The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer Scienc

A Simple and Scalable Static Analysis for Bound Analysis and Amortized Complexity Analysis

We present the first scalable bound analysis that achieves amortized complexity analysis. In contrast to earlier work, our bound analysis is not based on general purpose reasoners such as abstract interpreters, software model checkers or computer algebra tools. Rather, we derive bounds directly from abstract program models, which we obtain from programs by comparatively simple invariant generation and symbolic execution techniques. As a result, we obtain an analysis that is more predictable and more scalable than earlier approaches. Our experiments demonstrate that our analysis is fast and at the same time able to compute bounds for challenging loops in a large real-world benchmark. Technically, our approach is based on lossy vector addition systems (VASS). Our bound analysis first computes a lexicographic ranking function that proves the termination of a VASS, and then derives a bound from this ranking function. Our methodology achieves amortized analysis based on a new insight how lexicographic ranking functions can be used for bound analysis

A CORPUS STUDY OF THE DEVELOPMENT OF THE ADJECTIVE PHRASE IN FRENCH CHILDREN

In this thesis I attempt to answer three questions: H1) Do children use proportionally more prenominal or post-nominal placement of adjectives than adults? H2) Are children more conservative or more creative in their behavior in alternating prenominal and post-nominal placement of adjectives? H3) If colored terms are more frequent in child speech will they pattern more like prenominal adjectives or more like post nominal adjectives, as in adult speech? To do this, I examine two general semantic viewpoints, opting to use Scontras & Goodman (2017) subjectivity hypothesis. Next, I provide a general overview of First Language Acquisition research and then I turn to specifics of French adjective semantics and syntax, paying particular attention to factors that influence the preferential placement of an individual adjective. I next turn to some psychological factors, making certain types of adjectives especially difficult or easy to learn. I conclude by extending the work of Fox (2012). All this information is to provide the reader theoretical background to understand children’s adjective placement. The real answers come through a corpus investigation of how French children are treating adjectives in the earliest stages of development. Methodologically I answer my three questions by using three corpora from the CHILDES database (MacWhinney 2000). I also create an adult control group from a spoken French corpus. I run mixed effects models to project the behavior of adjectives past the sampling age using R. In the end, I discover that children are more conservative at this early stage. This can be seen by the greater number of post-nominal adjectives. I define conservative behavior as sticking more closely to either position (prenominal or post-nominal) than adults. For example, if a child uses an adjective more closely to 100% prenominal or 0% prenominal than adults, the child is being more conservative than an adult. I also find that children use proportionally more color terms than adults and are more creative with some common color terms. Size and color terms were found to be quickly learned

Computing Long-Distance Dependencies in Vowel Harmony

This article develops an explicit procedural model of vowel harmony, and takes steps toward finding a lower bound on the computational power of phonological rules. The focus on formalization and procedural computation allows for simplification in models of representation and the discovery of interesting interactions involving the conditions in rules. It is shown that locality principles are derivable, which motivates the elimination of iterative rule application advocated here. Along the way, a novel analysis of neutral vowels in harmony processes is also provided

Effective Theories for Circuits and Automata

Abstracting an effective theory from a complicated process is central to the study of complexity. Even when the underlying mechanisms are understood, or at least measurable, the presence of dissipation and irreversibility in biological, computational and social systems makes the problem harder. Here we demonstrate the construction of effective theories in the presence of both irreversibility and noise, in a dynamical model with underlying feedback. We use the Krohn-Rhodes theorem to show how the composition of underlying mechanisms can lead to innovations in the emergent effective theory. We show how dissipation and irreversibility fundamentally limit the lifetimes of these emergent structures, even though, on short timescales, the group properties may be enriched compared to their noiseless counterparts.Comment: 11 pages, 9 figure

Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

On past participle agreement in transitive clauses in French

This paper provides a Minimalist analysis of past participle agreement in French in transitive clauses. Our account posits that the head v of vP in such structures carries an (accusativeassigning) structural case feature which may apply (with or without concomitant agreement) to case-mark a clause-mate object, the subject of a defective complement clause, or an intermediate copy of a preposed subject in spec-CP. In structures where a goal is extracted from vP (e.g. via wh-movement) v also carries an edge feature, and may also carry a specificity feature and a set of (number and gender) agreement features. We show how these assumptions account for agreement of a participle with a preposed specific clausemate object or defective-clause subject, and for the absence of agreement with an embedded object, with the complement of an impersonal verb, and with the subject of an embedded (finite or nonfinite) CP complement. We also argue that the absence of agreement marking (in expected contexts) on the participles faitmade and laissélet in infinitive structures is essentially viral in nature. Finally, we claim that obligatory participle agreement with reflexive and reciprocal objects arises because the derivation of reflexives involves A-movement and concomitant agreement