Pushing the envelope of Optimization Modulo Theories with Linear-Arithmetic Cost Functions

In the last decade we have witnessed an impressive progress in the expressiveness and efficiency of Satisfiability Modulo Theories (SMT) solving techniques. This has brought previously-intractable problems at the reach of state-of-the-art SMT solvers, in particular in the domain of SW and HW verification. Many SMT-encodable problems of interest, however, require also the capability of finding models that are optimal wrt. some cost functions. In previous work, namely "Optimization Modulo Theory with Linear Rational Cost Functions -- OMT(LAR U T )", we have leveraged SMT solving to handle the minimization of cost functions on linear arithmetic over the rationals, by means of a combination of SMT and LP minimization techniques. In this paper we push the envelope of our OMT approach along three directions: first, we extend it to work also with linear arithmetic on the mixed integer/rational domain, by means of a combination of SMT, LP and ILP minimization techniques; second, we develop a multi-objective version of OMT, so that to handle many cost functions simultaneously; third, we develop an incremental version of OMT, so that to exploit the incrementality of some OMT-encodable problems. An empirical evaluation performed on OMT-encoded verification problems demonstrates the usefulness and efficiency of these extensions.Comment: A slightly-shorter version of this paper is published at TACAS 2015 conferenc

Optimization Modulo Theories with Linear Rational Costs

In the contexts of automated reasoning (AR) and formal verification (FV), important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, little work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any previous work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In the work described in this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of linear rational cost functions, combining SMT with standard minimization techniques. We have implemented the procedures within the MathSAT SMT solver. Due to the absence of competitors in the AR, FV and SMT domains, we have experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic, currently under revision. arXiv admin note: text overlap with arXiv:1202.140

Optimization in SMT with LA(Q) Cost Functions

In the contexts of automated reasoning and formal verification, important decision problems are effectively encoded into Satisfiability Modulo Theories (SMT). In the last decade efficient SMT solvers have been developed for several theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors). Surprisingly, very few work has been done to extend SMT to deal with optimization problems; in particular, we are not aware of any work on SMT solvers able to produce solutions which minimize cost functions over arithmetical variables. This is unfortunate, since some problems of interest require this functionality. In this paper we start filling this gap. We present and discuss two general procedures for leveraging SMT to handle the minimization of LA(Q) cost functions, combining SMT with standard minimization techniques. We have implemented the proposed approach within the MathSAT SMT solver. Due to the lack of competitors in AR and SMT domains, we experimentally evaluated our implementation against state-of-the-art tools for the domain of linear generalized disjunctive programming (LGDP), which is closest in spirit to our domain, on sets of problems which have been previously proposed as benchmarks for the latter tools. The results show that our tool is very competitive with, and often outperforms, these tools on these problems, clearly demonstrating the potential of the approach.Comment: A shorter version is currently under submissio

Controlled and effective interpolation

Model checking is a well established technique to verify systems, exhaustively and automatically. The state space explosion, known as the main difficulty in model checking scalability, has been successfully approached by symbolic model checking which represents programs using logic, usually at the propositional or first order theories level. Craig interpolation is one of the most successful abstraction techniques used in symbolic methods. Interpolants can be efficiently generated from proofs of unsatisfiability, and have been used as means of over-approximation to generate inductive invariants, refinement predicates, and function summaries. However, interpolation is still not fully understood. For several theories it is only possible to generate one interpolant, giving the interpolation-based application no chance of further optimization via interpolation. For the theories that have interpolation systems that are able to generate different interpolants, it is not understood what makes one interpolant better than another, and how to generate the most suitable ones for a particular verification task. The goal of this thesis is to address the problems of how to generate multiple interpolants for theories that still lack this flexibility in their interpolation algorithms, and how to aim at good interpolants. This thesis extends the state-of-the-art by introducing novel interpolation frameworks for different theories. For propositional logic, this work provides a thorough theoretical analysis showing which properties are desirable in a labeling function for the Labeled Interpolation Systems framework (LIS). The Proof-Sensitive labeling function is presented, and we prove that it generates interpolants with the smallest number of Boolean connectives in the entire LIS framework. Two variants that aim at controlling the logical strength of propositional interpolants while maintaining a small size are given. The new interpolation algorithms are compared to previous ones from the literature in different model checking settings, showing that they consistently lead to a better overall verification performance. The Equalities and Uninterpreted Functions (EUF)-interpolation system, presented in this thesis, is a duality-based interpolation framework capable of generating multiple interpolants for a single proof of unsatisfiability, and provides control over the logical strength of the interpolants it generates using labeling functions. The labeling functions can be theoretically compared with respect to their strength, and we prove that two of them generate the interpolants with the smallest number of equalities. Our experiments follow the theory, showing that the generated interpolants indeed have different logical strength. We combine propositional and EUF interpolation in a model checking setting, and show that the strength of the interpolation algorithms for different theories has to be aligned in order to generate smaller interpolants. This work also introduces the Linear Real Arithmetic (LRA)-interpolation system, an interpolation framework for LRA. The framework is able to generate infinitely many interpolants of different logical strength using the duality of interpolants. The strength of the LRA interpolants can be controlled by a normalized strength factor, which makes it straightforward for an interpolationbased application to choose the level of strength it wants for the interpolants. Our experiments with the LRA-interpolation system and a model checker show that it is very important for the application to be able to fine tune the strength of the LRA interpolants in order to achieve optimal performance. The interpolation frameworks were implemented and form the interpolation module in OpenSMT2, an open source efficient SMT solver. OpenSMT2 has been integrated to the propositional interpolation-based model checkers FunFrog and eVolCheck, and to the first order interpolation-based model checkerHiFrog. This thesis presents real life model checking experiments using the novel interpolation frameworks and the tools aforementioned, showing the viability and strengths of the techniques

