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  • 期刊名称:

    Journal of Automated Reasoning

  • 中文名称: 自动推理杂志
  • 刊频: 1.926
  • ISSN: 0168-7433
  • 出版社: -
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  • 机译 Isabelle / HOL中代数数的经过验证的实现
    摘要: We formalize algebraic numbers in Isabelle/HOL. Our development serves as a verified implementation of algebraic operations on real and complex numbers. We moreover provide algorithms that can identify all the real or complex roots of rational polynomials, and two implementations to display algebraic numbers, an approximative version and an injective precise one. We obtain verified Haskell code for these operations via Isabelle's code generator. The development combines various existing formalizations such as matrices, Sturm's theorem, and polynomial factorization, and it includes new formalizations about bivariate polynomials, unique factorization domains, resultants and subresultants.
  • 机译 一阶设置中有限的二阶功能
    摘要: We describe how we have defined in ACL2 a weak version of the Common Lisp functional apply, which takes a function and list of actuals and applies the function to the actuals. Our version, called apply$, does not operate on functions but on ordinary objects-symbols and lists representing lambda expressions-some of which are interpreted as functions. We define a syntactic notion of "tameness" to identify the interpretable objects. This makes our apply$ weaker than a true second-order functional but we believe apply$ is powerful enough for many uses in ACL2. To maintain soundness and the conservativity of our Definitional Principle we require that certain hypotheses, called "warrants", be present in any theorem relying on the behavior of apply$ on non-primitives. Within these constraints we can define "functionals" such as sum and foldr which map tame "functions" over lists and accumulate the results. This allows the ACL2 user to avoid defining specialized recursive functions for each such application. We can prove and use general-purpose lemmas about these "functionals." We describe the formalization, explain how we keep the Definitional Principle conservative, show examples of useful functions using apply$ and theorems about them, sketch the proof that there is a model of any extension of the system using the new primitives, discuss issues arising in making these functions executable, and show some preliminary performance results.
  • 机译 OptiMathSAT:优化模理论的工具
    摘要: Optimization Modulo Theories (OMT) is an extension of SMT which allows for finding models that optimize given objectives. OptiMathSAT is an OMT solver which allows for solving a list of optimization problems on SMT formulas with linear objective functions-on the Boolean, the rational and the integer domains, and on their combination thereof-including (partial weighted) MaxSMT . Multiple and heterogeneous objective functions can be combined together and handled either independently, or lexicographically, or in linear or min-max /max-min combinations. OptiMathSAT provides an incremental interface, it supports both an extended version of the SMT-LIBv2 language and a subset of the FlatZinc language, and can be interfaced via an API. In this paper we describe OptiMathSAT and its usage in full detail.
  • 机译 K_SP K_n的基于分辨率的定理证明:体系结构,细化,策略和实验
    摘要: In this paper we describe the implementation of KSP, a resolution-based prover for the basic multimodal logic K-n. The prover implements a resolution-based calculus for both local and global reasoning. The user can choose different normal forms, refinements of the basic resolution calculus, and strategies. We describe these options in detail and discuss their implications. We provide experiments comparing some of these options and comparing the prover with other provers for this logic.
  • 机译 可扩展的细粒度证明,用于公式处理
    摘要: We present a framework for processing formulas in automatic theorem provers, with generation of detailed proofs. The main components are a generic contextual recursion algorithm and an extensible set of inference rules. Clausification, skolemization, theory-specific simplifications, and expansion of 'let' expressions are instances of this framework. With suitable data structures, proof generation adds only a linear-time overhead, and proofs can be checked in linear time. We implemented the approach in the SMT solver veriT. This allowed us to dramatically simplify the code base while increasing the number of problems for which detailed proofs can be produced, which is important for independent checking and reconstruction in proof assistants. To validate the framework, we implemented proof reconstruction in Isabelle/HOL.
  • 机译 前言:2017年CADE精选论文
    摘要:
  • 机译 高效认证(UN)SAT证书检查
    • 作者:Lammich, Peter;
    • 刊名:Journal of Automated Reasoning
    • 2020年第3期
    摘要: SAT solvers decide the satisfiability of Boolean formulas in conjunctive normal form. They are commonly used for software and hardware verification. Modern SAT solvers are highly complex and optimized programs. As a single bug in the solver may invalidate the verification of many systems, SAT solvers output certificates for their answer, which are then checked independently. However, even certificate checking requires highly optimized non-trivial programs. This paper presents the first SAT solver certificate checker that is formally verified down to the integer sequence representing the formula. Our tool supports the full DRAT standard, and is even faster than the unverified state-of-the-art tool drat-trim, on a realistic set of benchmarks drawn from the 2016 and 2017 SAT competitions. An optional multi-threaded mode further reduces the runtime, in particular for big certificates.
