{"id":{"repo_id":"brock","oai_identifier":"oai:brocku.scholaris.ca:10464/3953"},"canonical_url":"https://search.dev.ndltd.org/etd/brock/oai:brocku.scholaris.ca:10464/3953","repository":{"repo_id":"brock","name":"Brock University","base_url":"https://brocku.scholaris.ca/server/oai/request"},"display":{"title":"Response curves of deterministic and probabilistic cellular automata in one and two dimensions","abstract":"One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules.","abstract_html":"One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules.","abstract_has_math":false,"creators":["Skelton, Andrew"],"institution":"Brock University","degree_name":"M.Sc. Mathematics and Statistics","degree_level":"Masters","degree_discipline":"Faculty of Mathematics and Science","degree_department":"Department of Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-04-02","date_published":"2012-04-02","updated_at":"2026-07-24T01:23:16Z","subjects":["Cellular automata"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10464/3953","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Department of Mathematics"]},{"key":"dc:creator","label":"Author","values":["Skelton, Andrew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2012-04-02T19:28:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2012-04-02T19:28:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2012-04-02"]},{"key":"dc:type","label":"Dc Type","values":["Electronic Thesis or Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Faculty of Mathematics and Science"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Masters"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.Sc. Mathematics and Statistics"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Brock University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Cellular automata"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10464/3953"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules."]},{"key":"dc:title","label":"Title","values":["Response curves of deterministic and probabilistic cellular automata in one and two dimensions"]}]}],"canonical_facts":{"dc:contributor.department":["Department of Mathematics"],"dc:creator":["Skelton, Andrew"],"dc:date.accessioned":["2012-04-02T19:28:28Z"],"dc:date.available":["2012-04-02T19:28:28Z"],"dc:date.issued":["2012-04-02"],"dc:description.abstract":["One of the most important problems in the theory of cellular automata (CA) is determining the proportion of cells in a specific state after a given number of time iterations. We approach this problem using patterns in preimage sets - that is, the set of blocks which iterate to the desired output. This allows us to construct a response curve - a relationship between the proportion of cells in state 1 after niterations as a function of the initial proportion. We derive response curve formulae for many two-dimensional deterministic CA rules with L-neighbourhood. For all remaining rules, we find experimental response curves. We also use preimage sets to classify surjective rules. In the last part of the thesis, we consider a special class of one-dimensional probabilistic CA rules. We find response surface formula for these rules and experimental response surfaces for all remaining rules."],"dc:identifier.uri":["http://hdl.handle.net/10464/3953"],"dc:subject":["Cellular automata"],"dc:title":["Response curves of deterministic and probabilistic cellular automata in one and two dimensions"],"dc:type":["Electronic Thesis or Dissertation"],"thesis:degree_discipline":["Faculty of Mathematics and Science"],"thesis:degree_level":["Masters"],"thesis:degree_name":["M.Sc. Mathematics and Statistics"],"thesis:institution_name":["Brock University"]},"updated_at":"2026-07-24T01:23:16Z"}