Satisfiability Research Articles

Reactive synthesis from interval temporal logic specifications

In this paper, we deal with the synthesis problem for Halpern and Shoham's interval temporal logic HS extended with an equivalence relation ∼ over time points (HS▪ for short). The definition of the problem is analogous to that for MSO(ω,<). Given an HS▪ formula φ and a finite set Σφ⋄ of proposition letters and temporal requests, it consists of determining, whether or not, for all possible valuations of elements in Σφ⋄ in every interval structure, there is a valuation of the remaining proposition letters and temporal requests such that the resulting structure is a model for φ.We focus on the decidability and complexity of the problem for some meaningful fragments of HS▪, whose modalities are drawn from the set {A(meets),A¯(metby),B(begunby),B¯(begins)} interpreted over finite linear orders and N. We prove that, over finite linear orders, the problem is decidable (Ackermann-hard) for ▪ and undecidable for AA¯BB¯. Moreover, we show that if we replace finite linear orders by N, then it becomes undecidable even for ABB¯. Finally, we study the generalization of ▪ to ▪, where k is the number of distinct equivalence relations. Despite the fact that the satisfiability problem for ▪, with k>1, over finite linear orders, is already undecidable, we prove that, under a natural semantic restriction (refinement condition), the synthesis problem turns out to be decidable.

Analysis and Transformation of Constrained Horn Clauses for Program Verification

AbstractThis paper surveys recent work on applying analysis and transformation techniques that originate in the field of constraint logic programming (CLP) to the problem of verifying software systems. We present specialization-based techniques for translating verification problems for different programming languages, and in general software systems, into satisfiability problems for constrained Horn clauses (CHCs), a term that has become popular in the verification field to refer to CLP programs. Then, we describe static analysis techniques for CHCs that may be used for inferring relevant program properties, such as loop invariants. We also give an overview of some transformation techniques based on specialization and fold/unfold rules, which are useful for improving the effectiveness of CHC satisfiability tools. Finally, we discuss future developments in applying these techniques.

An Parallel FPGA SAT Solver Based on Multi‐Thread and Pipeline

The Boolean Satisfiability (SAT) problem is the key problem in computer theory and application. A parallel multi-thread SAT solver named pprobSAT+ on a configurable hardware is proposed. In the algorithm, multithreads are executed simultaneously to hide the circuit stagnate. In order to improve the working frequency and throughput of the SAT solver, the deep pipeline strategy is adopted. When all data stored in block random access memory of the field programmable gate array, the solver can achieve maximum performance. If partial data are stored in the external memory, the size of the problem instances the SAT solver can be greatly improved. The experimental results show that the speedup of three-thread SAT solver is approximately 2.4 times with single thread, and shows that the pprobSAT+ have achieved substantial improvement while a solution is found.

A Workflow Criticality-Based Approach to Bypass the Workflow Satisfiability Problem

Workflow management systems are very important for any organization to manage and model complex business processes. However, significant work is needed to keep a workflow resilient and secure. Therefore, organizations apply a strict security policy and enforce access control constraints. As a result, the number of available and authorized users for the workflow execution decreases drastically. Thus, in many cases, such a situation leads to a workflow deadlock situation, where there no available authorized user-task assignments for critical tasks to accomplish the workflow execution. In the literature, this problem has gained interest of security researchers in the recent years, and is known as the workflow satisfiability problem (WSP). In this paper, we propose a new approach to bypass the WSP and to ensure workflow resiliency and security. For this purpose, we define workflow criticality, which can be used as a metric during run-time to prevent WSP. We believe that the workflow criticality value will help workflow managers to make decisions and start a mitigation solution in case of a critical workflow. Moreover, we propose a delegation process algorithm (DP) as a mitigation solution that uses workflow instance criticality, delegation, and priority concepts to find authorized and suitable users to perform the critical task with low-security risks.

A New Method for 3-Satisfiability Problem Solving Space Structure on Structural Entropy

Analyzing the solution space structure and evolution of 3-satisfiability (3-SAT) problem is an important way to study the difficulty of the solving satisfiability (SAT) problem. However, there is no unified analysis model for the spatial structure and evolution of solutions under different constraint densities. The analysis of different phase transition points and solution regions is based on different metric analysis models. The solution space of 3-SAT problem is obtained by planting strategy and belief propagation. According to the distribution of the influence of frozen variables on the solution, a label propagation algorithm based on planting strategy is proposed, is used to find the solution cluster, and then the structure entropy is used to measure its structure information. The structure entropy analysis model of 3-SAT problem solution space is established, and the unified analysis framework of solution space evolution and satisfiability phase transition is given. The experimental results show that the model is effective and can accurately analyze the evolution process of solution space and satisfiability phase transition, and verify the accuracy of interference phase transition point threshold predicted by long-range frustration theory.

