Results on the Propositional µ-Calculus

Kozen, Dexter

doi:10.7146/dpb.v11i146.7420

Cited by 329 publications

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“…One of the most famous temporal logics is Kozen's modal µ-calculus L µ [15] which extends multi-modal logic with extremal fixpoint quantifiers. L µ subsumes many other temporal logics like PDL [11] as well as CTL * [9], and with it CTL [8] and LTL [23].…”

Section: Introductionmentioning

confidence: 99%

The Complexity of Model Checking Higher-Order Fixpoint Logic

Axelsson¹,

Lange²,

Somla³

2007

Logical Methods in Computer Science

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Abstract. Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed λ-calculus and the modal µ-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal µ-calculus.This paper provides complexity results for its model checking problem. In particular, we consider those fragments of HFL that are built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment.The lower bound is established by a reduction from the word problem for alternating (k − 1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.

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Section: Introductionmentioning

confidence: 99%

The Complexity of Model Checking Higher-Order Fixpoint Logic

Axelsson¹,

Lange²,

Somla³

2007

Logical Methods in Computer Science

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“…which cannot be expressed in PDL (see [15]). More complex properties such as "on some path p occurs infinitely often" (EF 00 p in GT L* notation) can also be expressed (we assume that I'0 = {a}):…”

Section: µX[a]x = ((Ad)*)lmentioning

confidence: 99%

“…The µ-calculus introduces fixpoint operators into the modal language, yielding a logic which is strictly more expressive than PDL (see [15] …”

Section: Definition 3 (Bisimulation For Kripke Models) Let I= (Smentioning

confidence: 99%

From Programs to Games: Invariance and Safety for Bisimulation

Pauly

2000

Computer Science Logic

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Abstract. Bisimulation is generalized from process models to game models which are described using Game Logic (GL), a logic which extends Propositional Dynamic Logic by an additional operator dual which allows for the construction of complex 2-player games. It is shown that bisimilar states satisfy the same GIAormulas (invariance), and that an atomic bisimulation can be lifted to non-atomic GL-garnes (safety). Over process models, GL forms a highly expressive fragment of the modal µ-calculus, and within first-order logic, the game operations of GL are complete: they suffice to construct all first-order definable games which are monotonic and safe for bisimulation.

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“…Such correctness properties are formulated in the regular alternation free µ-calculus [9,8]. Informally, the modalities in the logic are relaxed to sequences of actions characterised by regular expressions.…”

Section: Model Checking Properties In Fault-tolerant Systemsmentioning

confidence: 99%

Verification of Language Based Fault-Tolerance

Earle

Fredlund

2005

Lecture Notes in Computer Science

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In this paper we target the verification of fault tolerant aspects of distributed applications written in the Erlang programming language. Erlang programmers mostly work with ready-made language components. Our approach to verification of fault tolerance is to verify systems built using a central component of most Erlang software, a generic server component with fault tolerance handling.To verify such Erlang programs we automatically translate them into processes of the µCRL process algebra, generate their state spaces, and use a model checker to determine whether they satisfy correctness properties specified in the µ-calculus.The key observation of this paper is that, due to the usage of these higher-level design patterns, the state space generated from a Erlang program, even with failures occurring, is relatively small, and can be generated automatically.

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Results on the Propositional µ-Calculus

Abstract: We define a propositional version of the µ-calculus, and give an exponential-time decision procedure, small model property, and complete deductive system. We also show that it is strictly more expressive than PDL. Finally we give an algebraic semantics and prove a representation theorem.

Cited by 329 publications

References 0 publications

The Complexity of Model Checking Higher-Order Fixpoint Logic

The Complexity of Model Checking Higher-Order Fixpoint Logic

From Programs to Games: Invariance and Safety for Bisimulation

Verification of Language Based Fault-Tolerance

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