scholarly journals Partitioning Techniques in LTLf Synthesis

2019 ◽  
Author(s):  
Lucas Martinelli Tabajara ◽  
Moshe Y. Vardi
Keyword(s):  
Model Checking ◽  
State Space ◽  
Reactive Systems ◽  
Computational Thinking ◽  
Standard Technique ◽  
The State ◽  
Problem Instance ◽  
State Spaces ◽  
Fundamental Limitations ◽  
Depth Analysis

Decomposition is a general principle in computational thinking, aiming at decomposing a problem instance into easier subproblems. Indeed, decomposing a transition system into a partitioned transition relation was critical to scaling BDD-based model checking to large state spaces. Since then, it has become a standard technique for dealing with related problems, such as Boolean synthesis. More recently, partitioning has begun to be explored in the synthesis of reactive systems. LTLf synthesis, a finite-horizon version of reactive synthesis with applications in areas such as robotics, seems like a promising candidate for partitioning techniques. After all, the state of the art is based on a BDD-based symbolic algorithm similar to those from model checking, and partitioning could be a potential solution to the current bottleneck of this approach, which is the construction of the state space. In this work, however, we expose fundamental limitations of partitioning that hinder its effective application to symbolic LTLf synthesis. We not only provide evidence for this fact through an extensive experimental evaluation, but also perform an in-depth analysis to identify the reason for these results. We trace the issue to an overall increase in the size of the explored state space, caused by an inability of partitioning to fully exploit state-space minimization, which has a crucial effect on performance. We conclude that more specialized decomposition techniques are needed for LTLf synthesis which take into account the effects of minimization.

On the Construction of Relativistic Representation Spaces in Functional Quantum Theory of the Nonlinear Spinor Field

1975 ◽  
Vol 30 (11) ◽  
pp. 1361-1371 ◽  
Author(s):  
H. Stumpf ◽  
K. Scheerer
Keyword(s):  
Quantum Theory ◽  
State Space ◽  
Functional State ◽  
Spinor Field ◽  
Function Spaces ◽  
The State ◽  
Nonlinear Spinor ◽  
Spinor Theory ◽  
State Spaces ◽  
Representation Spaces

Functional quantum theory is defined by an isomorphism of the state space H of a conventional quantum theory into an appropriate functional state space D It is a constructive approach to quantum theory in those cases where the state spaces H of physical eigenstates cannot be calculated explicitly like in nonlinear spinor field quantum theory. For the foundation of functional quantum theory appropriate functional state spaces have to be constructed which have to be representation spaces of the corresponding invariance groups. In this paper, this problem is treated for the spinor field. Using anticommuting source operator, it is shown that the construction problem of these spaces is tightly connected with the construction of appropriate relativistic function spaces. This is discussed in detail and explicit representations of the function spaces are given. Imposing no artificial restrictions it follows that the resulting functional spaces are indefinite. Physically the indefiniteness results from the inclusion of tachyon states. It is reasonable to assume a tight connection of these tachyon states with the ghost states introduced by Heisenberg for the regularization of the nonrenormalizable spinor theory

scholarly journals Model Checking Coloured Petri Nets - Exploiting Strongly Connected Components

1997 ◽  
Vol 26 (519) ◽  
Author(s):  
Allan Cheng ◽  
Søren Christensen ◽  
Kjeld Høyer Mortensen
Keyword(s):  
Model Checking ◽  
Petri Nets ◽  
The State ◽  
Connected Components ◽  
Coloured Petri Nets ◽  
Transition Systems ◽  
Strongly Connected Components ◽  
State Spaces ◽  
Labelled Transition Systems ◽  
Strongly Connected

In this paper we present a CTL-like logic which is interpreted over the state spaces of Coloured Petri Nets. The logic has been designed to express properties of both state and transition information. This is possible because the state spaces are labelled transition systems. We compare the expressiveness of our logic with CTL's. Then, we present a model checking algorithm which for efficiency reasons utilises strongly connected components and formula reduction rules. We present empirical results for non-trivial examples and compare the performance of our algorithm with that of Clarke, Emerson, and Sistla.

scholarly journals Modular State Space Analysis of Coloured Petri Nets

1997 ◽  
Vol 26 (524) ◽  
Author(s):  
Søren Christensen ◽  
Laure Petrucci
Keyword(s):  
Petri Nets ◽  
State Space ◽  
The State ◽  
Total System ◽  
Entire System ◽  
State Spaces ◽  
Space Analysis ◽  
State Space Analysis ◽  
Full State ◽  
Local Properties

