Deterministic Temporal Logics and Interval Constraints

In temporal logics, a central question is about the choice of modalities and their relative expressive power, in comparison to the complexity of decision problems such as satisfiability. In this tutorial, we will illustrate the study of such questions over finite word models, first with logics for Unambiguous Starfree Regular Languages (UL), originally defined by Schutzenberger, and then for extensions with constraints, which appear in interval logics. We present Deterministic temporal logics, with diverse sets of modalities, which also characterize UL. The tools and techniques used go under the name of "Turtle Programs" or "Rankers". These are simple kinds of automata. We use properties such as Ranker Directionality and Ranker Convexity to show that all these logics have NP satisfiability. A recursive extension of some of these modalities gives us the full power of first-order logic over finite linear orders. We also discuss Interval Constraint modalities extending Deterministic temporal logics, with intermediate expressiveness. These allow counting or simple algebraic operations on paths. The complexity of these extended logics is PSpace, as of full temporal logic (and ExpSpace when using binary notation).Comment: In Proceedings M4M9 2017, arXiv:1703.0173

Deterministic Logics for UL

The class of Unambiguous Star-Free Regular Languages (UL) was defined by Schutzenberger as the class of languages defined by Unambiguous Polynomials. UL has been variously characterized (over finite words) by logics such as TL[X_a,Y_a], UITL, TL[F,P], FO2[<], the variety DA of monoids, as well as partially-ordered two-way DFA (po2DFA). We revisit this language class with emphasis on notion of unambiguity and develop on the concept of Deterministic Logics for UL. The formulas of deterministic logics uniquely parse a word in order to evaluate satisfaction. We show that several deterministic logics robustly characterize UL. Moreover, we derive constructive reductions from these logics to the po2DFA automata. These reductions also allow us to show NP-complete satisfaction complexity for the deterministic logics considered. Logics such as TL[F,P], FO2[<] are not deterministic and have been shown to characterize UL using algebraic methods. However there has been no known constructive reduction from these logics to po2DFA. We use deterministic logics to bridge this gap. The language-equivalent po2DFA for a given TL[F,P] formula is constructed and we analyze its size relative to the size of the TL[F,P] formula. This is an efficient reduction which gives an alternate proof to NP-complete satisfiability complexity of TL[F,P] formulas

The Unary Fragments of Metric Interval Temporal Logic: Bounded versus Lower bound Constraints (Full Version)

We study two unary fragments of the well-known metric interval temporal logic MITL[U_I,S_I] that was originally proposed by Alur and Henzinger, and we pin down their expressiveness as well as satisfaction complexities. We show that MITL[F_\inf,P_\inf] which has unary modalities with only lower-bound constraints is (surprisingly) expressively complete for Partially Ordered 2-Way Deterministic Timed Automata (po2DTA) and the reduction from logic to automaton gives us its NP-complete satisfiability. We also show that the fragment MITL[F_b,P_b] having unary modalities with only bounded intervals has \nexptime-complete satisfiability. But strangely, MITL[F_b,P_b] is strictly less expressive than MITL[F_\inf,P_\inf]. We provide a comprehensive picture of the decidability and expressiveness of various unary fragments of MITL.Comment: Presented at ATVA, 201

On Expressive Powers of Timed Logics: Comparing Boundedness, Non-punctuality and Deterministic Freezing

Timed temporal logics exhibit a bewildering diversity of operators and the resulting decidability and expressiveness properties also vary considerably. We study the expressive power of timed logics TPTL[U,S] and MTL[U,S] as well as of their several fragments. Extending the LTL EF games of Etessami and Wilke, we define MTL Ehrenfeucht-Fraisse games on a pair of timed words. Using the associated EF theorem, we show that, expressively, the timed logics BoundedMTL[U,S], MTL[F,P] and MITL[U,S] (respectively incorporating the restrictions of boundedness, unary modalities and non-punctuality), are all pairwise incomparable. As our first main result, we show that MTL[U,S] is strictly contained within the freeze logic TPTL[U,S] for both weakly and strictly monotonic timed words, thereby extending the result of Bouyer et al and completing the proof of the original conjecture of Alur and Henziger from 1990. We also relate the expressiveness of a recently proposed deterministic freeze logic TTL[X,Y] (with NP-complete satisfiability) to MTL. As our second main result, we show by an explicit reduction that TTL[X,Y] lies strictly within the unary, non-punctual logic MITL[F,P]. This shows that deterministic freezing with punctuality is expressible in the non-punctual MITL[F,P].Comment: Major revision of the pape

DCSYNTH: Guided Reactive Synthesis with Soft Requirements for Robust Controller and Shield Synthesis

DCSYNTH is a tool for the synthesis of controllers from safety and bounded liveness requirements given in interval temporal logic QDDC. It investigates the role of soft requirements (with priorities) in obtaining high quality controllers. A QDDC formula specifies past time properties. In DCSYNTH synthesis, hard requirements must be invariantly satisfied whereas soft requirements may be satisfied "as much as possible" in a best effort manner by the controller. Soft requirements provide an invaluable ability to guide the controller synthesis. In the paper, using DCSYNTH, we show the application of soft requirements in obtaining robust controllers with various specifiable notions of robustness. We also show the use of soft requirements to specify and synthesize efficient runtime enforcement shields which can correct burst errors. Finally, we discuss the use of soft requirements in improving the latency of controlled system

