Mykola Nikitchenko

Satisfiability in composition-nominative logics

Composition-nominative logics are algebra-based logics of partial predicates constructed in a semantic-syntactic style on the methodological basis, which is common with programming. They can be considered as generalizations of traditional logics on classes of partial predicates that do not have fixed arity. In this paper we present and investigate algorithms for solving the satisfiability problem in various classes of composition-nominative logics. We consider the satisfiability problem for logics of the propositional, renominative, and quantifier levels and prove the reduction of the problem to the satisfiability problem for classical logics. The method developed in the paper enables us to leverage existent state-of-the-art satisfiability checking procedures for solving the satisfiability problem in composition-nominative logics, which could be crucial for handling industrial instances coming from domains such as program analysis and verification. The reduction proposed in the paper requires extension of logic language and logic models with an infinite number of unessential variables and with a predicate of equality to a constant.

Extending Floyd-Hoare Logic for Partial Pre- and Postconditions

Traditional (classical) Floyd-Hoare logic is defined for a case of total pre- and postconditions while programs can be partial. In the chapter we propose to extend this logic for partial conditions. To do this we first construct and investigate special program algebras of partial predicates, functions, and programs. In such algebras program correctness assertions are presented with the help of a special composition called Floyd-Hoare composition. This composition is monotone and continuous. Considering the class of constructed algebras as a semantic base we then define an extended logic – Partial Floyd-Hoare Logic – and investigate its properties. This logic has rather complicated soundness constraints for inference rules, therefore simpler sufficient constraints are proposed. The logic constructed can be used for program verification.

On a Decidable Formal Theory for Abstract Continuous-Time Dynamical Systems

We propose a decidable formal theory which describes high-level properties of abstract continuous-time dynamical systems called Nondeterministic Complete Markovian Systems (NCMS). NCMS is a rather general class of systems which can represent discrete and/or continuous evolutions in continuous time and which is sufficient for modeling a wide range of real-time information processing and cyber-physical systems (CPS). We illustrate the obtained results with a proof of the mutual exclusion property of a CPS which implements Peterson’s algorithm.

On Algebraic Properties of Nominative Data and Functions

In the chapter basic properties of nominative data and functions over nominative data (nominative functions) are investigated from the perspective of abstract algebra. A set of all nominative data over arbitrary fixed sets of names and values together with basic operations which include naming, denaming, and overlapping is considered as an algebraic structure and its main properties are studied. Nominative data with complex names satisfy the principle of associative naming and processing. For such data a natural equivalence relation is introduced. Properties of nominative functions (mathematical models of programs over nominative data) and predicates are studied. A notion of nominative stability of nominative functions and predicates is considered. A two-sorted algebra of nominative functions and predicates which generalizes Glushkov algorithmic algebras is introduced and it is proved that the set of nominative stable functions and the set of nominative stable predicates constitute its sub-algebra. The obtained results form a mathematical basis for nominative program logic construction.

Satisfiability and Validity Problems in Many-Sorted Composition-Nominative Pure Predicate Logics

We propose methods for solving the satisfiability and validity problems in many-sorted composition-nominative pure predicate logics (without functions and with equality). These logics are algebra-based logics of many-sorted partial predicates constructed in a semantic-syntactic style on the methodological basis that is common with programming; they can be considered as generalizations of traditional many-sorted logics on classes of partial predicates that do not have fixed arity. We show the reduction of the satisfiability problem to the same problem for many-sorted classical first-order pure predicate logic with equality. As validity is dual to satisfiability, the method proposed can be adopted to the validity problem. This enables us to use existent satisfiability and validity checking procedures developed for classical logic also for solving these problems in composition-nominative pure predicate logics with equality.

Formalization of the algebra of nominative data in mizar

In the paper we describe a formalization of the notion of a nominative data with simple names and complex values in the Mizar proof assistant. Such data can be considered as a partial variable assignment which allows arbitrarily deep nesting and can be useful for formalizing semantics of programs that operate in real time environment and/or process complex data structures and for reasoning about the behavior of such programs.

Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm

Summary Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. Analysis of the literature [1], [7], [12], [9], [13], [14], [15] shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems. In the paper [2] we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson’s algorithm [11]. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in [8]. Using Mizar [3], in this paper we give a formal proof of the mutual exclusion property of the Peterson’s algorithm in Mizar on the basis of the event-based approach proposed in [2]. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation ⩽ on the set of events which turns it into a linear preorder (events are considered simultaneous, if e1 ⩽ e2 and e2 ⩽ e1), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function “val of” on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) [2]. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in [2]). After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson’s algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events e1 and e2 in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson’s algorithm (setting the flag of the current process, writing the identifier of the opposite process to the “turn” shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the “turn” variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the “turn” variable, then either the events e1 and e2 coincide, or they are not simultaneous (mutual exclusion property).

Composition-Nominative Logics as Institutions

Composition-nominative logics (CNL) are program-oriented logics. They are based on algebras of partial predicates which do not have fixed arity. The aim of this work is to present CNL as institutions. Homomorphisms of first-order CNL are introduced, satisfaction condition is proved. Relations with institutions for classical first-order logic are considered. Directions for further investigation are outlined.

On Formalization of Semantics of Real-Time and Cyber-Physical Systems

In the article we describe theoretical foundations of a framework for formalizing semantics of real-time and cyber-physical systems in interactive theorem proving environments. The framework is based on viewing a system as a predicate on system’s executions which are modeled as functions from the continuous time domain to a set of states. We consider how it can be applied to the safety verification problems. The proposed framework may be useful in verification of software for real-time and cyber-physical systems and of the corresponding development tools.

Formalization of the Nominative Algorithmic Algebra in Mizar

We describe a formalization of the nominative algorithmic algebra in the Mizar proof assistant. This algebra is a generalization of Glushkov algorithmic algebras which is well suited for representing semantics and specifying properties of programs on complex data structures which interact with external environment in convenient and unified way. We describe formal definitions and formally proven and checked statements about the carrier sets and operations (program compositions) of this algebra. The obtained results are useful for implementing methods of automated verification of software, for example, for the Internet of Things (IoT) and ensuring safety of IoT systems.