Cores of Countably Categorical Structures

A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure has a core, i.e., has an endomorphism such that the structure induced by its image is a core; moreover, the core is unique up to isomorphism. Weprove that every \omega -categorical structure has a core. Moreover, every \omega-categorical structure is homomorphically equivalent to a model-complete core, which is unique up to isomorphism, and which is finite or \omega -categorical. We discuss consequences for constraint satisfaction with \omega -categorical templates

Processing underspecified semantic representations in the constraint language for lambda structures

The constraint language for lambda structures (CLLS) is an expressive language of tree descriptions which combines dominance constraints with powerful parallelism and binding constraints. CLLS was introduced as a uniform framework for defining underspecified semantics representations of natural language sentences, covering scope, ellipsis, and anaphora. This article presents saturation-based algorithms for processing the complete language of CLLS. It also gives an overview of previous results on questions of processing and complexity.Liegt nicht vor

The Universal Homogenous Binary Tree

A partial order is called semilinear if the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable existentially closed semilinear order, which we denote by (S 2 ;≤). We study the reducts of (S 2 ;≤), that is, the relational structures with domain S 2, all of whose relations are first-order definable in (S 2 ;≤)⁠. Our main result is a classification of the model-complete cores of the reducts of S 2. From this, we also obtain a classification of reducts up to first-order interdefinability, which is equivalent to a classification of all subgroups of the full symmetric group on S 2 that contain the automorphism group of (S 2 ;≤) and are closed with respect to the pointwise convergence topology

Constraint Network Satisfaction for Finite Relation Algebras

Network satisfaction problems (NSPs) for finite relation algebras are computational decision problems, studied intensively since the 1990s. The major open research challenge in this field is to understand which of these problems are solvable by polynomial-time algorithms. Since there are known examples of undecidable NSPs of finite relation algebras it is advisable to restrict the scope of such a classification attempt to well-behaved subclasses of relation algebras. The class of relation algebras with a normal representation is such a well-behaved subclass. Many well-known examples of relation algebras, such as the Point Algebra, RCC5, and Allen’s Interval Algebra admit a normal representation. The great advantage of finite relation algebras with normal representations is that their NSP is essentially the same as a constraint satisfaction problem (CSP). For a relational structure B the problem CSP(B) is the computational problem to decide whether a given finite relational structure C has a homomorphism to B. The study of CSPs has a long and rich history, culminating for the time being in the celebrated proofs of the Feder-Vardi dichotomy conjecture. Bulatov and Zhuk independently proved that for every finite structure B the problem CSP(B) is in P or NP-complete. Both proofs rely on the universal-algebraic approach, a powerful theory that connects algebraic properties of structures B with complexity results for the decision problems CSP(B). Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures. The CSPs that emerge from NSPs are typically of the form CSP(B) for an infinite structure B and therefore do not fall into the scope of the dichotomy result for finite structures. In this thesis we study NSPs of finite relation algebras with normal representations by the universal algebraic methods which were developed for the study of finite and infinite-domain CSPs. We additionally make use of model theory and a Ramsey-type result of Nešetril and Rödl. Our contributions to the field are divided into three parts. Firstly, we provide two algebraic criteria for NP-hardness of NSPs. Our second result is a complete classification of the complexity of NSPs for symmetric relation algebras with a flexible atom; these problems are in P or NP-complete. Our result is obtained via a decidable condition on the relation algebra which implies polynomial-time tractability of the NSP. As a third contribution we prove that for a large class of NSPs the containment in P implies that the problems can even be solved by Datalog programs, unless P = NP. As a third contribution we prove that for a large class of NSPs, non-hardness implies that the problems can even be solved by Datalog programs, unless P = NP. This result can be used to strengthen the dichotomy result for NSPs of symmetric relation algebras with a flexible atom: every such problem can be solved by a Datalog program or is NP-complete. Our proof relies equally on known results and new observations in the algebraic analysis of finite structures

Processing underspecified semantic representations in the constraint language for lambda structures

The constraint language for lambda structures (CLLS) is an expressive language of tree descriptions which combines dominance constraints with powerful parallelism and binding constraints. CLLS was introduced as a uniform framework for defining underspecified semantics representations of natural language sentences, covering scope, ellipsis, and anaphora. This article presents saturation-based algorithms for processing the complete language of CLLS. It also gives an overview of previous results on questions of processing and complexity.Liegt nicht vor

Pure Dominance Constraints

We present an efficient algorithm that checks the satisfiability of pure dominance constraints, which is a tree description language contained in several constraint languages studied in computational linguistics. Pure dominance constraints partially describe unlabeled rooted trees. For arbitrary pairs of nodes they specify sets of admissible relative positions in a tree. The task is to find a tree structure satisfying these constraints. Our algorithm constructs such a solution in time O(m^2) where m is the number of constraints. This solves an essential part of an open problem posed by Cornell