# HG changeset patch # User Christian Urban # Date 1331806048 0 # Node ID 907b1fff563752bf8d75d1d73d43cd5f8a0e620e # Parent 345dd18f020e857236774a63ac805e51d64aa74f added phd and prover diff -r 345dd18f020e -r 907b1fff5637 PhD/index.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/PhD/index.html Thu Mar 15 10:07:28 2012 +0000 @@ -0,0 +1,85 @@ + + +Homepage of Christian Urban + + + + + + + + + + + + + + + + + + +

+ +Links
+Home
+Publications
+Recent Talks +
+
+ +Handy Information
+People in Logic
+Programming Languages
+Miscellaneous +

+ + +

+ +

+Postscript file of the thesis: +ps.gz +

The Dinner after the Viva

+organised by Gavin in The Three Horseshoes in Maddingley. Very nice.

+ +
+My examiners, +Andy Pitts and +Henk Barendregt, are next to me. +Nick Benton, Mrs. Barendregt-Kessing (Henk's mother), +Gavin Bierman, +Mateja Jamnik, +Martin Hyland and +Glynn Winskel +are standing behind.

+ +
+There are Susan Pitts and Alice Benton on the left.

+ +

The Ceremony

+ +

+ +Last modified: Thu Feb 28 20:23:06 CET 2008 + +[Validate this page.] + + diff -r 345dd18f020e -r 907b1fff5637 PhD/phd-1.jpg Binary file PhD/phd-1.jpg has changed diff -r 345dd18f020e -r 907b1fff5637 PhD/phd-2.jpg Binary file PhD/phd-2.jpg has changed diff -r 345dd18f020e -r 907b1fff5637 PhD/phd-3.jpg Binary file PhD/phd-3.jpg has changed diff -r 345dd18f020e -r 907b1fff5637 Prover/Contexts.pizza --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Prover/Contexts.pizza Thu Mar 15 10:07:28 2012 +0000 @@ -0,0 +1,70 @@ +package G4ip; + +import java.util.Vector; +import G4ip.Form.*; + +/** Context acts as a multiset.

+ * Typical contexts in logical rules are Gamma, Delta, etc. + * @author Christian Urban + */ +public class Context extends Vector { + + public Context() { super(); } + + public Context(Context init_context) { + super(); + for (int i=0;i 0) { // add last formula + s=s.concat(elementAt(i).toString()); + } + return s; + } + + /** returns a context with one additional formula at the beginning + * @param new_formula a formula + */ + public Context add(Form new_formula) { + Context new_c = new Context(this); + new_c.insertElementAt(new_formula,0); + return new_c; + } + + /** returns a context with two additional formulae at the beginning + * @param new_formula1,new_formula2 two formulae + */ + public Context add(Form new_formula1, Form new_formula2) { + Context new_c = new Context(this); + new_c.insertElementAt(new_formula1,0); + new_c.insertElementAt(new_formula2,0); + return new_c; + } + + /** tests whether a context contains a specific atom + * @param a a formula + */ + public boolean includes(Form a) { + if (a instanceof Atm) { + for (int i=0;i + + + + + +public class G4ip.Context + + + + ++[all packages] +[package G4ip] +[class hierarchy] +[index] + +

public class G4ip.Context

+(source file: Contexts.pizza)
+

+java.lang.Object
+   |
+   +----java.util.Vector
+           |
+           +----G4ip.Context
+

+public class Context
+  extends Vector
+

Contexts act as multiset.

+ Typical contexts in logical rules are Gamma, Delta, etc.


Author:: Christian Urban

Context(Context) +: +
Context() +: +

add(Form, Form) +: returns a contexts with two additional formulae at the beginning +
add(Form) +: returns a context with one additional formula at the beginning +
includes(Form) +: tests whether a context contains a specific atom +
makeString() +: should be toString, but this is a "final" method in Vector +

Context
+

+public Context();
+

+ Context
+

+public Context(Context init_context);
+

makeString
+

+public String makeString();
+

should be toString, but this is a "final" method in Vector

Overrides:

makeString in class Context

+ add
+

+public Context add(Form new_formula);
+

returns a context with one additional formula at the beginning

Parameters:

new_formula - a formula

Overrides:

add in class Context

+ add
+

+public Context add(Form new_formula1,
+                   Form new_formula2);
+

returns a contexts with two additional formulae at the beginning

Parameters:

new_formula1,new_formula2 - two formulae

Overrides:

add in class Context

+ includes
+

+public boolean includes(Form a);
+

tests whether a context contains a specific atom

Parameters:

a - a formula

Overrides:

includes in class Context


+[all packages] 
+[package G4ip] 
+[class hierarchy] 
+[index] 
+

G4ip.Context.html

+ + diff -r 345dd18f020e -r 907b1fff5637 Prover/Doc/G4ip.Form.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Prover/Doc/G4ip.Form.html Thu Mar 15 10:07:28 2012 +0000 @@ -0,0 +1,108 @@ + + + + + + +public class G4ip.Form + + + + +


+[all packages] 
+[package G4ip] 
+[class hierarchy] 
+[index] 
+

public class G4ip.Form

+(source file: Formulae.pizza)
+

+java.lang.Object
+   |
+   +----G4ip.Form
+

+public class Form
+

Class for propositional formulae.


Author:: Christian Urban

And(Form, Form) +: +
Atm(String) +: +
Equ(Form, Form) +: +
False() +: +
Imp(Form, Form) +: +
Or(Form, Form) +: +

toString() +: +

False
+

+public case False();
+

+ And
+

+public case And(Form c1,
+                Form c2);
+

+ Or
+

+public case Or(Form c1,
+               Form c2);
+

+ Imp
+

+public case Imp(Form c1,
+                Form c2);
+

+ Equ
+

+public case Equ(Form c1,
+                Form c2);
+

+ Atm
+

+public case Atm(String c);
+

toString
+

+public String toString();
+


+[all packages] 
+[package G4ip] 
+[class hierarchy] 
+[index] 
+

G4ip.Form.html

+ + diff -r 345dd18f020e -r 907b1fff5637 Prover/Doc/G4ip.Parser.html --- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/Prover/Doc/G4ip.Parser.html Thu Mar 15 10:07:28 2012 +0000 @@ -0,0 +1,133 @@ + + + + + + +public class G4ip.Parser + + + + +


+[all packages] 
+[package G4ip] 
+[class hierarchy] 
+[index] 
+

public class G4ip.Parser

+(source file: Parser.pizza)
+

+java.lang.Object
+   |
+   +----G4ip.Parser
+

+public class Parser
+

A left-to-right, rightmost-derivation parser.

+ The following grammar is implemented:

Formula ::= Id

| false

| ( Formula )

| Formula Binop Formula

Id is a string of lower case letters

Binop is either &, v, -> or <->

FormulaList ::= empty

| [ FormulaList ,]* Formula

+ The parser uses a stack where two actions are performed: +

shift moves the next token to the top of the stack (getNextToken)

reduce chooses a grammar rule, X -> A B C; pops A, B, C from the + top of the stack and pushes X onto the stack +


Author:: Christian Urban

Parser(String) +: +

getNextToken() +: tokens are: identifiers( ) , & v -> <-> false + and EOI +
parseFormula() +: parses a single formula +
parseFormulae() +: parses a list of formulae separated by commas +
reduce() +: Implements the grammar rules. +

Parser
+

+public Parser(String init_in);
+

getNextToken
+

+public Token getNextToken()
+  throws Exception;
+

tokens are: identifiers( ) , & v -> <-> false + and EOI (end of input)

Overrides:

getNextToken in class Parser

+ reduce
+

+public void reduce();
+

Implements the grammar rules.

