Functors on relational structures which admit both left and right adjoints

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors $\Lambda$ and $\Gamma$ between thin categories of relational structures are adjoint if for all structures $\mathbf A$ and $\mathbf B$, we have that $\Lambda(\mathbf A)$ maps homomorphically to $\mathbf B$ if and only if $\mathbf A$ maps homomorphically to $\Gamma(\mathbf B)$. If this is the case $\Lambda$ is called the left adjoint to $\Gamma$ and $\Gamma$ the right adjoint to $\Lambda$. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems

Improved hardness for H-colourings of G-colourable graphs

We present new results on approximate colourings of graphs and, more generally, approximate H-colourings and promise constraint satisfaction problems. First, we show NP-hardness of colouring $k$-colourable graphs with $\binom{k}{\lfloor k/2\rfloor}-1$ colours for every $k\geq 4$. This improves the result of Bul\'in, Krokhin, and Opr\v{s}al [STOC'19], who gave NP-hardness of colouring $k$-colourable graphs with $2k-1$ colours for $k\geq 3$, and the result of Huang [APPROX-RANDOM'13], who gave NP-hardness of colouring $k$-colourable graphs with $2^{k^{1/3}}$ colours for sufficiently large $k$. Thus, for $k\geq 4$, we improve from known linear/sub-exponential gaps to exponential gaps. Second, we show that the topology of the box complex of H alone determines whether H-colouring of G-colourable graphs is NP-hard for all (non-bipartite, H-colourable) G. This formalises the topological intuition behind the result of Krokhin and Opr\v{s}al [FOCS'19] that 3-colouring of G-colourable graphs is NP-hard for all (3-colourable, non-bipartite) G. We use this technique to establish NP-hardness of H-colouring of G-colourable graphs for H that include but go beyond $K_3$, including square-free graphs and circular cliques (leaving $K_4$ and larger cliques open). Underlying all of our proofs is a very general observation that adjoint functors give reductions between promise constraint satisfaction problems.Comment: Mention improvement in Proposition 2.5. SODA 202

Quantaloidal approach to constraint satisfaction

The constraint satisfaction problem (CSP) is a computational problem that includes a range of important problems in computer science. We point out that fundamental concepts of the CSP, such as the solution set of an instance and polymorphisms, can be formulated abstractly inside the 2-category $\mathcal{P}\mathbf{FinSet}$ of finite sets and sets of functions between them. The 2-category $\mathcal{P}\mathbf{FinSet}$ is a quantaloid, and the formulation relies mainly on structure available in any quantaloid. This observation suggests a formal development of generalisations of the CSP and concomitant notions of polymorphism in a large class of quantaloids. We extract a class of optimisation problems as a special case, and show that their computational complexity can be classified by the associated notion of polymorphism.Comment: 17 page

Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used. We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming

Mathematical Foundations for a Compositional Account of the Bayesian Brain

This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain. Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference. To do so, we define on the `syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern. Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections. In the process, we introduce a new notion of `copy-composition'. On the `semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors). We use these opindexed categories to define monoidal bicategories of cilia: dynamical systems which control lenses, and which supply the target for our functorial semantics. Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex. Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment of statistical games. A number of errors also fixed, and some presentation improved. Comments most welcom

Superseded: Grammatical theory: From transformational grammar to constraint-based approaches. Second revised and extended edition.

This book is superseded by the third edition, available at http://langsci-press.org/catalog/book/255. This book introduces formal grammar theories that play a role in current linguistic theorizing (Phrase Structure Grammar, Transformational Grammar/Government & Binding, Generalized Phrase Structure Grammar, Lexical Functional Grammar, Categorial Grammar, Head-​Driven Phrase Structure Grammar, Construction Grammar, Tree Adjoining Grammar). The key assumptions are explained and it is shown how the respective theory treats arguments and adjuncts, the active/passive alternation, local reorderings, verb placement, and fronting of constituents over long distances. The analyses are explained with German as the object language. The second part of the book compares these approaches with respect to their predictions regarding language acquisition and psycholinguistic plausibility. The nativism hypothesis, which assumes that humans posses genetically determined innate language-specific knowledge, is critically examined and alternative models of language acquisition are discussed. The second part then addresses controversial issues of current theory building such as the question of flat or binary branching structures being more appropriate, the question whether constructions should be treated on the phrasal or the lexical level, and the question whether abstract, non-visible entities should play a role in syntactic analyses. It is shown that the analyses suggested in the respective frameworks are often translatable into each other. The book closes with a chapter showing how properties common to all languages or to certain classes of languages can be captured. The book is a translation of the German book Grammatiktheorie, which was published by Stauffenburg in 2010. The following quotes are taken from reviews: With this critical yet fair reflection on various grammatical theories, Müller fills what was a major gap in the literature. Karen Lehmann, Zeitschrift für Rezen­sio­nen zur ger­man­is­tis­chen Sprach­wis­senschaft, 2012 Stefan Müller’s recent introductory textbook, Gram­matik­the­o­rie, is an astonishingly comprehensive and insightful survey for beginning students of the present state of syntactic theory. Wolfgang Sternefeld und Frank Richter, Zeitschrift für Sprach­wissen­schaft, 2012 This is the kind of work that has been sought after for a while [...] The impartial and objective discussion offered by the author is particularly refreshing. Werner Abraham, Germanistik, 2012   This book is a new edition of http://langsci-press.org/catalog/book/25

