Goal: a tableau prover for propositional logic, the basic modal logic K and propositional dynamic logic (PDL). The prover should also find interpolants, following the method described by Manfred Borzechowski in 1988. See https://malv.in/2020/borzechowski-pdl/ for the original German text and an English translation.
There are two different provers: one based on lists. and a proper tableau.
Interpolation is also implemented twice: once replacing atoms by constants as described in Wikipedia: Craig interpolation, and using the tableaux.
stack ghci src/Logic/BasicModal/Prove/Tree.hs
λ> provable (Box (p --> q) --> (Box p --> Box q))
True
λ> provable (p --> Box p)
False
stack ghci src/Logic/BasicModal/Interpolation/ProofTree.hs
λ> let (f,g) = ( Box ((At 'p') --> (At 'q')) , (Neg (Box ((At 's') --> (At 'q')))) --> (Neg (Box (At 'p'))) )
λ> mapM_ (putStrLn .ppForm) [f, g]
☐¬(p & ¬q)
¬(¬☐¬(s & ¬q) & ¬¬☐p)
λ> interpolateShow (f,g)
Showing tableau with GraphViz ...
Interpolant: ☐¬(¬¬p & ¬¬¬q)
Simplified interpolant: ☐¬(p & ¬q)
The last command will also show this tableau:
See the test file for more examples, including interpolation and consistency checks.
Public web interface: https://w4eg.de/malvin/illc/tapdleau/
To run the web interface locally do stack build and then stack exec tapdleau.
For developing you can recompile and restart the web interface on any code changes Like this:
stack build --file-watch --exec "bash -c \"pkill tapdleau; stack exec tapdleau &\""
Use stack test to run all tests from the test folder.
For PDL we also use the files formulae_exp_unsat.txt and formulae_exp_sat.txt from http://users.cecs.anu.edu.au/~rpg/PDLComparisonBenchmarks/. Note: The files have been modified to use star as a postfix operator.
Prover:
- restricted language with Con, not Imp as primitive, as Borzechowski does
- proper search: extend -> extensions (as in BasicModal)
- add M+ rule
- Only allow (At) on marked formulas (page 24)
- Check priorities / preferences of all rules.
- mark active formula (as in BasicModal), needed for interpolation
- also mark active formula for extra condition 4, and history-path for 6
- more uniform encoding of rules and closing conditions
- Implement extra conditions:
- when reaching an atomic Box or NegBox, go back from n to *
- instead of X;[a^n]P reach X
- apply rules to n-formula whenever possible / prioritise them!
- Never apply a rule to a ¬[a^n] node (and mark those as end nodes!)
- directly after M+ do not apply M-
- close normal nodes with critical same-set predecessors (if loaded when whole path is loaded)
- every loaded node that is not an end node by 6, has a successor.
Interpolation:
- copy from BasicModal and get it to compile
- expose in web interface
- fillIPs: all local rules
- fillIPs: atomic rule
- add interpolate tests for star-free formulas
- Find test cases that fail due to the empty-side edge cases for (At)-interpolants?
- Correct definition of (At)-interpolants in the empty-side edge cases.
- TI, TJ, TK, canonical programs, interpolants for TK
- use TI etc. to find interpolants for loaded sub-tableaux, iterate!
- fillIPs: end notes due to extra condition 4 ??
Bonus Information:
These are not needed to define interpolants, but part of the proof the definition is correct.
- J and K sets from Definition 34
- extended tableau from Lemma 25 and 26.
Testing:
- self-testing: generate random implication, check if provable, compute interpolant, check if it is indeed an interpolant
- check for agreement with other PDL provers
- http://rsise.anu.edu.au/~rpg/PDLProvers/
- https://www.irit.fr/Lotrec/ see also https://github.com/bilals/lotrec
- http://www.cs.man.ac.uk/~schmidt/pdl-tableau/ (NOTE: prototype only, not working for nested stars)
Semantics:
- Kripke models
- satisfaction
- generate random arbitrary models
- build counter model from open tableaux? (done for BML)
Nice to have:
- option to start with a given set/partition instead of a single (to be proven and thus negated) formula
- use exact same syntax as Borzechowski (A for atomic programs etc.)
- add applicable extra conditions to rule annotation
- web interface
- use Graphviz HTML labels for better readability, e.g. highlight the active formula.
- color the loading part of formulas/nodes
- store current/submitted formula in (base64?) URL hash, provide perma-link for sharing
- Use better data structures, zipper instead of
(TableauIP, Path)pairs?
- https://github.com/m4lvin/tablean A formalization of the same method in Lean. Only covering Basic Modal Logic at this time.
-
Rajeev Goré and Florian Widmann (2009): An Optimal On-the-Fly Tableau-Based Decision Procedure for PDL-Satisfiability. https://doi.org/10.1007/978-3-642-02959-2_32
-
Roman Kuznets (2015): Craig Interpolation via Hypersequents. https://sites.google.com/site/kuznets/interpol_hyper_v2.pdf
-
Francesca Perin (2019): Implementing Maehara's Method for Star-Free Propositional Dynamic Logic. https://fse.studenttheses.ub.rug.nl/20770/ https://github.com/FrancescaPerin/BScProject
-
Other PDL provers are mentioned at http://www.cs.man.ac.uk/~schmidt/tools/.