Overrides:

reduce in class Parser

+ parseFormula
+

+public Form parseFormula()
+  throws Exception;
+

parses a single formula

Overrides:

parseFormula in class Parser

+ parseFormulae
+

+public Context parseFormulae()
+  throws 
+
+
+
+
+
+package G4ip
+
+
+
+
+
+[all packages] 
+[class hierarchy] 
+[index] 
+
+
+package G4ip
+
+
+
+
+
+ public class Context
Contexts act as multiset.
+
 public class Form
Class for propositional formulae.
+
 public class Parser
A left-to-right, rightmost-derivation parser.
+
 public class ProofDisplay
Draws a frame on the screen containing a proof.
+
 public class Prover
The Gi4p prover.
+
 public class ProverApplet
The applet for the prover.
+
 public class Sequent
An intuitionistic sequent of the form     
+  "Gamma => G".
+
 public class Token

+
+
+
+
+
+
+
+
+[all packages] 
+[class hierarchy] 
+[index] 
+
+G4ip.html
+
+
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diff -r 345dd18f020e -r 907b1fff5637 Prover/Formulae.pizza
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Formulae.pizza	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,48 @@
+
+package G4ip;
+
+/** Class for propositional formulae.
+  * @author      Christian Urban
+  */
+public class Form {
+
+  public case False();         
+  public case And(Form c1,Form c2);
+  public case Or(Form c1,Form c2);
+  public case Imp(Form c1,Form c2);
+  //public case Equ(Form c1,Form c2);
+  public case Atm(String c);
+
+  public String toString() {
+    switch(this) {
+    case False(): 
+      return "false"; 
+    case And(Form c1,Form c2): 
+      return "(" + c1.toString() + " & "   + c2.toString() + ")"; 
+    case Or(Form c1,Form c2):  
+      return "(" + c1.toString() + " v "   + c2.toString() + ")"; 
+    case Imp(Form c1,Form c2): 
+      return "(" + c1.toString() + " -> "  + c2.toString() + ")"; 
+    case Atm(String c): 
+      return c; 
+    }
+  }
+
+  // for testing
+  public String internalString() {
+    switch(this) {
+    case False(): 
+      return "false"; 
+    case And(Form c1,Form c2): 
+      return "And(" + c1.internalString() + ","   + c2.internalString() + ")"; 
+    case Or(Form c1,Form c2):  
+      return "Or(" + c1.internalString() + ","   + c2.internalString() + ")"; 
+    case Imp(Form c1,Form c2): 
+      return "Imp(" + c1.internalString() + ","  + c2.internalString() + ")"; 
+    case Atm(String c): 
+      return "Atm(\"p"+ c + "\")"; 
+    }
+  }
+
+}
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.bbl
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.bbl	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,9 @@
+\begin{thebibliography}{}
+
+\bibitem[Dyckhoff, 1992]{Dyckhoff92}
+Dyckhoff, R.
+\newblock {C}ontraction-{F}ree {S}equent {C}alculi for {I}ntuitionistic
+  {L}ogic.
+\newblock {\em Journal of Symbolic Logic}, 57(3):795--807.
+
+\end{thebibliography}
diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.blg
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diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.html	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,27 @@
+
+
+   
+   G4ip Infernce Rules
+
+
+
+
The Inference Rules of G4ip (LJT)  
+
+A dvi and ps version; the rules can be found in 
+[Dyckhoff, 1992].
+
+
+
+
+
+
+
+Christian Urban
+
+
+
+
+Last modified: Wed Jan 28 22:14:28 GMT 1998
+
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.mod
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.mod	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,118 @@
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% An Implementation of G4ip for Terzo
+% 	author: Christian.Urban@cl.cam.ac.uk 
+% 
+% some solvable sample queries:
+%
+% prove (nil |- p imp p).
+% prove (nil |- (p imp p) imp (p imp p)).
+% prove (nil |- ((((p imp q) imp p) imp p) imp q) imp q).
+% prove (nil |- (a imp (b imp c)) imp ((a imp b) imp (a imp c))).
+% prove (nil |- (a or (a imp b)) imp (((a imp b) imp a) imp a)).
+%
+% two non-solvable queries
+%
+% prove (nil |- a or (a imp false)).
+% prove (nil |- ((a imp b) imp a) imp a).
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+module G4ip.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% atomic formulae
+kind form type.
+
+type p form. 
+type q form.
+type a form.
+type b form.
+type c form.
+
+type isatomic form -> o.
+
+isatomic p.
+isatomic q.
+isatomic a.
+isatomic b.
+isatomic c.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% logical operators
+type false  form.
+type and    form -> form -> form.    
+type or     form -> form -> form.       
+type imp    form -> form -> form.
+ 
+infixr and   9.
+infixr or    9.
+infixr imp   9.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% sequent constructor; sequents are of the form: (list |- formula) 
+kind seq type.
+
+type   |-  list form -> form -> seq. 
+infixl |-   4.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% prove predicate; prints "solvable" if seq is provable, otherwise "no" 
+type prove seq -> o.
+
+prove (Gamma |- G) :- ( membNrest P Gamma Gamma',
+                        left (P::Gamma' |- G) );
+                        right   (Gamma  |- G).   
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% rightrules
+type right seq -> o.
+
+right (Gamma |- B and C) :- prove (Gamma |- B),          %% and-R   
+                            prove (Gamma |- C).          
+
+right (Gamma |- B imp C) :- prove (B::Gamma |- C).       %% imp-R
+
+right (Gamma |- B or  C) :- prove (Gamma |- B);          %% or-R 
+                            prove (Gamma |- C).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% leftrules
+type left  seq -> o.
+
+left (false :: Gamma |- G).                              %% false-L
+
+left (A :: Gamma |- A)     :- isatomic A.                %% axiom 
+
+left (B and C::Gamma |- G) :- prove (B::C::Gamma |- G).  %% and-L
+ 
+left (B  or C::Gamma |- G) :- prove (B::Gamma |- G),     %% or-L 
+                              prove (C::Gamma |- G).
+
+
+left (A imp B::Gamma |- G) :-                            %% imp-L1
+	isatomic A, ismember A Gamma, prove (B::Gamma |- G).
+
+left ((B and C) imp D::Gamma |- G) :-                    %% imp-L2
+	prove (B imp (C imp D)::Gamma |- G).
+
+left ((B or  C) imp D::Gamma |- G) :-                    %% imp-L3
+	prove (B imp D::C imp D::Gamma |- G).
+
+left ((B imp C) imp D::Gamma |- G) :-                    %% imp-L4
+	prove (C imp D::Gamma |- B imp C),	                              
+	prove (D::Gamma |- G).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% returns a member and the remainder of a list
+type membNrest     A -> list A -> list A -> o.
+
+membNrest X (X::Rest) Rest.
+membNrest X (Y::Tail) (Y::Rest) :- membNrest X Tail Rest.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% succeeds only once if A is element in the list 
+type ismember     A -> list A -> o.
+
+ismember X (X::Rest) :- !.
+ismember X (Y::Tail) :- ismember X Tail.
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.ps
