Interactions of Computability, Complexity and Group theory Computability from the Point of View of Geometric Group Theory Paul Schupp University of Illinois at Urbana-Champaign December, 2014 Shanghai Jiao Tong University Paul Schupp (UIUC) December, 2014 1 / 1
Some History The 1912 paper of Max Dehn on finitely presented groups is remarkable. First, he posed the word, conjugacy and isomorphism problems for finitely presented groups, realizing the importance of questions of computability for group theory. Second, for the fundamental groups − 1 ... x 2 g − 1 � S g = � x 1 , ..., x 2 g ; x 1 ... x 2 g x 1 of compact surfaces with g ≥ 2, Dehn gave a linear time algorithm for the word problem, now called Dehn’s Algorithm: If w = 1 in S g then w contains more than half of a cyclic permutation of the defining relator or its inverse, Note that this paper is thirty-four years before the development of the theory of computability and sixty-nine years before the theory of computational complexity. It is very far ahead of its time. Paul Schupp (UIUC) December, 2014 2 / 1
Some History The 1912 paper of Max Dehn on finitely presented groups is remarkable. First, he posed the word, conjugacy and isomorphism problems for finitely presented groups, realizing the importance of questions of computability for group theory. Second, for the fundamental groups − 1 ... x 2 g − 1 � S g = � x 1 , ..., x 2 g ; x 1 ... x 2 g x 1 of compact surfaces with g ≥ 2, Dehn gave a linear time algorithm for the word problem, now called Dehn’s Algorithm: If w = 1 in S g then w contains more than half of a cyclic permutation of the defining relator or its inverse, Note that this paper is thirty-four years before the development of the theory of computability and sixty-nine years before the theory of computational complexity. It is very far ahead of its time. Paul Schupp (UIUC) December, 2014 2 / 1
Some History The 1912 paper of Max Dehn on finitely presented groups is remarkable. First, he posed the word, conjugacy and isomorphism problems for finitely presented groups, realizing the importance of questions of computability for group theory. Second, for the fundamental groups − 1 ... x 2 g − 1 � S g = � x 1 , ..., x 2 g ; x 1 ... x 2 g x 1 of compact surfaces with g ≥ 2, Dehn gave a linear time algorithm for the word problem, now called Dehn’s Algorithm: If w = 1 in S g then w contains more than half of a cyclic permutation of the defining relator or its inverse, Note that this paper is thirty-four years before the development of the theory of computability and sixty-nine years before the theory of computational complexity. It is very far ahead of its time. Paul Schupp (UIUC) December, 2014 2 / 1
Some History The 1912 paper of Max Dehn on finitely presented groups is remarkable. First, he posed the word, conjugacy and isomorphism problems for finitely presented groups, realizing the importance of questions of computability for group theory. Second, for the fundamental groups − 1 ... x 2 g − 1 � S g = � x 1 , ..., x 2 g ; x 1 ... x 2 g x 1 of compact surfaces with g ≥ 2, Dehn gave a linear time algorithm for the word problem, now called Dehn’s Algorithm: If w = 1 in S g then w contains more than half of a cyclic permutation of the defining relator or its inverse, Note that this paper is thirty-four years before the development of the theory of computability and sixty-nine years before the theory of computational complexity. It is very far ahead of its time. Paul Schupp (UIUC) December, 2014 2 / 1
Turing Machines and the Halting Problem One of the great accomplishments of twentieth century mathematics was to give a precise definition of what it means to be computable . The model usually used is “computable by a Turing machine”. One can just think of a Turing machine as an idealized computer: the machine has an infinite memory and we do not care how many operations a computation takes. A machine started on a particular input may or may not eventually halt. The Halting Problem for Turing machines is the following decision problem: Given a Turing machine M and a word w on the input alphabet of M , does M halt when started with input w ? Turing showed in 1936 that the Halting Problem is not computable. In the 1930’s the Halting Problem was of course a very abstract mathematical problem. Now it is a daily problem faced by hundreds of people: If one finally gets a long computer program to compile and then starts it, what does it do? Paul Schupp (UIUC) December, 2014 3 / 1
