A proof system for the first-order relational calculus

Besides explicit axioms, an algebraic specification language contains model-theoretic constraints such as initiality. For proving properties of specifications and refining them to programs, an axiomatization of these constraints is needed; unfortunately, no effective, sound and complete proof system can be constructed for initial models, and a fortiori for their extensions.

Research by myself and by others has shown that there are natural programming language control structures that are impossible to describe adequately by means of Hoare axioms. Specifically, we have shown that there are control structures for which it is impossible to obtain axiom systems that are sound and relatively complete in the sense of Cook. These constructs include procedures with procedure parameters under standard ALGOL 60 scope rules and coroutines in a language with parameterless recursive procedures. A natural question to ask is whether it is possible to characterize those programming languages for which sound and complete proof systems can be obtained. For a wide class of programming languages and interpretations, it can be shown that P has a sound and relatively complete proof system for every expressive interpretation iff the halting problem for language P is decidable for all finite interpretations. Nevertheless, we are still far from a completely satisfactory characterization of the programming languages that can be axiomatized in this manner. The proof system that is generated in proving the above result does not have the property of being ‘syntax-directed’, which is distinctive of the Hoare axioms. Moreoever, theoretical considerations suggest that good axioms for total correctness may exist for a wider spectrum of languages than for partial correctness. In this paper we discuss these questions and others that still need to be addressed before the characterization problem can be considered solved.

Semantics and reasoning with free procedures

Incompleteness of first-order temporal logic with until

Actor Networks are a modeling framework for cyber-physical system protocols based on Latour’s actor-network theory that addresses the way we now create and exploit the power of computational networks. We advance a logic for modeling and reasoning about such actor networks, which is obtained through a two-stage constrained hybridization process. The first stage results in a logic that captures the structure of actor networks and the way knowledge flows across them; the second addresses the dynamic aspects of actor networks, that is the way they can evolve as a result of the interactions that occur within them. For each of these stages, we develop a sound and complete proof system.

We present a new model of knowledge, belief, desire and intention, called the interpreted KBDI-system model (or KBDI-model for short). The key point of the interpreted KBDI-system model is that we express an agent's knowledge, belief, desire and intention as a set of runs (computing paths), which is exactly a system in the interpreted system model, a well-known agent model due to Halpern and his colleagues. Our KBDI-model is computationally grounded in that we are able to associate a KBDI-model with a computer program, and formulas, involving agents' knowledge, belief, desire (goal) and intention, can be understood as properties of program computations. We present a sound and complete proof system with respect to our KBDI-model and explore how symbolic model checking techniques can be applied to model checking multi-agent systems with KBDI-models.

Sound over-approximation methods are effective for proving the absence of errors, but inevitably produce false alarms that can hamper programmers. In contrast, under-approximation methods focus on bug detection and are free from false alarms. In this work, we present two novel proof systems designed to locate the source of errors via backward under-approximation, namely Sufficient Incorrectness Logic (SIL) and its specialization for handling memory errors, called Separation SIL. The SIL proof system is minimal, sound and complete for Lisbon triples, enabling a detailed comparison of triple-based program logics across various dimensions, including negation, approximation, execution order, and analysis objectives. More importantly, SIL lays the foundation for our main technical contribution, by distilling the inference rules of Separation SIL, a sound and (relatively) complete proof system for automated backward reasoning in programs involving pointers and dynamic memory allocation. The completeness result for Separation SIL relies on a careful crafting of both the assertion language and the rules for atomic commands.

The main goal of this paper is to provide an abstract framework for constructing proof systems for various many-valued logics. Using the framework it is possible to generate strongly complete proof systems with respect to any finitely valued deterministic and non-deterministic logic. I provide a couple of examples of proof systems for well-known many-valued logics and prove the completeness of proof systems generated by the framework.

Completing the temporal picture

The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general structure of proofs in the original calculus in a way ensuring preservation of the weak cut elimination theorem under the transformation. The described transformation metod is illustrated on several concrete examples of many-valued logics, including a new application to information sources logics.

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

A complete modal proof system for HAL: the Herbrand agent language

The linear time μ-calculus extends LTL with arbitrary least and greatest fixpoint operators. This gives it the power to express all ω-regular languages, i.e. strictly more than LTL. The validity problem is PSPACE-complete for both LTL and the linear time μ-calculus. In practice it is more difficult for the latter because of nestings of fixpoint operators and variables with several occurrences. We present a simple sound and complete infinitary proof system for the linear time μ-calculus and then present two decision procedures for provability in the system, hence validity of formulas. One uses nondeterministic Büchi automata, the other one a generalisation of size-change termination analysis (SCT) known from functional programming. The main novelties of this paper are the connection with SCT and the fact that both decision procedures have a better asymptotic complexity than earlier ones and have been implemented.

Proof systems for structured specifications with observability operators

Fuzzy lattice operations on first-order terms over signatures with similar constructors: A constraint-based approach