The Complexity of Second-order Hyperlogic HyperLTL

Keskeiset käsitteet

The complexity of second-order HyperLTL satisfiability and model-checking is as hard as truth in third-order arithmetic.

Tiivistelmä

The paper investigates the complexity of second-order HyperLTL, an extension of the temporal logic HyperLTL that allows quantification over sets of traces. The key findings are:

There exists a satisfiable Hyper2LTL sentence that only has models of cardinality equal to the cardinality of the continuum. This is in contrast to HyperLTL, where every satisfiable sentence has a countable model.

The Hyper2LTL satisfiability problem is polynomial-time equivalent to truth in third-order arithmetic. This is much harder than HyperLTL satisfiability, which is Σ^1_1-complete.

The Hyper2LTL finite-state satisfiability problem is also polynomial-time equivalent to truth in third-order arithmetic. This is in contrast to HyperLTL, where finite-state satisfiability is much simpler (Σ^0_1-complete) than general satisfiability.

The Hyper2LTL model-checking problem is also polynomial-time equivalent to truth in third-order arithmetic, matching the complexity of the satisfiability problems.

The paper shows that the addition of second-order quantification to HyperLTL significantly increases the complexity of the decision problems, reaching the level of third-order arithmetic.

Tilastot

There exists a satisifiable Hyper2LTL sentence that only has models of cardinality equal to the cardinality of the continuum.
Hyper2LTL satisfiability is polynomial-time equivalent to truth in third-order arithmetic.
Hyper2LTL finite-state satisfiability is polynomial-time equivalent to truth in third-order arithmetic.
Hyper2LTL model-checking is polynomial-time equivalent to truth in third-order arithmetic.

Lainaukset

"The complexity of undecidable problems is typically captured in terms of the arithmetical and analytical hierarchy, where decision problems (encoded as subsets of N) are classified based on their definability by formulas of higher-order arithmetic, namely by the type of objects one can quantify over and by the number of alternations of such quantifiers."
"HyperLTL satisfiability is Σ^1_1-complete, HyperLTL finite-state satisfiability is Σ^0_1-complete, and Hyper2LTL model-checking is Σ^1_1-hard, but no upper bounds are known."

Tärkeimmät oivallukset

The Complexity of Second-order HyperLTL

by Hadar Frenke... klo arxiv.org 04-30-2024

https://arxiv.org/pdf/2311.15675.pdf

Syvällisempiä Kysymyksiä

What are some practical applications of second-order HyperLTL that motivate the study of its complexity

Practical applications of second-order HyperLTL that motivate the study of its complexity include scenarios where properties need to be expressed that involve quantification over sets of traces. For example, in multi-agent systems, common knowledge properties can be effectively captured using second-order quantification, which is not possible in standard HyperLTL. Additionally, asynchronous properties in distributed systems, which require reasoning about sets of traces, can also be expressed using second-order HyperLTL. By allowing quantification over sets of traces, second-order HyperLTL provides a more expressive language for specifying complex system properties that involve interactions between multiple system executions.

How do the complexity results for Hyper2LTL compare to other extensions of HyperLTL, such as those with fixed-point operators

The complexity results for Hyper2LTL show that the satisfiability and model-checking problems are as hard as truth in third-order arithmetic. This places Hyper2LTL in a complexity class that is higher than other extensions of HyperLTL, such as those with fixed-point operators. While HyperLTL with fixed-point operators may have its own complexities, the introduction of second-order quantification in Hyper2LTL significantly increases the computational complexity of reasoning about properties expressed in this logic. This highlights the trade-off between expressiveness and computational complexity in different extensions of HyperLTL.

Can the complexity of Hyper2LTL be reduced by further restricting the syntax or semantics of the logic, while still capturing interesting properties

The complexity of Hyper2LTL may potentially be reduced by further restricting the syntax or semantics of the logic while still capturing interesting properties. One approach could be to limit the range of second-order quantifiers to subsets of the universe of discourse, rather than allowing quantification over arbitrary sets of traces. By constraining the quantification in this way, the logic may become more tractable while still being able to capture essential properties. Additionally, introducing specific syntactic restrictions or semantic constraints that simplify the logic without sacrificing its expressive power could also help in reducing the complexity of Hyper2LTL. These modifications would need to be carefully designed to balance between complexity and expressiveness in the logic.