On Hoare logic and Kleene algebra with tests

Kleene algebra with tests (KAT) was introduced by Kozen as an extension of Kleene algebra (KA). The decidability of equational formulas p = q and Horn formulas ∧ i p i = q i → p = q in KAT has been studied so far by several researchers. Continuing this line of research, this paper studies the decidability of existentially quantified equational formulas ∃ q ∈ P. (p = q) in KAT, where P is a fixed collection of KAT terms. A new completeness theorem of KAT is proved, and via the completeness theorem, the decision problem of ∃ q ∈ P. (p = q) is reduced to a certain membership problem of regular languages, to which a pseudo-identity-based decision method is applicable. Based on this reduction, an instance of the problem is studied and shown to be decidable.

The modern software engineering process is evolutionary, with commits/patches begetting new versions of code, progressing steadily toward improved systems. In recent years, program analysis and verification tools have exploited version-based reasoning, where new code can be seen in terms of how it has changed from the previous version. When considering program versions, refinement seems a natural fit and, in recent decades, researchers have weakened classical notions of concrete refinement and program equivalence to capture similarities as well as differences between programs. For example, Benton, Yang and others have worked on state-based refinement relations . In this paper, we explore a form of weak refinement based on trace relations rather than state relations. The idea begins by partitioning traces of a program C 1 into trace classes, each identified via a restriction r 1 . For each class, we specify similar behavior in the other program C 2 via a separate restriction r 2 on C 2 . Still, these two trace classes may not yet be equivalent so we further permit a weakening via a binary relation A on traces, that allows one to, for instance disregard unimportant events, relate analogous atomic events, etc. We address several challenges that arise. First, we explore one way to specify trace refinement relations by instantiating the framework to Kleene Algebra with Tests (KAT) due to Kozen. We use KAT intersection for restriction, KAT hypotheses for A , KAT inclusion for refinement, and have proved compositionality. Next, we present an algorithm for automatically synthesizing refinement relations, based on a mixture of semantic program abstraction, KAT inclusion, a custom edit-distance algorithm on counterexamples, and case-analysis on nondeterministic branching. We have proved our algorithm to be sound. Finally, we implemented our algorithm as a tool called Knotical, on top of Interproc and Symkat. We demonstrate promising first steps in synthesizing trace refinement relations across a hand-crafted collection of 37 benchmarks that include changing fragments of array programs, models of systems code, and examples inspired by the thttpd and Merecat web servers.

Variants of Kleene algebra have been used to provide foundations of reasoning about programs, for instance by representing Hoare Logic (HL) in algebra. That work has generally emphasised program correctness, i.e., proving the absence of bugs. Recently, Incorrectness Logic (IL) has been advanced as a formalism for the dual problem: proving the presence of bugs. IL is intended to underpin the use of logic in program testing and static bug finding. Here, we use a Kleene algebra with diamond operators and countable joins of tests, which embeds IL, and which also is complete for reasoning about the image of the embedding. Next to embedding IL, the algebra is able to embed HL, and allows making connections between IL and HL specifications. In this sense, it unifies correctness and incorrectness reasoning in one formalism.

We present several results on de Morgan algebras and Kleene algebras of fuzzy logic. The main results include (1) a necessary and sufficient condition for a de Morgan algebra to be a Kleene algebra, (2) equivalence of some expressions, (3) some properties about de Morgan algebras with sup W/sup +/ and inf W/sup -/, (4) some properties of the de Morgan algebras which satisfy certain special conditions, (5) a method to obtain all fixed points of a de Morgan algebra and (6) extensions of some theorems on complete de Morgan algebras that have fixed points to the general case.

Kleene algebras with tests (KATs) offer sound, complete, and decidable equational reasoning about regularly structured programs. Interest in KATs has increased greatly since NetKAT demonstrated how well extensions of KATs with domain-specific primitives and extra axioms apply to computer networks. Unfortunately, extending a KAT to a new domain by adding custom primitives, proving its equational theory sound and complete, and coming up with an efficient implementation is still an expert’s task. Abstruse metatheory is holding back KAT’s potential.

A new first-order logic, functional logic, was proposed recently by Staples, Robinson and Hazel. The logic provides a formal means of describing and reasoning about dependence on an implicit parameter, a prime motivation being the unification of the Hoare logic used to reason about procedural programs with the powerful and well established techniques of classical logic. Viewed more abstractly, independently of possible applications, functional logic may be described as a logic with primitives, axioms and inference rules appropriate for reasoning about the properties of mathematical functions. In this paper, the completeness of functional logic is proved; that is, it is shown that any term of a theory in the logic which is true in all models is a theorem.

