Syntax without Abstract Objects

In line with the nominalistic denial of the existence of abstract objects, a basic theory of syntax for formal languages is developed and shown to satisfy certain fundamental requirements

First-Order Concatenation Theory with Bounded Quantifiers

We study first-order concatenation theory with bounded quantifiers. We give axiomatizations with interesting properties, and we prove some normal-form results. Finally, we prove a number of decidability and undecidability results.Comment: arXiv admin note: text overlap with arXiv:1804.0636

The Punctator\u27s World: A Discursion (Part Nine)

In the writing ofauthors Henryjames, Robert Louis Stevenson, D. H. Lawrence, Virginia Woolf, james joyce, E. E. Cummings, Ezra Pound, George Orwell, and Ernest Hemingway, Robinson traces the development in the twentieth century of two rival styles, one plaindealing and the other complected. In the literary skirmish between the two, the latter may be losing-perhaps at the expense of our reasoning powers

Finding the limit of incompleteness I

In this paper, we examine the limit of applicability of G\"{o}del's first incompleteness theorem ($\sf G1$ for short). We first define the notion "$\sf G1$ holds for the theory $T$". This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\sf G1$ holds. To approach this question, we first examine the following question: is there a theory $T$ such that Robinson's $\mathbf{R}$ interprets $T$ but $T$ does not interpret $\mathbf{R}$ (i.e. $T$ is weaker than $\mathbf{R}$ w.r.t. interpretation) and $\sf G1$ holds for $T$? In this paper, we show that there are many such theories based on Je\v{r}\'{a}bek's work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle$, we can construct a r.e. theory $U_{\langle A,B\rangle}$ such that $U_{\langle A,B\rangle}$ is weaker than $\mathbf{R}$ w.r.t. interpretation and $\sf G1$ holds for $U_{\langle A,B\rangle}$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf{0}< \mathbf{d}<\mathbf{0}^{\prime}$, there is a theory $T$ with Turing degree $\mathbf{d}$ such that $\sf G1$ holds for $T$ and $T$ is weaker than $\mathbf{R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield's work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\sf G1$ holds.Comment: 18 pages. Accepted and to appear in Bulletin of Symbolic Logi

Bases for Structures and Theories I

Sometimes structures or theories are formulated with different sets of primitives and yet are definitionally equivalent. In a sense, the transformations between such equivalent formulations are rather like basis transformations in linear algebra or co-ordinate transformations in geometry. Here an analogous idea is investigated. Let a relational signature $P = \{P_i\}_{i \in I_P}$ be given. For a set $\Phi = \{\phi_i\}_{i \in I_{\Phi}}$ of $L_P$-formulas, we introduce a corresponding set $Q = \{Q_i\}_{i \in I_{\Phi}}$ of new relation symbols and a set of explicit definitions of the $Q_i$ in terms of the $\phi_i$. This is called a definition system, denoted $d_{\Phi}$. A definition system $d_{\Phi}$ determines a \emph{translation function} $\tau_{\Phi} : L_Q \to L_P$. Any $L_P$-structure $A$ can be uniquely definitionally expanded to a model $A^{+} \models d_{\Phi}$, called $A + d_{\Phi}$. The reduct $A + d_{\Phi}$ to the $Q$-symbols is called the \emph{definitional image} $D_{\Phi}A$ of $A$. Likewise, a theory $T$ in $L_P$ may be extended a definitional extension $T + d_{\Phi}$; the restriction of this extension $T + d_{\Phi}$ to $L_Q$ is called the \emph{definitional image} $D_{\Phi}T$ of $T$. If $T_1$ and $T_2$ are in disjoint signatures and $T_1 + d_{\Phi} \equiv T_2 + d_{\Theta}$, we say that $T_1$ and $T_2$ are \emph{definitionally equivalent} (wrt the definition systems $d_{\Phi}$ and $d_{\Theta}$). Some results relating these notions are given, culminating in two characterization theorems for the definitional equivalence of structures and theories

Natural deduction and coherence for weakly distributive categories

AbstractThis paper examines coherence for certain monoidal categories using techniques coming from the proof theory of linear logic, in particular making heavy use of the graphical techniques of proof nets. We define a two sided notion of proof net, suitable for categories like weakly distributive categories which have the two-tensor structure (times/par) of linear logic, but lack a negation operator. Representing morphisms in weakly distributive categories as such nets, we derive a coherence theorem for such categories. As part of this process, we develop a theory of expansion-reduction systems with equalities and a term calculus for proof nets, each of which is of independent interest. In the symmetric case the expansion-reduction system on the term calculus yields a decision procedure for the equality of maps for free weakly distributive categories.The main results of this paper are these. First we have proved coherence for the full theory of weakly distributive categories, extending similar results for monoidal categories to include the treatment of the tensor units. Second, we extend these coherence results to the full theory of ∗-autonomous categories — providing a decision procedure for the maps of free symmetric ∗-autonomous categories. Third, we derive a conservative extension result for the passage from weakly distributive categories to ∗-autonomous categories. We show strong categorical conservativity, in the sense that the unit of the adjunction between weakly distributive and ∗-autonomous categories is fully faithful

Interpreting Attention-Based Models for Natural Language Processing

Large pre-trained language models (PLMs) such as BERT and XLNet have revolutionized the field of natural language processing (NLP). The interesting thing is that they are pre- trained through unsupervised tasks, so there is a natural curiosity as to what linguistic knowledge these models have learned from only unlabeled data. Fortunately, these models’ architectures are based on self-attention mechanisms, which are naturally interpretable. As such, there is a growing body of work that uses attention to gain insight as to what linguistic knowledge is possessed by these models. Most attention-focused studies use BERT as their subject, and consequently the field is sometimes referred to as BERTology. However, despite surpassing BERT in a large number of NLP tasks, XLNet has yet to receive the same level of attention (pun intended). Additionally, there is an interest in their field in how these pre-trained models change when fine-tuned for supervised tasks. This paper details many different attention-based interpretability analyses and performs each on BERT, XLNet, and a version of XLNet fine-tuned for a Twitter hate-speech-spreader detection task. The purpose of doing so is 1. to be a comprehensive summary of the current state of BERTology 2. to be the first to do many of these in-depth analyse on XLNet and 3. to study how PLMs’ attention patterns change over fine-tuning. I find that most identified linguistic phenomenon present in the attention patterns of BERT are also present in those of XLNet to similar extents. Further, it is shown that much about the internal organization and function of PLMs, and how they change over fine-tuning, can be understood through attention