Anthony Widjaja Lin

Michael Benedikt, Anthony Widjaja Lin, Di-De Yen

The study of graph queries in database theory has spanned more than three decades, resulting in a multitude of proposals for graph query languages. We can identify three main families of languages, with the canonical representatives being: (1) regular path queries, (2) walk logic, and (3) first-order logic with transitive closure operators. This paper provides a complete picture of the expressive power of these languages in the context of data graphs. Specifically, we consider a graph data model that supports querying over both data and topology. For example, ``Does there exist a path between two different persons in a social network with the same last name?''. We also show that an extension of (1) with regular path comparisons, augmented with transitive closure operators, can unify the expressivity of (1)--(3).

Pablo Barcelo, Alexander Kozachinskiy, Anthony Widjaja Lin et al.

We contribute to the study of formal languages that can be recognized by transformer encoders. We focus on two self-attention mechanisms: (1) UHAT (Unique Hard Attention Transformers) and (2) AHAT (Average Hard Attention Transformers). UHAT encoders are known to recognize only languages inside the circuit complexity class ${\sf AC}^0$, i.e., accepted by a family of poly-sized and depth-bounded boolean circuits with unbounded fan-ins. On the other hand, AHAT encoders can recognize languages outside ${\sf AC}^0$), but their expressive power still lies within the bigger circuit complexity class ${\sf TC}^0$, i.e., ${\sf AC}^0$-circuits extended by majority gates. We first show a negative result that there is an ${\sf AC}^0$-language that cannot be recognized by an UHAT encoder. On the positive side, we show that UHAT encoders can recognize a rich fragment of ${\sf AC}^0$-languages, namely, all languages definable in first-order logic with arbitrary unary numerical predicates. This logic, includes, for example, all regular languages from ${\sf AC}^0$. We then show that AHAT encoders can recognize all languages of our logic even when we enrich it with counting terms. We apply these results to derive new results on the expressive power of UHAT and AHAT up to permutation of letters (a.k.a. Parikh images).

Andy Yang, Anej Svete, Jiaoda Li et al.

Most expressivity results for transformers treat them as language recognizers (which accept or reject strings), and not as they are used in practice, as language models (which generate strings autoregressively and probabilistically). Here, we characterize the probability distributions that transformer language models can express. We show that making transformer language recognizers autoregressive can sometimes increase their expressivity, and that making them probabilistic can break equivalences that hold in the non-probabilistic case. Our overall contribution is to tease apart what functions transformers are capable of expressing, in their most common use-case as language models.

Hongjian Jiang, Matthew Hague, Philipp Rümmer et al.

C-RASP is a simple programming language that was recently shown to capture concepts expressible by transformers. In this paper, we develop new algorithmic techniques for automatically verifying C-RASPs. To this end, we establish a connection to the verification of synchronous dataflow programs in Lustre, which enables us to exploit state-of-the-art model checkers utilizing highly optimized SMT-solvers. Our second contribution addresses learning a C-RASP program in the first place. To this end, we provide a new algorithm for learning a C-RASP from examples using local search. We demonstrate efficacy of our implementation for benchmarks of C-RASPs in the literature, in particular in connection to the following applications: (1) transformer program optimization, and (2) constrained learning of transformer programs (based on a partial specification).

Hongjian Jiang, Michael Hahn, Georg Zetzsche et al.

Hard attention Chain-of-Thought (CoT) transformers are known to be Turing-complete. However, it is an open problem whether softmax attention Chain-of-Thought (CoT) transformers are Turing-complete. In this paper, we prove a stronger result that length-generalizable softmax CoT transformers are Turing-complete. More precisely, our Turing-completeness proof goes via the CoT extension of the Counting RASP (C-RASP), which correspond to softmax CoT transformers that admit length generalization. We prove Turing-completeness for CoT C-RASP with causal masking over a unary alphabet (more generally, for letter-bounded languages). While we show this is not Turing-complete for arbitrary languages, we prove that its extension with relative positional encoding is Turing-complete for arbitrary languages. We empirically validate our theory by training transformers for languages requiring complex (non-linear) arithmetic reasoning.

Matthew Hague, Anthony Widjaja Lin, Luke Ong

HTML5 applications normally have a large set of CSS (Cascading Style Sheets) rules for data display. Each CSS rule consists of a node selector (given in an XPath-like query language) and a declaration block (assigning values to selected nodes' display attributes). As web applications evolve, maintaining CSS files can easily become problematic. Some CSS rules will be replaced by new ones, but these obsolete (hence redundant) CSS rules often remain in the applications. Not only does this "bloat" the applications, but it also significantly increases web browsers' processing time. Most works on detecting redundant CSS rules in HTML5 applications do not consider the dynamic behaviors of HTML5 (specified in JavaScript); in fact, the only proposed method that takes these into account is dynamic analysis (a.k.a. testing), which cannot soundly prove redundancy of CSS rules. In this paper, we introduce an abstraction of HTML5 applications based on monotonic tree-rewriting and study its "redundancy problem". We establish the precise complexity of the problem and various subproblems of practical importance (ranging from P to EXP). In particular, our algorithm relies on an efficient reduction to an analysis of symbolic pushdown systems (for which highly optimised solvers are available), which yields a fast method for checking redundancy in practice. We implemented our algorithm and demonstrated its efficacy in detecting redundant CSS rules in HTML5 applications.