scholarly journals Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

2021 ◽  
Vol 22 (2) ◽  
pp. 1-37
Author(s):  
Christopher H. Broadbent ◽  
Arnaud Carayol ◽  
C.-H. Luke Ong ◽  
Olivier Serre
Keyword(s):  
Model Checking ◽  
Free Variable ◽  
Higher Order ◽  
Second Order ◽  
Effective Solution ◽  
Parity Games ◽  
Transition Graphs ◽  
Second Order Logic ◽  
Pushdown Automata ◽  
Monadic Second Order Logic

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

scholarly journals Automated Logical Verification based on Trace Abstractions

1995 ◽  
Vol 2 (53) ◽  
Author(s):  
Nils Klarlund ◽  
Mogens Nielsen ◽  
Kim Sunesen
Keyword(s):  
Distributed Systems ◽  
Model Checking ◽  
State Space ◽  
Temporal Logic ◽  
Second Order ◽  
Order Logic ◽  
Verification Problem ◽  
Finite State ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We propose a new and practical framework for integrating the behavioral<br />reasoning about distributed systems with model-checking methods.<br />Our proof methods are based on trace abstractions, which relate the<br />behaviors of the program and the specification. We show that for finite-state<br />systems such symbolic abstractions can be specified conveniently in<br />Monadic Second-Order Logic (M2L). Model-checking is then made possible<br />by the reduction of non-determinism implied by the trace abstraction.<br />Our method has been applied to a recent verification problem by Broy<br />and Lamport. We have transcribed their behavioral description of a distributed<br />program into temporal logic and verified it against another distributed<br />system without constructing the global program state space. The<br />reasoning is expressed entirely within M2L and is carried out by a decision<br />procedure. Thus M2L is a practical vehicle for handling complex temporal<br />logic specifications, where formulas decided by a push of a button are as<br />long as 10-15 pages.

Monadic second-order logic on finite sequences

2017 ◽  
Vol 52 (1) ◽  
pp. 232-245
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Finite Sequences ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Existential monadic second order logic on random rooted trees

2019 ◽  
Vol 342 (1) ◽  
pp. 152-167
Author(s):  
Alexander E. Holroyd ◽  
Avi Levy ◽  
Moumanti Podder ◽  
Joel Spencer
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Rooted Trees ◽  
Second Order Logic ◽  
Monadic Second Order Logic
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