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.ps	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,427 @@
+%!PS-Adobe-2.0
+%%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software
+%%Title: G4ip.dvi
+%%Pages: 1
+%%PageOrder: Ascend
+%%BoundingBox: 0 0 596 842
+%%EndComments
+%DVIPSCommandLine: dvips G4ip.dvi
+%DVIPSParameters: dpi=300, comments removed
+%DVIPSSource:  TeX output 1997.11.23:1951
+%%BeginProcSet: tex.pro
+/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
+/X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72
+mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1}
+ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
+isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div
+hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul
+TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if}
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diff -r 345dd18f020e -r 907b1fff5637 Prover/G4ip.tex
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip.tex	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,116 @@
+\documentclass[11pt]{article}
+\usepackage{dina4}
+\usepackage{my}
+\usepackage{latexsym,amssymb}
+%\usepackage[final]{graphics}
+
+
+\newcommand{\FN}[1]{FN(#1)} % free name
+\newcommand{\FC}[1]{FC(#1)} % free co-name
+\renewcommand{\refname}{\-}
+
+\begin{document}
+\section*{Inference Rules of G4ip (LJT)}
+
+\begin{figure}[h]
+  \begin{center}
+  \begin{FramedTitled}{\-}
+  \centering
+  \def\arraystretch{3.0}
+  \begin{tabular}{cc}
+
+   \infer[Axiom]{\sequ{A,\Gamm}{A}}{} &
+
+   \infer[${\em false}-L$]{\sequ{\mbox{\em false},\Gamm}{G}}{}\\[-8mm]
+
+   ($A$ being atomic) \\
+
+   \infer[\&$-L$]{\sequ{B \& C,\Gamm}{G}}
+                 {\sequ{B,C,\Gamm}{G}} &
+
+   \infer[\&$-R$]{\sequ{\Gamm}{B \& C}}
+                 {  \sequ{\Gamm}{B}
+                  & \sequ{\Gamm}{C}}\\
+
+   \infer[\OR$-L$]
+              {\sequ{B\OR C,\Gamm}{G}}
+              { \sequ{B,\Gamm}{G}
+               &\sequ{C,\Gamm}{G}} &
+
+   \infer[\OR$-R$_1]
+              {\sequ{\Gamm}{B\OR C}}
+              {\sequ{\Gamm}{B}}\hspace{3mm}
+   \infer[\OR$-R$_2]
+              {\sequ{\Gamm}{B\OR C}}
+              {\sequ{\Gamm}{C}}\\
+
+   \infer[\mbox{\tt ->}$-L$_1]
+                    {\sequ{A\mbox{\tt ->} B,A,\Gamm}{G}}
+                    {\sequ{B,A,\Gamm}{G}} &
+
+   \infer[\mbox{\tt ->}$-R$]
+                 {\sequ{\Gamm}{B\mbox{\tt ->} C}}
+                 {  \sequ{B,\Gamm}{C}}\\[-8mm]
+      
+   ($A$ being atomic)\\           
+       
+   \multicolumn{2}{c}{
+   \infer[\mbox{\tt ->}$-L$_2]
+              {\sequ{(C \& D)\mbox{\tt ->} B,\Gamm}{G}}
+              {\sequ{C \mbox{\tt ->} (D \mbox{\tt ->} B)\Gamm}{G}}} \\
+
+   \multicolumn{2}{c}{
+   \infer[\mbox{\tt ->}$-L$_3]
+              {\sequ{(C\OR D)\mbox{\tt ->} B\Gamm}{G}}
+              {\sequ{C\mbox{\tt ->}B,D\mbox{\tt ->}B,\Gamm}{G}}}\\
+
+   \multicolumn{2}{c}{
+   \infer[\mbox{\tt ->}$-L$_4]
+              {\sequ{(C\mbox{\tt ->} D)\mbox{\tt ->} B\Gamm}{G}}
+              {\sequ{D\mbox{\tt ->}B,\Gamm}{C\mbox{\tt ->} D} &
+               \sequ{B,\Gamm}{G}} }\\
+
+   %\infer[\mbox{\tt <->}$-L$]
+   %                 {\sequ{B\mbox{\tt <->} C,\Gamm}{G}}
+   %                 {\sequ{B\mbox{\tt ->} C, C\mbox{\tt ->} B,\Gamm}{G} 
+   %                 } &
+
+   %\infer[\mbox{\tt <->}$-R$]
+   %              {\sequ{\Gamm}{B\mbox{\tt <->} C}}
+   %              {  \sequ{\Gamm}{B\mbox{\tt ->} C} &
+   %                 \sequ{\Gamm}{C\mbox{\tt ->} B} }\\
+
+   \multicolumn{2}{c}
+     { $B\mbox{\tt <->}C$ is defined as $B\mbox{\tt ->}C \& C\mbox{\tt ->}B$
+     }           
+  \end{tabular}
+  \end{FramedTitled}
+  \caption{Inference rules for G4ip; see for example \cite{Dyckhoff92}.}
+  \label{fig:CL}
+  \end{center}
+\end{figure}
+
+\bibliography{/home/cu200/tex/bib/all}
+\bibliographystyle{fullnames}
+
+
+\end{document}
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Local Variables:
+%%% mode: latex
+%%% TeX-master: "G4ip.tex"
+%%% End:
+
+
+
+
+
+
+
+
+
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+++ b/Prover/G4ip/.svn/prop-base/Form$False.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,5 @@
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Form$Imp.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,5 @@
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+++ b/Prover/G4ip/.svn/prop-base/Form$Or.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,5 @@
+K 13
+svn:mime-type
+V 24
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+++ b/Prover/G4ip/.svn/prop-base/Form.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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+K 13
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+V 24
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+++ b/Prover/G4ip/.svn/prop-base/MyLabel.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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+++ b/Prover/G4ip/.svn/prop-base/ProofDisplay.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Prover$$closures.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Prover.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/ProverApplet.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Sequent.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Token$BINOP.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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+K 13
+svn:mime-type
+V 24
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Token$COMMA.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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+++ b/Prover/G4ip/.svn/prop-base/Token$EOI.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/G4ip/.svn/prop-base/Token$FALSE.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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+++ b/Prover/G4ip/.svn/prop-base/Token$ID.class.svn-base	Thu Mar 15 10:07:28 2012 +0000
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diff -r 345dd18f020e -r 907b1fff5637 Prover/GettingStarted.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
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@@ -0,0 +1,1 @@
+Not yet available!
diff -r 345dd18f020e -r 907b1fff5637 Prover/Parser.pizza
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Parser.pizza	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,183 @@
+package G4ip;
+
+import java.util.*;
+import pizza.lang.Pair;
+import G4ip.Form.*;
+import G4ip.Token.*;
+
+public class Token {
+  public static final int AND = 0;
+  public static final int OR  = 1;
+  public static final int IMP = 2;
+  public static final int EQU = 3;
+
+  /** End-of-Input */
+  public case EOI();             // end-of-input
+  public case LPAREN();
+  public case RPAREN();
+  public case ID(String id);
+  public case FALSE();
+  public case BINOP(int op);     // binary operation
+  public case COMMA();
+}
+
+/** A left-to-right, rightmost-derivation parser.
 
