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**21 - 26**of**26**### ALPS’07 -- Groups and Complexity

, 2007

"... The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-compl ..."

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The connection between groups and recursive (un)decidability has a long history, going back to the early 1900s. Also, various polynomial-time algorithms have been known in group theory for a long time. However the impact of more general computational complexity (e.g., NP-completeness or PSpace-completeness) has been relatively small and recent. These lectures review a sampling of older facts about algorithmic problems in group theory, and then present more recent results about the connection with complexity: isoperimetric functions and NP; Thompson groups, boolean circuits, and coNP; Thompson monoids and circuit complexity; Thompson groups, reversible computing, and #P; distortion of Thompson groups within Thompson monoids, and one-way permutations. We are especially interested in deep connections between computational complexity and group theory. By “connection ” we do not just mean analyzing the computational complexity of algorithms about groups. We are more interested in algebraic characterizations of complexity classes in terms of group theory, i.e., in finding a “mirror image” of all of complexity theory within group theory. Conversely, we are interested in the computational nature of concepts that appear at first purely algebraic.

### Quantum Computing: Lecture Notes

, 2011

"... These lecture notes were formed in small chunks during my “Quantum computing ” course at the University of Amsterdam, Feb-May 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 × 45 minutes, with an additional 45-minute lecture for exercises and homework. The fir ..."

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These lecture notes were formed in small chunks during my “Quantum computing ” course at the University of Amsterdam, Feb-May 2011, and compiled into one text thereafter. Each chapter was covered in a lecture of 2 × 45 minutes, with an additional 45-minute lecture for exercises and homework. The first half of the course (Chapters 1–7) covers quantum algorithms, the second half covers quantum complexity (Chapters 8–9), stuff involving Alice and Bob (Chapters 10–13), and error-correction (Chapter 14). A 15th lecture about physical implementations and general outlook was more sketchy, and I didn’t write lecture notes for it. These chapters may also be read as a general introduction to the area of quantum computation and information from the perspective of a theoretical computer scientist. While I made an effort to make the text self-contained and consistent, it may still be somewhat rough around the edges; I hope to continue polishing and adding to it. Comments & constructive criticism are very welcome, and can be sent to rdewolf@cwi.nl

### Circuits, coNP-completeness, and the groups of Richard

, 2003

"... We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuit ..."

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We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson-Higman group G3,1. 1

### The R- and L-orders of the Thompson-Higman monoid Mk,1 and their complexity

, 812

"... We study the monoid generalization Mk,1 of the Thompson-Higman groups, and we characterize the R- and the L-preorder of Mk,1. Although Mk,1 has only one non-zero J-class and k−1 non-zero D-classes, the R- and the L-preorder are complicated; in particular,

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We study the monoid generalization Mk,1 of the Thompson-Higman groups, and we characterize the R- and the L-preorder of Mk,1. Although Mk,1 has only one non-zero J-class and k−1 non-zero D-classes, the R- and the L-preorder are complicated; in particular, <R is dense (even within an L-class), and <L is dense (even within an R-class). We study the computational complexity of the R- and the L-preorder. When inputs are given by words over a finite generating set of Mk,1, the R- and the L-preorder decision problems are in P. The main result of the paper is that over a “circuit-like ” generating set, the R-preorder decision problem of Mk,1 is Π P 2-complete, whereas the L-preorder decision problem is coNP-complete. We also prove related results about circuits: For combinational circuits, the surjectiveness problem is Π P 2-complete, whereas the injectiveness problem is coNP-complete. 1

### The Thompson-Higman monoids Mk,i: the J-order, the D-relation, and their complexity

, 904

"... The Thompson-Higman groups Gk,i have a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,i and Invk,i and investigate the computational complexity of related decision problems. The main interest of these monoids is their close ..."

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The Thompson-Higman groups Gk,i have a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,i and Invk,i and investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of Mk,1 and Invk,1 are isomorphic to the groups Gk,j (1 ≤ j ≤ k − 1); so we rediscover all the Thompson-Higman groups within Mk,1. Deciding the Green relations ≤J and ≡D of Mk,1, when the inputs are words over a finite generating set of Mk,1, is in P. When a circuit-like generating set is used for Mk,1 then deciding ≤J is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for ≤J is xNPsearch-complete, whereas the multiplier search problems of ≤R and ≤L are not in xNPsearch unless NP = coNP. We introduce the class of search problems xNPsearch as a slight generalization of NPsearch. Deciding ≡D for Mk,1 when the inputs are words over a circuit-like generating set, is ⊕k−1•NPcomplete; for any h ≥ 2, ⊕h•NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # • NPcomplete), and the image size modulo h problem (which is ⊕h •NP-complete). For Invk,1 over a circuit-like generating set, deciding ≡D is ⊕k−1P-complete. It is interesting that the little known complexity classes coDP and ⊕k−1•NP play a central role in Mk,1. 1