The Proof Tree Visualiser By David Alexander Supervisor: Rajeev Gor - - PowerPoint PPT Presentation

The Proof Tree Visualiser

By David Alexander

Supervisor: Rajeev Goré

Summer Scholar, RSISE

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What is it?

• A graphical interface for...

– ...constructing proof trees, and... – ...visualising proof trees generated by automated

provers.

• What kind of proof trees?

– Proof trees in the tableau and sequent proof

systems.

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Background

• Propositional Logic

– Connectives: , , ¬,

, ∧ ∨ → ↔

– Theorems can be proven using resolution.

• Modal Logic

– ◊p means “possibly p”;

p ◻ means “necessarily p”

• (other interpretations also exist)

– Example: ◻(a b)

¬ ∧ → ◊¬b

– A different approach is required for theorem-

proving.

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Background

• The Tableau Method

– To prove/disprove a statement:

• Create a tree with the statement at the root.
• Choose an applicable tableau rule and use it to generate

the child nodes.

• Repeat until a contradiction is found, or no more rules

can be applied.

• Sequents are similar, but usually drawn with the

root at the bottom.

• Very large proof trees can result – automated

provers often used.

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Sample Prover Output

$ ./k.twb --trace --verbose <(echo "[] (<> ~ p0 v p0) v <> [] Falsum v <> ([] p0 & <> <> ~ p0) v <> ([] <> p0 & <> <> [] ~ p0) v <> (p0 & <> [] ~ p0) v <> (~ p0 & <> [] p0)") Proving: [] (<> ~ p0 v p0) v <> [] Falsum v <> ([] p0 & <> <> ~ p0) v <> ([] <> p0 & <> <> [] ~ p0) v <> (p0 & <> [] ~ p0) v <> (~ p0 & <> [] p0) Start Node: ((((((<> (([ ] p0) & (~ p0))) & ([ ] (<> Verum))) & ([ ] ((<> (~ p0)) v ([ ] ([ ] p0))))) & ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))))) & ([ ] ((~ p0) v ([ ] (<> p0))))) & ([ ] (p0 v ([ ] (<> (~ p0)))))) And ( 0 -> 1 ) ([ ] (p0 v ([ ] (<> (~ p0))))) ; (((((<> (([ ] p0) & (~ p0))) & ([ ] (<> Verum))) & ([ ] ((<> (~ p0)) v ([ ] ([ ] p0))))) & ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))))) & ([ ] ((~ p0) v ([ ] (<> p0))))) And ( 1 -> 2 ) ((((<> (([ ] p0) & (~ p0))) & ([ ] (<> Verum))) & ([ ] ((<> (~ p0)) v ([ ] ([ ] p0))))) & ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))))) ; ([ ] ((~ p0) v ([ ] (<> p0)))) ; ([ ] (p0 v ([ ] (<> (~ p0))))) And ( 2 -> 3 ) ([ ] ((~ p0) v ([ ] (<> p0)))) ; ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0))))) ; ([ ] (p0 v ([ ] (<> (~ p0))))) ; (((<> (([ ] p0) & (~ p0))) & ([ ] (<> Verum))) & ([ ] ((<> (~ p0)) v ([ ] ([ ] p0))))) And ( 3 -> 4 ) ((<> (([ ] p0) & (~ p0))) & ([ ] (<> Verum))) ; ([ ] ((~ p0) v ([ ] (<> p0)))) ; ([ ] ((<> (~ p0)) v ([ ] ([ ] p0)))) ; ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0))))) ; ([ ] (p0 v ([ ] (<> (~ p0))))) And ( 4 -> 5 ) (<> (([ ] p0) & (~ p0))) ; ([ ] (<> Verum)) ; ([ ] ((~ p0) v ([ ] (<> p0)))) ; ([ ] ((<> (~ p0)) v ([ ] ([ ] p0)))) ; ([ ] ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0))))) ; ([ ] (p0 v ([ ] (<> (~ p0))))) K ( 5 -> 6 ) (<> Verum) ; ((~ p0) v ([ ] (<> p0))) ; ((<> (~ p0)) v ([ ] ([ ] p0))) ; ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))) ; (p0 v ([ ] (<> (~ p0)))) ; (([ ] p0) & (~ p0)) ; And ( 6 -> 7 ) (~ p0) ; ((~ p0) v ([ ] (<> p0))) ; ((<> (~ p0)) v ([ ] ([ ] p0))) ; ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))) ; (p0 v ([ ] (<> (~ p0)))) ; (<> Verum) ; ([ ] p0) ; Or ( 7 -> 8 ) ((<> (~ p0)) v ([ ] ([ ] p0))) ; ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))) ; (p0 v ([ ] (<> (~ p0)))) ; ; (<> Verum) ; ([ ] p0) ; (~ p0) Or ( 8 -> 9 ) ((<> ([ ] (~ p0))) v ([ ] ([ ] (<> p0)))) ; (p0 v ([ ] (<> (~ p0)))) ; (<> Verum) ; (<> (~ p0)) ; ; ([ ] p0) ; (~ p0) Or ( 9 -> 10 ) (p0 v ([ ] (<> (~ p0)))) ; ; (<> Verum) ; (<> (~ p0)) ; (<> ([ ] (~ p0))) ; ([ ] p0) ; (~ p0) Or ( 10 -> 11 ) (<> Verum) ; (<> (~ p0)) ; (<> ([ ] (~ p0))) ; ; p0 ; ([ ] p0) ; (~ p0) Id ( 11 -> 12 )