+  * The following grammar is implemented:

+  * 

+  * Formula ::= Id                             
+  * | false                                     
+  * | ( Formula )                                 
+  * | Formula Binop Formula  
+  * 

+  * 

+  * Id is a string of lower case letters
+  * 

+  * 

+  * Binop is either &, v, -> or <->
+  * 

+  * 

+  * FormulaList ::=  empty
+  * | [ FormulaList ,]* Formula
+  * 
+  * The parser uses a stack where two actions are performed:
+  * 

+  * shift moves the next token to the top of the stack (getNextToken)
+  * reduce chooses a grammar rule, X -> A B C; pops A, B, C from the 
+  *                   top of the stack and pushes X onto the stack  
+  * 
+  * @author      Christian Urban
+  */       
+public class Parser {
+  final Pair keywords[] = 
+          { new Pair("(",    LPAREN()),
+	    new Pair(")",    RPAREN()),
+	    new Pair(",",    COMMA()),
+	    new Pair("&",    BINOP(Token.AND)),
+	    new Pair("v ",   BINOP(Token.OR)),
+	    new Pair("->",   BINOP(Token.IMP)),
+            new Pair("<->",  BINOP(Token.EQU)),
+	    new Pair("false",FALSE()) };
+  String in;     // string to be parsed
+  int index;     // position of the current input
+  Stack stack;   // stack for the left-to-right parser 'LR(0)'
+  
+  public Parser(String init_in){
+    index = 0;
+    in = init_in;
+    stack = new Stack();
+  }
+
+  /** tokens are: identifiers( ) , & v -> <-> false 
+    *             and EOI (end of input) 
+    */
+  public Token getNextToken() throws Exception {
+    while ( index < in.length() && Character.isSpace(in.charAt(index)) ) 
+          { index++; }                                //delete  white-spaces  
+    if (index == in.length()) { return EOI(); }       //end-of-string  
+    for (int i=0;i Atm(string)  */
+      if (stack.size() > 0 && 
+	 (stack.elementAt(stack.size()-1) instanceof ID))
+	    { ID id = (ID)stack.pop();
+	      stack.push(Atm(id.id));
+	      again = true;
+	    } 
+      /*  FALSE -> False()   */
+      if (stack.size() > 0 && 
+	  (stack.elementAt(stack.size()-1) instanceof FALSE))
+	    { stack.pop();
+	      stack.push(False());
+	      again = true;
+	    } 
+      /*   ( Formula )  -> Formula  */ 
+      if (stack.size() > 2 &&
+	  (stack.elementAt(stack.size()-3) instanceof LPAREN) &&
+	  (stack.elementAt(stack.size()-2) instanceof Form)   &&
+	  (stack.elementAt(stack.size()-1) instanceof RPAREN))
+	    {  stack.pop();
+               Form form = (Form)stack.pop(); 
+               stack.pop();
+               stack.push(form);
+               again = true;
+            }   
+      /*   Formula BINOP Formula  -> Formula  */
+      if (stack.size() > 2 &&
+	  (stack.elementAt(stack.size()-3) instanceof Form)  &&
+	  (stack.elementAt(stack.size()-2) instanceof BINOP) &&
+	  (stack.elementAt(stack.size()-1) instanceof Form))
+	    {  Form  c2 = (Form)stack.pop();
+               BINOP op = (BINOP)stack.pop();
+               Form  c1 = (Form)stack.pop(); 
+               switch(op.op) {
+	       case Token.AND: stack.push(new And(c1,c2)); again = true;break;
+               case Token.OR:  stack.push(new Or(c1,c2));  again = true;break; 
+	       case Token.IMP: stack.push(new Imp(c1,c2)); again = true;break;
+               case Token.EQU: stack.push(new And(Imp(c1,c2),Imp(c2,c1))); again = true;break;
+	       }
+            }  
+      if (again == true) { reduce(); }  // do as many "reduces" as possible 
+    }
+
+  /** parses a single formula
+    */
+  public Form parseFormula() throws Exception {
+    Token tok;
+    while (!((tok = getNextToken()) instanceof EOI)) {
+      stack.push(tok);
+      reduce();
+    }
+    if (stack.size() == 1 &&
+       (stack.elementAt(stack.size()-1) instanceof Form))
+       { return (Form)stack.pop(); } 
+    else throw new Exception("Grammar error");
+    return null;  
+  }
+
+  /** parses a list of formulae separated by commas
+    */
+  public Context parseFormulae() throws Exception {
+    Token tok;
+    Context erg = new Context();
+    while (!((tok = getNextToken()) instanceof EOI)) {
+      stack.push(tok);
+      reduce();
+    }
+    if (stack.empty()) return erg;  // LHS can be empty  !!
+    stack.push(new COMMA());
+    for(int i=0;i proof;      // contains the indexed sequents 
+
+  /** The proof is represented in the hashtable.
+    * @param proof is a hashtable; the structure of the proof (or tree) 
+    * is mapped onto a sequence of integers; each integer is a key for
+    * a sequent stored in the hashtable.
+    */
+  public ProofDisplay(Hashtable iproof) {
+    super("Proof");
+    ok = new Button("Close");
+    proof = iproof;
+
+    setLayout(new BorderLayout());
+    // SOUTH
+    Panel p = new Panel();
+    p.add(ok);
+    add("South",p);
+
+    //CENTER
+    Panel display = new Panel();
+    printproof(display,1);
+    add("Center", display);
+
+    pack();
+    show();
+  }
+
+  /** Recovers the structure of the proofs from the hashtable representation.
+    */
+  public void printproof(Panel p,int index) {
+    MyLabel l = new MyLabel(proof.get(index).toString());
+    l.setAlignment(MyLabel.CENTER);
+    p.setLayout(new BorderLayout());
+    p.add("South",l); 
+    // axiom (do nothing)
+    if ((proof.get(2*index) == null) && (proof.get(2*index+1) == null)) { 
+      return;    
+    }
+    // rule with a single premise
+    if ((proof.get(2*index) != null) && (proof.get(2*index+1) == null)) { 
+      Panel np = new Panel();
+      printproof(np,2*index);
+      p.add("Center",np);
+      return;  
+    }
+    // rule with two premises
+    if ((proof.get(2*index) != null) && (proof.get(2*index+1) != null)) { 
+      Panel np1 = new Panel();
+      Panel np2 = new Panel();
+      printproof(np1,2*index);
+      printproof(np2,2*index+1);
+      p.add("West",np1);
+      p.add("East",np2);
+      return;
+    }
+    
+  }
+
+  /** If either the "Close" or the "Window-destroy" button
+    * is pressed, then close the window.
+    */
+  public boolean handleEvent(Event e)  {
+    if (e.id == Event.ACTION_EVENT && e.target == ok) {
+      this.finalize(); 
+      return true;
+    }
+    if (e.id == Event.WINDOW_DESTROY) {
+      this.finalize(); 
+      return true;
+    }
+    return false;
+  }
+
+  /** Closes the window.
+    */
+  public void finalize()
+    { this.dispose();  }
+  
+}
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/Prover.pizza
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Prover.pizza	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,223 @@
+package G4ip;
+
+
+import pizza.util.Hashtable;
+import java.util.Vector;
+import G4ip.Form.*;
+import G4ip.ProofDisplay.*;
+import G4ip.Sequent.*;
+
+
+/** The Gi4p prover.
+  * The prover is a thread in order to suspend the proof search
+  * after one proof is found and to let the user make a choice 
+  * of whether to stop or to continue with the proof search.
+  * @author Christian Urban
+  */
+public class Prover extends Thread
+{ ProverApplet  parent;  // the parent of the prover;
+  boolean once;          // if true then at least one proof was found
+  Sequent sequ;          // the root sequent to be proved
+  Vector  frames;        // to record all frames which have been opened
+  int index ;
+  Hashtable proof;      // keeps a record of the proof
+             
+ 
+  /** The constructor creates a prover which proves a specific sequent.

