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

Temporal Logic with Recursion

We introduce extensions of the standard temporal logics CTL and LTL with a recursion operator that takes propositional arguments. Unlike other proposals for modal fixpoint logics of high expressive power, we obtain logics that retain some of the appealing pragmatic advantages of CTL and LTL, yet have expressive power beyond that of the modal ?-calculus or MSO. We advocate these logics by showing how the recursion operator can be used to express interesting non-regular properties. We also study decidability and complexity issues of the standard decision problems

The Digital Logic of Death

In The Digital Logic of Death, Steven Pustay skillfully makes visible the immensely important but often overlooked role that moving images play in shaping our understanding of mortality. This relationship, he argues, is made all the more urgent by the technologies of the digital age, which have profoundly altered our ability to represent and contemplate death through moving images, resulting in an entirely new cultural logic of death. To draw out this new logic, Pustay presents accessible readings of otherwise dense and difficult philosophical approaches to death – such as those found in existentialism, psychoanalysis, and critical theory – by reading them through the lens of contemporary media. From art-house films like Irréversible and The Fountain to blockbusters like the Matrix trilogy, from television commercials for M&M's to pay-cable dramas like The Sopranos and Breaking Bad, from first-person shooters like Bioshock to indie-games like LIMBO, Pustay shows how moving images have shifted our understanding of death in general and our recognition of our own finiteness in particular

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Verification of Non-Regular Program Properties

Most temporal logics which have been introduced and studied in the past decades can be embedded into the modal mu-calculus. This is the case for e.g. PDL, CTL, CTL*, ECTL, LTL, etc. and entails that these logics cannot express non-regular program properties. In recent years, some novel approaches towards an increase in expressive power have been made: Fixpoint Logic with Chop enriches the mu-calculus with a sequential composition operator and thereby allows to characterise context-free processes. The Modal Iteration Calculus uses inflationary fixpoints to exceed the expressive power of the mu-calculus. Higher-Order Fixpoint Logic (HFL) incorporates a simply typed lambda-calculus into a setting with extremal fixpoint operators and even exceeds the expressive power of Fixpoint Logic with Chop. But also PDL has been equipped with context-free programs instead of regular ones. In terms of expressivity there is a natural demand for richer frameworks since program property specifications are simply not limited to the regular sphere. Expressivity however usually comes at the price of an increased computational complexity of logic-related decision problems. For instance are the satisfiability problems for the above mentioned logics undecidable. We investigate in this work the model checking problem of three different logics which are capable of expressing non-regular program properties and aim at identifying fragments with feasible model checking complexity. Firstly, we develop a generic method for determining the complexity of model checking PDL over arbitrary classes of programs and show that the border to undecidability runs between PDL over indexed languages and PDL over context-sensitive languages. It is however still in PTIME for PDL over linear indexed languages and in EXPTIME for PDL over indexed languages. We present concrete algorithms which allow implementations of model checkers for these two fragments. We then introduce an extension of CTL in which the UNTIL- and RELEASE- operators are adorned with formal languages. These are interpreted over labeled paths and restrict the moments on such a path at which the operators are satisfied. The UNTIL-operator is for instance satisfied if some path prefix forms a word in the language it is adorned with (besides the usual requirement that until that moment some property has to hold and at that very moment some other property must hold). Again, we determine the computational complexities of the model checking problems for varying classes of allowed languages in either operator. It turns out that either enabling context-sensitive languages in the UNTIL or context-free languages in the RELEASE- operator renders the model checking problem undecidable while it is EXPTIME-complete for indexed languages in the UNTIL and visibly pushdown languages in the RELEASE- operator. PTIME-completeness is a result of allowing linear indexed languages in the UNTIL and deterministic context-free languages in the RELEASE. We do also give concrete model checking algorithms for several interesting fragments of these logics. Finally, we turn our attention to the model checking problem of HFL which we have already studied in previous works. On finite state models it is k-EXPTIME-complete for HFL(k), the fragment of HFL obtained by restricting functions in the lambda-calculus to order k. Novel in this work is however the generalisation (from the first-order case to the case for functions of arbitrary order) of an idea to improve the best and average case behaviour of a model checking algorithm by using partial functions during the fixpoint iteration guided by the neededness of arguments. This is possible, because the semantics of a closed HFL formula is not a total function but the value of a function at some argument. Again, we give a concrete algorithm for such an improved model checker and argue that despite the very high model checking complexity this improvement is very useful in practice and gives feasible results for HFL with lower order fuctions, backed up by a statistical analysis of the number of needed arguments on a concrete example. Furthermore, we show how HFL can be used as a tool for the development of algorithms. Its high expressivity allows to encode a wide variety of problems as instances of model checking already in the first-order fragment. The rather unintuitive -- yet very succinct -- problem encoding together with an analysis of the behaviour of the above sketched optimisation may give deep insights into the problem. We demonstrate this on the example of the universality problem for nondeterministic finite automata, where a slight variation of the optimised model checking algorithm yields one of the best known methods so far which was only discovered recently. We do also investigate typical model-theoretic properties for each of these logics and compare them with respect to expressive power

