László Egri

Symmetric Datalog and Constraint Satisfaction Problems in Logspace

We introduce symmetric Datalog, a syntactic restriction of linear Datalog and show that its expressive power is exactly that of restricted symmetric Krom monotone SNP. The deep result of Reingold [17] on the complexity of undirected connectivity suffices to show that symmetric Datalog queries can be evaluated in logarithmic space. We show that for a number of constraint languages Gamma, the complement of the constraint satisfaction problem CSP(Gamma) can be expressed in symmetric Datalog. In particular, we show that if CSP(Gamma) is first-order definable and Lambda is a finite subset of the relational clone generated by Gamma then notCSP(Lambda) is definable in symmetric Datalog. Over the two-element domain and under standard complexity-theoretic assumptions, expressibility of notCSP(Gamma) in symmetric Datalog corresponds exactly to the class of CSPs computable in logarithmic space. Finally, we describe a fairly general subclass of implicational (or 0/1/all) constraints for which the complement of the corresponding CSP is also definable in symmetric Datalog. Our results provide preliminary evidence that symmetric Datalog may be a unifying explanation for families of CSPs lying in L.

COULD KNOWLEDGE-BASED NEURAL LEARNING BE USEFUL IN DEVELOPMENTAL ROBOTICS? THE CASE OF KBCC

The new field of developmental robotics faces the formidable challenge of implementing effective learning mechanisms in complex, dynamic environments. We make a case that knowledge-based learning algorithms might help to meet this challenge. A constructive neural learning algorithm, knowledge-based cascade-correlation (KBCC), autonomously recruits previously-learned networks in addition to the single hidden units recruited by ordinary cascade-correlation. This enables learning by analogy when adequate prior knowledge is available, learning by induction from examples when there is no relevant prior knowledge, and various combinations of analogy and induction. A review of experiments with KBCC indicates that recruitment of relevant existing knowledge typically speeds learning and sometimes enables learning of otherwise impossible problems. Some additional domains of interest to developmental robotics are identified in which knowledge-based learning seems essential. The characteristics of KBCC in relation to other knowledge-based neural learners and analogical reasoning are summarized as is the neurological basis for learning from knowledge. Current limitations of this approach and directions for future work are discussed.

The Complexity of the List Homomorphism Problem for Graphs

We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H, the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

The Complexity of the List Homomorphism Problem for Graphs.

We completely classify the computational complexity of the list

Constraint-Satisfaction Models

\bH

On Maltsev digraphs

-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph

Space complexity of listH-colouring: a dichotomy

\bH

List H-Coloring a Graph by Removing Few Vertices

the problem is either NP-complete, NL-complete, L-complete or is first-order definable; descriptive complexity equivalents are given as well via Datalog and its fragments. Our algebraic characterisations match important conjectures in the study of constraint satisfaction problems.

Directed st-Connectivity Is Not Expressible in Symmetric Datalog

Constraint satisfaction is the art of finding values for variables in a way that satisfies various constraints. Many everyday problems can be usefully couched in these terms. This survey of constraint-satisfaction models covers both symbolic approaches from artificial intelligence (AI) and neural network approaches from psychology and cognitive science. Initial symbolic AI approaches attempted to solve constraint-satisfaction problems in a hard fashion – either the constraints were perfectly satisfied or there was no solution. Due to its limited modeling capacity, this approach was not particularly appealing to computational modelers in psychology who intuited that the problems people face often have no perfect solutions that satisfy all constraints. Hence, psychological approaches more often turned to the soft constraint satisfaction of artificial neural networks. Newer AI work on constraint satisfaction has made considerable progress in building systems that optimize solutions to hard problems even when it is impossible to satisfy all constraints. The most dominant trends in constraint satisfaction are reviewed and integrated, and forecasts are made for future research and modeling applications.

Descriptive Complexity of List H-Coloring Problems in Logspace: A Refined Dichotomy

We study digraphs preserved by a Maltsev operation, Maltsev digraphs. We show that these digraphs retract either onto a directed path or to the disjoint union of directed cycles, showing that the constraint satisfaction problem for Maltsev digraphs is in logspace, L. (This was observed in [19] using an indirect argument.) We then generalize results in [19] to show that a Maltsev digraph is preserved not only by a majority operation, but by a class of other operations (e.g., minority, Pixley) and obtain a O(VG4)-time algorithm to recognize Maltsev digraphs. We also prove analogous results for digraphs preserved by conservative Maltsev operations which we use to establish that the list homomorphism problem for Maltsev digraphs is in L. We then give a polynomial time characterisation of Maltsev digraphs admitting a conservative 2-semilattice operation. Finally, we give a simple inductive construction of directed acyclic digraphs preserved by a Maltsev operation.