Terminating Tableaux for Mini-PDL

Transcription

1 Terminating Tableaux for Mini-PDL Sigurd Schneider Bachelor s Thesis Proposal Talk Advisors: Mark Kaminski, Gert Smolka Responsible Professor: Gert Smolka April 30, 2009 Terminating Tableaux for Mini-PDL (Sigurd Schneider) 1/39

2 Propositional Dynamic Logic Propositional Dynamic Logic Mini-PDL Semantics Goal Statement t ::= p t t t t t ρt ρt ρ ::= r ρ ρ ; ρ ρ ρ t? Small Model Property [Fisher and Ladner, 1979] Robustly decidable [Giacomo and Massacci, 2000][Abate, Goré, and Widmann, 2009] Terminating Tableaux for Mini-PDL (Sigurd Schneider) 2/39

3 Mini-PDL Propositional Dynamic Logic Mini-PDL Semantics Goal Statement t ::= p t t t t t ρt ρt ρ ::= r r Most important aspect covered: kleene star Thus still complex enough Terminating Tableaux for Mini-PDL (Sigurd Schneider) 3/39

4 Semantics Propositional Dynamic Logic Mini-PDL Semantics Goal Statement y rpy Terminating Tableaux for Mini-PDL (Sigurd Schneider) 4/39

5 Semantics Propositional Dynamic Logic Mini-PDL Semantics Goal Statement y r py Terminating Tableaux for Mini-PDL (Sigurd Schneider) 5/39

6 Where do we want to go? Propositional Dynamic Logic Mini-PDL Semantics Goal Statement Complete and Terminating Tabelau System for Mini-PDL incorporating key ideas from literature elegant proofs pattern-based blocking Terminating Tableaux for Mini-PDL (Sigurd Schneider) 6/39

7 Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation Approach to a complete tableau system Start with tableaux system for K. R ( p)x px A R (t 1 t 2 )x t 1 x, t 2 x R (t 1 t 2 )x t 1 x t 2 x R rtx ty N A R rtx, ty y / N A Add the following rules: R r tx tx, r( r t)x R r tx tx r( r t)x Terminating Tableaux for Mini-PDL (Sigurd Schneider) 7/39

8 Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation Approach to a complete tableau system Start with tableaux system for K. R ( p)x px A R (t 1 t 2 )x t 1 x, t 2 x R (t 1 t 2 )x t 1 x t 2 x R rtx ty N A R rtx, ty y / N A Add the following rules: R r tx tx, r( r t)x R r tx tx r( r t)x Terminating Tableaux for Mini-PDL (Sigurd Schneider) 8/39

9 An Infinite Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py R r tx tx r( r t)x r( r p)y R rtx, ty y / N A r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 9/39

10 An Infinite Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py R r tx tx r( r t)x r( r p)y R rtx, ty y / N A r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 10/39

11 An Infinite Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py R r tx tx r( r t)x r( r p)y R rtx, ty y / N A r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 11/39

12 An Infinite Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py R r tx tx r( r t)x r( r p)y R rtx, ty y / N A r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 12/39

13 An Infinite Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py R r tx tx r( r t)x r( r p)y R rtx, ty y / N A r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 13/39

14 Pattern Based Blocking Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y The pattern P rtx A of a formula rtx A is defined as { rt} { rt rtx A}. P r( r p)x A {} = P r( r p)y A = r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 14/39

15 Pattern Based Blocking Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y The pattern P rtx A of a formula rtx A is defined as { rt} { rt rtx A}. P r( r p)x A {} = P r( r p)y A = r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 15/39

16 Pattern Based Blocking Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y The pattern P rtx A of a formula rtx A is defined as { rt} { rt rtx A}. P r( r p)x A {} = P r( r p)y A = r pz... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 16/39

17 A Blocked Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y Derivation is blocked before witness is generated. r pz pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 17/39

18 A Blocked Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y Derivation is blocked before witness is generated. r pz pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 18/39

19 A Blocked Derivation Tableau Rules An Infinite Derivation Pattern Based Blocking A Blocked Derivation r py r( r p)y Derivation is blocked before witness is generated. r pz pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 19/39

20 Necessity Definition Existence Different Kinds of Maximal Branches There are three kinds of maximal derivations: Verifying derivations that yield a model. Falsifying derivations that are inconsistent. Blocked derivations, that may or may not be extended to an verifying derivation. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 20/39

21 Necessity Definition Existence Different Kinds of Maximal Branches r py py There are three kinds of maximal derivations: Verifying derivations that yield a model. Falsifying derivations that are inconsistent. Blocked derivations, that may or may not be extended to an verifying derivation. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 21/39

22 Necessity Definition Existence Different Kinds of Maximal Branches There are three kinds of maximal derivations: px px Verifying derivations that yield a model. Falsifying derivations that are inconsistent. Blocked derivations, that may or may not be extended to an verifying derivation. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 22/39

23 Necessity Definition Existence Different Kinds of Maximal Branches r py r( r p)y r pz pz There are three kinds of maximal derivations: Verifying derivations that yield a model. Falsifying derivations that are inconsistent. Blocked derivations, that may or may not be extended to an verifying derivation. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 23/39

24 Proof Strategy Necessity Definition Existence Usually, a completeness proof for a tableaux system shows: System terminates. If system terminates and branch is consistent, we can construct a model. By refutation soundness, completeness follows. New strategy Show that if a set of formulas is satisfiable, there exists a verifying derivation in the blocked system. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 24/39

