Tight Size-Degree Bounds for Sums-of-Squares Proofs

We exhibit families of $4$-CNF formulas over $n$ variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) $d$ but require SOS proofs of size $n^{\Omega(d)}$ for values of $d = d(n)$ from constant all the way up to $n^{\delta}$ for some universal constant$\delta$. This shows that the $n^{O(d)}$ running time obtained by using the Lasserre semidefinite programming relaxations to find degree-$d$ SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining $\mathsf{NP}$-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Kraj\'i\v{c}ek '04] and [Dantchev and Riis'03], and then applying a restriction argument as in [Atserias, M\"uller, and Oliva '13] and [Atserias, Lauria, and Nordstr\"om '14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively

Rank Lower Bounds in Propositional Proof Systems Based on Integer Linear Programming Methods

The work of this thesis is in the area of proof complexity, an area which looks to uncover the limitations of proof systems. In this thesis we investigate the rank complexity of tautologies for several of the most important proof systems based on integer linear programming methods. The three main contributions of this thesis are as follows: Firstly we develop the first rank lower bounds for the proof system based on the Sherali-Adams operator and show that both the Pigeonhole and Least Number Principles require linear rank in this system. We also demonstrate a link between the complexity measures of Sherali-Adams rank and Resolution width. Secondly we present a novel method for deriving rank lower bounds in the well-studied Cutting Planes proof system. We use this technique to show that the Cutting Plane rank of the Pigeonhole Principle is logarithmic. Finally we separate the complexity measures of Resolution width and Sherali-Adams rank from the complexity measures of Lovasz and Schrijver rank and Cutting Planes rank

Complexity Measures, Separations, and Automatability in Proof Complexity

With the rise of computer science, questions like ``can we solve a problem?'' got a quantitative counterpart: ``how hard is it to solve a problem?''. Proof complexity deals with the quantitative version of ``can we prove a theorem?'', namely, the question ``how hard is it to prove a theorem?''. The systematic study of the latter question for propositional proof systems started with Cook and Reckhow~\cite{CookReck74,CookReck79}, initiating a line of research which investigates how different proof systems for propositional logic compare with each other with respect to the minimum size needed to prove tautological formulas. A question that had remained open in the classification of Cook and Reckhow is whether cut-free sequent calculus can polynomially simulate resolution, in other words, whether adding atomic cuts to cut-free sequent calculus can super-polynomially decrease the size of proofs. We answer this question negatively, by showing a super-polynomial separation between cut-free sequent calculus as a system for refuting sets of clauses, and resolution. Now, while size is certainly the most well studied, and arguably the most important measure of the complexity of proofs, other complexity measures have emerged, along with a line of study about relations between them, lack of relations thereof, and the inherent impossibility of optimizing two different measures at once. We contribute to this line of research by identifying two new clusters of known proof complexity measures equal up to polynomial and $\log n$ factors. The first cluster contains the logarithm of tree-like resolution size, regularized clause and monomial space, and clause space, ordinary and regularized, in regular and tree-like resolution. Consequently, separating clause or monomial space from the logarithm of tree-like resolution size is equivalent to showing strong trade-offs between clause space and length, and equivalent to showing super-critical trade-offs between clause space and depth. The second cluster contains resolution width, $\Sigma_2$ space (a generalization of clause space to depth-2 Frege systems), ordinary and regularized, and the logarithm of tree-like $R(\log)$ size. As an application, we improve a known size-space trade-off for polynomial calculus with resolution. We further show a quadratic lower bound on tree-like resolution size for formulas refutable in clause space $4$, and introduce a measure intermediate between depth and the logarithm of tree-like resolution size that might be of independent interest. A third contribution of the dissertation is on the topic of automatability, that is the topic of how hard it is to find short proofs of a statement in a proof system. Significantly refining earlier results, Atserias and M{\" u}ller \cite{AtseMull20} showed that the problem of automatability is $\mathsf{NP}$-hard for resolution. Subsequently, this result was extended to stronger proof systems, including cutting planes, Res($k$), and algebraic proof systems. We extend it to bounded-depth Frege, showing that for any $d>0$, depth-$d$ Frege systems are $\mathsf{NP}$-hard to automate