  • 机译 强大的无扩展证明系统
    摘要: We introduce proof systems for propositional logic that admit short proofs of hard formulas as well as the succinct expression of most techniques used by modern SAT solvers. Our proof systems allow the derivation of clauses that are not necessarily implied, but which are redundant in the sense that their addition preserves satisfiability. To guarantee that these added clauses are redundant, we consider various efficiently decidable redundancy criteria which we obtain by first characterizing clause redundancy in terms of a semantic implication relationship and then restricting this relationship so that it becomes decidable in polynomial time. As the restricted implication relation is based on unit propagation-a core technique of SAT solvers-it allows efficient proof checking too. The resulting proof systems are surprisingly strong, even without the introduction of new variables-a key feature of short proofs presented in the proof-complexity literature. We demonstrate the strength of our proof systems on the famous pigeon hole formulas by providing short clausal proofs without new variables.
  • 机译 使用循环证明自动验证指针程序的时间特性
    摘要: In this article, we investigate the automated verification of temporal properties of heap-aware programs. We propose a deductive reasoning approach based on cyclic proof. Judgements in our proof system assert that a program has a certain temporal property over memory state assertions, written in separation logic with user-defined inductive predicates, while the proof rules of the system unfold temporal modalities and predicate definitions as well as symbolically executing programs. Cyclic proofs in our system are, as usual, finite proof graphs subject to a natural, decidable soundness condition, encoding a form of proof by infinite descent. We present a proof system tailored to proving CTL properties of nondeterministic pointer programs, and then adapt this system to handle fair execution conditions. We show both versions of the system to be sound, and provide an implementation of each in the Cyclist theorem prover, yielding an automated tool that is capable of automatically discovering proofs of (fair) temporal properties of pointer programs. Experimental evaluation of our tool indicates that our approach is viable, and offers an interesting alternative to traditional model checking techniques.
  • 机译 使用Weil建立的关系来证明运营终止
    • 作者:Lucas, Salvador;
    • 刊名:Journal of Automated Reasoning
    • 2020年第2期
    摘要: In this paper, we study operational termination, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be characterized at different levels by means of well-founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.
  • 机译 自底向上模型生成方法的功能块和其他增强功能
    摘要: Model generation is a problem complementary to theorem proving and is important for fault analysis and debugging of formal specifications of security protocols, programs and terminological definitions, for example. This paper discusses several ways of enhancing the paradigm of bottom-up model generation, with the two main contributions being a new range-restriction transformation and generalized blocking techniques. The range-restriction transformation refines existing transformations to range-restricted clauses by carefully limiting the creation of domain terms. The blocking techniques are based on simple transformations of the input set together with standard equality reasoning and redundancy elimination techniques, and allow for finding small, finite models. All possible combinations of the introduced techniques and a classical range-restriction technique were tested on the clausal problems of the TPTP Version 6.0.0 with an implementation based on the SPASS theorem prover using a hyperresolution-like refinement. Unrestricted domain blocking gave best results for satisfiable problems, showing that it is an indispensable technique for bottom-up model generation methods, that yields good results in combination with both new and classical range-restricting transformations. Limiting the creation of terms during the inference process by using the new range-restricting transformation has paid off, especially when using it together with a shifting transformation. The experimental results also show that classical range restriction with unrestricted blocking provides a useful complementary method. Overall, the results show bottom-up model generation methods are good for disproving theorems and generating models for satisfiable problems, but less efficient for unsatisfiable problems.
  • 机译 归纳和基于饱和的定理证明相结合
    摘要: A method is devised to integrate reasoning by mathematical induction into saturation-based proof procedures based on resolution or superposition. The obtained calculi are capable of handling formulas in which some of the quantified variables range over inductively defined domains (which, as is well-known, cannot be expressed in first-order logic). The procedure is defined as a set of inference rules that generate inductive invariants incrementally and prove their validity. Although the considered logic itself is incomplete, it is shown that the invariant generation rules are complete, in the sense that if an invariant (of some specific form) is deducible from the considered clauses, then it is eventually generated.
  • 机译 解决有限集和二元关系上的无量词一阶约束
    摘要: In this paper we present a solver for a first-order logic language where sets and binary relations can be freely and naturally combined. The language can express, at least, any full set relation algebra on finite sets. It provides untyped, hereditarily finite sets, whose elements can be variables, and basically all the classic set and relational operators used in formal languages such as B and Z. Sets are first-class entities in the language, thus they are not encoded in lower level theories. Relations are just sets of ordered pairs. The solver exploits set unification and set constraint solving as primitive features. The solver is proved to be a sound semi-decision procedure for the accepted language. A Prolog implementation is presented and an extensive empirical evaluation provides evidence of its usefulness.