Tractability Frontier of Data Complexity in Team Semantics

We study the data complexity of model checking for logics with team semantics. We focus on dependence, inclusion, and independence logic formulas under both strict and lax team semantics. Our results delineate a clear tractability/intractability frontiers in data complexity of both quantifier-free and quantified formulas for each of the logics. For inclusion logic under the lax semantics, we reduce the model-checking problem to the satisfiability problem of so-called dual-Horn Boolean formulas. Via this reduction, we give an alternative proof for the known result that the data complexity of inclusion logic is in PTIME.

Stable Model Semantics for Guarded Existential Rules and Description Logics: Decidability and Complexity

This work investigates the decidability and complexity of database query answering under guarded existential rules with nonmonotonic negation according to the classical stable model semantics. In this setting, existential quantification is interpreted via Skolem functions, and the unique name assumption is adopted. As a first result, we show the decidability of answering first-order queries based on such rules by a translation into the satisfiability problem for guarded second-order formulas having the tree-model property. To obtain precise complexity results for unions of conjunctive queries, we transform the original problem in polynomial time into an intermediate problem that is easier to analyze: query answering for guarded disjunctive existential rules with stratified negation. We obtain precise bounds for the general setting and for various restricted settings. We also consider extensions of the original formalism with negative constraints, keys, and the possibility of negated atoms in queries. Finally, we show how the above results can be used to provide decidability and complexity results for a natural adaptation of the stable model semantics to description logics such as ELHI and the DL-Lite family.

Resolvable Networks—A Graphical Tool for Representing and Solving SAT

In this paper, we introduce the notion of resolvable networks. A resolvable network is a digraph of subnetworks, where subnetworks may overlap, and the inner structure of subnetworks are not interesting from the viewpoint of the network. There are two special subnetworks, Source and Sink, with the following properties: there is no incoming edge to Source, and there is no outgoing edge from Sink. Any resolvable network can be represented by a satisfiability problem in Boolean logic (shortly, SAT problem), and any SAT problem can be represented by a resolvable network. Because of that, the resolution operation is valid also for resolvable networks. We can use resolution to find out or refine the inner structure of subnetworks. We give also a pessimistic and an optimistic interpretation of subnetworks. In the pessimistic case, we assume that inside a subnetwork, all communication possibilities are represented as part of the resolvable network. In the optimistic case, we assume that each subnetwork is strongly connected. We show that any SAT problem can be visualized using the pessimistic interpretation. We show that transitivity is very limited in the pessimistic interpretation, and in this case, transitivity corresponds to resolution of clauses. In the optimistic interpretation of subnetworks, we have transitivity without any further condition, but not all SAT problems can be represented in this case; however, any such network can be represented as a SAT problem. The newly introduced graphical concept allows to use terminology and tools from directed graphs in the field of SAT and also to give graphical representations of various concepts of satisfiability problems. A resolvable network is also a suitable data structure to study, for example, wireless sensor networks. The visualization power of resolvable networks is demonstrated on some pigeon hole SAT problems. Another important application field could be modeling the communication network of an information bank. Here, a subnetwork represents a dataset of a user which is secured by a proxy. Any communication should be done through the proxy, and this constraint can be checked using our model.

Power aware floorplanning in multiple supply voltage domain

SummaryFloorplanning in this nano‐scale era involves consideration of various other objectives apart from minimizing wirelength in a fixed outlined region. In recent era of electronic devices, reducing power consumption is an important objective which can be achieved by multiple supply voltage island aware floorplanning. Optimizing power routing resource to simplify power planning while reducing voltage drop or IR drop (represented by maximum power length) is the major goal of the proposed work. Moreover, it has been observed that locations of blocks and power pads are important considerations for IR drop optimization. The proposed work hence concentrates on generating efficient floorplan with respect to wirelength, power routing resource, and IR drop. The proposed technique uses effective heuristics along with a SAT (Boolean satisfiability)‐based framework to generate efficient floorplan considering all factors. The experimental validation depicts good results with respect to power resource, wirelength, and voltage drop in a fixed outline framework.

SAT encodings for Pseudo-Boolean constraints together with at-most-one constraints

When solving a combinatorial problem using propositional satisfiability (SAT), the encoding of the problem is of vital importance. We study encodings of Pseudo-Boolean (PB) constraints, a common type of arithmetic constraint that appears in a wide variety of combinatorial problems such as timetabling, scheduling, and resource allocation. In some cases PB constraints occur together with at-most-one (AMO) constraints over subsets of their variables (forming PB(AMO) constraints). Recent work has shown that taking account of AMOs when encoding PB constraints using decision diagrams can produce a dramatic improvement in solver efficiency. In this paper we extend the approach to other state-of-the-art encodings of PB constraints, developing several new encodings for PB(AMO) constraints. Also, we present a more compact and efficient version of the popular Generalized Totalizer encoding, named Reduced Generalized Totalizer. This new encoding is also adapted for PB(AMO) constraints for a further gain. Our experiments show that the encodings of PB(AMO) constraints can be substantially smaller than those of PB constraints. PB(AMO) encodings allow many more instances to be solved within a time limit, and solving time is improved by more than one order of magnitude in some cases. We also observed that there is no single overall winner among the considered encodings, but efficiency of each encoding may depend on PB(AMO) characteristics such as the magnitude of coefficient values.

