What's after EXPSPACE?












4












$begingroup$


As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE.



Thinking about this question, I came up with a couple thoughts of my own:




  1. Maybe EXPSPACE = the set of all Recursively Enumerable problems, i.e. $Delta_1$ in the arithmetical hierarchy. In this case, the rest of the arithmetical hierarchy would come after EXPSPACE.


  2. Maybe there are problems with space requirements that need to be described by non-computatable functions. For example, take the Busy Beaver function, $BB(n)$, which returns the maximum number of steps an $n$-state Turing Machine could take before halting. $BB$ grows faster than any computable function -- so if a problem required $BB(n)$ space to solve, maybe that would put it in a complexity class after EXPSPACE? In other words, maybe after EXPSPACE there is one or more sets of problems that require a non-computable amount of space to solve.











share|cite|improve this question









$endgroup$

















    4












    $begingroup$


    As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE.



    Thinking about this question, I came up with a couple thoughts of my own:




    1. Maybe EXPSPACE = the set of all Recursively Enumerable problems, i.e. $Delta_1$ in the arithmetical hierarchy. In this case, the rest of the arithmetical hierarchy would come after EXPSPACE.


    2. Maybe there are problems with space requirements that need to be described by non-computatable functions. For example, take the Busy Beaver function, $BB(n)$, which returns the maximum number of steps an $n$-state Turing Machine could take before halting. $BB$ grows faster than any computable function -- so if a problem required $BB(n)$ space to solve, maybe that would put it in a complexity class after EXPSPACE? In other words, maybe after EXPSPACE there is one or more sets of problems that require a non-computable amount of space to solve.











    share|cite|improve this question









    $endgroup$















      4












      4








      4





      $begingroup$


      As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE.



      Thinking about this question, I came up with a couple thoughts of my own:




      1. Maybe EXPSPACE = the set of all Recursively Enumerable problems, i.e. $Delta_1$ in the arithmetical hierarchy. In this case, the rest of the arithmetical hierarchy would come after EXPSPACE.


      2. Maybe there are problems with space requirements that need to be described by non-computatable functions. For example, take the Busy Beaver function, $BB(n)$, which returns the maximum number of steps an $n$-state Turing Machine could take before halting. $BB$ grows faster than any computable function -- so if a problem required $BB(n)$ space to solve, maybe that would put it in a complexity class after EXPSPACE? In other words, maybe after EXPSPACE there is one or more sets of problems that require a non-computable amount of space to solve.











      share|cite|improve this question









      $endgroup$




      As far as I'm aware, EXPSPACE is the most inclusive computational complexity class. I was wondering if/how people conceptualize supersets of EXPSPACE.



      Thinking about this question, I came up with a couple thoughts of my own:




      1. Maybe EXPSPACE = the set of all Recursively Enumerable problems, i.e. $Delta_1$ in the arithmetical hierarchy. In this case, the rest of the arithmetical hierarchy would come after EXPSPACE.


      2. Maybe there are problems with space requirements that need to be described by non-computatable functions. For example, take the Busy Beaver function, $BB(n)$, which returns the maximum number of steps an $n$-state Turing Machine could take before halting. $BB$ grows faster than any computable function -- so if a problem required $BB(n)$ space to solve, maybe that would put it in a complexity class after EXPSPACE? In other words, maybe after EXPSPACE there is one or more sets of problems that require a non-computable amount of space to solve.








      complexity-theory computability decision-problem space-complexity






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 6 hours ago









      Joel MillerJoel Miller

      553




      553






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).



          The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$. If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE. So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE. And so on. There is an infinite hierarchy of complexity classes, each bigger than the previous.



          This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).






          share|cite|improve this answer











          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "419"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: false,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: null,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f104835%2fwhats-after-expspace%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            4












            $begingroup$

            EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).



            The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$. If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE. So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE. And so on. There is an infinite hierarchy of complexity classes, each bigger than the previous.



            This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).






            share|cite|improve this answer











            $endgroup$


















              4












              $begingroup$

              EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).



              The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$. If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE. So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE. And so on. There is an infinite hierarchy of complexity classes, each bigger than the previous.



              This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).






              share|cite|improve this answer











              $endgroup$
















                4












                4








                4





                $begingroup$

                EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).



                The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$. If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE. So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE. And so on. There is an infinite hierarchy of complexity classes, each bigger than the previous.



                This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).






                share|cite|improve this answer











                $endgroup$



                EXPSPACE is not the most inclusive computational complexity class. There's a huge number of complexity classes; see, e.g., the Complexity Zoo for some that have been studied (and in principle there are an unlimited number one could define).



                The space hierarchy theorem shows that for each $f(n)$, there are languages that can be decided in space $f(n)$ but cannot be decided in space $o(f(n))$. If we take $f(n)=2^{2^n}$ (say), then we find there are problems that can be decided in space $2^{2^n}$, but not in space $2^n$, and indeed, are not in EXPSPACE. So, we can "make up" a complexity class, call it 2EXPSPACE, of problems that can be solved in doubly-exponential space; it follows that 2EXPSPACE is bigger than EXPSPACE. And so on. There is an infinite hierarchy of complexity classes, each bigger than the previous.



                This also shows that EXPSPACE is not equal to RE (the recursively enumerable problems).







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 1 hour ago

























                answered 5 hours ago









                D.W.D.W.

                99.7k12121286




                99.7k12121286






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Computer Science Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fcs.stackexchange.com%2fquestions%2f104835%2fwhats-after-expspace%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    How to label and detect the document text images

                    Vallis Paradisi

                    Tabula Rosettana