Turing Machines and the Halting Problem One of the great accomplishments of twentieth century mathematics was to give a precise definition of what it means to be computable . The model usually used is “computable by a Turing machine”. One can just think of a Turing machine as an idealized computer: the machine has an infinite memory and we do not care how many operations a computation takes. A machine started on a particular input may or may not eventually halt. The Halting Problem for Turing machines is the following decision problem: Given a Turing machine M and a word w on the input alphabet of M , does M halt when started with input w ? Turing showed in 1936 that the Halting Problem is not computable. In the 1930’s the Halting Problem was of course a very abstract mathematical problem. Now it is a daily problem faced by hundreds of people: If one finally gets a long computer program to compile and then starts it, what does it do? Paul Schupp (UIUC) December, 2014 3 / 1
Turing Machines and the Halting Problem One of the great accomplishments of twentieth century mathematics was to give a precise definition of what it means to be computable . The model usually used is “computable by a Turing machine”. One can just think of a Turing machine as an idealized computer: the machine has an infinite memory and we do not care how many operations a computation takes. A machine started on a particular input may or may not eventually halt. The Halting Problem for Turing machines is the following decision problem: Given a Turing machine M and a word w on the input alphabet of M , does M halt when started with input w ? Turing showed in 1936 that the Halting Problem is not computable. In the 1930’s the Halting Problem was of course a very abstract mathematical problem. Now it is a daily problem faced by hundreds of people: If one finally gets a long computer program to compile and then starts it, what does it do? Paul Schupp (UIUC) December, 2014 3 / 1
Turing Machines and the Halting Problem One of the great accomplishments of twentieth century mathematics was to give a precise definition of what it means to be computable . The model usually used is “computable by a Turing machine”. One can just think of a Turing machine as an idealized computer: the machine has an infinite memory and we do not care how many operations a computation takes. A machine started on a particular input may or may not eventually halt. The Halting Problem for Turing machines is the following decision problem: Given a Turing machine M and a word w on the input alphabet of M , does M halt when started with input w ? Turing showed in 1936 that the Halting Problem is not computable. In the 1930’s the Halting Problem was of course a very abstract mathematical problem. Now it is a daily problem faced by hundreds of people: If one finally gets a long computer program to compile and then starts it, what does it do? Paul Schupp (UIUC) December, 2014 3 / 1
The Undecidability of the Word Problem In the 1950’s P .S. Novikov and W.W. Boone independently constructed finitely presented groups with unsolvable word problem by coding the Halting Problem into the groups. This fundamental result is the basis of all undecidability results in group theory and topology. Almost all sufficiently general problems are unsolvable. Isomorphism to any fixed finitely presented group is undecidable. In particular, there is no algorithm deciding whether or not finite presentations are presentations of the trivial group. Paul Schupp (UIUC) December, 2014 4 / 1
The Undecidability of the Word Problem In the 1950’s P .S. Novikov and W.W. Boone independently constructed finitely presented groups with unsolvable word problem by coding the Halting Problem into the groups. This fundamental result is the basis of all undecidability results in group theory and topology. Almost all sufficiently general problems are unsolvable. Isomorphism to any fixed finitely presented group is undecidable. In particular, there is no algorithm deciding whether or not finite presentations are presentations of the trivial group. Paul Schupp (UIUC) December, 2014 4 / 1
Another basic problem about finitely presented groups is the Membership Problem : Fix a finitely presented group G . Given a finite set of S of words and a word w , does w belong to the subgroup H generated by S ? This problem is already unsolvable in F 2 × F 2 , the direct product of two free groups of rank 2 (Mikhailova”s Theorem) and in many hyperbolic groups (the Rips Construction). A.A. Markov used the undecidability of the Isomorphism Problem to show that for n ≥ 4, there is no algorithm which decides, when given two n -manifolds, whether or not they are homeomorphic. Paul Schupp (UIUC) December, 2014 5 / 1
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