Classical Floyd-Hoare logic is sound when total pre- and post-conditions are considered. In the case of partial conditions (predicates) the logic becomes unsound. This situation may be corrected by introducing additional constraints to the rules of the logic. But such constraints, in particular, for the sequence and while rules, are rather complicated. In this paper we propose new simpler rules formulated in a program algebra extended with the composition of predicate complement. The obtained logic is called the Complemented Partial Floyd-Hoare Logic (CPFHL). The predicate component of this logic is related to three-valued logic. We prove the soundness theorem for CPFHL and discuss further investigations of the problem. The obtained results can be useful for software verification.

Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning tasks about qualitative properties. The use of logic provides a general framework which improves the capacity of reasoning. This way, we can infer additional information by using axioms and the logic apparatus. In this paper we present sound and complete relational dual tableau that can be used for verification of validity of formulas of the logic in question.Keywordsqualitative reasoningpropositional dynamic logicrelational logicshybrid qualitative reasoning

This chapter addresses the problem of the acquisition of the syntax of propositional logic. An approach based on general purpose cognitive capacities such as invention, adoption, parsing, generation, and induction is proposed. Self-organisation principles are used to show how a shared set of preferred lexical entries and grammatical constructions, that is, a language, can emerge in a population of autonomous agents which do not have any initial linguistic knowledge. Experiments in which a population of autonomous agents constructs a grammar that allows communicating the formulas of a propositional logic language are presented. These experiments extend previous work by considering a larger population and a much larger search space of grammar rules. In particular, the agents are allowed to order the expressions associated with the constituents of a logical formula in arbitrary order. Previous work assumed that the expressions associated with the connectives should be placed in the first position of the sentence.

First-order logic augments the expressive power of propositional logic as it links the logical assertions to properties of objects of some non-empty universe: the domain of discourse. This is achieved by allowing the propositional symbols to take arguments that range over elements of the domain of discourse. These are now called predicate symbols and are interpreted as relations on the domain. Elements of the domain of discourse are denoted by terms built up from variables, constants, and functions applied to other terms. First-order logic also expands the lexicon of propositional logic with the quantifiers “for all” and “there exists” that are interpreted consistently with their natural language meaning.This chapter is devoted to classical first-order logic. Our presentation is similar to the one conducted for propositional logic. We first define the syntax of first-order logic, followed by its semantics. Next we define a proof system for it and present the fundamental theoretical results of soundness and completeness. We also discuss the decision problems related to this logic. The remaining sections of the chapter cover variations and extensions of first-order logic, as well as first-order theories.

In this paper, we apply the concept of similarity operations to equality algebras and introduce similarity equality algebras. We investigate some related properties of them. In particular, we prove that there is a one to one correspondence between similarity filters and similarity congruence relations on a similarity equality algebra. Furthermore, we prove the first isomorphism theorem on similarity equality algebras. We use the similarity filters to establish the uniform structure on similarity equality algebras and obtain some properties of this topology. In addition, we introduce and study a corresponding propositional logic for similarity equality algebras and prove the soundness and completeness of this propositional logic based on similarity equality algebras.

Over the past few years, CL diagrams have gained popularity in diagrammatic reasoning, drawing inspiration from Lange’s Cubus Logicus. The intuitive understanding of CL diagrams is based on simple structures that are straightforward both to draw and to comprehend. This structure supports embedding of information and inferencing. Furthermore, these diagrams are more than just heuristic tools; CL diagrams can actually be extended to full blown formal systems. The present paper shows that a formal system for CL diagrams can have an expressivity similar to standard Boolean algebra of propositions or propositional logic without losing their intuitive properties. It is also possible to combine this interpretation of CL with elements from the class calculus. The syntax and semantics are presented separately, and the system’s soundness and completeness are proven using a variant of bitstring semantics.

In classical deterministic planning, solutions to planning tasks are simply sequences of actions, but that is not sufficient for contingent plans in non-deterministic environments. Contingent plans are often expressed through policies that map states to actions. An alternative is to specify contingent plans as programs, e.g. in the syntax of Propositional Dynamic Logic (PDL). PDL is a logic for reasoning about programs with sequential composition, test and non-deterministic choice. However, as we show in the paper, none of the existing PDL modalities directly captures the notion of a solution to a planning task under non-determinism. We add a new modality to star-free PDL correctly capturing this notion. We prove the appropriateness of the new modality by showing how to translate back and forth between policies and PDL programs under the new modality. More precisely, we show how a policy solution to a planning task gives rise to a program solution expressed via the new modality, and vice versa. We also provide an axiomatisation of our PDL extension through reduction axioms into standard star-free PDL.

In this paper we give an abstract representation of propositional Horn logic (finite or infinitary) in terms of set-valued functions F : (sic) (PR) -> (sic) (PR); here PR is the collection of atomic propositions. We investigate the properties of set-valued functions and establish that they satisfy the axioms of weak lazy Kleene algebra. These axioms and other properties are used to prove uniform left interpolation for propositional Horn logic. Then we introduce thin set-valued functions, which enable us to prove polynomial interpolation for propositional Horn logic. For the infinite case, the Axiom of Choice is used.

Some Remarks on Discussive Logic