Wittgenstein and Poetry: Negotiations of the Inexpressible

This study performs a reading of Wittgenstein’s thought that integrates his sometimes sidelined remarks on aesthetics and belief, and emphasises consideration of language use on the level of practice. It analyses the many ways that Wittgenstein engages with the inexpressible or the limits of expression through comparison with poetry as a practice. The potential of a Wittgensteinian method of literary analysis concentrating on grammatical structures, exemplary forms of expression and quotidian meaning-making is shown by viewing several poets’ work in connection with specific forms of the inexpressible. This thesis consists of three parts. The first chapter surveys previous applications of Wittgenstein to aesthetic appreciation and analysis, and considers common interpretations of his earlier and later work. Incorporating a wide range of Wittgenstein sources allows a new reading to emerge that gives appropriate weight to his hitherto under-researched writings. This reading is tested in Chapters 2-5, in each case studying a poet or poets alongside a philosophical text or topic. Chapter 2 uses the negative theology of Pseudo-Dionysius to probe the ineffable; through Cora Diamond’s resolute reading of the Tractatus, Kei Miller’s ‘Church Women’ series and John Burnside’s intimate ineffable of ‘Parousia’, a grammatical understanding of inexpressibility emerges. Chapter 3 compares John McDowell’s minimal realism in Mind and World with Wallace Steven’s Supreme Fiction, demonstrating how Stevens’ – and Wittgenstein’s – rich conception of experience can close off a number of philosophical lacunae. Chapter 4 concentrates on the poetry of Jorie Graham, whose conception of the self is saturated with language. Parallels with Wittgenstein’s methodology are drawn, and some reminders issued to curb the excesses of postmodern accounts of subjectivity. The focus in Chapter 5 moves to the use of cartographical metaphor in Philosophical Investigations and Kei Miller’s poetry. The constraints of specific discourses on our thinking are examined, together with poetry’s potential for laying bare or reinvigorating the pictures by which we navigate. Finally, Chapter 6 discusses a selection of poetic projects completed alongside my research, to extend the reading of Wittgenstein into the area of creative practice. This thesis demonstrates Wittgenstein’s prolonged engagement with the limits of expression and with poetry, as well as the profit of a Wittgensteinian approach to poetry. It thereby questions a number of current responses to Wittgenstein’s work, and displays its own original creative outcomes.AHRC sponsored PhD studentshi

Showtime: the phenomenology of film consciousness

The thesis argues that the notion of film consciousness deepens a wide-range of philosophical issues in ways which are only accessible through film experience. These issues, directly related to the continental tradition, deal with consciousness, experience, intentionally and meaning. We look to the implications of the initial acts of film reproduction as it creates 'images' of the world which reconceptualise vision in terms of space, time and dimension. We move from ontology to experience and examine an aesthetic form with radical implications for spectator consciousness. These issues are explored from two philosophical positions. Firstly, phenomenology, especially Edmund Husserl and Maurice Merleau-Ponty. Secondly, the work of Gilles Deleuze who presents the most penetrating insights to date into film consciousness and its repercussions for thought and affectivity. The focus of this study is to draw together these two philosophical positions, showing their fundamental differences but also similarities where they exist. This approach is rarely attempted but the belief running through this thesis is that film is one arena which is invaluable for making such comparisons. It is argued philosophically that film writes large key phenomenological concepts on intentionality, time-consciousness and the relation of the lifeworld to the predicative. In terms of Deleuze, film is shown as a unique artform which in allowing us to link otherwise casts light on Deleuze's own complex system of thought. Chapters 1-3 are concerned with phenomenology and detail the role of film in terms of the lifeworld, intentionally, reduction and the transcendental in a way which has not been attempted elsewhere. The linking chapter on time (4) is used to introduce the work of Henri Bergson and its influence both on phenomenology's inner time-consciousness and Deleuze's fundamental categories of film movement and time imagery. The final two chapters look at the way film is reconfigured through montage and the implications of this for film's unique expression of movement and time