+    * The parameter parent is for access to the applet buttons.
+    */
+  public Prover(Sequent init_sequ, ProverApplet init_parent) {
+    super("Prover");
+    parent = init_parent;
+    once   = false;
+    sequ   = new Sequent(init_sequ);
+    frames = new Vector();
+    index  = 1;
+    proof  = new Hashtable();
+  }
+   
+  /** Starts the thread; calls prove and stops the thread when 
+    * the proof search is finished.
+    */
+  public void run()
+    { prove(1,sequ,initial_sc); 
+      if (once == true) { 
+	parent.messages.setText("No more proofs.");
+	parent.messages.repaint();
+	parent.repaint();
+      }
+      else { 
+	parent.messages.setText("Not provable.");
+	parent.messages.repaint();
+        parent.repaint();	
+      }
+      parent.switch_into_input_mode();
+      this.stop();     // everything is done, stop the thread
+    } 
+
+  /** Closes all frames which have been opened.
+    */
+  public void finalize() { 
+    for (int i=0; i
+    * Method prove first enumerates all formulae on the LHS        
+    * as being the principal formula and attempts to apply left-rules,  
+    * subsequently it analyses the goal formula.                     
+    */ 
+  public void prove(int index, Sequent is,(() -> void) sc)  {
+    Form principal;
+    proof.put(index,new Sequent(is));      // add sequence to proof
+    rightrules(index,is,sc);
+    for (int i=0;i void) sc)  {
+    Context Gamma = sequ.Gamma;
+    switch(sequ.G) { 
+                          /** And-R
+			    *  Gamma => A    Gamma => B  
+			    * --------------------------   
+			    *      Gamma => A and B     
+			    */
+    case And(Form A,Form B): 
+      prove(2*index,new Sequent(Gamma,A),
+            fun() -> void { prove(2*index+1,new Sequent(Gamma,B),sc); }  ); 
+      break;
+
+                          /** Imp-R  
+			    *   Gamma,A => B   
+			    * ----------------- 
+			    *  Gamma => A imp B     
+			    */ 
+    case Imp(Form A,Form B):
+      prove(2*index,new Sequent(Gamma.add(A),B),sc); break;
+                           
+                           /** Or-R 
+			     *   Gamma  => A           Gamma => B   
+			     * ----------------- or -----------------  
+			     *  Gamma => A or B      Gamma => A or B   
+			     */
+    case Or(Form A,Form B):
+      prove(2*index,new Sequent(Gamma,A),sc);
+      prove(2*index,new Sequent(Gamma,B),sc); break;
+                           
+                           /** Equ-R
+			     *  Gamma => A -> B  Gamma => B -> A 
+			     * -------------------------------- 
+			     *          Gamma => A <-> B         
+			     */ 
+    }
+  }
+
+  /** Analyses the principal formula on the LHS. 
+    */
+  public void leftrules(int index,Form principal,Sequent sequ,(() -> void) sc){
+    Context Gamma = sequ.Gamma;
+    Form    G     = sequ.G;
+    switch(principal) {
+                          /** false-L
+			    * --------------------  
+                            *  false, Gamma => G     
+			    */
+    case False():  sc(); break;
+
+                           /** Axiom
+      			     *  ---------------                  
+      			     *   Gamma, G => G     G being atomic 
+      			     */
+    case Atm(String c):
+      if (G instanceof Atm) {
+	if (((Atm)G).c.compareTo(c) == 0) { sc(); }
+      }
+      break;
+
+                          /** And-L
+			    *   Gamma, A, B => G    
+                            * -------------------- 
+                            *  Gamma, A and B => G     
+			    */
+      
+    case And(Form A, Form B): 
+      prove(2*index,new Sequent(Gamma.add(A,B),G),sc); break;
+
+                          /** Or-R
+			    *  Gamma, A => G   Gamma, B => G 
+                            * ------------------------------- 
+                            *      Gamma, A or B => G      
+			    */ 
+    case Or(Form A, Form B):
+      prove(2*index,new Sequent(Gamma.add(A),G),
+	    fun() -> void {prove(2*index+1,new Sequent(Gamma.add(B),G),sc);});
+      break;
+
+                          /** Imp-L 2         
+			    *  Gamma, A imp (B imp C) => G  
+                            * ------------------------------
+                            *  Gamma, (A and B) imp C => G    
+                            */
+    case Imp(And(Form A, Form B), Form C):
+      prove(2*index,new Sequent(Gamma.add(Imp(A,Imp(B,C))),G),sc); break; 
+
+
+                          /** Imp-L 3       
+                            *  Gamma, (A imp C), (B imp C) => G 
+                            * ----------------------------------
+                            *   Gamma, (A or B) imp C => G       
+			    */
+    case Imp(Or(Form A, Form B), Form C):  
+      prove(2*index,new Sequent(Gamma.add(Imp(A,C),Imp(B,C)),G),sc); break; 
+
+                          /** Imp-L 4 
+			    *  Gamma, (B imp C) => (A imp B)    Gamma, C => G 
+                            * ------------------------------------------------
+                            *         Gamma, (A imp B) imp C => G                
+			    */
+    case Imp(Imp(Form A, Form B), Form C): 
+      prove(2*index,new Sequent(Gamma.add(Imp(B,C)),Imp(A,B)),
+	    fun() -> void { prove(2*index+1,new Sequent(Gamma.add(C),G),sc);});
+      break;  
+                          /** Imp-L 1         
+			    *   Gamma(A), B => G          A being atomic       
+                            * ------------------------    Gamma(A) means:
+                            *  Gamma(A), A imp B => G     Gamma contains A
+			    */ 
+    case Imp(Form A, Form B):
+      if (A instanceof Atm) { 
+	if (Gamma.includes(A)) {
+	  { prove(2*index,new Sequent(Gamma.add(B),G),sc); }
+	}
+      }
+      break;    
+    }
+
+  }
+
+
+  /** The initial success continuation. 
+    * Suspends the thread when a proof is found. 
+    */
+  public void initial_sc()  {  
+    once = true;
+    ProofDisplay p = new ProofDisplay(proof);
+    frames.addElement(p);       // keep a record for later disposal
+    try {                       // suspend the proof search, 
+    suspend(); }           // it might be resumed later on  by the user
+    catch(SecurityException sec_exc)
+      { /* this catch is neccessary for Netscape 3.0 (Linux) */ }
+  }  
+  
+}
diff -r 345dd18f020e -r 907b1fff5637 Prover/ProverApplet.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverApplet.html	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,108 @@
+
+
+   
+   G4ip Prover
+
+
+
+
Prover for G4ip (LJT)  
+
+Readme
+
+
+
+Formulae:    
+    F ::= false 
+    | A 
+    | F & F 
+    | F v F 
+    | F -> F 
+    | F <-> F
+    | (F)

+Sequents:     => F     or 
+                     [F,]* F => F
+
+
+Inference Rules: 
+html,
+dvi or 
+ps.
+
+
+Start: starts the proof search, 
+Clear: clears the text fields,

+Next:  searches for the next proof,
+Stop:  stops the proof search.
+
+
+
+
+
+
+
+
+
+Provable Examples:
+
+
+             => (p v (q v r)) <-> ((p v q) v r)
+             => (p & (q & r)) <-> ((p & q) & r)
+             => (p & (q v r)) <-> ((p & q) v (p & r))
+             => (p v (q & r)) <-> ((p v q) & (p v r))
+             => (p -> p) -> (p -> p)
+             => (a -> (b -> c)) -> ((a -> b) -> (a -> c))
+             => a -> ((a -> b) -> a)
+             => b -> ((a -> b) -> b)
+             => (a & b) -> (b & a)
+             => (a -> (a -> b)) -> (a -> b)
+             => ((((p -> q) -> p) -> p) -> q) -> q
+             => (a v (a -> b)) -> (((a -> b) -> a)-> a)
+             => (a -> (b -> false)) -> (b -> (a -> false))
+             => ((a v (a -> false))-> false)-> false
+a & b, c & d => b & c
+(a v (a-> false)) -> false => false
+
+
+
+Non-Provable Examples:
+
+
+             => ((a -> b) -> a) -> a
+             => a v (a -> false)
+             => (a & b v (((( a-> flase) -> false) -> q) v (b -> q)) -> q) -> q
+
+
+
+
+Some advanced examples
+
+
+Source Code:
+Formulae,
+Contexts,
+Sequents,
+ProofDisplay,
+Prover,
+Parser,
+Applet

