Abstract

We describe a set of simple features that are sufficient in order to make the satisfiability problem of logics interpreted on trees Tower-hard. We exhibit these features through an Auxiliary Logic on Trees (ALT), a modal logic that essentially deals with reachability of a fixed node inside a forest and features modalities from sabotage modal logic to reason on submodels. After showing that ALT admits a Tower-complete satisfiability problem, we prove that this logic is captured by four other logics that were independently found to be Tower-complete: two-variables separation logic, quantified computation tree logic, modal logic of heaps and modal separation logic. As a by-product of establishing these connections, we discover strict fragments of these logics that are still non-elementary.

Highlights

  • In mathematical logic there is a well-known trade-off between expressive power and complexity, where weaker languages cannot capture interesting properties of complex systems, whereas finding solutions of a given problem is infeasible for richer languages

  • Among the various temporal logics, from the classical linear temporal logic (LTL) [39] and computation tree logic (CTL) [13], as well as their fragments [2,33], to the more recently developed interval temporal logics [7,8], the main common feature of this framework is perhaps the ability to check whether the system can evolve to a certain configuration, i.e. a reachability query

  • We studied an Auxiliary Logic on Trees (ALT), a quite simple formalism that admits a TOWER-complete satisfiability problem

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Summary

Introduction

In mathematical logic there is a well-known trade-off between expressive power and complexity, where weaker languages cannot capture interesting properties of complex systems, whereas finding solutions of a given problem is infeasible for richer languages. Among the various temporal logics, from the classical linear temporal logic (LTL) [39] and computation tree logic (CTL) [13], as well as their fragments [2,33], to the more recently developed interval temporal logics [7,8], the main common feature of this framework is perhaps the ability to check whether the system can evolve to a certain configuration, i.e. a reachability query In this context, we recall the landmark result on the satisfiability of CTL, shown EXPTIME-complete by Fisher and Ladner [23]. Similar ideas are developed in sabotage modal logics, where the formula ⧫ , headed by the sabotage modality ⧫, states that must hold in a graph obtained by removing one edge from the current model [4,21] Within these logics, we highlight the quantifier-free fragment of separation logic restricted to the ∗ operator, denoted here with (∗) and whose satisfiability problem is proved to be PSPACE-complete in [12].

The definition of an Auxiliary Logic on Trees
On the complexity and expressive power of ALT
Towards the TOWER-hardness of SAT(ALT): how to encode finite words
Inexpressibility results via the Ehrenfeucht-Fraïssé games for ALT
PITL on marked words and the TOWER-hardness of SAT(ALT)
Revisiting TOWER-hard logics with ALT
From ALT to First-Order Separation Logic
From ALT to Quantified Computation Tree Logic
From ALT to Modal Logic of Heaps and Modal Separation Logic
Conclusions

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