<p>State Space Analysis is one of the most developed analysis methods for Petri Nets. The main problem of state space analysis is the size of the state spaces. Several ways to reduce it have been proposed but cannot yet handle industrial size systems.</p><p>Large models often consist of a set of modules. Local properties of each module can be checked separately, before checking the validity of the entire system. We want to avoid the construction of a single state space of the entire system.</p><p>When considering transition sharing, the behaviour of the total system can be capture by the state spaces of modules combined with a Synchronisation Graph. To verify that we do not lose information we show how the full state space can be conctructed.</p><p>We show how it is possible to determine usual Petri Nets properites, without unfolding to the ordinary state space.</p>

On Improving Model Checking of Time Petri Nets and Its Application to the Formal Verification

2021 ◽  
Vol 12 (4) ◽  
pp. 68-84
Author(s):  
Naima Jbeli ◽  
Zohra Sbai
Keyword(s):  
Model Checking ◽  
Petri Nets ◽  
Formal Verification ◽  
State Space ◽  
Order Reduction ◽  
The State ◽  
Timing Constraints ◽  
Time Petri Nets ◽  
Explicit Timing ◽  
Class Graph

Time Petri nets (TPN) are successfully used in the specification and analysis of distributed systems that involve explicit timing constraints. Especially, model checking TPN is a hopeful method for the formal verification of such complex systems. For this, it is promising to lean to the construction of an optimized version of the state space. The well-known methods of state space abstraction are SCG (state class graph) and ZBG (graph based on zones). For ZBG, a symbolic state represents the real evaluations of the clocks of the TPN; it is thus possible to directly check quantitative time properties. However, this method suffers from the state space explosion. To attenuate this problem, the authors propose in this paper to combine the ZBG approach with the partial order reduction technique based on stubborn set, leading thus to the proposal of a new state space abstraction called reduced zone-based graph (RZBG). The authors show via case studies the efficiency of the RZBG which is implemented and integrated within the 〖TPN-TCTL〗_h^∆ model checking in the model checker Romeo.

scholarly journals Card-Based Cryptography Meets Formal Verification

2021 ◽  
Author(s):  
Alexander Koch ◽  
Michael Schrempp ◽  
Michael Kirsten
Keyword(s):  
Model Checking ◽  
Formal Verification ◽  
State Space ◽  
Lower Bounds ◽  
Bounded Model Checking ◽  
Minimal Length ◽  
State Spaces ◽  
Bounded State ◽  
Sat Solver ◽  
General Translation

AbstractCard-based cryptography provides simple and practicable protocols for performing secure multi-party computation with just a deck of cards. For the sake of simplicity, this is often done using cards with only two symbols, e.g., $$\clubsuit $$ ♣ and $$\heartsuit $$ ♡ . Within this paper, we also target the setting where all cards carry distinct symbols, catering for use-cases with commonly available standard decks and a weaker indistinguishability assumption. As of yet, the literature provides for only three protocols and no proofs for non-trivial lower bounds on the number of cards. As such complex proofs (handling very large combinatorial state spaces) tend to be involved and error-prone, we propose using formal verification for finding protocols and proving lower bounds. In this paper, we employ the technique of software bounded model checking (SBMC), which reduces the problem to a bounded state space, which is automatically searched exhaustively using a SAT solver as a backend. Our contribution is threefold: (a) we identify two protocols for converting between different bit encodings with overlapping bases, and then show them to be card-minimal. This completes the picture of tight lower bounds on the number of cards with respect to runtime behavior and shuffle properties of conversion protocols. For computing AND, we show that there is no protocol with finite runtime using four cards with distinguishable symbols and fixed output encoding, and give a four-card protocol with an expected finite runtime using only random cuts. (b) We provide a general translation of proofs for lower bounds to a bounded model checking framework for automatically finding card- and run-minimal (i.e., the protocol has a run of minimal length) protocols and to give additional confidence in lower bounds. We apply this to validate our method and, as an example, confirm our new AND protocol to have its shortest run for protocols using this number of cards. (c) We extend our method to also handle the case of decks on symbols $$\clubsuit $$ ♣ and $$\heartsuit $$ ♡ , where we show run-minimality for two AND protocols from the literature.

scholarly journals Graph Inclusion and Matching Algorithms for Programs Manipulating Singly linked Heaps

2021 ◽  
Vol 9 (1) ◽  
pp. 30-37
Author(s):  
Muhsin H. Atto
Keyword(s):  
State Space ◽  
Data Structures ◽  
Graph Matching ◽  
The State ◽  
Memory Cells ◽  
Dynamic Data Structures ◽  
Dynamic Data ◽  
State Spaces ◽  
As Graph

