A tableau-based decision procedure for a branching-time interval - PowerPoint PPT Presentation

A tableau-based decision procedure for a branching-time interval temporal logic Davide Bresolin Angelo Montanari Dipartimento di Matematica e Informatica Università degli Studi di Udine { bresolin, montana } @dimi.uniud.it M4M-4, 1-2 December 2005 D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 1 / 28

Outline Introduction 1 A branching-time interval temporal logic 2 A Tableau for BTNL[R] − 3 Future work 4 D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 2 / 28

Outline Introduction 1 A branching-time interval temporal logic 2 A Tableau for BTNL[R] − 3 Future work 4 D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 3 / 28

Interval temporal logics Interval temporal logics (HS, CDT, PITL) are very expressive simple syntax and semantics; can naturally express statements that refer to time intervals and continuous processes; the most expressive ones (HS and CDT) are strictly more expressive than every point-based temporal logic. Interval temporal logics are (highly) undecidable The validity problem for HS is not recursively axiomatizable. Problem Find expressive, but decidable, fragments of interval temporal logics. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 4 / 28

Halpern and Shoam’s HS HS features four basic unary operators: � B � ( begins ) and � E � ( ends ), and their transposes � B � ( begun by ) and � E � ( ended by ). � B � ϕ � E � ϕ � �� � � �� � begins: ends: d 0 d 2 d 1 d 0 d 2 d 1 � �� � � �� � ϕ ϕ � B � ϕ � E � ϕ � �� � � �� � begun by: ended by: d 0 d 1 d 2 d 2 d 0 d 1 � �� � � �� � ϕ ϕ Given a formula ϕ and an interval [ d 0 , d 1 ] , � B � ϕ holds over [ d 0 , d 1 ] if ϕ holds over [ d 0 , d 2 ] , for some d 0 ≤ d 2 < d 1 , and � E � ϕ holds over [ d 0 , d 1 ] if ϕ holds over [ d 2 , d 1 ] , for some d 0 < d 2 ≤ d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 5 / 28

Some interesting fragments of HS The � B �� E � fragment ( undecidable ); The � B �� B � and � E �� E � fragments ( decidable ); Goranko, Montanari, and Sciavicco’s PNL: ◮ based on the derived neighborhood operators � A � ( meets ) and � A � ( met by ); � A � ϕ � A � ϕ ϕ ϕ � �� � � �� � � �� � � �� � met by: meets: d 0 d 1 d 2 d 2 d 0 d 1 ◮ decidable (by reduction to 2FO [ < ] ), but no tableau methods. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 6 / 28

The linear case: Right PNL (RPNL) future-only fragment of PNL; interpreted over natural numbers; decidable, doubly exponential tableau-based decision procedure for RPNL (TABLEAUX 2005); recently, we devised an optimal (NEXPTIME) tableau-based decision procedure for RPNL. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 7 / 28

The branching case We developed a branching-time propositional interval temporal logic. Such a logic combines: ◮ interval quantifiers � A � and [ A ] from RPNL; ◮ path quantifiers A and E from CTL. We devised a tableau-based decision procedure for it, combining: ◮ the tableau for RPNL (TABLEAUX 2005); ◮ Emerson and Halpern’s tableau for CTL (J. of Computer and System Sciences, 1985). D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 8 / 28

Outline Introduction 1 A branching-time interval temporal logic 2 A Tableau for BTNL[R] − 3 Future work 4 D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 9 / 28

Branching Time Right-Neighborhood Logic Syntax of BTNL[R] − ϕ = p | ¬ ϕ | ϕ ∨ ϕ | E � A � ϕ | E [ A ] ϕ | A � A � ϕ | A [ A ] ϕ. Interpreted over infinite trees . Combines path quantifiers A (for all paths) and E (for any path) with the interval modalities � A � and [ A ] . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 10 / 28

BTNL[R] − semantics: E � A � ψ . . . . . . . . . . . . E � A � ψ . . . d 0 d 1 . . . . . . . . . E � A � ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for some d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 11 / 28

BTNL[R] − semantics: E � A � ψ . . . . . . . . . . . . E � A � ψ . . . d 0 d 1 . . . ψ d 2 . . . . . . E � A � ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for some d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 11 / 28