Virtuoso Beasts: Modernist Fables and the Vitality of Style

This dissertation examines the pivotal function of animals in modernist writing, particularly where modernist style confronts inherited moral codes. Classic accounts of modernism emphasize â??impersonalityâ?? as the prime method for artists seeking cultural and ethical authority in the period after 1880. This project digs into what I argue is ultimately the more palatable capacity of literary beasts to animate a similar poetics of authority. Where doctrines of impersonality often resorted to figures of the inorganic in order to simultaneously disavow and indulge the expression of authorial intention, modernists such as Gerard Manley Hopkins, Virginia Woolf, James Joyce, Rudyard Kipling, and Marianne Moore instead avowed their didactic ambitions by appealing not to traditional expressive and explicit methods nor, on the other hand, to the complete evacuation of personality, but to the vitality and instinct of animals. More than any platinum filament (T.S. Eliotâ??s famous catalyst), animals offered modernists a vocabulary for the bodily and behavioral mechanisms by which individuals become ethical and historical subjects. In my chapters, I examine specific formal effects that require animal energyâ??from the modeling of queer poetic virtuosities upon animal instinct (as in Hopkinsâ?? windhover, for example, or Mooreâ??s slapstick critters) and the casting of fictional characters along evolutionary-typological lines (as in Woolfâ??s The Waves) to the anticolonial implementation, even, of a â??bestialâ?? prose style resistant to modernismâ??s self-authorization (exemplified by the indifferent creatures we find in Finnegans Wake)

A Victorian Christmas in Hell: Yuletide Ghosts and Necessary Pleasures in the Age of Capital

This dissertation explores how the cultural and literary development – one might argue, creation – of a specifically Victorian Christmas arose in response to social anxieties related to the expansion of industrial capitalism, Darwinian theories of evolution, and the increasingly problematic definition of the family during the nineteenth century in Britain. Using a Lacanian psychoanalytic lens, the dissertation explores how the liminal figure of the ghost pervades the literary narrativization of the Christmas holiday, and how such ghosts provided uncanny comforts to a reading public increasingly horrified by social, economic, and natural forces seemingly beyond their control. The dissertation argues that the success of the literary institution of Christmas as a necessary assurance of cultural pleasure in the face of social anxiety has made it perhaps the most long-lived of Victorian artifacts. Indeed, the importance of Christmas to our late capitalist present suggests post-modernity is not so far removed from Victorian sensibilities as we might suspect. The first chapter of A Victorian Christmas in Hell examines the seminal role of Charles Dickens’ Christmas Carol in constructing the Victorian Christmas narrative, and its astonishing longevity as a living cultural touchstone to the present day. The chapter argues that A Christmas Carol’s success is due in large part to its construction of a seductive injunction to enjoy that deflects and transcends social anxiety. The second chapter examines the role of mourning and grief that pervades Victorian Christmas texts, and how this focus on melancholy paradoxically underwrites the command to enjoy Christmas. The third chapter focuses on how the uncertainty of the role of women in Victorian culture helped lend a volatility, urgency, and horror to the genre of the Christmas ghost story – a genre developed in large part by women writers. The fourth chapter explores the role of children in Christmas ghost stories, and argues that the social guilt of child abuse gave rise to the figure of the child ghost: a transcendent figure with the power to believe in the fantastic and phantasmic aspects of Christmas for an increasingly skeptical, and guilt-ridden adult population. A brief postscript considers the possible evolution of the Christmas narrative beyond its capitalist trappings. What would a post-Victorian Christmas look like after capitalism? Is it even possible, or desirable, to exorcise our Victorian Christmas ghosts