# HG changeset patch # User Christian Urban # Date 1331826618 0 # Node ID e10eb390c1035e996fb3328a90658ae8c9bbd818 # Parent 3a2a47f967d48c45a3ac2d5dd06ff3f2a8ad6042 added Cut diff -r 3a2a47f967d4 -r e10eb390c103 Cut/CL-plugin.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/CL-plugin.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,169 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
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+LJT
+ND

+ +Plugin Versions
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+ +
+
+ + + + +

 + +

Cut-Elimination in Classical Logic

+ +If you use this applet it is strongly(!) recommended to have read the the paper on +strong normalisation of cut-elimination in classical logic by Urban and Bierman +[ps.gz, + pdf]. +Mayor difference between the standard sequent calculus and the sequent calculus +implemented by the applet is that the rules contraction and weakening are completely +implicit. This means that sequents consist of two sets of (labelled) formulae, +as opposed to lists or multisets of formulae. Although this is very simple, it +needs some time to get used to. +
+How to use it?  +If your browser is playing game with Java, then you will see a window +with a number of examples. When pressing on one of the buttons, a new +window will pop up. What follows is a brief explanation of all +the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        and +        are for + commuting cuts, which can slide up either in the right or left proof branch. +
  • Pressing on     +  ,      or +        + `fires' a logical cut. The latter two buttons correspond to the two possible + nestings of multiplicative logical cuts. +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct(s). +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Auxiliary Substitution  If you have read the paper referred + above, you know what this does. +
  • Gentzen's D.N.Transl. and Kolmogorov's D.N.Transl. + perform double negation translations on the current proof. +
+
+
+ + + + + + + + + + + +<p><blink><FONT COLOR="#800000"> +Applet should appear in a new window, but you don't have Java enabled in your browser. +</FONT></blink><p> + + + + + + + + + + + +
+

+ +Last modified: Sat Mar 3 05:17:41 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/CL.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/CL.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,142 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +

 + +

Cut-Elimination in Classical Logic

+ +If you use this applet it is strongly(!) recommended to have read the the paper on +strong normalisation of cut-elimination in classical logic by Urban and Bierman +[ps.gz, + pdf]. +Mayor difference between the standard sequent calculus and the sequent calculus +implemented by the applet is that the rules contraction and weakening are completely +implicit. This means that sequents consist of two sets of (labelled) formulae, +as opposed to lists or multisets of formulae. Although this is very simple, it +needs some time to get used to. +
+How to use it?  +If your browser is playing game with Java, then you will see a window +with a number of examples. When pressing on one of the buttons, a new +window will pop up. What follows is a brief explanation of all +the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        and +        are for + commuting cuts, which can slide up either in the right or left proof branch. +
  • Pressing on     +  ,      or +        + `fires' a logical cut. The latter two buttons correspond to the two possible + nestings of multiplicative logical cuts. +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct(s). +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Auxiliary Substitution  If you have read the paper referred + above, you know what this does. +
  • Gentzen's D.N.Transl. and Kolmogorov's D.N.Transl. + perform double negation translations on the current proof. +
+
+

+ + +

+Applet should appear in a new window, but you don't have Java enabled in your browser. +

+

 + +
+

+ +Last modified: Sat Mar 3 05:17:31 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/LJT-plugin.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/LJT-plugin.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,161 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +

 + +

Cut-Elimination in Herbelin's calculus for intuitionistic logic

+ +Recommended reading for this applet is a paper by Dyckhoff and Urban, +[ps, + pdf]. +Mayor difference between the standard sequent calculus and Herbelin's sequent calculus +is that in the cut-free fragment "there are no semantically +trivial permutations of the inference rules, where by 'semantically trivial' we mean +'interpreted in NJ as true equations between to (normal) deductions'" +(from [ps]). +
+How to use it?  +If your browser is working correctly, then you will see a window +with two examples (more added if you suggest some). When pressing on +one of the buttons, a new window will pop up. What follows +is a brief explanation of all the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        + `fire' cuts (not all cuts necessarily have such buttons). +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct. +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Compact printing  In the paper referred above weakening + is done only in axioms. If this radiobutton is switched on (default), + then, in order to save space, formulae are weakened as soon as possible. +
+
+