Beyond 'Individualism' : personhood and transformation in the reclaiming pagan community of San Francisco

Many social scientists have sought to understand the dynamics of personhood in Western modernity, asking in particular whether it can be said that personhood in 'the West' is more individualistic than is typical elsewhere. Following Marcel Mauss, a number of anthropologists have suggested that the dominance of commodity exchange in modern Western societies lays a basis for individualised social relations over and above the relational patterns of gift exchange prevalent in many smaller-scale societies. Theorists from Weber to Foucault have likewise suggested that rationalised institutions in Western modernity condition an individualisation of subjectivity. Members of the San Francisco Reclaiming Pagan tradition seek to challenge the individualism, atomisation and rationalisation of social life they associate with wider US society, through ritual magic, activism and community-building. At times, they are able to create numinous worlds of beauty and interconnection against what Weber calls the "disenchantment of the world" (Weber [1919]1991 :155), helping to forge, in part, a more relational basis to their sociality. In doing so, they foreground many sites of relationality that exist in US society under a veneer of individualism, from gift exchange among kin networks to corporeal dissolution in crowds. Yet, their theories and cosmologies also valorise a particular type of artistic, expressive individualism, while their practices absorb and mirror some of the individualising and rationalising tendencies of wider systems and discourses they seek to resist. As a result, patterns of personhood and sociality in Reclaiming illustrate some of the complexities obtaining in US sociality more broadly. Examining these complexities highlights the individualising effects modern Euro-American institutions can have on subjectivity, while calling into question any overly-simplistic link between Western societies and 'individualism'. As such, this study can contribute to the project other anthropologists of personhood have begun: of problematising the dichotomy of 'Western-individualism' and 'non-Western-sociocentrism' which has at times underpinned anthropological studies of personhood

Synergy, inteligibility and revelation in neighbourhood places

In architectural and urban design the notion of place is highly desired, or in its absence, strongly criticised. Yet what is place and how might it be engendered by design? Over the last 30 years an extensive body of research on place has emerged, largely based on phenomenological approaches. This work gives rise to the question of whether place is a purely social concept completely divorced from physical space, or is linked to space and therefore amenable to design based intervention. Talen and Relph, for example, assert that there is no link between space and the social notion of place. This thesis attempts to approach place from a highly empirical and positivist methodology grounded in the theories known as space syntax but inspired by phenomenological approaches to place. The hypothesis presented here is that neighbourhoodplace, or sense of the genius loci of a place, is partially dependent on the global homogeneity of the relationships between spaces defining a region (the neighbourhood) combined with a local heterogeneity of the spatial properties that create a place’s identity. Results from a study show that a measure of total revelation (a measure of the difference in information content between a space and its immediately adjacent spaces) is consistent with the degree to which participants would locate a café/place, reinforcing other work done in the area and by environmental psychologists such as Kaplan and Kaplan. Total revelation serves as a powerful measure of the local heterogeneity of a location and hence a place’s identity. In further experiments presented in this thesis, neighbourhood boundaries were compared to the areas reported by inhabitants and against new measures of point synergy and point intelligibility, as well as a number of methods suggested by Raford and Hillier, Read, Yang and Hillier, and Peponis, along with a ‘null’ control measure. Evidence is presented suggesting that point synergy is the most effective method for predicting a neighbourhood’s extent from its spatial configuration, hence making it a suitable method to define the global homogeneity of a named district. This work concludes by suggesting that that while place may be unrelated to geographic location there is evidence to suggest that it is related to space (in the configurational or architectural sense) which would appear to contradict those who assert that the notion of place is wholly unrelated to the physical aspects of space. From an architectural perspective this thesis suggests that certain key aspects of spatial design are present in the affordance of social neighbourhoods