BTNL[R] − semantics: E [ A ] ψ . . . . . . . . . . . . E [ A ] ψ . . . d 0 d 1 . . . . . . . . . E [ A ] ψ holds over [ d 0 , d 1 ] if there exists an infinite path d 1 , d 2 , . . . such that ψ holds over [ d 1 , d i ] , for all d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 12 / 28

BTNL[R] − semantics: E [ A ] ψ . . . . . . d 2 . . . ψ . . . E [ A ] ψ . . . d 0 d 1 . . . . . . . . . E [ A ] ψ holds over [ d 0 , d 1 ] if there exists an infinite path d 1 , d 2 , . . . such that ψ holds over [ d 1 , d i ] , for all d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 12 / 28

BTNL[R] − semantics: E [ A ] ψ . . . . . . d 2 . . . d 3 ψ . . . E [ A ] ψ . . . d 0 d 1 . . . . . . . . . E [ A ] ψ holds over [ d 0 , d 1 ] if there exists an infinite path d 1 , d 2 , . . . such that ψ holds over [ d 1 , d i ] , for all d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 12 / 28

BTNL[R] − semantics: E [ A ] ψ . . . . . . d 2 . . . d 3 ψ . . . E [ A ] ψ d 4 . . . d 0 d 1 . . . . . . . . . E [ A ] ψ holds over [ d 0 , d 1 ] if there exists an infinite path d 1 , d 2 , . . . such that ψ holds over [ d 1 , d i ] , for all d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 12 / 28

BTNL[R] − semantics: A [ A ] ψ . . . . . . . . . . . . A [ A ] ψ . . . d 0 d 1 . . . . . . . . . A [ A ] is the dual of E � A � : A [ A ] ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for all d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 13 / 28

BTNL[R] − semantics: A [ A ] ψ . . . . . . . . . d 2 ψ . . . A [ A ] ψ . . . d 0 d 1 ψ . . . . . . d 3 . . . A [ A ] is the dual of E � A � : A [ A ] ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for all d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 13 / 28

BTNL[R] − semantics: A [ A ] ψ . . . ψ . . . d 4 . . . ψ . . . A [ A ] ψ d 5 . . . d 0 d 1 ψ . . . d 6 . . . ψ . . . d 7 A [ A ] is the dual of E � A � : A [ A ] ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for all d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 13 / 28

BTNL[R] − semantics: A [ A ] ψ . . . ψ d 8 . . . ψ d 9 . . . d 10 ψ . . . A [ A ] ψ d 11 . . . d 0 d 1 ψ d 12 . . . ψ d 13 . . . d 14 ψ . . . d 15 A [ A ] is the dual of E � A � : A [ A ] ψ holds over [ d 0 , d 1 ] if ψ holds over [ d 1 , d 2 ] , for all d 2 < d 1 . D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 13 / 28

BTNL[R] − semantics: A � A � ψ . . . . . . . . . . . . A � A � ψ . . . d 0 d 1 . . . . . . . . . A � A � is the dual of E [ A ] : A � A � ψ holds over [ d 0 , d 1 ] if, for all infinite paths d 1 , d 2 , . . . , ψ holds over [ d 1 , d i ] , for some d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 14 / 28

BTNL[R] − semantics: A � A � ψ . . . . . . . . . d 2 ψ . . . A � A � ψ . . . d 0 d 1 . . . . . . . . . A � A � is the dual of E [ A ] : A � A � ψ holds over [ d 0 , d 1 ] if, for all infinite paths d 1 , d 2 , . . . , ψ holds over [ d 1 , d i ] , for some d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 14 / 28

BTNL[R] − semantics: A � A � ψ . . . . . . . . . d 2 . . . A � A � ψ . . . d 0 d 1 ψ . . . d 3 . . . . . . A � A � is the dual of E [ A ] : A � A � ψ holds over [ d 0 , d 1 ] if, for all infinite paths d 1 , d 2 , . . . , ψ holds over [ d 1 , d i ] , for some d i > d 1 in the path. D. Bresolin, A. Montanari (Univ. of Udine) A Tableau for BTNL[R] − M4M-4, 1-2 December 2005 14 / 28