Superseded: Grammatical theory: From transformational grammar to constraint-based approaches. Second revised and extended edition.

This book is superseded by the third edition, available at http://langsci-press.org/catalog/book/255. This book introduces formal grammar theories that play a role in current linguistic theorizing (Phrase Structure Grammar, Transformational Grammar/Government & Binding, Generalized Phrase Structure Grammar, Lexical Functional Grammar, Categorial Grammar, Head-​Driven Phrase Structure Grammar, Construction Grammar, Tree Adjoining Grammar). The key assumptions are explained and it is shown how the respective theory treats arguments and adjuncts, the active/passive alternation, local reorderings, verb placement, and fronting of constituents over long distances. The analyses are explained with German as the object language. The second part of the book compares these approaches with respect to their predictions regarding language acquisition and psycholinguistic plausibility. The nativism hypothesis, which assumes that humans posses genetically determined innate language-specific knowledge, is critically examined and alternative models of language acquisition are discussed. The second part then addresses controversial issues of current theory building such as the question of flat or binary branching structures being more appropriate, the question whether constructions should be treated on the phrasal or the lexical level, and the question whether abstract, non-visible entities should play a role in syntactic analyses. It is shown that the analyses suggested in the respective frameworks are often translatable into each other. The book closes with a chapter showing how properties common to all languages or to certain classes of languages can be captured. The book is a translation of the German book Grammatiktheorie, which was published by Stauffenburg in 2010. The following quotes are taken from reviews: With this critical yet fair reflection on various grammatical theories, Müller fills what was a major gap in the literature. Karen Lehmann, Zeitschrift für Rezen­sio­nen zur ger­man­is­tis­chen Sprach­wis­senschaft, 2012 Stefan Müller’s recent introductory textbook, Gram­matik­the­o­rie, is an astonishingly comprehensive and insightful survey for beginning students of the present state of syntactic theory. Wolfgang Sternefeld und Frank Richter, Zeitschrift für Sprach­wissen­schaft, 2012 This is the kind of work that has been sought after for a while [...] The impartial and objective discussion offered by the author is particularly refreshing. Werner Abraham, Germanistik, 2012   This book is a new edition of http://langsci-press.org/catalog/book/25

Superseded: Grammatical theory: From transformational grammar to constraint-based approaches. Second revised and extended edition.

This book is superseded by the third edition, available at http://langsci-press.org/catalog/book/255. This book introduces formal grammar theories that play a role in current linguistic theorizing (Phrase Structure Grammar, Transformational Grammar/Government & Binding, Generalized Phrase Structure Grammar, Lexical Functional Grammar, Categorial Grammar, Head-​Driven Phrase Structure Grammar, Construction Grammar, Tree Adjoining Grammar). The key assumptions are explained and it is shown how the respective theory treats arguments and adjuncts, the active/passive alternation, local reorderings, verb placement, and fronting of constituents over long distances. The analyses are explained with German as the object language. The second part of the book compares these approaches with respect to their predictions regarding language acquisition and psycholinguistic plausibility. The nativism hypothesis, which assumes that humans posses genetically determined innate language-specific knowledge, is critically examined and alternative models of language acquisition are discussed. The second part then addresses controversial issues of current theory building such as the question of flat or binary branching structures being more appropriate, the question whether constructions should be treated on the phrasal or the lexical level, and the question whether abstract, non-visible entities should play a role in syntactic analyses. It is shown that the analyses suggested in the respective frameworks are often translatable into each other. The book closes with a chapter showing how properties common to all languages or to certain classes of languages can be captured. The book is a translation of the German book Grammatiktheorie, which was published by Stauffenburg in 2010. The following quotes are taken from reviews: With this critical yet fair reflection on various grammatical theories, Müller fills what was a major gap in the literature. Karen Lehmann, Zeitschrift für Rezen­sio­nen zur ger­man­is­tis­chen Sprach­wis­senschaft, 2012 Stefan Müller’s recent introductory textbook, Gram­matik­the­o­rie, is an astonishingly comprehensive and insightful survey for beginning students of the present state of syntactic theory. Wolfgang Sternefeld und Frank Richter, Zeitschrift für Sprach­wissen­schaft, 2012 This is the kind of work that has been sought after for a while [...] The impartial and objective discussion offered by the author is particularly refreshing. Werner Abraham, Germanistik, 2012   This book is a new edition of http://langsci-press.org/catalog/book/25