+ + + + + + + + + + + +<p><blink><FONT COLOR="#800000"> +Applet should appear in a new window, but you don't have Java enabled in your browser. +</FONT></blink><p> + + + + + + + + + + + +
+

+ +Last modified: Sat Mar 3 05:17:20 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/LJT.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/LJT.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,134 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +

 + +

Cut-Elimination in Herbelin's calculus for intuitionistic logic

+ +Recommended reading for this applet is a paper by Dyckhoff and Urban, +[ps, + pdf]. +Mayor difference between the standard sequent calculus and Herbelin's sequent calculus +is that in the cut-free fragment "there are no semantically +trivial permutations of the inference rules, where by 'semantically trivial' we mean +'interpreted in NJ as true equations between to (normal) deductions'" +(from [ps]). +
+How to use it?  +If your browser is working correctly, then you will see a window +with two examples (more added if you suggest some). When pressing on +one of the buttons, a new window will pop up. What follows +is a brief explanation of all the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        + `fire' cuts (not all cuts necessarily have such buttons). +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct. +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Compact printing  In the paper referred above weakening + is done only in axioms. If this radiobutton is switched on (default), + then, in order to save space, formulae are weakened as soon as possible. +
+
+

+ + +

+Applet should appear in a new window, but you don't have Java enabled in your browser. +

+

 + +
+

+ +Last modified: Sat Mar 3 05:17:09 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/Main.zip Binary file Cut/Main.zip has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/MyApplet.zip Binary file Cut/MyApplet.zip has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ND-plugin.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/ND-plugin.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,157 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +

 + +

Normalisation in intuitionistic natural deduction

+ +This applet implements the bog-standard version of intuitionistic +natural deduction. If you are familiar with any textbook on proof +theory (e.g. this one), then most things will look very familiar to you. +Note that natural deduction is here implemented in `sequent style'. +
+How to use it?  +If your browser is working correctly, then you will see a window +with two examples (more added if I have time or you suggest one). +When you press on one of the buttons, a new window will pop up. What +follows is a brief explanation of all the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        + `fire' beta-redexes (nothing surprising here). +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct. +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Compact printing  Usually formulae are weakened only in axiom. + If this radiobutton is switched on (default), then, in order to save space, + formulae are weakened as soon as possible. +
+
+

+ + + + + + + + + + + +<p><blink><FONT COLOR="#800000"> +Applet should appear in a new window, but you don't have Java enabled in your browser. +</FONT></blink><p> + + + + + + + + + + + +
+

+ +Last modified: Sat Mar 3 05:16:58 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ND.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/ND.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,130 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +

 + +

Normalisation in intuitionistic natural deduction

+ +This applet implements the bog-standard version of intuitionistic +natural deduction. If you are familiar with any textbook on proof +theory (e.g. this one), then most things will look very familiar to you. +Note that natural deduction is here implemented in `sequent style'. +
+How to use it?  +If your browser is working correctly, then you will see a window +with two examples (more added if I have time or you suggest one). +When you press on one of the buttons, a new window will pop up. What +follows is a brief explanation of all the features available in this window. +
+Buttons and Keys +
    +
  • The buttons +        + `fire' beta-redexes (nothing surprising here). +
  • If you use the left mouse button for pressing on those buttons, + then the new proof will appear inside the window. Whereas if you use + the right mouse button, a new window will pop up and + the new proof will be displayed in this window. Use the right mouse + button if you want to compare a proof and its reduct. +
  • The keys Page-up and Page-down zoom in or out of a proof + respectively. +
+
+Menu Functions + +
    +
  • Zoom in and Zoom out work like Page-up and Page-down, + respectively. +
  • Unicode  If this radiobutton is switched on, logic symbols + are displayed in unicode, otherwise in ascii. (This is for the + poor guys who have a browser and/or operating system which cannot handle + unicode.) +
  • Labels  As mentioned earlier the sequents are composed of + two sets of labelled + formulae, not multisets of formulae as in the standard formulation + of sequent calculus. Enabling this radiobutton causes that labels are + drawn. Use this when you are unsure where implicit contractions are. +
  • Compact printing  Usually formulae are weakened only in axiom. + If this radiobutton is switched on (default), then, in order to save space, + formulae are weakened as soon as possible. +
+
+

