What is modal logic

ing WS1S Systems to Verify Parameterized Networks . . . . . . . . . . . . 188 Kai Baukus, Saddek Bensalem, Yassine Lakhnech and Karsten Stahl FMona: A Tool for Expressing Validation Techniques over Infinite State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 J.-P. Bodeveix and M. Filali Transitive Closures of Regular Relations for Verifying Infinite-State Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Bengt Jonsson and Marcus Nilsson Diagnostic and Test Generation Using Static Analysis to Improve Automatic Test Generation . . . . . . . . . . . . . 235 Marius Bozga, Jean-Claude Fernandez and Lucian Ghirvu Efficient Diagnostic Generation for Boolean Equation Systems . . . . . . . . . . . . 251 Radu Mateescu Efficient Model-Checking Compositional State Space Generation with Partial Order Reductions for Asynchronous Communicating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Jean-Pierre Krimm and Laurent Mounier Checking for CFFD-Preorder with Tester Processes . . . . . . . . . . . . . . . . . . . . . . . 283 Juhana Helovuo and Antti Valmari Fair Bisimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Thomas A. Henzinger and Sriram K. Rajamani Integrating Low Level Symmetries into Reachability Analysis . . . . . . . . . . . . . 315 Karsten Schmidt Model-Checking Tools Model Checking Support for the ASM High-Level Language . . . . . . . . . . . . . . 331 Giuseppe Del Castillo and Kirsten Winter Table of

Tools and Algorithms for the Construction and Analysis of Systems. Proc. TACAS 2009

http://user.it.uu.se/~jarst116/slides/week4.pdf

Graph-Based Algorithms for Boolean Function Manipulation

We consider a fully SAT-based method of unbounded symbolic model checking based on computing Craig interpolants. In benchmark studies using a set of large industrial circuit verification instances, this method is greatly more efficient than BDD-based symbolic model checking, and compares favorably to some recent SAT-based model checking methods on positive instances.

Interpolation and SAT-based model checking

We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

Fourier meets Möbius: fast subset convolution

In this paper we study a version of constructive linear-time temporal logic (LTL) with the ''next'' temporal operator The logic is originally due to Davies, who has shown that the proof system of the logic corresponds to a type system for binding-time analysis via the Curry-Howard isomorphism However, he did not investigate the logic itself in detail; he has proved only that the logic augmented with negation and classical reasoning is equivalent to (the ''next'' fragment of) the standard formulation of classical linear-time temporal logic We give natural deduction, sequent calculus and Hilbert-style proof systems for constructive LTL with conjunction, disjunction and falsehood, and show that the sequent calculus enjoys cut elimination Moreover, we also consider Kripke semantics and prove soundness and completeness One distinguishing feature of this logic is that distributivity of the ''next'' operator over disjunction ''@?(A@?B)@?@?A@?@?B'' is rejected in view of a type-theoretic interpretation

http://www.fos.kuis.kyoto-u.ac.jp/~igarashi/papers/pdf/cltl-full.pdf

Constructive linear-time temporal logic: Proof systems and Kripke semantics

We investigate semantics for an intuitionistic modal logic in which the “possibility” modality does not distribute over disjunction. In particular, the main aim of this paper is to study such intuitionistic modal logic as a variant of classical non-normal modal logic. We first give a neighborhood semantics together with a sound and complete axiomatization. Next, we study relationships between our approach and the relational (Kripke-style) semantics considered in the literature. It is shown that a relational model can be represented as a neighborhood model, and the converse direction holds under a slight restriction. Also, by considering degenerate cases of neighborhood and relational semantics, we demonstrate that a certain classical monotone modal logic has relational semantics, and can be embedded into a classical normal bimodal logic.

Relational and neighborhood semantics for intuitionistic modal logic

We investigate several intuitionistic modal logics (IMLs), mainly from semantical viewpoint. In these decades, in the area of type theory for programming language, various type systems corresponding (via Curry-Howard correspondence) to IMLs have been considered. However, IMLs arising from such a context are not extensively studied before. We try to understand the logical meaning of such IMLs, and what kind of structure are behind IMLs motivated from computation. The technical materials appearing in the thesis are divided into three parts: (1) proof systems and Kripke semantics for an intuitionistic version of linear-time temporal logic (LTL), which corresponds to a lambda-calculus for binding-time analysis; (2) correspondence theory in a certain Kripke semantics for IML; and (3) a new representation of existing Kripke semantics for IML by using neighborhood semantics (also called Scott-Montague semantics), and investigations of the relationship between them. Through these technical developments, the following observation arises: there are actually two possible meanings of the assertion “a formula A is true at a possible world x.” This explains the difference between the traditional modal logic and IMLs considered in this thesis, and several difficulties that emerge in establishing meta-theoretical results such as completeness and cut-elimination. In the thesis we also show that the existing Kripke model can be transformed into an equivalent neighborhood model, and each neighborhood model satisfying a certain condition can be transformed into an equivalent Kripke model. This mutual translation clearly explains how “non-standard” behavior of modalities (captured well by neighborhood semantics) has been simulated in the existing Kripke semantics.

/pdf/semantical-study-of-intuitionistic-modal-logics-14pejifj4l.pdf

Semantical study of intuitionistic modal logics

We study a Hoare Logic to reason about GPU kernels, which are parallel programs executed on GPUs. We consider the SIMT (Single Instruction Multiple Threads) execution model, in which multiple threads execute in lockstep (that is, execute the same instruction at a time). When control branches both branches are executed sequentially but during the execution of each branch only those threads that take it are enabled; after the control converges, all threads are enabled and execute in lockstep again. In this paper we adapt Hoare Logic to the SIMT setting, by adding an extra component representing the set of enabled threads to the usual Hoare triples. It turns out that soundness and relative completeness do not hold for all programs; a difficulty arises from the fact that one thread can invalidate the loop termination condition of another thread through shared memory. We overcome this difficulty by identifying an appropriate class of programs for which soundness and relative completeness hold.

/pdf/a-hoare-logic-for-simt-programs-3lc4xkdjdk.pdf

A Hoare Logic for SIMT Programs

We study a Hoare Logic to reason about parallel programs executed on graphics processing units (GPUs), called GPU kernels. During the execution of GPU kernels, multiple threads execute in lockstep, that is, execute the same instruction simultaneously. When the control branches, the two branches are executed sequentially, but during the execution of each branch only those threads that take it are enabled; after the control converges, all the threads are enabled and again execute in lockstep. In this article, we first consider a semantics in which all threads execute in lockstep (this semantics simplifies the actual execution model of GPUs) and adapt Hoare Logic to this setting by augmenting the usual Hoare triples with an additional component representing the set of enabled threads. It is determined that the soundness and relative completeness of the logic do not hold for all programs; a difficulty arises from the fact that one thread can invalidate the loop termination condition of another thread through shared memory. We overcome this difficulty by identifying an appropriate class of programs for which the soundness and relative completeness hold. Additionally, we discuss thread interleaving, which is present in the actual execution of GPUs but not in the lockstep semantics mentioned above. We show that if a program is race free, then the lockstep and interleaving semantics produce the same result. This implies that our logic is sound and relatively complete for race-free programs, even if the thread interleaving is taken into account.

Kensuke Kojima

Papers

Constructive linear-time temporal logic: Proof systems and Kripke semantics

Relational and neighborhood semantics for intuitionistic modal logic

Semantical study of intuitionistic modal logics

A Hoare Logic for SIMT Programs

A Hoare Logic for GPU Kernels