... 104 lines omitted ...

Id ( 70 -> 71 ) Or ( 64 -> 72 ) (p0 v ([ ] (<> (~ p0)))) ; (<> Verum) ; ; ([ ] (<> p0)) ; ([ ] ([ ] (<> p0))) ; ([ ] ([ ] p0)) ; ([ ] p0) ; (~ p0) Or ( 72 -> 73 ) (<> Verum) ; p0 ; ([ ] (<> p0)) ; ([ ] ([ ] (<> p0))) ; ([ ] ([ ] p0)) ; ([ ] p0) ; (~ p0) Id ( 73 -> 74 ) Or ( 72 -> 75 ) (<> Verum) ; ; ([ ] (<> (~ p0))) ; ([ ] (<> p0)) ; ([ ] ([ ] (<> p0))) ; ([ ] ([ ] p0)) ; ([ ] p0) ; (~ p0) K ( 75 -> 76 ) ([ ] (<> p0)) ; ([ ] p0) ; p0 ; Verum ; ; ; ; (<> (~ p0)) ; (<> p0) K ( 76 -> 77 ) p0 ; ; ; (~ p0) ; ; (<> p0) Id ( 77 -> 78 ) Time:0.0080 Result:Close Total Rules applications:78 Cache results: Total queries:0 Hits:0 Miss:0 Elements in the cache:0

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PTV Features

• Layout according to size of subtrees.

– Compact structure – formulae only shown on

request.

• Subtrees can be collapsed and expanded.
• Tableau/sequent rules displayed in full.

– Assignments of formulae to variables shown using

colours.

• Formulae can be “traced” up the tree.

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Demonstration

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Implementation Details

• Written in Haskell.
• Prover runs on server.

– Nodes loaded over network as user views them.

• Provers use a special format to specify the

proof tree.

– Each rule application is specified as a set of

assignments of formulae to variables.

– Exact details of format still in development.

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Further Work

• Web browser-embeddable version.
• More powerful tree editing

– Allow users to adjust the layout manually if desired.

• Adapt more provers to work with the PTV.

– Allow use of provers stored on the user's computer,

rather than on the server.

• Set up build systems to provide Mac and

Windows binaries.

• Code refactoring and optimisation.

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More Information

• PTV is free software.

– Licensed under the GNU GPL.

• Source code, Linux binaries and user guide

available at:

http://users.rsise.anu.edu.au/~dalexander/ptv

• This presentation will also be posted on the above

website.