Computational Structure of Human Language

The central thesis of this report is that human language is NP-complete. That is, the process of comprehending and producing utterances is bounded above by the class NP, and below by NP-hardness. This constructive complexity thesis has two empirical consequences. The first is to predict that a linguistic theory outside NP is unnaturally powerful. The second is to predict that a linguistic theory easier than NP-hard is descriptively inadequate. To prove the lower bound, I show that the following three subproblems of language comprehension are all NP-hard: decide whether a given sound is possible sound of a given language; disambiguate a sequence of words; and compute the antecedents of pronouns. The proofs are based directly on the empirical facts of the language user's knowledge, under an appropriate idealization. Therefore, they are invariant across linguistic theories. (For this reason, no knowledge of linguistic theory is needed to understand the proofs, only knowledge of English.) To illustrate the usefulness of the upper bound, I show that two widely-accepted analyses of the language user's knowledge (of syntactic ellipsis and phonological dependencies) lead to complexity outside of NP (PSPACE-hard and Undecidable, respectively). Next, guided by the complexity proofs, I construct alternate linguisitic analyses that are strictly superior on descriptive grounds, as well as being less complex computationally (in NP). The report also presents a new framework for linguistic theorizing, that resolves important puzzles in generative linguistics, and guides the mathematical investigation of human language.

A Virtual Machine for a Type-omega Denotational Proof Language

In this thesis, I designed and implemented a virtual machine (VM) for a monomorphic variant of Athena, a type-omega denotational proof language (DPL). This machine attempts to maintain the minimum state required to evaluate Athena phrases. This thesis also includes the design and implementation of a compiler for monomorphic Athena that compiles to the VM. Finally, it includes details on my implementation of a read-eval-print loop that glues together the VM core and the compiler to provide a full, user-accessible interface to monomorphic Athena. The Athena VM provides the same basis for DPLs that the SECD machine does for pure, functional programming and the Warren Abstract Machine does for Prolog.

Procedures as a Representation for Data in a Computer Program for Understanding Natural Language

This paper describes a system for the computer understanding of English. The system answers questions, executes commands, and accepts information in normal English dialog. It uses semantic information and context to understand discourse and to disambiguate sentences. It combines a complete syntactic analysis of each sentence with a "heuristic understander" which uses different kinds of information about a sentence, other parts of the discourse, and general information about the world in deciding what the sentence means. It is based on the belief that a computer cannot deal reasonably with language unless it can "understand" the subject it is discussing. The program is given a detailed model of the knowledge needed by a simple robot having only a hand and an eye. We can give it instructions to manipulate toy objects, interrogate it about the scene, and give it information it will use in deduction. In addition to knowing the properties of toy objects, the program has a simple model of its own mentality. It can remember and discuss its plans and actions as well as carry them out. It enters into a dialog with a person, responding to English sentences with actions and English replies, and asking for clarification when its heuristic programs cannot understand a sentence through use of context and physical knowledge.

The Logical Problem of Language Change

This paper considers the problem of language change. Linguists must explain not only how languages are learned but also how and why they have evolved along certain trajectories and not others. While the language learning problem has focused on the behavior of individuals and how they acquire a particular grammar from a class of grammars ${cal G}$, here we consider a population of such learners and investigate the emergent, global population characteristics of linguistic communities over several generations. We argue that language change follows logically from specific assumptions about grammatical theories and learning paradigms. In particular, we are able to transform parameterized theories and memoryless acquisition algorithms into grammatical dynamical systems, whose evolution depicts a population's evolving linguistic composition. We investigate the linguistic and computational consequences of this model, showing that the formalization allows one to ask questions about diachronic that one otherwise could not ask, such as the effect of varying initial conditions on the resulting diachronic trajectories. From a more programmatic perspective, we give an example of how the dynamical system model for language change can serve as a way to distinguish among alternative grammatical theories, introducing a formal diachronic adequacy criterion for linguistic theories.

A Dynamical Systems Model for Language Change

Formalizing linguists' intuitions of language change as a dynamical system, we quantify the time course of language change including sudden vs. gradual changes in languages. We apply the computer model to the historical loss of Verb Second from Old French to modern French, showing that otherwise adequate grammatical theories can fail our new evolutionary criterion.

Complexity of Human Language Comprehension

The goal of this article is to reveal the computational structure of modern principle-and-parameter (Chomskian) linguistic theories: what computational problems do these informal theories pose, and what is the underlying structure of those computations? To do this, I analyze the computational complexity of human language comprehension: what linguistic representation is assigned to a given sound? This problem is factored into smaller, interrelated (but independently statable) problems. For example, in order to understand a given sound, the listener must assign a phonetic form to the sound; determine the morphemes that compose the words in the sound; and calculate the linguistic antecedent of every pronoun in the utterance. I prove that these and other subproblems are all NP-hard, and that language comprehension is itself PSPACE-hard.