25 Necessity Definition Existence r py r py r( r p)y py r pz pz Minimal Derivation On every path from the root over a diamond-star formula to its witness, no pattern occurs twice. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 25/39

26 Necessity Definition Existence r py r py r( r p)y py r pz pz Minimal Derivation On every path from the root over a diamond-star formula to its witness, no pattern occurs twice. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 26/39

27 Necessity Definition Existence r py r py r( r p)y py r pz pz Minimal Derivation On every path from the root over a diamond-star formula to its witness, no pattern occurs twice. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 27/39

28 Necessity Definition Existence r py r py r( r p)y py r pz pz Minimal Derivation On every path from the root over a diamond-star formula to its witness, no pattern occurs twice. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 28/39

29 Existence of Necessity Definition Existence Desired Theorem Let A be a set of Mini-PDL formulas. For every verifying derivation starting from A, there is a minimal verifying derivation starting from A which is obtained by shortening. Proof Idea: r pz pz r py r( r p)y Terminating Tableaux for Mini-PDL (Sigurd Schneider) 29/39

30 Existence of Necessity Definition Existence Desired Theorem Let A be a set of Mini-PDL formulas. For every verifying derivation starting from A, there is a minimal verifying derivation starting from A which is obtained by shortening. Proof Idea: r py py Terminating Tableaux for Mini-PDL (Sigurd Schneider) 30/39

31 Roadmap and Open Problems Proving completeness of unconstrained system. Formalizing derivations as graphs. R r tx tx, r( r t)x tx r tx r( r t)x Proving minimal derivation existence theorem using derivation graphs. If that does not work, prove existence for every satisfiable set and analyse confluency of derivation relation. Does the approach scale to PDL? Terminating Tableaux for Mini-PDL (Sigurd Schneider) 31/39

32 References David Harel and Dexter Kozen and Jerzy Tiuryn Dynamic Logic MIT Press, 2000 Fischer, M.J. and R. E. Ladner (1979) Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18(2), Abate, P., Goré, R., and Widmann, F. (2009) An On-the-fly Tableau-based Decision Procedure for PDL-satisfiability. Electron. Notes Theor. Comput. Sci. 231 (Mar. 2009), de Giacomo, G. and Massacci, F. (2000) Combining deduction and model checking into Tableaux and algorithms for converse-pdl. Inf. Comput. 162, 1/2 (Oct. 2000), Terminating Tableaux for Mini-PDL (Sigurd Schneider) 32/39

33 Does it work for full PDL? r( (r ; r) p)x (r ; r) px (r ; r) py (r ; r)( (r ; r) p)y r( r( (r ; r) p))y (r ; r)( (r ; r) p)x r( r( (r ; r) p))x r( (r ; r) p)y r( (r ; r) p)z ry z (r ; r) pz rza (r ; r) pa pa pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 33/39

34 Does it work for full PDL? r( (r ; r) p)x (r ; r) px (r ; r) py (r ; r)( (r ; r) p)y r( r( (r ; r) p))y (r ; r)( (r ; r) p)x r( r( (r ; r) p))x r( (r ; r) p)y r( (r ; r) p)z ry z (r ; r) pz rza (r ; r) pa pa pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 34/39

35 Does it work for full PDL? r( (r ; r) p)x (r ; r) px (r ; r) py (r ; r)( (r ; r) p)y r( r( (r ; r) p))y (r ; r)( (r ; r) p)x r( r( (r ; r) p))x r( (r ; r) p)y r( (r ; r) p)z ry z (r ; r) pz rza (r ; r) pa pa pz Terminating Tableaux for Mini-PDL (Sigurd Schneider) 35/39

36 Semantics = λpx. px : (IB)IB = λpqx. px qx : (IB)(IB)IB = λpqx. px qx = λrpx. y. py = λrpx. y. = py 0 = λ. x = y n = λ. z. rxz r n 1 zy = λ. n N.r n xy : (IB)(IB)IB : (IIB)(IB)IB : (IIB)(IB)IB 0 : (IIB)IIB n : (IIB)IIB n N,n > 0 : (IIB)IIB Terminating Tableaux for Mini-PDL (Sigurd Schneider) 36/39

37 A derivation with a box r ( rp)x r( r ( rp))x rpx py r ( rp)y r( r ( rp))y r py pz r ( rp)z... Terminating Tableaux for Mini-PDL (Sigurd Schneider) 37/39

38 Why does the construction of a minimal derivation terminate? Scan through each path from the root in some order. If a processed pattern occurs, cut everthing beyond off. Examine every section between duplicate patterns: If there is no witness path running over the second occurence, cut the direct witness off. If there is a witness path running over the second occurence, shorten the derivation. Patterns occuring on a processed path can be cut off instantly. The stock of patterns is finite. Terminating Tableaux for Mini-PDL (Sigurd Schneider) 38/39

39 Tableau Rules R ( p)x px A R (t 1 t 2 )x t 1 x, t 2 x R (t 1 t 2 )x t 1 x t 2 x R r tx tx r( r t)x R r tx tx, r( r t)x R rtx, ty y / N A R rtx ty N A Terminating Tableaux for Mini-PDL (Sigurd Schneider) 39/39