Grammatical theory

This book introduces formal grammar theories that play a role in current linguistic theorizing (Phrase Structure Grammar, Transformational Grammar/Government & Binding, Generalized Phrase Structure Grammar, Lexical Functional Grammar, Categorial Grammar, Head-​Driven Phrase Structure Grammar, Construction Grammar, Tree Adjoining Grammar). The key assumptions are explained and it is shown how the respective theory treats arguments and adjuncts, the active/passive alternation, local reorderings, verb placement, and fronting of constituents over long distances. The analyses are explained with German as the object language. The second part of the book compares these approaches with respect to their predictions regarding language acquisition and psycholinguistic plausibility. The nativism hypothesis, which assumes that humans posses genetically determined innate language-specific knowledge, is critically examined and alternative models of language acquisition are discussed. The second part then addresses controversial issues of current theory building such as the question of flat or binary branching structures being more appropriate, the question whether constructions should be treated on the phrasal or the lexical level, and the question whether abstract, non-visible entities should play a role in syntactic analyses. It is shown that the analyses suggested in the respective frameworks are often translatable into each other. The book closes with a chapter showing how properties common to all languages or to certain classes of languages can be captured. The book is a translation of the German book Grammatiktheorie, which was published by Stauffenburg in 2010. The following quotes are taken from reviews: With this critical yet fair reflection on various grammatical theories, Müller fills what was a major gap in the literature. Karen Lehmann, Zeitschrift für Rezen­sio­nen zur ger­man­is­tis­chen Sprach­wis­senschaft, 2012 Stefan Müller’s recent introductory textbook, Gram­matik­the­o­rie, is an astonishingly comprehensive and insightful survey for beginning students of the present state of syntactic theory. Wolfgang Sternefeld und Frank Richter, Zeitschrift für Sprach­wissen­schaft, 2012 This is the kind of work that has been sought after for a while [...] The impartial and objective discussion offered by the author is particularly refreshing. Werner Abraham, Germanistik, 201

Russian verbal prefixation: A frame semantic analysis

This book addresses the complexity of Russian verbal prefixation system that has been extensively studied but yet not explained. Traditionally, different meanings have been investigated and listed in the dictionaries and grammars and more recently linguists attempted to unify various prefix usages under more general descriptions. The existent semantic approaches, however, do not aim to use semantic representations in order to account for the problems of prefix stacking and aspect determination. This task has been so far undertaken by syntactic approaches to prefixation, that divide verbal prefixes in classes and limit complex verb formation by restricting structural positions available for the members of each class. I show that these approaches have two major drawbacks: the implicit prediction of the non-existence of complex biaspectual verbs and the absence of uniformly accepted formal criteria for the underlying prefix classification. In this book the reader can find an implementable formal semantic approach to prefixation that covers five prefixes: za-, na-, po-, pere-, and do-. It is shown how to predict the existence, semantics, and aspect of a given complex verb with the help of the combination of an LTAG and frame semantics. The task of identifying the possible affix combinations is distributed between three modules: syntax, which is kept simple (only basic structural assumptions), frame semantics, which ensures that the constraints are respected, and pragmatics, which rules out some prefixed verbs and restricts the range of available interpretations. For the purpose of the evaluation of the theory, an implementation of the proposed analysis for a grammar fragment using a metagrammar description is provided. It is shown that the proposed analysis delivers more accurate and complete predictions with respect to the existence of complex verbs than the most precise syntactic account