  • 机译 通过Isabelle / HOL中的柯西指数评估绕组数并计算复数根
    摘要: In complex analysis, the winding number measures the number of times a path (counter-clockwise) winds around a point, while the Cauchy index can approximate how the path winds. We formalise this approximation in the Isabelle theorem prover, and provide a tactic to evaluate winding numbers through Cauchy indices. By further combining this approximation with the argument principle, we are able to make use of remainder sequences to effectively count the number of complex roots of a polynomial within some domains, such as a rectangular box and a half-plane.
  • 机译 多电源约束的冲突驱动解决程序
    摘要: This paper studies the satisfiability problem of poly-power constraints (conjunctions of poly-power equations and inequalities), in which poly-powers are univariate nonlinear functions that extend integer exponents of polynomials to real algebraic exponents. To solve the poly-power constraint, we present a sound and complete procedure that incorporates conflict-driven learning with the exclusion algorithm for isolating positive roots of poly-powers. Furthermore, we introduce a kind of optimal interval-splitting, based on the Stern-Brocot tree and on binary rational numbers respectively, so that the operands occurring in the execution are chosen to be as simple as possible. The solving procedure, thereby, turns out to be promisingly efficient on randomly generated examples.
  • 机译 具有全局高速缓存的ExpTime Tableaux,用于混合PDL
    摘要: We present the first direct tableau decision procedure with the ExpTime complexity for HPDL (Hybrid Propositional Dynamic Logic). It checks whether a given ABox (a finite set of assertions) in HPDL is satisfiable. Technically, it combines global caching with checking fulfillment of eventualities and dealing with nominals. Our procedure contains enough details for direct implementation and has been implemented for the TGC2 (Tableaux with Global Caching) system. As HPDL can be used as a description logic for representing and reasoning about terminological knowledge, our procedure is useful for practical applications.
  • 机译 在HOL中自动化自由逻辑,并在类别理论中进行实验应用
    摘要: A shallow semantical embedding of free logic in classical higher-order logic is presented, which enables the off-the-shelf application of higher-order interactive and automated theorem provers for the formalisation and verification of free logic theories. Subsequently, this approach is applied to a selected domain of mathematics: starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. As a side-effect of this work some (minor) issues in a prominent category theory textbook have been revealed. The purpose of this article is not to claim any novel results in category theory, but to demonstrate an elegant way to "implement" and utilize interactive and automated reasoning in free logic, and to present illustrative experiments.
  • 机译 优先继承协议证明正确
    摘要: In real-time systems with threads, resource locking and priority scheduling, one faces the problem of Priority Inversion. This problem can make the behaviour of threads unpredictable and the resulting bugs can be hard to find. The Priority Inheritance Protocol is one solution implemented in many systems for solving this problem, but the correctness of this solution has never been formally verified in a theorem prover. As already pointed out in the literature, the original informal investigation of the Property Inheritance Protocol presents a correctness "proof" for an incorrect algorithm. In this paper we fix the problem of this proof by making all notions precise and implementing a variant of a solution proposed earlier. We also generalise the scheduling problem to the practically relevant case where critical sections can overlap. Our formalisation in Isabelle/HOL is based on Paulson's inductive approach to protocol verification. The formalisation not only uncovers facts overlooked in the literature, but also helps with an efficient implementation of this protocol. Earlier implementations were criticised as too inefficient. Our implementation builds on top of the small PINTOS operating system used for teaching.
  • 机译 具有桥接功能的理论的礼貌和组合方法
    摘要: The Nelson-Oppen combination method is ubiquitous in Satisfiability Modulo Theories solvers. However, one of its major drawbacks is to be restricted to disjoint unions of theories. We investigate the problem of extending this combination method to particular non-disjoint unions of theories defined by connecting disjoint theories via bridging functions. A possible application is to solve verification problems expressed in a combination of data structures connected to arithmetic with bridging functions such as the length of lists and the size of trees. We present a sound and complete combination method a la Nelson-Oppen for the theory of absolutely free data structures, including lists and trees. This combination procedure is then refined for standard interpretations. The resulting theory has a nice politeness property, enabling combinations with arbitrary decidable theories of elements. In addition, we have identified a class of polite data structure theories for which the combination method remains sound and complete. This class includes all the subtheories of absolutely free data structures (e.g, the empty theory, injectivity, projection). Again, the politeness property holds for any theory in this class, which can thus be combined with bridging functions and arbitrary decidable theories of elements. This illustrates the significance of politeness in the context of non-disjoint combinations of theories.
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