Proving Finite Satisfiability of Deductive Databases

It is shown how certain refutation methods can be extended into semi-decision procedures that are complete for both unsatisfiability and finite satisfiability. The proposed extension is justified by a new characterization of finite satisfiability. This research was motivated by a database design problem: Deduction rules and integrity constraints in definite databases have to be finitely satisfiabl

Satisfiability Games for Branching-Time Logics

The satisfiability problem for branching-time temporal logics like CTL*, CTL and CTL+ has important applications in program specification and verification. Their computational complexities are known: CTL* and CTL+ are complete for doubly exponential time, CTL is complete for single exponential time. Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems. In this paper we present a uniform game-theoretic framework for the satisfiability problem of these branching-time temporal logics. We define satisfiability games for the full branching-time temporal logic CTL* using a high-level definition of winning condition that captures the essence of well-foundedness of least fixpoint unfoldings. These winning conditions form formal languages of \omega-words. We analyse which kinds of deterministic {\omega}-automata are needed in which case in order to recognise these languages. We then obtain a reduction to the problem of solving parity or B\"uchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics. This approach provides a uniform, yet complexity-theoretically optimal treatment of satisfiability for branching-time temporal logics. It separates the use of temporal logic machinery from the use of automata thus preserving a syntactical relationship between the input formula and the object that represents satisfiability, i.e. a winning strategy in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner closure of the input formula only. Last but not least, the games presented here come with an attempt at providing tool support for the satisfiability problem of complex branching-time logics like CTL* and CTL+

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

SAT-based Explicit LTL Reasoning

We present here a new explicit reasoning framework for linear temporal logic (LTL), which is built on top of propositional satisfiability (SAT) solving. As a proof-of-concept of this framework, we describe a new LTL satisfiability tool, Aalta\_v2.0, which is built on top of the MiniSAT SAT solver. We test the effectiveness of this approach by demonnstrating that Aalta\_v2.0 significantly outperforms all existing LTL satisfiability solvers. Furthermore, we show that the framework can be extended from propositional LTL to assertional LTL (where we allow theory atoms), by replacing MiniSAT with the Z3 SMT solver, and demonstrating that this can yield an exponential improvement in performance

Efficient Certified Resolution Proof Checking

We present a novel propositional proof tracing format that eliminates complex processing, thus enabling efficient (formal) proof checking. The benefits of this format are demonstrated by implementing a proof checker in C, which outperforms a state-of-the-art checker by two orders of magnitude. We then formalize the theory underlying propositional proof checking in Coq, and extract a correct-by-construction proof checker for our format from the formalization. An empirical evaluation using 280 unsatisfiable instances from the 2015 and 2016 SAT competitions shows that this certified checker usually performs comparably to a state-of-the-art non-certified proof checker. Using this format, we formally verify the recent 200 TB proof of the Boolean Pythagorean Triples conjecture

Applying Formal Methods to Networking: Theory, Techniques and Applications

Despite its great importance, modern network infrastructure is remarkable for the lack of rigor in its engineering. The Internet which began as a research experiment was never designed to handle the users and applications it hosts today. The lack of formalization of the Internet architecture meant limited abstractions and modularity, especially for the control and management planes, thus requiring for every new need a new protocol built from scratch. This led to an unwieldy ossified Internet architecture resistant to any attempts at formal verification, and an Internet culture where expediency and pragmatism are favored over formal correctness. Fortunately, recent work in the space of clean slate Internet design---especially, the software defined networking (SDN) paradigm---offers the Internet community another chance to develop the right kind of architecture and abstractions. This has also led to a great resurgence in interest of applying formal methods to specification, verification, and synthesis of networking protocols and applications. In this paper, we present a self-contained tutorial of the formidable amount of work that has been done in formal methods, and present a survey of its applications to networking.Comment: 30 pages, submitted to IEEE Communications Surveys and Tutorial

A System for Deduction-based Formal Verification of Workflow-oriented Software Models

The work concerns formal verification of workflow-oriented software models using deductive approach. The formal correctness of a model's behaviour is considered. Manually building logical specifications, which are considered as a set of temporal logic formulas, seems to be the significant obstacle for an inexperienced user when applying the deductive approach. A system, and its architecture, for the deduction-based verification of workflow-oriented models is proposed. The process of inference is based on the semantic tableaux method which has some advantages when compared to traditional deduction strategies. The algorithm for an automatic generation of logical specifications is proposed. The generation procedure is based on the predefined workflow patterns for BPMN, which is a standard and dominant notation for the modeling of business processes. The main idea for the approach is to consider patterns, defined in terms of temporal logic,as a kind of (logical) primitives which enable the transformation of models to temporal logic formulas constituting a logical specification. Automation of the generation process is crucial for bridging the gap between intuitiveness of the deductive reasoning and the difficulty of its practical application in the case when logical specifications are built manually. This approach has gone some way towards supporting, hopefully enhancing our understanding of, the deduction-based formal verification of workflow-oriented models.Comment: International Journal of Applied Mathematics and Computer Scienc

A Survey of Satisfiability Modulo Theory

Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories. In this survey, we explain the combination of propositional satisfiability and decision procedures for conjunctions known as DPLL(T), and the alternative "natural domain" approaches. We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest, Romania. 201