Modal context restriction for multiagent BDI logics

We present and discuss a novel language restriction for modal logics for multiagent systems, called modal context restriction, that reduces the complexity of the satisfiability problem from EXPTIME complete to NPTIME complete. We focus on BDI multimodal logics that contain fix-point modalities like common beliefs and mutual intentions together with realism and introspection axioms. We show how this combination of modalities and axioms affects complexity of the satisfiability problem and how it can be reduced by restricting the modal context of formulas.

ScanSAT: Unlocking Static and Dynamic Scan Obfuscation

While financially advantageous, outsourcing key steps, such as testing, to potentially untrusted Outsourced Assembly and Test (OSAT) companies may pose a risk of compromising on-chip assets. Obfuscation of scan chains is a technique that hides the actual scan data from the untrusted testers; logic inserted between the scan cells, driven by a secret key, hides the transformation functions that map the scan-in stimulus (scan-out response) and the delivered scan pattern (captured response). While static scan obfuscation utilizes the same secret key, and thus, the same secret transformation functions throughout the lifetime of the chip, dynamic scan obfuscation updates the key periodically. In this paper, we propose ScanSAT: an attack that transforms a scan obfuscated circuit to its logic-locked version and applies the Boolean satisfiability (SAT) based attack, thereby extracting the secret key. We implement our attack, apply on representative scan obfuscation techniques, and show that ScanSAT can break both static and dynamic scan obfuscation schemes with 100 percent success rate. Moreover, ScanSAT is effective even for large key sizes and in the presence of scan compression.

SMT-Based Verification of Hybrid Systems

Hybrid automata networks (HAN) are a powerful formalism to model complex embedded systems. In this paper, we survey the recent advances in the application of Satisfiability Modulo Theories (SMT) to the analysis of HAN. SMT can be seen as an extended form of Boolean satisfiability (SAT), where literals are interpreted with respect to a background theory (e.g. linear arithmetic). HAN can be symbolically represented by means of SMT formulae, and analyzed by generalizing to the case of SMT the traditional model checking algorithms based on SAT.

Study of Fine-grained Nested Parallelism in CDCL SAT Solvers

Boolean satisfiability (SAT) is an important performance-hungry problem with applications in many problem domains. However, most work on parallelizing SAT solvers has focused on coarse-grained, mostly embarrassing, parallelism. Here, we study fine-grained parallelism that can speed up existing sequential SAT solvers, which all happen to be of the so-called Conflict-Directed Clause Learning variety. We show the potential for speedups of up to 382× across a variety of problem instances. We hope that these results will stimulate future research, particularly with respect to a computer architecture open problem we present.

Belief Functions on Distributive Lattices

The Dempster-Shafer theory of belief functions is an important approach to deal with uncertainty in AI.In the theory, belief functions are defined on Boolean algebras of events. In many applications of belief functions in real world problems, however, the objects that we manipulateis no more a Boolean algebra but a distributive lattice. In this paper, we extend the Dempster-Shafer theory to the setting of distributive lattices, which has a mathematical theory as attractive as in that of Boolean algebras.Moreover, we apply this more general theory to a simple epistemic logic the first-degree-entailment fragment of relevance logic R, provide a sound and complete axiomatization for reasoning about belief functions for this logic and show that the complexity of the satisfiability problem of a belief formula with respect to the class of the corresponding Dempster-Shafer structures is NP-complete.

Adding the Relation Meets to the Temporal Logic of Prefixes and Infixes makes it EXPSPACE-Complete

The choice of the right trade-off between expressiveness and complexity is the main issue in interval temporal logic. In their seminal paper, Halpern and Shoham showed that the satisfiability problem for HS (the temporal logic of Allen's relations) is highly undecidable over any reasonable class of linear orders. In order to recover decidability, one can restrict the set of temporal modalities and/or the class of models. In the following, we focus on the satisfiability problem for HS fragments under the homogeneity assumption, according to which any proposition letter holds over an interval if only if it holds at all its points. The problem for full HS with homogeneity has been shown to be non-elementarily decidable, but its only known lower bound is EXPSPACE (in fact, EXPSPACE-hardness has been shown for the logic of prefixes and suffixes BE, which is a very small fragment of it. The logic of prefixes and infixes BD has been recently shown to be PSPACE-complete. In this paper, we prove that the addition of the Allen relation Meets to BD makes it EXPSPACE-complete.