PEF: Python Error Finder

Tesis (Lic. en Cs. de la Computación)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2014.La verificación del correcto funcionamiento de los sistemas es uno de los aspectos más importante del desarrollo de software, y a su vez, uno de los más costosos de llevar a cabo. El testing tradicional y la verificación de modelos requiere mucho esfuerzo para lograr una buena cobertura de las posibles ejecuciones. En este trabajo, utilizamos y extendemos una técnica que combina ejecución simbólica, el poder de los razonadores recientes y la versatilidad de lenguajes puramente orientados a objetos, para crear una librería que explora y describe los caminos de un programa, detectando errores de manera automática y facilitando posteriormente, la generación de tests. Presentaremos a PEF, una herramienta novedosa que hace uso de la técnica mencionada para detectar errores en programas escritos en Python 3 con muy poco esfuerzo por parte del usuario. Mostraremos la arquitectura del motor de ejecución simbólica y los aspectos fundamentales del lenguaje necesarios para construir el motor en forma de librería, sin tener que re-implementar el intérprete, como es usual en proyectos similares. También explicaremos el sistema de contratos que incorporamos para indicar pre y post condiciones que se desean aplicar y validar en los programas a ser explorados por PEF

The Temporal Booleanization Theorem: realizability checking over numerical-LTL industrial requirements

Industrial systems are getting more complex every year, and due to that complexity growth, the languages to specify them are becoming increasingly more expressive: so that they can properly model both controllable parts and the environment, or even temporal events. As a result, a great effort has been made over the last few years to move forward in the area of realizability checking of complex requirements. However, there is a blocking problem in the state-of-the-art realizability checkers: they only accept requirements that only contain Boolean variables on them. Therefore, these checkers cannot handle many real-industrial requirements, like those requirements containing numerical variables. One approach is to Booleanize numeric requirements and convert them into equivalent Boolean requirements. This problem has been researched and it has been discovered that its solution is not trivial. Thus, the main contribution of this thesis is that we have proved a theorem which verifies that (1) a correct Booleanization exists for all requirements that use theories with a decidable Ǝ*∀* fragment; and, thus (2) we have a realizability checking procedure for numeric requirements based on safety two-player turn-based LTL games. In addition, an algorithm that performs this has been proposed and implemented in OCaml

Modeling and Intelligent Control for Spatial Processes and Spatially Distributed Systems

Dynamical systems are often characterized by their time-dependent evolution, named temporal dynamics. The space-dependent evolution of dynamical systems, named spatial dynamics, is another important domain of interest for many engineering applications. By studying both the spatial and temporal evolution, novel modeling and control applications may be developed for many industrial processes. One process of special interest is additive manufacturing, where a three-dimensional object is manufactured in a layer-wise fashion via a numerically controlled process. The material is printed over a spatial domain in each layer and subsequent layers are printed on top of each other. The spatial dynamics of the printing process over the layers is named the layer-to-layer spatial dynamics. Additive manufacturing provides great flexibility in terms of material selection and design geometry for modern manufacturing applications, and has been hailed as a cornerstone technology for smart manufacturing, or Industry 4.0, applications in industry. However, due to the issues in reliability and repeatability, the applicability of additive manufacturing in industry has been limited. Layer-to-layer spatial dynamics represent the dynamics of the printed part. Through the layer-to-layer spatial dynamics, it is possible to represent the physical properties of the part such as dimensional properties of each layer in the form of a heightmap over a spatial domain. Thus, by considering the spatial dynamics, it is possible to develop models and controllers for the physical properties of a printed part. This dissertation develops control-oriented models to characterize the spatial dynamics and layer-to-layer closed-loop controllers to improve the performance of the printed parts in the layer-to-layer spatial domain. In practice, additive manufacturing resources are often utilized as a fleet to improve the throughput and yield of a manufacturing system. An additive manufacturing fleet poses additional challenges in modeling, analysis, and control at a system-level. An additive manufacturing fleet is an instance of the more general class of spatially distributed systems, where the resources in the system (e.g., additive manufacturing machines, robots) are spatially distributed within the system. The goal is to efficiently model, analyze, and control spatially distributed systems by considering the system-level interactions of the resources. This dissertation develops a centralized system-level modeling and control framework for additive manufacturing fleets. Many monitoring and control applications rely on the availability of run-time, up-to-date representations of the physical resources (e.g., the spatial state of a process, connectivity and availability of resources in a fleet). Purpose-driven digital representations of the physical resources, known as digital twins, provide up-to-date digital representations of resources in run-time for analysis and control. This dissertation develops an extensible digital twin framework for cyber-physical manufacturing systems. The proposed digital twin framework is demonstrated through experimental case studies on abnormality detection, cyber-security, and spatial monitoring for additive manufacturing processes. The results and the contributions presented in this dissertation improve the performance and reliability of additive manufacturing processes and fleets for industrial applications, which in turn enables next-generation manufacturing systems with enhanced control and analysis capabilities through intelligent controllers and digital twins.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169635/1/baltaefe_1.pd