The graph edit distance (GED) measures the dissimilarity between two graphs as the minimal cost of a sequence of elementary operations transforming one graph into another. This measure is fundamental in many areas such as structural pattern recognition or classification. However, exactly computing GED is NP-hard. Among different classes of heuristic algorithms that were proposed to compute approximate solutions, local search based algorithms provide the tightest upper bounds for GED. In this paper, we present K-REFINE and RANDPOST. K-REFINE (...) generalizes and improves an existing local search algorithm and performs particularly well on small graphs. RANDPOST is a general warm start framework that stochastically generates promising initial solutions to be used by any local search based GED algorithm. It is particularly efficient on large graphs. An extensive empirical evaluation demonstrates that both K-REFINE and RANDPOST perform excellently in practice. (shrink)

We introduce the equational notion of a monadic bounded algebra (MBA), intended to capture algebraic properties of bounded quantification. The variety of all MBA's is shown to be generated by certain algebras of two-valued propositional functions that correspond to models of monadic free logic with an existence predicate. Every MBA is a subdirect product of such functional algebras, a fact that can be seen as an algebraic counterpart to semantic completeness for monadic free logic. The analysis involves the (...) representation of MBA's as powerset algebras of certain directed graphs with a set of "marked" points. It is shown that there are only countably many varieties of MBA's, all are generated by their finite members, and all have finite equational axiomatisations classifying them into fourteen kinds of variety. The universal theory of each variety is decidable. Finitely generated MBA's are shown to be finite, with the free algebra on r generators having exactly $2^{3.2^{r.} } ^{2^{{\rm{2}}^{{\rm{r - 1}}} } } $ elements. An explicit procedure is given for constructing this freely generated algebra as the powerset algebra of a certain marked graph determined by the number r. (shrink)

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.

We define theories of bounded arithmetic, whose definable functions and relations are exactly those in certain complexity classes. Based on a recursion-theoretic characterization of NC in Clote , the first-order theory TNC, whose principal axiom scheme is a form of short induction on notation for nondeterministic polynomial-time computable relations, has the property that those functions having nondeterministic polynomial-time graph Θ such that TNC x y Θ are exactly the functions in NC, computable on a parallel random-access machine in (...) polylogarithmic parallel time with a polynomial number of processors.0 We then define three theories of weak second-order arithmetic which respectively characterize relations in the classes of alternating logarithmic time, logspace and nondeterministic logspace. (shrink)

A cardinal characteristic can often be described as the smallest size of a family of sequences which has a given property. Instead of this traditional concern for a smallest realization of the given property, a basically new approach, taken in [4] and [5], asks for a realization whose members are sequences of labels that correspond to 1-way infinite paths in a labelled graph. We study this approach as such, establishing tools that are applicable to all these cardinal characteristics. As (...) an application, we demonstrate the power of the tools developed by presenting a short proof of the bounded graph conjecture [4]. (shrink)

We develop a combinatorial model to study the evolution of graphs underlying proofs during the process of cut elimination. Proofs are two-dimensional objects and differences in the behavior of their cut elimination can often be accounted for by differences in their two-dimensional structure. Our purpose is to determine geometrical conditions on the graphs of proofs to explain the expansion of the size of proofs after cut elimination. We will be concerned with exponential expansion and we give upper and lower bounds (...) which depend on the geometry of the graphs. The lower bound is computed passing through the notion of universal covering for directed graphs. In this paper we present ground material for the study of cut elimination and structure of proofs in purely combinatorial terms. We develop a theory of duplication for directed graphs and derive results on graphs of proofs as corollaries. (shrink)

A function is diagonally non-computable if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function implies Ramsey-type weak König’s lemma. In this paper, we prove that for every computable order h, there exists an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -model of h-DNR which is not a (...) not model of the Ramsey-type graph coloring principle for two colors and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -models. (shrink)

Analogues of Ramsey’s Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author’s recent result for the triangle-free Henson graph, we prove (...) that for each [Formula: see text], the k-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey’s Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker. (shrink)

We prove that a finitely generated group is context-free whenever its Cayley-graph has a decidable monadic second-order theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of context-free groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is context-free whenever its monadic second-order theory is decidable. Further, it is shown (...) that the word problem of a finitely generated group is decidable if and only if the first-order theory of its Cayley-graph is decidable. (shrink)

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and (...) culminating in work of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

This article presents a simplified proof of the result that bounded depth propositional proofs of the pigeonhole principle are exponentially large. The proof uses the new techniques for proving switching lemmas developed by Razborov and Beame. A similar result is also proved for some examples based on graphs.