Abduction of trap invariants in parameterized systems

In a previous paper we have presented a CEGAR approach for the verification of parameterized systems with an arbitrary number of processes organized in an array or a ring. The technique is based on the iterative computation of parameterized invariants, i.e., infinite families of invariants for the infinitely many instances of the system. Safety properties are proved by checking that every global configuration of the system satisfying all parameterized invariants also satisfies the property; we have shown that this check can be reduced to the satisfiability problem for Monadic Second Order on words, which is decidable. A strong limitation of the approach is that processes can only have a fixed number of variables with a fixed finite range. In particular, they cannot use variables with range [0,N-1], where N is the number of processes, which appear in many standard distributed algorithms. In this paper, we extend our technique to this case. While conducting the check whether a safety property is inductive assuming a computed set of invariants becomes undecidable, we show how to reduce it to checking satisfiability of a first-order formula. We report on experiments showing that automatic first-order theorem provers can still perform this check for a collection of non-trivial examples. Additionally, we can give small sets of readable invariants for these checks.

A new branch-and-filter exact algorithm for binary constraint satisfaction problems

A binary constraint satisfaction problem (BCSP) consists in determining an assignment of values to variables that is compatible with a set of constraints. The problem is called binary because the constraints involve only pairs of variables. The BCSP is a cornerstone problem in Constraint Programming (CP), appearing in a very wide range of real-world applications. In this work, we develop a new exact algorithm which effectively solves the BCSP by reformulating it as a k-clique problem on the underlying microstructure graph representation. Our new algorithm exploits the cutting-edge branching scheme of the state-of-the-art maximum clique algorithms combined with two filtering phases in which the domains of the variables are reduced. Our filtering phases are based on colouring techniques and on heuristically solving an associated boolean satisfiability (SAT) problem. In addition, the algorithm initialization phase performs a reordering of the microstructure graph vertices that produces an often easier reformulation to solve. We carry out an extensive computational campaign on a benchmark of almost 2000 instances, encompassing numerous real and synthetic problems from the literature. The performance of the new algorithm is compared against four SAT-based solvers and three general purpose CP solvers. Our tests reveal that the new algorithm significantly outperforms all the others in several classes of BCSP instances.

Mining Closed High Utility Itemsets based on Propositional Satisfiability

A high utility itemset mining problem is the question of recognizing a set of items that have utility values greater than a given user utility threshold. This generalization of the classical problem of frequent itemset mining is a useful and well-known task in data analysis and data mining, since it is used in a wide range of real applications. In this paper, we first propose to use symbolic Artificial Intelligence for computing the set of all closed high utility itemsets from transaction databases. Our approach is based on reduction to enumeration problems of propositional satisfiability. Then, we enhance the efficiency of our SAT-based approach using the weighted clique cover problem. After that, in order to improve scalability, a decomposition technique is applied to derive smaller and independent sub-problems in order to capture all the closed high utility itemsets. Clearly, our SAT-based encoding can be constantly enhanced by integrating the last improvements in powerful SAT solvers and models enumeration algorithms. Finally, through empirical evaluations on different real-world datasets, we demonstrate that the proposed approach is very competitive with state-of-the-art specialized algorithms for high utility itemsets mining, while being sufficiently flexible to take into account additional constraints to finding closed high utility itemsets.

Design of Hardware IP Core Security Protection Based on Multi-Level Co-obfuscation

Most of the reported hardware obfuscations are single-level ones focusing on physical level, logical level or behavior level, in which the lack of synergy among different levels commonly results in limited security performance. Based on study of the relationships among circuit layout, logic and states transition, a multi-level co-obfuscation scheme is proposed to protect hardware IP cores. In bottom-up collaborative confusion design, dummy vias are introduced into camouflage gates layout to perform physical-logic obfuscation, and via-PUF (Physical Unclonable Fuction) are utilized in state transition control to realize physical-behavior obfuscation. Then, in top-down collaborative obfuscation design, logic locks are used to perform behavior-logic obfuscation, and parallel-branch obfuscation wire technique is designed to complete the behavior-physical confusion. Finally, a substitution algorithm of the obfuscation gates into the circuit’s netlist is proposed, and the three-level cooperative obfuscation is realized to achieve IP core security protection. ISCAS-89 Benchmarks and a typical cryptogram algorithm are used to verify the correctness and efficiency of the proposed IP core protection scheme. The test results show that under TSMC 65nm process, the average area cost percentage of the proposed co-obfuscation in large-scale circuits is 11.7%, the average power consumption accounts for 5.1%, The difference of register toggle between correct and wrong keys is less than 10%, and the proposed scheme can effectively resist violence attack, reverse engineering, boolean SATisfiability (SAT) attack.