LTL f synthesis is the process of finding a strategy that satisfies a linear temporal specification over finite traces. An existing solution to this problem relies on a reduction to a DFA game. In this paper, we propose a symbolic framework for LTL f synthesis based on this technique, by performing the computation over a representation of the DFA as a boolean formula rather than as an explicit graph. This approach enables strategy generation by utilizing the mechanism of boolean synthesis. We implement this symbolic synthesis method in a tool called Syft, and demonstrate by experiments on scalable benchmarks that the symbolic approach scales better than the explicit one.

Temporal synthesis is the automated design of a system that interacts with an environment, using the declarative specification of the system's behavior. A popular language for providing such a specification is Linear Temporal Logic, or LTL. LTL synthesis in the general case has remained, however, a hard problem to solve in practice. Because of this, many works have focused on developing synthesis procedures for specific fragments of LTL, with an easier synthesis problem. In this work, we focus on Safety LTL, defined here to be the Until-free fragment of LTL in Negation Normal Form (NNF), and shown to express a fragment of safe LTL formulas. The intrinsic motivation for this fragment is the observation that in many cases it is not enough to say that something "good" will eventually happen, we need to say by when it will happen. We show here that Safety LTL synthesis is significantly simpler algorithmically than LTL synthesis. We exploit this simplicity in two ways, first by describing an explicit approach based on a reduction to Horn-SAT, which can be solved in linear time in the size of the game graph, and then through an efficient symbolic construction, allowing a BDD-based symbolic approach which significantly outperforms extant LTL-synthesis tools.

We present here a new explicit reasoning framework for linear temporal logic (LTL), which is built on top of propositional satisfiability (SAT) solving. As a proof-of-concept of this framework, we describe a new LTL satisfiability algorithm. We implemented the algorithm in a tool, Aalta v2.0, which is built on top of the Minisat SAT solver. We tested the effectiveness of this approach by demonstrating that Aalta v2.0 significantly outperforms all existing LTL satisfiability solvers. arXiv:1507.02519v3 [cs.LO] 4 Dec 2015 the capabilities of modern SAT solvers [19] as well as new SAT-based modelchecking algorithms [1, 3], while progress in explicit temporal reasoning is slower and does not fully leverage modern SAT solving. (It should be noted that several LTL satisfiability solvers, including Aalta [17] and ls4 [30] do employ SAT solvers, but they do so as an aid to the main reasoning engine, rather than serve as the main reasoning engine.) Our main aim in this paper is to study how SAT solving can be fully leveraged in explicit temporal reasoning. The key intuition is that explicit temporal reasoning consists of construction of states and transitions, subject to temporal constraints. Such temporal constraints can be reduced to a sequence of Boolean constraints, which enables the application of SAT solving. This idea underlies the complexitytheoretic analysis in [33], and has been explored in the context of modal logic [12], but not yet in the context of explicit temporal reasoning. Our belief is that SAT solving would prove to be superior to tableau in that context. We describe in this paper a general framework for SAT-based explicit temporal reasoning. The crux of our approach is a construction of temporal transition system that is based on SAT-solving rather than tableau to construct states and transitions. The obtained transition system can be used for LTL-satisfiability solving, LTL-to-automata translation, and runtime-monitor construction. As proof of concept for the new framework, we use it to develop a SAT-based algorithm for LTL-satisfiability checking. We also propose several heuristics to speed up the checking by leveraging SAT solvers. We implemented the algorithm and heuristics in an LTL-satisfiability solver Aalta v2.0. To evaluate its performance, we compared it against Aalta, the existing best-of-breed LTLsatisfiability solver [18,17], which is tableau-based. We also compare it against NuXmv, a symbolic LTL-satisfiability solver that is based on cutting-edge SATbased model-checking algorithms [1,3], which outperforms Aalta. We show that our explicit SAT-based LTL-satisfiability solver outperforms both. In summary, the contributions in this paper are as follows:-We propose a SAT-based explicit LTL-reasoning framework.-We show a successful application of the framework to LTL-satisfiability checking, by designing a novel algorithm and efficient heuristics. -We compare our new framework for LTL-satisfiability checking with existing approaches. The experimental results demonstrate that our tool signific...

This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. By relating a contemporary finite-dimensional linear algebra problem to a classical problem on infinite-dimensional linear integral operator, the paper shows how the spectral distribution of a kernel matrix relates to the smoothness of the underlying kernel function.The asymptotic behaviour of the eigenvalues of a infinite-dimensional kernel operator are studied from a perspective of low rank approximation-approximating an integral operator in terms of Fourier series or Chebyshev series truncations. Further, we study how the spectral distribution of interpolation matrices of an infinite smooth kernel with flat limit depends on the geometric property of the underlying interpolation points. In particularly, the paper discusses the analytic eigenvalue distribution of Gaussian kernels, which have important application on stable computing of Gaussian radial basis functions.

In synthesis, assumptions are constraints on the environment that rule out certain environment behaviors. A key observation here is that even if we consider systems with LTLƒ goals on finite traces, environment assumptions need to be expressed over infinite traces, since accomplishing the agent goals may require an unbounded number of environment action. To solve synthesis with respect to finite-trace LTLƒ goals under infinite-trace assumptions, we could reduce the problem to LTL synthesis. Unfortunately, while synthesis in LTLƒ and in LTL have the same worst-case complexity (both 2EXPTIME-complete), the algorithms available for LTL synthesis are much more difficult in practice than those for LTLƒ synthesis. In this work we show that in interesting cases we can avoid such a detour to LTL synthesis and keep the simplicity of LTLƒ synthesis. Specifically, we develop a BDD-based fixpoint-based technique for handling basic forms of fairness and of stability assumptions. We show, empirically, that this technique performs much better than standard LTL synthesis.

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Abstract-Formal-verification techniques, such as model checking, are becoming popular in hardware design. SAT-based model checking techniques, such as IC3/PDR, have gained a significant success in the hardware industry. In this paper, we present a new framework for SAT-based safety model checking, named Complementary Approximate Reachability (CAR). CAR is based on standard reachability analysis, but instead of maintaining a single sequence of reachable-state sets, CAR maintains two sequences of over-and under-approximate reachable-state sets, checking safety and unsafety at the same time. To construct the two sequences, CAR uses standard Boolean-reasoning algorithms, based on satisfiability solving, one to find a satisfying cube of a satisfiable Boolean formula, and one to provide a minimal unsatisfiable core of an unsatisfiable Boolean formula. We applied CAR to 548 hardware model-checking instances, and compared its performance with IC3/PDR. Our results show that CAR is able to solve 42 instances that cannot be solved by IC3/PDR. When evaluated against a portfolio that includes IC3/PDR and other approaches, CAR is able to solve 21 instances that the other approaches cannot solve. We conclude that CAR should be considered as a valuable member of any algorithmic portfolio for safety model checking.

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