Freeze Quantifier Research Articles

A Subclass of Mu-Calculus with the Freeze Quantifier Equivalent to Register Automata

Register automaton (RA) is an extension of finite automaton by adding registers storing data values. RA has good properties such as the decidability of the membership and emptiness problems. Linear temporal logic with the freeze quantifier (LTL↓) proposed by Demri and Lazić is a counterpart of RA. However, the expressive power of LTL↓ is too high to be applied to automatic verification. In this paper, we propose a subclass of modal µ-calculus with the freeze quantifier, which has the same expressive power as RA. Since a conjunction ψ1 ∧ ψ2 in a general LTL↓ formula cannot be simulated by RA, the proposed subclass prohibits at least one of ψ1 and ψ2 from containing the freeze quantifier or a temporal operator other than X (next). Since the obtained subclass of LTL↓ does not have the ability to represent a cycle in RA, we adopt µ-calculus over the subclass of LTL↓, which allows recursive definition of temporal formulas. We provide equivalent translations from the proposed subclass of µ-calculus to RA and vice versa and prove their correctness.

Runtime Verification over Out-of-order Streams

We present an approach for verifying systems at runtime. Our approach targets distributed systems whose components communicate with monitors over unreliable channels, where messages can be delayed, reordered, or even lost. Furthermore, our approach handles an expressive specification language that extends the real-time logic MTL with freeze quantifiers for reasoning about data values. The logic’s main novelty is a new three-valued semantics that is well suited for runtime verification as it accounts for partial knowledge about a system’s behavior. Based on this semantics, we present online algorithms that reason soundly and completely about streams where events can occur out of order. We also evaluate our algorithms experimentally. Depending on the specification, our prototype implementation scales to out-of-order streams with hundreds to thousands of events per second.

Expressiveness of Hybrid Temporal Logic on Data Words

Hybrid temporal logic (HTL) on data words can be considered as an extension of the logic LTL↓ introduced by Demri and Lazic [Stephane Demri and Ranko Lazic. LTL with the freeze quantifier and register automata. In LICSʼ06: Proceedings of the 21st Annual IEEE Symposium on Logic in Computer Science, pages 17–26, Washington, DC, USA, 2006. IEEE Computer Society]. The paper compares the expressive power of HTL on data words with that of LTL↓. It is shown that there are properties of data words that can be expressed in HTL with two variables but not in LTL↓. On the other hand, every property that can be expressed in HTL with one variable can also be expressed in LTL↓ with one variable. The paper further studies the succinctness of HTL in comparison with LTL↓ and shows that the number-of-variables hierarchy of HTL is infinite.

Temporal Logics of Repeating Values

Various logical formalisms with the freeze quantifier have been recently considered to model computer systems even though this is a powerful mechanism that often leads to undecidability. In this article, we study a linear-time temporal logic with past-time operators such that the freeze operator is only used to express that some value from an infinite set is repeated in the future or in the past. Such a restriction has been inspired by a recent work on spatio-temporal logics that suggests such a restricted use of the freeze operator. We show decidability of finitary and infinitary satisfiability by reduction into the verification of temporal properties in Petri nets by proposing a symbolic representation of models. This is a quite surprising result in view of the expressive power of the logic since the logic is closed under negation, contains future-time and past-time temporal operators and can express the nonce property and its negation. These ingredients are known to lead to undecidability with a more liberal use of the freeze quantifier. The article also contains developments about the relationships between temporal logics with the freeze operator and counter automata as well as reductions into first-order logics over data words.

On pebble automata for data languages with decidable emptiness problem

In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we show that the emptiness problem for weak 2-pebble automata is decidable, while the same problem for weak 3-pebble automata is undecidable. We also introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is still decidable. It is also robust: alternating, non-deterministic and deterministic top view weak PA have the same recognition power; and are strong enough to accept all data languages expressible in Linear Temporal Logic with the future-time operators, augmented with one register freeze quantifier.

LTL with the freeze quantifier and register automata

A data word is a sequence of pairs of a letter from a finite alphabet and an element from an infinite set, where the latter can only be compared for equality. To reason about data words, linear temporal logic is extended by the freeze quantifier, which stores the element at the current word position into a register, for equality comparisons deeper in the formula. By translations from the logic to alternating automata with registers and then to faulty counter automata whose counters may erroneously increase at any time, and from faulty and error-free counter automata to the logic, we obtain a complete complexity table for logical fragments defined by varying the set of temporal operators and the number of registers. In particular, the logic with future-time operators and 1 register is decidable but not primitive recursive over finite data words. Adding past-time operators or 1 more register, or switching to infinite data words, causes undecidability.

On the freeze quantifier in Constraint LTL: Decidability and complexity

Constraint LTL, a generalisation of LTL over Presburger constraints, is often used as a formal language to specify the behavior of operational models with constraints. The freeze quantifier can be part of the language, as in some real-time logics, but this variable-binding mechanism is quite general and ubiquitous in many logical languages (first-order temporal logics, hybrid logics, logics for sequence diagrams, navigation logics, logics with λ-abstraction, etc.). We show that Constraint LTL over the simple domain 〈N,=〉 augmented with the freeze quantifier is undecidable which is a surprising result in view of the poor language for constraints (only equality tests). Many versions of freeze-free Constraint LTL are decidable over domains with qualitative predicates and our undecidability result actually establishes Σ11-completeness. On the positive side, we provide complexity results when the domain is finite (ExpSpace-completeness) or when the formulae are flat in a sense introduced in the paper. Our undecidability results are sharp (i.e. with restrictions on the number of variables) and all our complexity characterisations ensure completeness with respect to some complexity class (mainly PSpace and ExpSpace).

Runtime Verification of Timed LTL using Disjunctive Normalized Equation Systems

In this paper we present a new framework for runtime verification of properties of real time systems such as financial systems or backend databases. Such a systems has a semantics which resemples that of timed traces, namely a sequence of states where each state consists of predicates true in this state and then a timestamp explaining when the state is valid. We present a logic, LTLt, which is an extension of LTL with time constraints and a freeze quantifier and show how formulae in this logic are able to express properties of bounded liveness and safety which are ideal for these systems. It is shown how a formula in LTLt may be rewritten to a certain disjunctive normal form suitable for checking a real time system at runtime. The normal form captures the essential part of runtime verification by a set of mutually defined formula identifiers, each expressing two things: What should hold now and which formula identifiers that will need to hold in the next state. As part of the theoretical foundation for this work we propose a characterization of Runtime Verification and address the challenges in developing a method which is both sound and complete while at the same time efficient.

On the expressivity and complexity of quantitative branching-time temporal logics

We investigate extensions of CTL allowing to express quantitative requirements about an abstract notion of time in a simple discrete-time framework, and study the expressive power of several relevant logics. When only subscripted modalities are used, polynomial-time model checking is possible even for the largest logic we consider, while the introduction of freeze quantifiers leads to a complexity blow-up.

Specification and validation of dynamic systems using temporal logic

A specification and validation technique for dynamic systems is proposed. In particular, a new temporal logic, called HDTL, is presented and the tableau method revised for automatic analysis. Using a freeze quantifier, HDTL with the revised tableau method makes it possible to specify the correctness requirements of dynamic systems and validate them. The proposed logic is rather generic, i.e. it has only a few assumptions on operational language. The authors introduce a simple dynamic modelling language and illustrate its experiment. The experiment shows that HDTL is suitable for specifying dynamic properties and the analysis technique is promising.