EPSRC Reference: 
EP/X03190X/1 
Title: 
Algebraic Methods for Quantified Constraints 
Principal Investigator: 
Martin, Dr B D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Computer Science 
Organisation: 
Durham, University of 
Scheme: 
Standard Research 
Starts: 
01 October 2023 
Ends: 
30 September 2026 
Value (£): 
520,766

EPSRC Research Topic Classifications: 
Fundamentals of Computing 


EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Panel History: 

Summary on Grant Application Form 
The constraint satisfaction problem (CSP) is a paradigm in which it is possible to express, in a natural way, a wide range of problems arising in many areas of Computer Science and beyond, e.g. Artificial Intelligence, Computational Linguistics, Computational Biology, Combinatorics and Databases. A CSP instance involves a finite set of variables, a set of values (the domain) and a finite set of constraints. The task is to assign the variables to the values so as to satisfy all of the constraints. The CSP can be seen as a modelchecking problem for the fragment of firstorder logic that has just existential quantification, conjunction and equality. When universal quantification is added, the new paradigm is the quantified constraint satisfaction problem (QCSP). When the problem is parameterised by the language of constraints permitted, one finds for some types of constraint, the task of finding a solution is computationally easy (i.e. requires a feasible amount of computational resources such as running time and memory), but for many types, it is computationally hard. The complexity classification across finite constraint languages for the CSP was completed in 2017, independently by Bulatov and Zhuk, and is now known to be a dichotomy between polynomial time and NPcomplete. The similar classification problem for the QCSP represents the only connectivebased syntactic fragment of firstorder logic where the outcome is not known. Recently, Zhuk and Martin (2019) refuted the Chen Conjecture that only complexities of polynomial time, NPcomplete and Pspacecomplete would in this classification. It is now known that exotic complexity classes such as DPcomplete, Theta^P_2complete and Pi^P_2complete can be realised in QCSPs. Zhuk and Martin (2019) completed the threeelement domain classification as a tetrachotomy between polynomial time, NPcomplete, coNPcomplete and Pspacecomplete. This proposal aims to take Zhuk's new methods beyond the threeelement case to larger domains. The proposal will consider all finite domains, as well as some infinite domains where the constraint languages model notions from temporal reasoning. The central objective of the proposal is to map out landscapes of computational complexity across these constraint languages.

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