Programs that manipulate heaps  such  as  singlylinked  lists,  doublylinked  lists,  skiplists,  and  treesare  ubiquitous,  and  hence ensuring their correctness is of utmost importance. Analysing correctness properties for such programs is not trivial since they induce dynamic data structures, leading to unbounded state spaces with intricate patterns. One approach that has been adopted to tackle this problem  is  the  use  of  symbolic  searching  techniques.  The  state  space  is  encoded  using  graphs  where  the  nodes represent memory cells, and the edges represent pointers between the cells. It is necessary to prune the search to avoid generating massive numbers of graphs, thus making the procedure unpractical. Pruning strategies are defined based on operations such as graph matching and inclusion. In this paper, a set of algorithms for performing these operations are presented. It is demonstrated that the proposed algorithms can handle typical graphs that arise in the verification of heap manipulating programs.

scholarly journals Sequential and Parallel Algorithms for the State Space Exploration

2016 ◽  
Vol 16 (1) ◽  
pp. 3-18 ◽  
Author(s):  
Lamia Allal ◽  
Ghalem Belalem ◽  
Philippe Dhaussy ◽  
Ciprian Teodorov
Keyword(s):  
Model Checking ◽  
Parallel Algorithms ◽  
State Space ◽  
Parallel Algorithm ◽  
Space Exploration ◽  
The State ◽  
State Space Exploration

Abstract In this article, we are interested in the exploration part of model checking which consists in traversing all the possible states of a system. We propose two approaches to exploration, parallel and sequential. We present a comparison between our parallel approach and the parallel algorithm proposed in SPIN.

scholarly journals Symbolic and Structural Model-Checking

2022 ◽  
Vol 183 (3-4) ◽  
pp. 319-342
Author(s):  
Yann Thierry-Mieg
Keyword(s):  
Model Checking ◽  
State Space ◽  
Structural Model ◽  
The State ◽  
Force Model ◽  
Deadlock Detection ◽  
Brute Force ◽  
General Applicability ◽  
State Space Explosion ◽  
Reachability Queries

Brute-force model-checking consists in exhaustive exploration of the state-space of a Petri net, and meets the dreaded state-space explosion problem. In contrast, this paper shows how to solve model-checking problems using a combination of techniques that stay in complexity proportional to the size of the net structure rather than to the state-space size. We combine an SMT based over-approximation to prove that some behaviors are unfeasible, an under-approximation using memory-less sampling of runs to find witness traces or counter-examples, and a set of structural reduction rules that can simplify both the system and the property. This approach was able to win by a clear margin the model-checking contest 2020 for reachability queries as well as deadlock detection, thus demonstrating the practical effectiveness and general applicability of the system of rules presented in this paper.

scholarly journals Distances between convex subsets of state spaces

1985 ◽  
Vol 32 (1) ◽  
pp. 109-117
Author(s):  
A.J. Ellis
Keyword(s):  
State Space ◽  
Hausdorff Space ◽  
Compact Convex ◽  
The State ◽  
Facial Structure ◽  
State Spaces ◽  
New Criteria ◽  
Uniformly Closed ◽  
Geometric Nature ◽  
Convex Subsets

Let L be a closed linear space of continuous real-valued functions, containing constants, on a compact Hausdorff space Ω. This paper gives some new criteria for a closed subset E of Ω to be an L-interpolation set, or more generally for L|E to be uniformly closed or simplicial, in terms of distances between certain compact convex subsets of the state space of L. These criteria involve the facial structure of the state space and hence are of a geometric nature. The results sharpen some standard results of Glicksberg.

Affine maps of state spaces and state spaces of K 0 groups

2013 ◽  
Vol 63 (6) ◽  
Author(s):  
Xiaosheng Zhu
Keyword(s):  
Abelian Group ◽  
State Space ◽  
Quotient Ring ◽  
The State ◽  
Noetherian Rings ◽  
Group Rings ◽  
State Spaces ◽  
Affine Map ◽  
Finite Algebras ◽  
Partially Ordered

AbstractLet φ be a homomorphism from the partially ordered abelian group (S, v) to the partially ordered abelian group (G, u) with φ(v) = u, where v and u are order units of S and G respectively. Then φ induces an affine map φ* from the state space St(G, u) to the state space St(S, v). Firstly, in this paper, we give some suitable conditions under which φ* is injective, surjective or bijective. Let R be a semilocal ring with the Jacobson radical J(R) and let π: R → R/J(R) be a canonical map. We discuss the affine map (K 0 π)*. Secondly, for a semiprime right Goldie ring R with the maximal right quotient ring Q, we consider the relations between St(R) and St(Q). Some results from [ALFARO, R.: State spaces, finite algebras, and skew group rings, J. Algebra 139 (1991), 134–154] and [GOODEARL, K. R.-WARFIELD, R. B., Jr.: State spaces of K 0 of noetherian rings, J. Algebra 71 (1981), 322–378] are extended.

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