Verification of Rewrite Rules for Computation Tree Logics

Abstract

A number of procedures for checking the satisfiability of formulas in the important branching time temporal logic CTL* have recently been proposed. This paper instead focuses on automatic generation and verification of rewrite rules for computation tree logics; shows that non-local computation tree logics can be used to verify rewrite rules, including for CTL*; presents an efficient tableau for the non-local bundled variant NL-BCTL*; and shows that NL-BCTL* is 2EXPTIME-complete.We show that such rules can quickly simplify CTL* formulas. These simplified formulas are shorter and easier to reason with using existing decision procedures for CTL*, as demonstrated by significant speed-ups across a wide range of benchmark formulas.While CTL* is not widely used due to the complexity of its reasoning tasks, it is strictly more expressive than LTL or CTL. Furthermore, there are applications for theorem-proving and model-checking.

Rewrite rules for CTL*

“…The tree tableau gives reasonable performance while it is known that constructing all hues (as in NL-BCTLTAB) gives unacceptable performance except in the most trivial cases. The implementation is available as a web application with downloadable source 1 [15].…”

“…In this section we present some simple benchmarks for bundled tableaux. Note that all these tableaux are included in our unified rewrite tool [15]. To gain a better understanding of the asymptotic performance we would include families of formulas as in [8].…”

“…For Table 2 we use the "BPATHUE" and "BPATH-NEW" implementations, based on the tree unwinding found in Section 4. All of these are available from the unified online tool [15].…”

“…The main contributions of this paper are: a tableau for non-local bundled CTL* (NL-BCTL*); a proof of its correctness; an interesting demonstration that the non-local and bundled variants of CTL* can make contributions to practical reasoning tools for the important CTL* logic; a simple valid test for candidate rewrite rules in CTL*; an implementation of these; a guaranteed correct online manual rewrite collection tool ( [15]); a sound automated rewrite learning tool; a combined tool to help decide satisfiability of CTL* formulas; presentation of a sound, confluent and terminating TRS for CTL*; and demonstration of significant speed-ups in decision tasks (on top of the best existing tools for these logics) over a large range of benchmarks.…”

“…A preliminary version of this paper has appeared in conference proceedings [16]. This journal version adds: an efficient tree shaped variant (Section 4) of the NL-BCTL* Tableau based on a similar tableau [17] for BCTL*; discussion and benchmarks of a new OCaml implementation designed to follow closely the formal definition of the simple graph tableau; more discussion with brief descriptions of previous (B)CTL* solvers used in the benchmarks (table 1); more benchmarks such as problems solvable within 60 seconds; algorithms (1,2) for our "trivial" rewrite system are presented; consideration of Associative Commutative rewrite systems, including benchmarks, now uses CiME3 instead of KBCV (interestingly with the former, nicer and more sophisticated rewrite systems performing worse); and presentation of a sound, confluent and terminating Associative Commutative TRS for CTL*.…”