Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.787827 
Title:  Symmetric circuits and modeltheoretic logics  
Author:  Wilsenach, Gregory Barnard 
ORCID:
0000000247972777
ISNI:
0000 0004 7972 9370


Awarding Body:  University of Cambridge  
Current Institution:  University of Cambridge  
Date of Award:  2019  
Availability of Full Text: 


Abstract:  
The question of whether there is a logic that characterises polynomialtime is arguably the most important open question in finite model theory. The study of extensions of fixedpoint logic are of central importance to this question. It was shown by Anderson and Dawar that fixedpoint logic with counting (FPC) has the same expressive power as uniform families of symmetric circuits over a basis with threshold functions. In this thesis we prove a farreaching generalisation of their result and establish an analogous circuit characterisation for each from a broad range of extensions of fixedpoint logic. In order to do so we fist develop a very general framework for defining and studying extensions of fixedpoint logics, which we call generalised operators. These operators generalise Lindström quantifiers as well as the counting and rank operators used to define FPC and fixedpoint logic with rank (FPR). We also show that in order to define a symmetric circuit model that goes beyond FPC we need to consider circuits with gates that are allowed to compute nonsymmetric functions. In order to do so we develop a far more general framework for studying circuits. We also show that key notions, such as the notion of a symmetric circuit, can be analogously defined in this more general framework. The characterisation of FPC in terms of symmetric circuits, and the treatment of circuits generally, relies heavily on the assumption that the gates in the circuit compute symmetric functions. We develop a broad range of new techniques and approaches in order to study these more general symmetric circuit models. As a corollary of our main result we establish a circuit characterisation of FPR. We also show that the question of whether there is a logic that characterises polynomialtime can be understood as a question about the symmetry property of circuits. We lastly propose a number of new approaches that might exploit this newfound connection between circuit complexity and descriptive complexity.


Supervisor:  Dawar, Anuj  Sponsor:  Gates Cambridge Scholarship  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.787827  DOI:  
Keywords:  Logic ; Model Theory ; Complexity Theory ; FixedPoint Logics ; Symmetric Circuits ; FixedPoint Logic with Rank ; Finite Model Theory ; Descriptive Complexity ; Circuits ; Circuit Complexity ; Generalised Operators ; Vectorised Operators ; FixedPoint Logic with Counting ; Extending Logics ; Symmetric Functions ; Structured Functions  
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