+
+
+Lambda Prolog Version: The programm 
+G4ip.mod can be executed using  
+Terzo
+Another implementation of G4ip by Joshua Hodas and Dale Miller written in 
+Lolli can be found
+here.
+
+
+
+
+Christian Urban
+
+
+
+
+Last modified: Sun May 10 18:18:13 GMT 1998
+
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/ProverApplet.pizza
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverApplet.pizza	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,150 @@
+package G4ip;
+
+import java.applet.*;
+import java.awt.*;
+import G4ip.Sequent.*;
+import G4ip.Prover.*;
+import G4ip.Parser.*;
+
+/** The applet for the prover.
+  * @author      Christian Urban
+  */
+public class ProverApplet extends Applet {
+  Prover prover;
+  Parser parser;
+
+  Button start;           
+  Button clear;           
+  Button next;            
+  Button stop;           
+  TextField lhs_text;     // place for the program formulae
+  TextField rhs_text;     // place for the goal formula
+  /** A field for various messages (e.g. error messages).
+    */
+  public Label messages;  // to be verbose   
+
+  public void init() {
+    setLayout(new BorderLayout());
+    // NORTH
+    Panel textfields = new Panel();
+    add("North", textfields);
+       // the parameters LHS_size and RHS_size are given in the html-file
+    int LHS_size = Integer.parseInt(this.getParameter("LHS_size"));
+    int RHS_size = Integer.parseInt(this.getParameter("RHS_size"));
+    lhs_text = new TextField(LHS_size);    // program textfield
+    rhs_text = new TextField(RHS_size);    // goal textfield
+    Label separation = new Label("=>");    // for separation
+
+    if (LHS_size > 0) {     // if someone prefears only single side sequents 
+    textfields.add(lhs_text);   }         
+    textfields.add(separation);
+    textfields.add(rhs_text);
+
+    // CENTER
+    Panel buttonfield = new Panel();
+    add("Center", buttonfield);
+    start = new Button("Start");         
+    clear = new Button("Clear");         
+    next  = new Button("Next");          
+    stop  = new Button("Stop");            
+
+    switch_into_input_mode();   // start in input mode
+    buttonfield.add(start);
+    buttonfield.add(clear);
+    buttonfield.add(next);
+    buttonfield.add(stop);
+
+    // SOUTH
+    messages = new Label("");   // for any kind of message
+    messages.setAlignment(Label.CENTER);
+    add("South",messages);
+  }
+  
+  /** When start is pressed; 
+    * start and clear button are disabled;
+    * next and stop button are enabled. 
+    */
+  public void switch_into_prove_mode() { 
+      start.disable();    next.enable();
+      clear.disable();    stop.enable();
+      repaint();
+    }
+
+  /** When start is pressed 
+    * and (hopefully) when prover finishes. 
+    */  
+  public void switch_into_input_mode() { 
+      start.enable();    next.disable(); 
+      clear.enable();    stop.disable();
+      repaint();
+    }
+
+  /** Initiates the parse and prove process. 
+    */  
+  void initiate_proving() {
+      Sequent sequ;
+      messages.setText(""); 
+      messages.repaint();
+      repaint(); 
+      Parser rhs_parser = new Parser(rhs_text.getText());
+      Parser lhs_parser = new Parser(lhs_text.getText());
+      try {                          // try to parse the input 
+	sequ = new Sequent(lhs_parser.parseFormulae(),
+			   rhs_parser.parseFormula());
+      }
+      catch (Exception exc) {        // there was a parser exception 
+	messages.setText(exc.getMessage()); 
+	messages.repaint();
+	repaint(); 
+	return;                     // in case there was an exception
+      }                             // don't run the following code
+      System.out.println(sequ.G.internalString()); 
+      switch_into_prove_mode();
+      prover = new Prover(sequ,this);
+      prover.start();
+  }
+
+  /** Forces the prover to dispose all open frames.
+    */
+  public void destroy() {  
+    if (prover != null && prover.isAlive()) { 
+      prover.stop(); 
+      prover.finalize();
+    }
+  }
+
+
+  /** The actions for the buttons:
+    * this bit of code would be different in Java 1.1
+    */
+  public boolean action(Event e, Object arg) {
+    // clear button has been pressed
+    if (e.target == clear) {
+      messages.setText(""); 
+      lhs_text.setText("");
+      rhs_text.setText(""); 
+      messages.repaint();
+      repaint(); 
+    }
+    // start button has been pressed
+    if (e.target == start) {
+      initiate_proving();
+      return true; 
+    }
+    // stop button has been pressed
+    if (e.target == stop) {
+      switch_into_input_mode();
+      if (prover != null && prover.isAlive()) { prover.stop();  }
+      return true; 
+    }
+    // next button has been pressed
+    if (e.target == next) {
+      if (prover != null && prover.isAlive()) { prover.resume();} 
+      return true; 
+    }
+    return false;
+  }
+
+}
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/ProverAppletJar.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/ProverAppletJar.html	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,110 @@
+
+
+   
+   G4ip Prover
+
+
+
+
Prover for G4ip (LJT)  
+
+Readme
+
+
+
+Formulae:    
+    F ::= false 
+    | A 
+    | F & F 
+    | F v F 
+    | F -> F 
+    | F <-> F
+    | (F)

+Sequents:     => F     or 
+                     [F,]* F => F
+
+
+Inference Rules: 
+html,
+dvi or 
+ps.
+
+
+Start: starts the proof search, 
+Clear: clears the text fields,

+Next:  searches for the next proof,
+Stop:  stops the proof search.
+
+
+
+
+
+
+
+
+
+
+Provable Examples:
+
+
+             => (p v (q v r)) <-> ((p v q) v r)
+             => (p & (q & r)) <-> ((p & q) & r)
+             => (p & (q v r)) <-> ((p & q) v (p & r))
+             => (p v (q & r)) <-> ((p v q) & (p v r))
+             => (p -> p) -> (p -> p)
+             => (a -> (b -> c)) -> ((a -> b) -> (a -> c))
+             => a -> ((a -> b) -> a)
+             => b -> ((a -> b) -> b)
+             => (a & b) -> (b & a)
+             => (a -> (a -> b)) -> (a -> b)
+             => ((((p -> q) -> p) -> p) -> q) -> q
+             => (a v (a -> b)) -> (((a -> b) -> a)-> a)
+             => (a -> (b -> false)) -> (b -> (a -> false))
+             => ((a v (a -> false))-> false)-> false
+a & b, c & d => b & c
+(a v (a-> false)) -> false => false
+
+
+
+Non-Provable Examples:
+
+
+             => ((a -> b) -> a) -> a
+             => a v (a -> false)
+             => (a & b v (((( a-> flase) -> false) -> q) v (b -> q)) -> q) -> q
+
+
+
+
+Some advanced examples
+
+
+Source Code:
+Formulae,
+Contexts,
+Sequents,
+ProofDisplay,
+Prover,
+Parser,
+Applet

