Re: [Haskell-beginners] Nested data types and bisimilarity

On Tue, Mar 8, 2011 at 3:11 AM, Berlin Brown wrote:

On Mon, Mar 7, 2011 at 4:30 PM, Brent Yorgey wrote:

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Actually, you are missing the point. ;) The point of bisimulations is that they are defined *coinductively*, so they let you work with potentially infinite data structures. In your example, proving that xs and ys are in the relation R really is that simple -- 1 = 1, and then to complete the proof we are allowed to use the coinduction hypothesis that xs and ys are in the relation R, since they are guarded by a constructor (:).

Dan, does this help answer your original question? If not I can try to give a more detailed answer in the morning.

I understand the coinduction principle for data structures like streams (e.g., Felipe's example) and finitely branching trees (from papers like "A calculus of binary trees"). In general, for lists and types constructed from arrow, product, and so on, it's easy to define conditions for a relation to be a bisimulation. For instance, I know that a relation *R* is a bisimulation over *n*-branching trees *t1 *and *t2* (for some *n*) if

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roots are equal and each of their subtrees are in *R*. My problem is, specifically, with the case of infinitely branching trees. In Haskell,

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are modeled by the data type

T a = T a [T a]

and the possibility arises, of course, that the list [T a] is a stream. Clearly, we can't just say that a relation *R* is a bisimulation on

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t1* and *t2* of type T a if their root values are equal and their *lists* of subtrees are equal. Because if the lists are infinite, we have to prove

On Sat, Mar 05, 2011 at 03:17:57AM -0800, dan portin wrote: their these trees * that

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they are bisimilar. And the coinduction principle for lists requires us to have established that the head of each list is equal. But this is what we're trying to prove!

I don't actually see a problem here, as long as we generalize the notion of "equality" to "bisimilarity" (which is of course the point of bisimilarity). We say that two trees are bisimilar if there is a relation R, for which

* if the roots of the two trees are equal * and their forest-streams are bisimilar

then the trees are in relation R.

It's perfectly fine that the notion of bisimilarity for the forest-streams is defined in terms of bisimilarity of trees. Perhaps to be completely rigorous we should say that we define the notions of bisimilarity for trees and for streams of trees by simultaneous coinduction.

-Brent

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I tried, nothing seems to work. But, here is my configuration for the other poor souls.

C:\Documents and Settings\Usr Berlin>echo %PATH% C:\Program Files\Gtk+\bin;C:\Program Files\Haskell\bin;C:\Program Files\Haskell Platform\2010.2.0.0\lib\extralibs\bin;C:\Program Files\Haskell Platform\2010.2.0 .0\bin; ... and other stuff

C:\Documents and Settings\Usr Berlin>echo %PKG_CONFIG_PATH% C:\Program Files\Gtk+\lib\pkgconfig;C:\Program Files\Gtk+\include\libglade-2.0;C :\Program Files\libxml2\libxml2-dev_2.7.7-1_win32\lib\pkgconfig

C:\Documents and Settings\Usr Berlin>echo %INCLUDE% C:\Program Files\Gtk+\include;C:\Program Files\libxml2\libxml2-dev_2.7.7-1_win32 \include;C:\Program Files\Gtk+\include\libglade-2.0;C:\Program Files\Gtk+\includ e

C:\Documents and Settings\Usr Berlin>pkg-config.exe --modversion gtk+-2.0 2.16.2

C:\Documents and Settings\Usr Berlin>pkg-config --cflags gtk+-2.0 Files/Gtk+/include/gtk-2.0 -mms-bitfields Files/Gtk+/lib/gtk-2.0/include Files/G tk+/include/atk-1.0 Files/Gtk+/include/cairo Files/Gtk+/include/pango-1.0 Files/ Gtk+/include/glib-2.0 Files/Gtk+/lib/glib-2.0/include Files/Gtk+/include/libpng1 2 -IC:/Program

C:\Documents and Settings\Usr Berlin>cabal update Downloading the latest package list from hackage.haskell.org

C:\Documents and Settings\Usr Berlin>cabal install gtk Resolving dependencies...

C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cairo-0.12.043564\cairo-0.12.0\Gtk2HsSetup.h :25: warning: #warning Setup.hs is guessing the version of Cabal. If compilatio of Setup.hs fails use -DCABAL_VERSION_MINOR=x for Cabal version 1.x.0 when bui ding (prefixed by --ghc-option= when using the 'cabal' command) [1 of 2] Compiling Gtk2HsSetup ( C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cairo- .12.043564\cairo-0.12.0\Gtk2HsSetup.hs, C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cair -0.12.043564\cairo-0.12.0\dist\setup\Gtk2HsSetup.o ) [2 of 2] Compiling Main ( C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cairo- .12.043564\cairo-0.12.0\Setup.hs, C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cairo-0.12 043564\cairo-0.12.0\dist\setup\Main.o ) Linking C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\cairo-0.12.043564\cairo-0.12.0\dist\ etup\setup.exe ... Configuring cairo-0.12.0... setup.exe: Missing dependencies on foreign libraries: * Missing C libraries: z, cairo This problem can usually be solved by installing the system packages that provide these libraries (you may need the "-dev" versions). If the libraries are already installed but in a non-standard location then you can use the flags --extra-include-dirs= and --extra-lib-dirs= to specify where they are.

C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0.12.043564\glib-0.12.0\Gtk2HsSetup.hs: 5: warning: #warning Setup.hs is guessing the version of Cabal. If compilation f Setup.hs fails use -DCABAL_VERSION_MINOR=x for Cabal version 1.x.0 when build ng (prefixed by --ghc-option= when using the 'cabal' command) [1 of 2] Compiling Gtk2HsSetup ( C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0 12.043564\glib-0.12.0\Gtk2HsSetup.hs, C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0 12.043564\glib-0.12.0\dist\setup\Gtk2HsSetup.o ) [2 of 2] Compiling Main ( C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0 12.043564\glib-0.12.0\Setup.hs, C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0.12.04 564\glib-0.12.0\dist\setup\Main.o ) Linking C:\DOCUME~1\UsrBER~1\LOCALS~1\Temp\glib-0.12.043564\glib-0.12.0\dist\se up\setup.exe ... Configuring glib-0.12.0... setup.exe: Missing dependencies on foreign libraries: * Missing C libraries: gobject-2.0, glib-2.0, intl This problem can usually be solved by installing the system packages that provide these libraries (you may need the "-dev" versions). If the libraries are already installed but in a non-standard location then you can use the flags --extra-include-dirs= and --extra-lib-dirs= to specify where they are. cabal: Error: some packages failed to install: cairo-0.12.0 failed during the configure step. The exception was: ExitFailure 1 gio-0.12.0 depends on glib-0.12.0 which failed to install. glib-0.12.0 failed during the configure step. The exception was: ExitFailure 1 gtk-0.12.0 depends on glib-0.12.0 which failed to install. pango-0.12.0 depends on glib-0.12.0 which failed to install.

-- Berlin Brown (berlin dot brown at gmail.com) http://botnode.com http://berlinbrowndev.blogspot.com/

I might have figured it out. The default install for gtk installed in "Program Files". I moved it to just c:\gtk without the + and the program files space and that seem to have corrected the problem. Weird. It is 2011 and that is still an issue apparently. -- Berlin Brown (berlin dot brown at gmail.com) http://botnode.com http://berlinbrowndev.blogspot.com/