Formalizing Timing Diagram Requirements in Discrete Duration Calulus

Several temporal logics have been proposed to formalise timing diagram requirements over hardware and embedded controllers. These include LTL, discrete time MTL and the recent industry standard PSL. However, succintness and visual structure of a timing diagram are not adequately captured by their formulae. Interval temporal logic QDDC is a highly succint and visual notation for specifying patterns of behaviours. In this paper, we propose a practically useful notation called SeCeCntnl which enhances negation free fragment of QDDC with features of nominals and limited liveness. We show that timing diagrams can be naturally (compositionally) and succintly formalized in SeCeCntnl as compared with PSL and MTL. We give a linear time translation from timing diagrams to SeCeCntnl. As our second main result, we propose a linear time translation of SeCeCntnl into QDDC. This allows QDDC tools such as DCVALID and DCSynth to be used for checking consistency of timing diagram requirements as well as for automatic synthesis of property monitors and controllers. We give examples of a minepump controller and a bus arbiter to illustrate our tools. Giving a theoretical analysis, we show that for the proposed SeCeCntnl, the satisfiability and model checking have elementary complexity as compared to the non-elementary complexity for the full logic QDDC

On the Decidability and Complexity of Some Fragments of Metric Temporal Logic

Metric Temporal Logic, \mtlfull is amongst the most studied real-time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval constraints $I$. \oomit{The classical results of Alur and Henzinger showed that \mtlfull is undecidable where as \mitl which uses only non-singular intervals $NS$ is decidable. In a surprizing result, Ouaknine and Worrell showed that the satisfiability of \mtl is decidable over finite pointwise models, albeit with NPR decision complexity, whereas it remains undecidable for infinite pointwise models or for continuous time.} In this paper, we sharpen the decidability results by showing that the satisfiability of \mtlsns (where $NS$ denotes non-singular intervals) is also decidable over finite pointwise strictly monotonic time. We give a satisfiability preserving reduction from the logic \mtlsns to decidable logic \mtl of Ouaknine and Worrell using the technique of temporal projections. We also investigate the decidability of unary fragment \mtlfullunary (a question posed by A. Rabinovich) and show that \mtlfut over continuous time as well as \mtlfullunary over finite pointwise time are both undecidable. Moreover, \mathsf{MTL}^{pw}[\fut_I] over finite pointwise models already has NPR lower bound for satisfiability checking. We also compare the expressive powers of some of these fragments using the technique of EF games for $\mathsf{MTL}$

Two-variable logics with some betweenness relations: Expressiveness, satisfiability and membership

We study two extensions of FO2[<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, "the letter $a$ appears between positions $x$ and $y$" and "the factor $u$ appears between positions $x$ and $y$". These are, in a sense, the simplest properties that are not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We give effective conditions, in terms of the syntactic monoid of a regular language, for a property to be expressible in these logics. This algebraic analysis allows us to prove, among other things, that our new logics have strictly less expressive power than full first-order logic FO[<]. Our proofs required the development of novel techniques concerning factorizations of words

Specification and Optimal Reactive Synthesis of Run-time Enforcement Shields

A system with sporadic errors (SSE) is a controller which produces high quality output but it may occasionally violate a critical requirement REQ(I,O). A run-time enforcement shield is a controller which takes (I,O) (coming from SSE) as its input, and it produces a corrected output O' which guarantees the invariance of requirement REQ(I,O'). Moreover, the output sequence O' must deviate from O "as little as possible" to maintain the quality. In this paper, we give a method for logical specification of shields using formulas of logic Quantified Discrete Duration Calculus (QDDC). The specification consists of a correctness requirement REQ as well as a hard deviation constraint HDC which must both be mandatorily and invariantly satisfied by the shield. Moreover, we also use quantitative optimization to give a shield which minimizes the expected value of cumulative deviation in an H-optimal fashion. We show how tool DCSynth implementing soft requirement guided synthesis can be used for automatic synthesis of shields from a given specification. Next, we give logical formulas specifying several notions of shields including the k-Stabilizing shield of Bloem "et al." as well as the Burst-error shield of Wu "et al.", and a new e,d-shield. Shields can be automatically synthesized for all these specifications using the tool DCSynth. We give experimental results showing the performance of our shield synthesis tool in relation to previous work. We also compare the performance of the shields synthesized under diverse hard deviation constraints in terms of their expected deviation and the worst case burst-deviation latency.Comment: In Proceedings GandALF 2019, arXiv:1909.05979. arXiv admin note: text overlap with arXiv:1905.1115

DCSYNTH: Guided Reactive Synthesis with Soft Requirements

In reactive controller synthesis, a number of implementations (controllers) are possible for a given specification because of the incomplete nature of specification. To choose the most desirable one from the various options, we need to specify additional properties which can guide the synthesis. In this paper, We propose a technique for guided controller synthesis from regular requirements which are specified using an interval temporal logic QDDC. We find that QDDC is well suited for guided synthesis due to its superiority in dealing with both qualitative and quantitative specifications. Our framework allows specification consisting of both hard and soft requirements as QDDC formulas. We have also developed a method and a tool DCSynth, which computes a controller that invariantly satisfies the hard requirement and it optimally meets the soft requirement. The proposed technique is also useful in dealing with conflicting i.e., unrealizable requirements, by making some of them as soft requirements. Case studies are carried out to demonstrate the effectiveness of the soft requirement guided synthesis in obtaining high-quality controllers. The quality of the synthesized controllers is compared using metrics measuring both the guaranteed and the expected case behaviour of the controlled system. Tool DCSynth facilitates such comparison