+
+
+
+Lambda Prolog Version: The programm 
+G4ip.mod can be executed using  
+Terzo
+Another implementation of G4ip by Joshua Hodas and Dale Miller written in 
+Lolli can be found
+here.
+
+
+
+
+Christian Urban
+
+
+
+
+Last modified: Sun May 10 18:18:28 GMT 1998
+
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/README
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/README	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,76 @@
+
+Write once, run everywhere
+==========================
+  	Good idea, but still wishful thinking.
+
+	From my experience I would like to add:
+        "Write once, run everywhere, show the same behaviour!"	
+
+
+Known bugs and failures:
+========================
+
+- Cut-and-paste
+	I have not included any code for cut-and-paste. However,
+	cut-and-paste should work fine under Unix. Cut-and-Paste
+	also works in MS IE 3.0 under Mickeysoft Windows. For most of 
+        the other platforms (MacOS etc.) and browsers (Netscape etc.)
+        cut-and-paste does not work.
+
+- Security exception
+	Various versions of Netscape throw a security exception when
+	I try to suspend the thread of the proof search. I have caught 
+	this exception, but my literature does not tell me how to resume 
+	safely from this exception. In effect the proof search enumerates 
+ 	all possible proofs for a sequent and subsequently stops. The user 
+	has no control over how many proofs are shown (the corresponding 
+	windows might appear over each other). This unintended behaviour 
+	has been encountered in:
+	
+	  - Netscape Navigator 3.0 Gold under MS Windows'45
+	  - Netscape Navigator 3.0 under MacOS
+          - Netscape Communicator 4.0 under MacOS
+	  - Netscape Navigator 3.01 Gold under Linux
+	      (But here a "catch" of this exception and a "do nothing" 
+	       helps. God knows why.)
+	  - possible others
+
+- Boxes around the sequents in proofs
+	I have modified the "Label" class such that the "MyLabel" class 
+	has a surrounding box. This improves the readability of the proofs.
+	However,  this feature works ONLY under Unix. The boxes do not 
+	appear in MS Windows and MacOS (reason unknown). 
+
+- Flickering messages
+	The applet prints some messages ("Grammar error", "No more proofs",
+	"Not provable", etc.) in a field below the control buttons. 
+	In some browsers the messages only flicker for a short time.
+	Other browsers show the messages correctly (reason unknown).
+
+
+The applet has been tested under (but see restrictions and bugs):
+=================================================================
+
+Linux:   
+        Netscape Navigator 3.01 Gold
+	Netscape Communicator 4.03, 4.04, 4.07 
+        HotJava 1.1 beta2 (Linux-port)
+        SUN's appletviewer for JDK 1.1.3
+        
+Win'95:
+        Netscape Navigator 3.0 Gold
+	Netscape Communicator 4.01a
+        MS Internet Explorer 3.0.2
+	MS Internet Explorer 4.0	
+
+MacOS:  
+	Netscape Navigator 3.0
+	Netscape Communicator 4.0
+	MS Internet Explorer 3.0.1b1 (PowerPC)
+
+
+Any comments on success or failure for other
+environments and platforms are appreciated.  
+
+Christian.Urban@cl.cam.ac.uk
+  
diff -r 345dd18f020e -r 907b1fff5637 Prover/Sequents.pizza
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/Sequents.pizza	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,29 @@
+package G4ip;
+
+import G4ip.Form.*;
+
+/** An intuitionistic sequent of the form     
+  *  "Gamma => G".
+  */
+public class Sequent  {
+  /** Goal formula on the right-hand side. */
+  public Form    G;       
+  /** Context (multiset of formulae) on the left-hand side. */
+  public Context Gamma;   
+
+  public Sequent(Context init_Gamma, Form init_G) {
+        G = init_G;
+    Gamma = new Context(init_Gamma);
+  }
+
+  public Sequent(Sequent sequ) {
+        G = sequ.G;
+    Gamma = new Context(sequ.Gamma);
+  }
+
+  /** Prints a sequent.  */
+  public String toString() {
+    String s = Gamma.makeString() +  " => " + G.toString();
+    return s;
+  }
+}
diff -r 345dd18f020e -r 907b1fff5637 Prover/Tableaux98.dvi.gz
Binary file Prover/Tableaux98.dvi.gz has changed
diff -r 345dd18f020e -r 907b1fff5637 Prover/Tableaux98.ps.gz
Binary file Prover/Tableaux98.ps.gz has changed
diff -r 345dd18f020e -r 907b1fff5637 Prover/execution.gif
Binary file Prover/execution.gif has changed
diff -r 345dd18f020e -r 907b1fff5637 Prover/fun.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Prover/fun.html	Thu Mar 15 10:07:28 2012 +0000
@@ -0,0 +1,32 @@
+
+
+   
+   Advanced Examples
+
+
+
+
Some Advanced Examples 
+
+Be cautious: some of the following formulae have big proofs!
+
+
+
+
+             => (((a & b) v (a->f) v (b->f))->f)->f
+             => (a & b & c v ((a->f) v (b->f) v (c->f))->f)->f
+             => d&(b->b->a)&(c->c->b)&(d->d->c)->a
+
+
+
+
+
+
+Christian Urban
+
+
+
+
+Last modified: Sun Nov 23 20:20:31 GMT 1997
+
+
+
diff -r 345dd18f020e -r 907b1fff5637 Prover/index.html
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
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+
+
+   
+   G4ip
+
+
+
+
An Implementation of G4ip in Pizza
+
+Warning: 
+This page is now rather old! While you might still be interested
+in the algorithms, Robert Macdonald reported that Pizza and the current Java 
+implementation (version 1.3.0) do not work together. This means you need to 
+install an older Java version if you want to recompile the files given below. 
+I am happy to answer all question concerning the prover, but be aware that 
+currently for any kind of Java stuff I am using MLJ, which as of writing 
+this note  has not yet been made available for the general audience (maybe
+in the future also its OCaml equivalent). So I am not very fluent in Pizza
+anymore. Update Pizza development is continued and starting from version 
+0.40 it should work
+with recent Java implementations. 
+
+
+Jump to the implementation.
+
+
+
Introduction
+
+
+A convenient representation of intuitionistic logic is Getzen's 
+sequent calculus LJ (also G1i). A sequent of LJ can be proved 
+by applying inference rules until one reaches axioms, or can make no further 
+progress in which case one must backtrack or even abandon the search. 
+Unfortunately an interpreter for LJ using this depth-first strategy cannot 
+guarantee termination of the proof search.  Several modifications can be 
+made to LJ's inference rules without loss of soundness and completeness.  
+As result an efficient depth-first proof search can be designed for the 
+propositional fragment of intuitionistic logic. The name G4ip has been 
+assigned to the corresponding calculus in 
+[Troelstra and Schwichtenberg, 1996]. 
+This calculus is also known as LJT which has been studied thoroughly 
+in [Dyckhoff, 1992]. The inference rules of 
+G4ip are given here.
+
+It is not very complicated to implement an interpreter for G4ip using a logic 
+programming language (backtracking is directly supported within the language).
+Our first implementation is written in the logic programming language
+Lambda Prolog 
+and can be found here. Another implementation by
+Hodas and Miller written in Lolli
+can be found here
+(see [Hodas and Miller, 1994]). These are simple and 
+straightforward implementations of G4ip's rules. On the other hand it seems 
+that imperative languages need a rather high overhead of code when implementing
+a logic calculus. For example choice points are usually implemented with stacks.
+We shall demonstrate the implementation technique of success 
+continuations which provides an equally simple method for implementing logic calculi 
+in imperative languages. This technique is not new: it has been introduced in 
+[Carlsson, 1984]. This paper presents a rather technical
+implementation of Prolog in LISP. Later an excellent paper, 
+[Elliot and Pfenning, 1991], appeared which 
+describes a full-fledged implementation of Lambda Prolog in SML.
+We demonstrate the technique of success continuations for G4ip in 
+Pizza.

+
+Pizza is an object-oriented programming language and an attractive extension of  
+Java. Although Pizza is a superset of
+Java, Pizza programs can be translated into Java or compiled into ordinary
+Java Byte Code (see [Odersky and Wadler, 1997]
+for a technical introduction to Pizza). We make use of the following two new 
+features of Pizza:
+

+ higher-order functions, i.e. functions may be passed as parameters or returned
+from methods,
+
 class cases and pattern matching: this allows much simpler and more readable code.
+
+
+These features are not directly present in Java, but Pizza makes them accessible by 
+translating them into Java. Pizza provides the programmer with the same 
+extensive libraries for graphic and network applications as Java. The higher-order 
+functions are essential for the technique of success continuations. The success 
+continuations are functions passed as parameters or returned as values.

+
+
+The Representation of Formulae and Sequents
+
+Amongst the new language features of Pizza are class cases and pattern
+matching, which provide a very pleasant syntax for algebraic data types. The
+formulae of G4ip are specified by the following grammar:
+
+F ::= false | A | F & F | F v F | F -> F

+
+
+The class cases allow a straightforward implementation of this specification;
+it is analogous to the SML implementation of 
+Lambda Prolog's 
+formulae in [Elliot and Pfenning, 1991]. The class 
+of formulae for G4ip is given below:

+
+
+

+public class Form {            
+      case False();                  
+      case Atm(String c);          
+      case And(Form c1,Form c2); 
+      case Or(Form c1,Form c2);  
+      case Imp(Form c1,Form c2); 
+}                              
+

+
+
+Two examples that illustrate the use of the representation are as follows:
+
+
+          p -> p   
+is represented as      Imp(Atm("p"),Atm("p"))

+a v (a -> false)   is represented as     Or(Atm("a"),Imp(Atm("a"),False()))

+
+The class cases of Pizza also support an implementation of formulae specified
+by a mutually recursive grammar.  This is required, for example, when
+implementing hereditary Harrop formulae.
 
+
+The sequents of G4ip, which have the form Gamma=>G, are represented
+by means of the class below. The left-hand side of each sequent is specified by a multiset 
+of formulae.  Therefore, we do not need to worry about the order in which the
+formulae occur.

