Formula Synthesis in Propositional Dynamic Logic with Shuffle

Abstract

We introduce the formula-synthesis problem for Propositional Dynamic Logic with Shuffle (PDL || ). This problem, which generalises the model-checking problem againsts PDL || is the following: given a finite transition system and a regular term-grammar that generates (possibly infinitely many) PDL || formulas, find a formula generated by the grammar that is true in the structure (or return that there is none). We prove that the problem is undecidable in general, but add certain restrictions on the input structure or on the input grammar to yield decidability. In particular, we prove that (1) if the grammar only generates formulas in PDL (without shuffle), then the problem is EXPTIME-complete, and a further restriction to linear grammars is PSPACE-complete, and a further restriction to non-recursive grammars is NP-complete, and (2) if one restricts the input structure to have only simple paths then the problem is in 2-EXPTIME. This work is motivated by and opens up connections to other forms of synthesis from hierarchical descriptions, including HTN problems in Planning and Attack-tree Synthesis problems in Security.