RECOGNIZABLE AND LOGICALLY DEFINABLE LANGUAGES OF INFINITE COMPUTATIONS IN CONCURRENT AUTOMATA

1998 ◽  
Vol 09 (03) ◽  
pp. 295-313 ◽  
Author(s):  
MANFRED DROSTE ◽  
DIETRICH KUSKE
Keyword(s):  
Concurrent Programs ◽  
Binary Relations ◽  
Ordered Sets ◽  
Transition Systems ◽  
Infinite Products ◽  
Verification Methods ◽  
Language L ◽  
Rational Expressions ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Automata with concurrency relations [Formula: see text], which occurred in formal verification methods of concurrent programs, are labeled transition systems with a collection of binary relations describing when two actions in a given state of the automaton can happen independently of each other. The concurrency monoid M($\mathcal{A}$) comprises all finite computation sequences of [Formula: see text], modulo a canonical congruence induced by the concurrency relations, with composition as monoid operation. Then M∞($\mathcal{A}$) denotes the set of all infinite products in M($\mathcal{A}$); its elements can be represented by labeled partially ordered sets. Under suitable assumptions on [Formula: see text], we show that a language L in M∞($\mathcal{A}$) is recognizable iff it is definable by a formula of monadic second order logic, and that it is recognizable iff it can be constructed from recognizable languages in M($\mathcal{A}$) using co-rational expressions. This generalizes various recent results in trace theory.

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 Monadic Second-Order Logic, Graphs and Unfoldings of Transition Systems

1995 ◽  
Vol 2 (44) ◽  
Author(s):  
Bruno Courcelle ◽  
Igor Walukiewicz
Keyword(s):  
Transition System ◽  
Second Order ◽  
Order Logic ◽  
Order Property ◽  
Transition Systems ◽  
Graph Coverings ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We prove that every monadic second-order property of the unfolding<br />of a transition system is a monadic second-order property of the<br />system itself. We prove a similar result for certain graph coverings.

scholarly journals BDD Algortihms and Cache Misses

1996 ◽  
Vol 3 (26) ◽  
Author(s):  
Nils Klarlund ◽  
Theis Rauhe
Keyword(s):  
Data Structures ◽  
Analytical Tool ◽  
Order Logic ◽  
General Interest ◽  
Concurrent Programs ◽  
Data Structures And Algorithms ◽  
Cache Miss ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Symbolic Algorithms

Within the last few years, CPU speed has greatly overtaken memory speed. For this reason, implementation of symbolic algorithms - with their extensive use of pointers and hashing - must be reexamined. In this paper, we introduce the concept of cache miss complexity<br />as an analytical tool for evaluating algorithms depending on pointer chasing. Such algorithms are typical of symbolic computation found in verification. We show how this measure suggests new data structures and algorithms<br />for multi-terminal BDDs. Our ideas have been implemented in<br />a BDD package, which is used in a decision procedure for the Monadic Second-order Logic on strings.<br />Experimental results show that on large examples involving e.g the verification of concurrent programs, our implementation runs 4 to 5 times faster than a widely used BDD implementation.<br />We believe that the method of cache miss complexity is of general interest to any implementor of symbolic algorithms used in verification.

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.

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
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