+ + +

+Applet should appear in a new window, but you don't have Java enabled in your browser. +

+

 + +
+

+ +Last modified: Sat Mar 3 05:16:48 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/START.txt --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/START.txt Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,1 @@ +java -classpath Main.zip Main diff -r 3a2a47f967d4 -r e10eb390c103 Cut/cutapplet.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/cutapplet.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,137 @@ + + + Christian Urban + + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +		 + +

Applet for Cut-Elimination and Normalisation

+ +Warning: +These pages are still under construction and designed for the conoscenti +only. If you encounter any problems, email me. Some general information +about the applet is given below.

+ +The applet is outcome of my interest in cut-elimination. It helps to +explore reduction trees in a slight variant of the sequent calculus LK, +and also reduction trees in Herbelin's calculus LJT (as presented +here) +and in the standard formulation of natural deduction for intuitionistic +logic. Given for example the following classical sequent proof

+ +

+ +

+ +you can produce by simply pressing the buttons L and R the following +two cut-free proofs.

+ +

+ + +

+ +If you ever have done cut-elimination by hand, then you +know how useful this applet is.

+ + The applet is particularly useful for doing +some calculations finding out what the (proof-theoretical) semantics for classical logic +should look like. Some informal notes are +here. +

+ + +The applet needs at least Java 1.2 including the +Swing libraries. Some browsers support these requirements +directly, and if you are one of the lucky to have such a browser +you can access the three calculi by pressing on one of the +following links. + +

+ +If not, then your browser can most probably run the applet with +the help of a Java plugin. In this case follow the links below. + +

+ +

+For the curious: the applet is written in MLJ, which is being +developed by +Benton, +Kennedy and +Russo. +MLJ is a dialect of SML that provides access to Java libraries; in my +opinion it is a really nifty language! I am using an improved version of +the MLJ-0.2 compiler. +
+

+ +Last modified: Sat Mar 3 05:16:36 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ex1.jpg Binary file Cut/ex1.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ex2.jpg Binary file Cut/ex2.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ex3.jpg Binary file Cut/ex3.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/l.jpg Binary file Cut/l.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/lafont1.jpg Binary file Cut/lafont1.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/lafont2.jpg Binary file Cut/lafont2.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/lafont3.jpg Binary file Cut/lafont3.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/new.gif Binary file Cut/new.gif has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/notes.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Cut/notes.html Thu Mar 15 15:50:18 2012 +0000 @@ -0,0 +1,91 @@ + + + Christian Urban + + + + + + + + + + + + + + + + +
+ +Links
+Home
+Applet Home

+ +Java Versions
+CL
+LJT
+ND

+ +Plugin Versions
+CL
+LJT
+ND
+ +
+
+ + + + +		 + +

What is the Semantics for Classical Logic?

+ +In the intuitionistic case, the answer to this question is well-understood - +essentially a semantics is given to all proofs in normal form and all other +proofs inherit this semantics from the normal forms they reduce to. This +implies that: + +
+(1) reductions preserving the semantics of a proof. +
+ + +It is widely known that this assumption causes any reasonable semantics +for classical logic to collaps. In the literature Lafont's example is +often cited for this observation.

+ +

+ +

+ + +

+ + +

+ + +

+

+ +Last modified: Sat Mar 3 05:17:53 CET 2007 + + + diff -r 3a2a47f967d4 -r e10eb390c103 Cut/r.jpg Binary file Cut/r.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/ribbon.gif Binary file Cut/ribbon.gif has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/x.jpg Binary file Cut/x.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/x1.jpg Binary file Cut/x1.jpg has changed diff -r 3a2a47f967d4 -r e10eb390c103 Cut/x2.jpg Binary file Cut/x2.jpg has changed