Abstract. We prove several decidability and undecidability results for the satisfiability and validity problems for languages that can express solutions of word equations with length constraints. The atomic formulas over this language are equality over string terms (word equations), linear inequality over the length function (length constraints), and membership in regular sets. These questions are important in logic, program analysis, and formal verification. Variants of these questions have been studied for many decades and practical satisfiability procedures (aka SMT solvers) for these formulas have become increasingly important in the analysis of string-manipulating programs such as web applications and scripts. We prove three main theorems. First, we give a new proof of undecidability for the validity problem for the set of sentences written as a ∀∃ quantifier alternation applied to positive word equations. A corollary of this undecidability result is that this set is undecidable even with sentences with at most two occurrences of a string variable. Second, we consider Boolean combinations of quantifier-free formulas constructed out of word equations and length constraints. We show that if word equations can be converted to a solved form, a form relevant in practice, then the satisfiability problem for Boolean combinations of word equations and length constraints is decidable. Third, we show that the satisfiability problem for quantifier-free formulas over word equations in regular solved form, length constraints, and the membership predicate over regular expressions is also decidable.

a b s t r a c tWe study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor-Bendixson ranks (of trees). We prove the following results: (1) The ordinal height of any automatic well-founded partial order is bounded by ω ω . (2) The ordinal heights of automatic well-founded relations are unbounded below ω CK 1 , the first non-computable ordinal. (3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ω CK 1 , ω CK 1 +1. (4) For any computable ordinal α, there is an automatic successor tree of Cantor-Bendixson rank α.

PrefaceThis paper grew out of three tutorial lectures on automatic structures given at the Logic Colloquium 2007 in Wroc law (Poland). The paper will follow the outline of the tutorial lectures, supplementing some material along the way. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Büchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Büchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Finite automata are the most commonly known in the general computer science community. Such an automaton reads finite input strings from left to right, making state transitions along the way. Depending on its last state after processing a given string, the automaton either accepts or rejects the input string. Automatic structures are mathematical objects which can be represented by (word, tree, Büchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research.We begin with some motivation and history for studying automatic structures. We introduce definitions of automatic structures, present examples, and discuss decidability and definability theorems. Next, we concentrate on finding natural isomorphism invariants for classes of automatic structures. These classes include well-founded partial orders, Boolean algebras, linear orders, trees, and finitely generated groups. Finally, we address the issue of complexity for automatic structures. In order to measure complexity of automatic structures we involve topology (via the Borel hierarchy), model theory (Scott ranks), computability theory (Σ 1 1 -completeness), and computational complexity (the class P).This paper consists of three sections based on the tutorial lectures. The first lecture provides motivating questions and historical context, formal definitions of the different types of automata involved, examples of automatic structures, and decidability and definability results about automatic structures. The second lecture considers tree and Rabin automatic structures, and outlines techniques for proving non-automaticity. We then turn our attention to the study of algorithmic and structural properties of automatic trees, Boolean algebras, and finitely generated groups. The final lecture presents a framework for reducing certain questions about computable structures to questions about automatic structures. These reductions have been used to show that, in some cases, the sharp bound on complexity of automatic structures is as high as the bounds for computable structures. We conclude by looking at Borel structures from descriptive set theory and connecting them to Büchi and Rabin automatic structures.1 Basics Motivating questionsThe study of structures has played a central role in the development of logic.In the course of this study, several themes have been pursued. We will see how questions related to each of these themes is addressed in the ...

No abstract

No abstract

No abstract

1This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced via such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another, and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognizes reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

Grading is a notoriously difficult and time-consuming part of teaching. For open-ended programming, mathematical, or design problems, assigning consistent scores and giving useful feedback can be very challenging. Large classes compound this difficulty. Adding TAs to the team can help parallelize the process but may impede grading consistency and quality. We present an adaptive rubric creation and application process to enable high-quality responses to student work, at scale. This process uses exploratory data analysis to discover common patterns in student responses to a problem, then tailors a rubric and feedback to address these patterns. Our method is supported by current grading tools, which allow calculation of the simple population-level statistics we need to extract meaningful features from a corpus of student work. In this case study, we describe using adaptive rubrics for a discrete math class for CS majors: the grading team found that this process produced concrete and transparent justifications of student scores and that it facilitated conversations around grading that were grounded in course learning objectives and values. CCS CONCEPTS • Social and professional topics → Student assessment; • Applied computing → Learning management systems;