+
+
+

+public class Sequent  {                       
+   Form  G;                                               
+   Context Gamma;                                
+   public Sequent(Context _Gamma, Form _G) {...};
+}                                             
+

+
+
+We have a constructor for generating new sequents during proof search. 
+Context is a class which represents multisets; it is a simple 
+extension of the class Vector available in the Java libraries. 
+This class provides methods for adding elements to a multiset (add), 
+taking out elements from a multiset (removeElement) and testing
+the membership of an element in a multiset (includes).
+
+
+The Technique of Success Continuations
+
+We have to distinguish between the concepts of proof obligations (which must 
+be proved) and choice points (which can be tried out to construct a proof). 
+The first argument of the method prove is the sequent being 
+proved; the second argument is an anonymous function. The function prove is now 
+of the form prove(sequent,sc).  Somewhat simplified the
+first argument is the leftmost premise and the second argument sc, 
+the success continuation, represents the other proof obligations. In case we
+succeed in proving the first premise we then can attempt to prove the other
+premises. The technique of success continuations will be explained using the following 
+proof (each sequent is marked with a number):
+
+

+

+
+The inference rules fall into three groups:
+
+

+  inference rules with a single premise (e.g. ->_R, &_L), 
+
  inference rules with two premises (e.g. v_L) and
+
  inference rules without premises (e.g. Axiom). 
+
+
+The following picture shows the order in which the sequents are being proved.
+
+
+

+
+Suppose we have called prove with a sequent s and a 
+success continuation is. The inference rules of the first
+group manipulate  s obtaining s' and call prove 
+again with the new sequent s' and the current success continuation 
+(Steps 1-2, 3-4 and 5-6).  The inference rules 
+of the second group have two premises, s1 and s2. 
+These rules call prove with s1 and a new success 
+continuation prove(s2,is) (Step 2-3).
+The third group of inference rules only invoke the success continuation 
+if the rule was applicable (Steps 4-5 and 6-7).

+
+
+We are going to give a detailed description of the code for the rules: &_L,
+->_R, v_Ri, v_L and Axiom.  The function prove receives as arguments 
+a sequent Sequent(Gamma,G) and a success continuation 
+sc.  It enumerates all formulae as being principal and 
+two switch statements select a corresponding case depending on the form 
+and the occurrence of the principal formula.
 
+
+The &_L rule is in the first group; it modifies the sequent being proved and calls
+prove again with the current success continuation sc. The code is as
+follows (Gamma stands for the set of formulae on the left-hand
+side of a sequent excluding the principal formula; G stands 
+for the goal formula of a sequent; B and C stand 
+for the two components of the principal formula).

+
+
+

+case And(Form B, Form C):
+   prove(new Sequent(Gamma.add(B,C),G),sc); break;
+

+
+
+The code for the ->_R rule is similar:
+
+
+

+case Imp(Form B, Form C):
+   prove(new Sequent(Gamma.add(A),B),sc); break;
+

+
+
+The v_Ri rule is an exception in the first group. It breaks up a goal
+formula of the form B1 v B2 and proceeds with one of its component. 
+Since we do not know in advance which component leads to a successful proof we have
+to try both. Therefore this rule acts as a choice point, which is encoded by a
+recursive call of prove for each case.
+
+
+
+case Or(Form B1,Form B2):
+   prove(new Sequent(Gamma,B1),sc);
+   prove(new Sequent(Gamma,B2),sc); break;
+
+
+
+The v_L rule falls into the second group where the current success
+continuation, sc, is modified. It calls prove with the first premise,
+B,Gamma=>G, and wraps up the success continuation with the 
+new proof obligation, C,Gamma=>G. The construction 
+fun()->void {...} defines an anonymous function: the new 
+success continuation. In case the sequent B,Gamma=>G can be 
+proved, this function is invoked.
+
+
+
+case Or(Form B,Form C):
+   prove(new Sequent(Gamma.add(B),G),
+           
+      fun()->void {prove(new Sequent(Gamma.add(C),G),sc);}
+        ); break
+
+
+
+The Axiom rule falls into the third group. It first checks if the
+principal formula (which is an atom) matches with the goal formula and
+then invokes the success continuation sc in order to prove all remaining
+proof obligations.
+
+
+
+case Atm(String c):
+   if (G instanceof Atm) { 
+      if (G.c.compareTo(c) == 0) { sc(); }
+   } break;
+
+
+
+The proof search is started with an initial success continuation is. 
+This initial success continuation is invoked when a proof has been found. 
+In this case we want to give some response to the user, an
+example for the initial success continuation could be as follows:
+
+
+
+ public void initial_sc()  { System.out.println("Provable!"); } 
+
+
+
+
+Suppose we attempt to start the proof search with prove(p,p => p,is). 
+We would find that the prover responds twice with "Provable!", because 
+it finds two proofs. In our implementation this problem is avoided by encoding 
+the proof search as a thread. Whenever a proof is found, the initial success 
+continuation displays the proof and suspends the thread. The user can 
+decide to resume with the proof search or abandon the search.
+
+
+Conclusion
+
+The implementation cannot be considered as optimal in terms of speed. 
+A much more efficient algorithm for G4ip (but less clear) has been 
+implemented by Dyckhoff in Prolog. Similar ideas can be encoded in our 
+Pizza implementation; but our point was not the efficiency but the clarity 
+of the implementation using success continuations.
+The technique is applicable elsewhere whenever backtracking is required.  We
+compared the code of our implementation with an implementation in
+Lambda Prolog: 
+the ratio of code is approximately 2 to 1. 
+(see LambdaProlog code and 
+Pizza code).
+This result is partly due to the fact that we had to implement a class for 
+multisets. In a future version of Java, we could have accessed a package 
+in the library. The technique of success continuation can also be applied 
+to a first-order calculus as shown in [Elliot and Pfenning, 1991], 
+but the required mechanism of substitution needs to be implemented separately. 
+However, we think the technique of success continuations provides a remarkable 
+simple implementation for logic calculi. 
+
+We had to make some compromises in order to support as many platforms
+as possible. This should change with the release of new browsers and a stable
+Java-specification (resp. Pizza-specification).
 
+
+A paper about the implementation appeared in the LNAI series No 1397,
+Automated Reasoning with Analytic Tableaux and Related Methods, 
+ed. Harry de Swart, International Conference Tableaux'98 in Oisterwijk, 
+The Netherlands. The title is: Implementation of Proof Search in 
+the Imperative Programming Language Pizza (pp. 313-319). The paper can be 
+found here: DVI, Postscript 
+(� Springer-Verlag LNCS).

+
+
+Acknowledgements: I am very grateful for Dr Roy Dyckhoff's constant
+encouragement and many comments on my work. I thank Dr Gavin Bierman who
+helped me to test the prover applet. 
+
+

+Implementation
+
+Readme
+
+Prover Applet

+Jar Version 
+(slightly faster, but requires Netscape 4 or MS Explorer 4).

+
+
+

+References 
+
+ 
+     [Carlsson, 1984]
 
+     M. Carlsson, On  Implementing Prolog in Functional Programming, 
+     New Generation Computing, pp 347-359.
+
 
+     [Dyckhoff, 1992]
 
+     R. Dyckhoff, 
+     Contraction-Free Sequent Calculi for Intuitionistic Logic, 
+     Journal of Symbolic Logic 57(3), pp 795-807.
+
 
+     [Elliot and Pfenning, 1991]
 
+     C. Elliot,
+     F. Pfenning, 
+     A Semi-Functional Implementation of a Higher-Order Programming Language, 
+     In Peter Lee, editor, Topics in Advanced Language Implementation, MIT Press,
+     pp 289-352. 
+     Available electronically.
+
 
+     [Hodas and Miller, 1994]
 
+     J. Hodas,
+     D. Miller,
+     Logic Programming in a Fragment of Intuitionistic Linear Logic, 
+     Information and Computation 110(2), pp 327-365.
+     Available electronically.
+
 
+     [Odersky and Wadler, 1997]
 
+     M. Odersky,
+     P. Wadler,
+     Pizza into Java: Translating Theory into Practice, 
+     In Proceedings of the 24th ACM Symposium on Principles of Programming Languages.
+     Available electronically.
+
 
+     [Troelstra and Schwichtenberg, 1996]

+     A. Troelstra, 
+     H. Schwichtenberg,  
+     Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science,
+     Cambridge University Press.
+
+
+
+
+
+Christian Urban
+
+
+
+
+Last modified: Sun Sep 23 12:04:47 BST 2001
+
+
+
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