We formalize the construction of Paterson’s variant of the Ajtai–Komlós–Szemerédi sorting network of logarithmic depth in the bounded arithmetical theory , under the assumption of the existence of suitable expander graphs. We derive a conditional p-simulation of the propositional sequent calculus in the monotone sequent calculus.

In this essay, we propose that Peirce’s Existential Graphs can derive the desired uniqueness implication (or in a weaker claim, the definite description readings) of donkey pronouns in conjunctive discourse (A man walks in the park. He whistles), without postulating a separate category of E-type pronouns.

In this article the structure of the models of decidable monadic theories of planar graphs is investigated. It is shown that if the monadic theory of a class K of planar graphs is decidable, then the tree-width in the sense of Robertson and Seymour of the elements of K is universally bounded and there is a class T of trees such that the monadic theory of K is interpretable in the monadic theory of T.

In every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first spanning tree. Applications are given to the characterization of the classes of graphs and hypergraphs having decidable monadic theories.

This work deals with the expressive power of logics on finite graphs with access to an additional "arbitrary" linear order. The queries that can be expressed this way are the order-invariant queries for the logic. For the standard logics used in computer science, such as first-order logic, it is known that access to an arbitrary linear order increases the expressiveness of the logic. However, when we look at the separating examples, we find that they have satisfying models whose Gaifman (...) class='Hi'>Graph is complex -unbounded in valence and in treewidth. We thus explore the expressiveness of order-invariant queries over well-behaved graphs. We prove that first-order order-invariant queries over strings and trees have no additional expressiveness over first-order logic in the original signature. We also prove new upper bounds on order-invariant queries over bounded treewidth and bounded valence graphs. Our results make use of a new technique of independent interest: the application of algebraic characterizations of definability to show collapse results. (shrink)

The Wiener index WI is one of the connectivity parameters used to know the biochemical and physicochemical properties of compounds depending upon their molecular structures. Intuitionistic fuzzy graphs IFG s are a convenient tool to represent the objects and relations between them with two types of information using truth membership degree and falsity membership degree. This research work presents the concept of WI under the structure IFG s, I F trees, and I F cycles. Some bounds on WI are investigated. (...) The relationship between WI and connectivity index CI is also studied. In the end of this study, an application of the WI in transport network flow is proposed. (shrink)

For automatic and recursive graphs, we investigate the following problems: (A) existence of a Hamiltonian path and existence of an infinite path in a tree (B) existence of an Euler path, bounding the number of ends, and bounding the number of infinite branches in a tree (C) existence of an infinite clique and an infinite version of set cover The complexity of these problems is determined for automatic graphs and, supplementing results from the literature, for recursive graphs. Our results show (...) that these problems (A) are equally complex for automatic and for recursive graphs ( $\Sigma _{1}^{1}\text{-complete}$ ), (B) are moderately less complex for automatic than for recursive graphs (complete for different levels of the arithmetic hierarchy), (C) are much simpler for automatic than for recursive graphs (decidable and $\Sigma _{1}^{1}\text{-complete}$ , resp.). (shrink)

We say that a first order sentence A defines a graph G if A is true on G but false on any graph non-isomorphic to G. Let L ) denote the minimum length of such a sentence. We define the succinctness function s ) to be the minimum L ) over all graphs on n vertices.We prove that s and q may be so small that for no general recursive function f we can have f)≥n for all n. (...) However, for the function q*=maxi≤nq, which is the least nondecreasing function bounding q from above, we have q*=)log*n, where log*n equals the minimum number of iterations of the binary logarithm sufficient to lower n to 1 or below.We show an upper bound qshrink)

We consider Choiceless Polynomial Time , a language introduced by Blass, Gurevich and Shelah, and show that it can express a query originally constructed by Cai, Fürer and Immerman to separate fixed-point logic with counting from image. This settles a question posed by Blass et al. The program we present uses sets of unbounded finite rank: we demonstrate that this is necessary by showing that the query cannot be computed by any program that has a constant bound on the rank (...) of sets used, even in image, an extension of image with counting. (shrink)

We define a new NP search problem, the “local improvement” principle, about labellings of an acyclic, bounded-degree graph. We show that, provably in , it characterizes the consequences of and that natural restrictions of it characterize the consequences of and of the bounded arithmetic hierarchy. We also show that over V0 it characterizes the consequences of V1 and hence that, in some sense, a miniaturized version of the principle gives a new characterization of the consequences of . (...) Throughout our search problems are “type-2” NP search problems, which take second-order objects as parameters. (shrink)

