The Complexity of Satisfiability for Fragments of CTL and Ctl⋆

Abstract

The satisfiability problems for CTL and CTL are known to be EXPTIME-complete, resp. 2EXPTIME-complete (Fischer and Ladner (1979), ). For fragments that use less temporal or propositional operators, the complexity may decrease. This paper undertakes a systematic study of satisfiability for CTL-and CTL -formulae over restricted sets of propositional and temporal operators. We show that restricting the temporal operators yields satisfiability problems complete for 2EXPTIME, EXPTIME, PSPACE, and NP. Restricting the propositional operators either does not change the complexity (as determined by the temporal operators), or yields very low complexity like NC 1 , TC 0 , or NLOGTIME.

Parameterised Complexity of Abduction in Schaefer’s Framework

“…The most prominent result under this approach is the dichotomy theorem of Lewis [18] which classifies propositional satisfiability into polynomialtime solvable cases and intractable ones depending merely on the existence of specific Boolean operators. This approach has been followed many times in a wealth of different contexts [1,2,6,10,19,20,29] as well as in the context of abduction itself [22,9]. Interestingly, in the scope of constraint satisfaction problems, the investigation of co-clones allows one to proceed a similar kind of classification.…”

Quirky Quantifiers: Optimal Models and Complexity of Computation Tree Logic

“…Many fragments of temporal logic have been investigated in the hope to find efficient algorithms. This includes restricted temporal operator sets, bounded operator nesting depth, bounded numbers of variables, and restricted sets of logical connectives [DS02,Hal95,MM+09,Sch02,SC85]. The results are not too optimistic: For no fragment of CTL or LTL the satisfiability problem becomes tractable, except for trivial combinations of Boolean connectives [MM+09].…”

“…This includes restricted temporal operator sets, bounded operator nesting depth, bounded numbers of variables, and restricted sets of logical connectives [DS02,Hal95,MM+09,Sch02,SC85]. The results are not too optimistic: For no fragment of CTL or LTL the satisfiability problem becomes tractable, except for trivial combinations of Boolean connectives [MM+09]. Restricting the CTL or LTL operators or the number of propositions does not decrease the computational complexity noteworthily; and also very low temporal depth already carries the complexity of LTL beyond that of propositional logic [DS02,Sch02,SC85].…”

“…This is a consequent continuation of the work of Lewis [Lew79], who showed that propositional satisfiability over is already NP-complete, whereas it is in P for all sets of Boolean connectives that are unable to express . In the setting of temporal logic, similarly the tractable Boolean fragments were investigated by Meier et al [MM+09].…”

LTL Fragments are Hard for Standard Parameterisations

“…After a decade of seminal work from Allen Emerson, Clarke, Halpern, Schnoebelen, and Sistla [4,8,19,20]-to name only a few-the most important concepts, e.g., satisfiability and model checking have been well understood and classified with respect to their computational complexity. Recently the mentioned decision problems have been investigated in the light of a study which considers fragments of the logics in the sense of allowed operators [1,14].…”