A simple example is given of the use of the Incremental Inference algorithm plus an overview of the integration of this mechanism within the SPECTRUM expert system for geological interpretation of imaging spectrometer data

Efficient instance and hypothesis space revision in Meta-Interpretive Learning

Inductive Logic Programming (ILP) is a form of Machine Learning. The goal of ILP is to induce hypotheses, as logic programs, that generalise training examples. ILP is characterised by a high expressivity, generalisation ability and interpretability. Meta-Interpretive Learning (MIL) is a state-of-the-art sub-field of ILP. However, current MIL approaches have limited efficiency: the sample and learning complexity respectively are polynomial and exponential in the number of clauses. My thesis is that improvements over the sample and learning complexity can be achieved in MIL through instance and hypothesis space revision. Specifically, we investigate 1) methods that revise the instance space, 2) methods that revise the hypothesis space and 3) methods that revise both the instance and the hypothesis spaces for achieving more efficient MIL. First, we introduce a method for building training sets with active learning in Bayesian MIL. Instances are selected maximising the entropy. We demonstrate this method can reduce the sample complexity and supports efficient learning of agent strategies. Second, we introduce a new method for revising the MIL hypothesis space with predicate invention. Our method generates predicates bottom-up from the background knowledge related to the training examples. We demonstrate this method is complete and can reduce the learning and sample complexity. Finally, we introduce a new MIL system called MIGO for learning optimal two-player game strategies. MIGO learns from playing: its training sets are built from the sequence of actions it chooses. Moreover, MIGO revises its hypothesis space with Dependent Learning: it first solves simpler tasks and can reuse any learned solution for solving more complex tasks. We demonstrate MIGO significantly outperforms both classical and deep reinforcement learning. The methods presented in this thesis open exciting perspectives for efficiently learning theories with MIL in a wide range of applications including robotics, modelling of agent strategies and game playing.Open Acces

Constructive problem solving : a model construction approach towards configuration

In this paper we give a formalisation of configuration as the task to construct for a given specification, which is understood as a finite set of logical formulas, a model that satisfies the specification. In this approach, a specification consists of two parts. One part describes the domain, the possible components, and their interdependencies. The other part specifies the particular object that is to be configured. The language that is used to represent knowledge about configuration problems integrates three sublanguages that allow one to express constraints, to build up taxonomies, and to define rules. We give a sound calculus by which one can compute solutions to configuration problems if they exist and that allows one to recognize that a specification is inconsistent. In particular, the calculus can be used in order to check whether a given configuration satisfies the specification

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Proceedings of the 21st Conference on Formal Methods in Computer-Aided Design – FMCAD 2021

The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing

Policy Search Based Relational Reinforcement Learning using the Cross-Entropy Method

Relational Reinforcement Learning (RRL) is a subfield of machine learning in which a learning agent seeks to maximise a numerical reward within an environment, represented as collections of objects and relations, by performing actions that interact with the environment. The relational representation allows more dynamic environment states than an attribute-based representation of reinforcement learning, but this flexibility also creates new problems such as a potentially infinite number of states. This thesis describes an RRL algorithm named Cerrla that creates policies directly from a set of learned relational “condition-action” rules using the Cross-Entropy Method (CEM) to control policy creation. The CEM assigns each rule a sampling probability and gradually modifies these probabilities such that the randomly sampled policies consist of ‘better’ rules, resulting in larger rewards received. Rule creation is guided by an inferred partial model of the environment that defines: the minimal conditions needed to take an action, the possible specialisation conditions per rule, and a set of simplification rules to remove redundant and illegal rule conditions, resulting in compact, efficient, and comprehensible policies. Cerrla is evaluated on four separate environments, where each environment has several different goals. Results show that compared to existing RRL algorithms, Cerrla is able to learn equal or better behaviour in less time on the standard RRL environment. On other larger, more complex environments, it can learn behaviour that is competitive to specialised approaches. The simplified rules and CEM’s bias towards compact policies result in comprehensive and effective relational policies created in a relatively short amount of time