Contemporary versions of Bell’s argument against local hidden variable (LHV) theories are based on the Clauser Horne Shimony and Holt (CHSH) inequality, and various attempts to generalize it. The amount of violation of these inequalities cannot exceed the bound set by the Grothendieck constants. However, if we go back to the original derivation by Bell, and use the perfect anticorrelation embodied in the singlet spin state, we can go beyond these bounds. In this paper we derive two-particle Bell inequalities for (...) traceless two-outcome observables, whose violation in the singlet spin state go beyond the Grothendieck constant both for the two and three dimensional cases. Moreover, creating a higher dimensional analog of perfect correlations, and applying a recent result of Alon and his associates (Invent. Math. 163 499 (2006)) we prove that there are two-particle Bell inequalities for traceless two-outcome observables whose violation increases to in…nity as the dimension and number of measurements grow. Technically these result are possible because perfect correlations (or anti-correlations) allow us to transport the indices of the inequality from the edges of a bipartite graph to those of the complete graph. Finally, it is shown how to apply these results to mixed Werner states, provided that the noise does not exceed 20%. (shrink)

The story of the body. Fay Bound Alberti takes the human body apart in order to put it back anew, telling the cultural history of our key organs and systems from the inside out, from blood to guts, brains to sex organs.

Many contemporary scholars argue that Schelling’s version of intellectual intuition retains certain central features of the Kantian and Fichtean conceptions. One of the common claims is that, as with Kant and Fichte, Schelling’s intellectual intuition is the power of the subject’s productive understanding. However, we show that for the Schelling of the Identitätsphilosophie period, intellectual intuition is the power not of an understanding that intuits, or a productive intellect, but of a receptive and penetrating sensibility that understands.

In this paper I draw from Martin Heidegger’s phenomenology of the 1920s to outline some basic features of his theory of intentionality that I believe have not been fully appreciated or utilized, and that allow for both novel and fruitful interventions in questions about meaning, the relationship between mind and the world, and epistemic justification, principally as they appear in John McDowell’s synoptic project in Mind and World. I argue that while elements of McDowell’s picture are ultimately unsatisfying and problematic, (...) much of his conceptual framework can and should be put into dialogue with Heidegger’s, and that in so doing we make available powerful resources for amending the McDowellian account. Moreover, these emendations have attractive implications for his distinctive desiderata. In particular, they provide original conceptions of normativity’s place in nature, of the boundaries of the space of reasons, and of the relationship between the answerability of thought both to the world and to human beings as a rational community. (shrink)

Loneliness is one of the most neglected aspects of emotion history, despite claims that the 21st century is the loneliest ever. This article argues against the widespread belief that modern-day loneliness is inevitable, negative, and universal. Looking at its language and etymology, it suggests that loneliness needs to be understood firstly as an “emotion cluster” composed of a variety of affective states, and secondly as a relatively recent invention, dating from around 1800. Loneliness can be positive, and as much a (...) part of the body as the mind. Using a longue durée approach, I argue that we cannot understand loneliness as a “modern epidemic” without considering its history, its meanings, its practice, and its links with the body. (shrink)

This essay explores small “ordinary” experiences of moral harm as problems of social injustice. Starting with two stories, we first argue against a dominant framework of personal responsibility that assigns responsibility to particular blameworthy agents. Instead we sketch an account of why structural responsibility for social harm must be considered, drawing on the work of Iris Marion Young and Pierre Bourdieu. Finally, drawing on Margaret Walker’s notion of moral repair and Christopher Marshall’s interpretation of the parable of the Good Samaritan, (...) we sketch an account of Christian moral responsibility grounded in restorative justice that seeks to address daily experiences of moral harm through the moral repair work of neighbor-love. (shrink)

In this paper, the notions of Fα-categorical and Gα-categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G2-categorical; in the latter case it is not Fα-categorical for (...) any recursive ordinal α. (shrink)

We study human performance in two classical NP‐hard optimization problems: Set Cover and Maximum Coverage. We suggest that Set Cover and Max Coverage are related to means selection problems that arise in human problem‐solving and in pursuing multiple goals: The relationship between goals and means is expressed as a bipartite graph where edges between means and goals indicate which means can be used to achieve which goals. While these problems are believed to be computationally intractable in general, they become (...) more tractable when the structure of the network resembles a tree. Thus, our main prediction is that people should perform better with goal systems that are more tree‐like. We report three behavioral experiments which confirm this prediction. Our results suggest that combinatorial parameters that are instrumental to algorithm design can also be useful for understanding when and why people struggle to choose between multiple means to achieve multiple goals. (shrink)

Let $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms (...) of forbidden induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results: – There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs. – Even if we only consider classes $\mathscr {C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that defines $\mathscr {C}$ and the size of the characterization cannot be bounded by $f(|\varphi |)$ for any computable function f. Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures. (shrink)

The logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with O(k n+l ) lines. In particular, there is a polynomial time algorithm which eliminates cycles from a proof. These results are motivated by the search for general methods on (...) proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search. (shrink)

We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.