On incorrectness logic and Kleene algebra with top and tests

Kleene algebra with tests (KAT) is a foundational equational framework for reasoning about programs, which has found applications in program transformations, networking and compiler optimizations, among many other areas. In his seminal work, Kozen proved that KAT subsumes propositional Hoare logic, showing that one can reason about the (partial) correctness of while programs by means of the equational theory of KAT. In this work, we investigate the support that KAT provides for reasoning about incorrectness, instead, as embodied by O'Hearn's recently proposed incorrectness logic. We show that KAT cannot directly express incorrectness logic. The main reason for this limitation can be traced to the fact that KAT cannot express explicitly the notion of codomain, which is essential to express incorrectness triples. To address this issue, we study Kleene Algebra with Top and Tests (TopKAT), an extension of KAT with a top element. We show that TopKAT is powerful enough to express a codomain operation, to express incorrectness triples, and to prove all the rules of incorrectness logic sound. This shows that one can reason about the incorrectness of while-like programs by means of the equational theory of TopKAT.

Completeness and Complexity of Reasoning about Call-by-Value in Hoare Logic

We provide a sound and relatively complete Hoare logic for reasoning about partial correctness of recursive procedures in presence of local variables and the call-by-value parameter mechanism and in which the correctness proofs support contracts and are linear in the length of the program. We argue that in spite of the fact that Hoare logics for recursive procedures were intensively studied, no such logic has been proposed in the literature.

Prophetischer oder marxistischer Sozialismus?

German philosopher and sociologist Max Scheler (1874–1928) puts forward the concept of “prophetic Christian socialism” as a means of political and ideological opposition to Marxism. The concept expresses his religious-philosophical views, developed in earlier works, primarily in the main work “Formalism in Ethics and Material Ethics of Values”. Scheler compares his own views on socialism, understanding of history, the possibility of foreseeing historical processes with the views of these realities of K. Marx. Scheler's criticism of Marx's teachings is interspersed with the recognition of its partial correctness.

Partial Correctness of a Fibonacci Algorithm

Summary In this paper we introduce some notions to facilitate formulating and proving properties of iterative algorithms encoded in nominative data language [19] in the Mizar system [3], [1]. It is tested on verification of the partial correctness of an algorithm computing n-th Fibonacci number: i := 0 s := 0 b := 1 c := 0 while (i <> n) c := s s := b b := c + s i := i + 1 return s This paper continues verification of algorithms [10], [13], [12] written in terms of simple-named complex-valued nominative data [6], [8], [17], [11], [14], [15]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [16], [18], [7], [5].

Reasoning about Partial Correctness Assertions in Isabelle/HOL

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented and concurrent programs. The purpose of this work is to provide a detailed and accessible exposition of the several ways the user can conduct, explore and write proofs of correctness of sequential imperative programs with Hoare logic and the ISABELLE proof assistant. With the proof language Isar, it is possible to write structured, readable proofs that are suitable for human understanding and communication.

Partial Correctness of a Power Algorithm

Summary This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[15],[11],[12],[13]. In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 b := val.3 n := val.4 s := val.5 while (i <> n) i := i + j s := s * b return s computing the natural n power of given complex number b, where variables i, b, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = b, loc/.4 = n and loc/.5 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [17]). The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [14],[16],[7],[5].

Partial Correctness of a Factorial Algorithm

Summary In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 n := val.3 s := val.4 while (i <> n) i := i + j s := s * i return s computing the factorial of given natural number n, where variables i, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = n and loc/.4 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [16]). This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[14],[10],[11],[12]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [13],[15],[7],[5].