Proof Calculus Research Articles

A Complete Uniform Substitution Calculus for Differential Dynamic Logic

This article introduces a relatively complete proof calculus for differential dynamic logic (dL) that is entirely based on uniform substitution, a proof rule that substitutes a formula for a predicate symbol everywhere. Uniform substitutions make it possible to use axioms instead of axiom schemata, thereby substantially simplifying implementations. Instead of subtle schema variables and soundness-critical side conditions on the occurrence patterns of logical variables to restrict infinitely many axiom schema instances to sound ones, the resulting calculus adopts only a finite number of ordinary dLformulas as axioms, which uniform substitutions instantiate soundly. The static semantics of differential dynamic logic and the soundness-critical restrictions it imposes on proof steps is captured exclusively in uniform substitutions and variable renamings as opposed to being spread in delicate ways across the prover implementation. In addition to sound uniform substitutions, this article introduces differential forms for differential dynamic logic that make it possible to internalize differential invariants, differential substitutions, and derivatives as first-class axioms to reason about differential equations axiomatically. The resulting axiomatization of differential dynamic logic is proved to be sound and relatively complete.

Logic of approximate entailment in quasimetric and in metric spaces

It is known that a quasimetric space can be represented by means of a metric space; the points of the former space become closed subsets of the latter one, and the role of the quasimetric is assumed by the Hausdorff quasidistance. In this paper, we show that, in a slightly more special context, a sharpened version of this representation theorem holds. Namely, we assume a quasimetric to fulfil separability in the original sense due to Wilson. Then any quasimetric space can be represented by means of a metric space such that distinct points are assigned disjoint closed subsets.This result is tailored to the solution of an open problem from the area of approximate reasoning. Following the lines of E. Ruspini’s work, the Logic of Approximate Entailment (mathsf {LAE}) is based on a graded version of the classical entailment relation. We present a proof calculus for mathsf {LAE} and show its completeness with regard to finite theories.

On-the-fly construction of provably correct service compositions – templates and proofs

Today, service compositions often need to be assembled or changed on-the-fly, which leaves only little time for quality assurance. Moreover, quality assurance is complicated by service providers only giving information on their services in terms of domain specific concepts with only limited semantic meaning.In this paper, we propose a method for constructing service compositions based on pre-verified templates. Templates, given as workflow descriptions, are typed over a (domain-independent) template ontology defining concepts and predicates. Their meaning is defined by an abstract semantics, leaving the specific meaning of ontology concepts open, however, only up to given ontology rules. Templates are proven correct using a Hoare-style proof calculus, extended by a specific rule for service calls. Construction of service compositions amounts to instantiation of templates with domain-specific services. Correctness of an instantiation can then simply be checked by verifying that the domain ontology (a) adheres to the rules of the template ontology, and (b) fulfills the constraints of the employed template.

Proof theory for hybrid(ised) logics

Hybridisation is a systematic process along which the characteristic features of hybrid logic, both at the syntactic and the semantic levels, are developed on top of an arbitrary logic framed as an institution. In a series of papers this process has been detailed and taken as a basis for a specification methodology for reconfigurable systems. The present paper extends this work by showing how a proof calculus (in both a Hilbert and a tableau based format) for the hybridised version of a logic can be systematically generated from a proof calculus for the latter. Such developments provide the basis for a complete proof theory for hybrid(ised) logics, and thus pave the way to the development of (dedicated) proof support.

Numerically-aided Deductive Safety Proof for a Powertrain Control System

The use of deductive techniques, such as theorem provers, has several advantages in safety verification of hybrid systems. There is often a gap, however, between the type of assistance that a theorem prover requires to make progress on a proof task and the assistance that a system designer is able to provide. To address this deficiency we present an extension to the deductive verification framework of differential dynamic logic that allows the theorem prover KeYmaera to locally reason about behaviors by leveraging forward invariant sets provided by external methods, such as numerical techniques and designer insights. Our key contribution is a new inference rule, the forward invariant cut rule, introduced into the proof calculus of KeYmaera. We demonstrate the cut rule in action on an example involving an automotive powertrain control systems, in which we make use of a simulation-driven numerical technique to compute a local barrier function.

Importing SMT and Connection proofs as expansion trees

Different automated theorem provers reason in various deductive systems and, thus, produce proof objects which are in general not compatible. To understand and analyze these objects, one needs to study the corresponding proof theory, and then study the language used to represent proofs, on a prover by prover basis. In this work we present an implementation that takes SMT and Connection proof objects from two different provers and imports them both as expansion trees. By representing the proofs in the same framework, all the algorithms and tools available for expansion trees (compression, visualization, sequent calculus proof construction, proof checking, etc.) can be employed uniformly. The expansion proofs can also be used as a validation tool for the proof objects produced.

Vehicle authentication via monolithically certified public key and attributes

Vehicular networks are used to coordinate actions among vehicles in traffic by the use of wireless transceivers (pairs of transmitters and receivers). Unfortunately, the wireless communication among vehicles is vulnerable to security threats that may lead to very serious safety hazards. In this work, we propose a viable solution for coping with Man-in-the-Middle attacks. Conventionally, Public Key Infrastructure is utilized for a secure communication with the pre-certified public key. However, a secure vehicle-to-vehicle communication requires additional means of verification in order to avoid impersonation attacks. To the best of our knowledge, this is the first work that proposes to certify both the public key and out-of-band sense-able static attributes to enable mutual authentication of the communicating vehicles. Vehicle owners are bound to preprocess (periodically) a certificate for both a public key and a list of fixed unchangeable attributes of the vehicle. Furthermore, the proposed approach is shown to be adaptable with regards to the existing authentication protocols. We illustrate the security verification of the proposed protocol using a detailed proof in Spi calculus.

Logic of approximate entailment in quasimetric spaces

The logic LAE q discussed in this paper is based on an approximate entailment relation. LAE q generalises classical propositional logic to the effect that conclusions can be drawn with a quantified imprecision. To this end, properties are modelled by subsets of a distance space and statements are of the form that one property implies another property within a certain limit of tolerance. We adopt the conceptual framework defined by E. Ruspini; our work is towards a contribution to the investigation of suitable logical calculi. LAE q is based on the assumption that the distance function is a quasimetric. We provide a proof calculus for LAE q and we show its soundness and completeness for finite theories. As our main tool for showing completeness, we use a representation of proofs by means of weighted directed graphs. • We propose an axiomatisation of Approximate Reasoning in the sense of E. Ruspini's paper “On the semantics of fuzzy logic”. • The Logic LAE q deals with statements of the form that a property implies another one within a specified limit of tolerance. • We present a proof calculus for LAE q and show its soundness and completeness.

Mathematics Majors’ Perceptions of the Admissibility of Graphical Inferences in Proofs

In this paper, we investigate mathematics majors’ perceptions of the admissibility of inferences based on graphical reasoning for calculus proofs. The main findings from our study is that the majority of mathematics majors did not think that graphical perceptual inferences (i.e., inferences based on the appearance of the graph) were permissible in a proof, but the majority of mathematics majors did believe that graphical deductive inferences (i.e., inferences based on what must necessarily be entailed by a graph having a certain property) were permissible.

A Denotational Semantics for Communicating Unstructured Code

An important property of programming language semantics is that they should be compositional. However, unstructured low-level code contains goto-like commands making it hard to define a semantics that is compositional. In this paper, we follow the ideas of Saabas and Uustalu to structure low-level code. This gives us the possibility to define a compositional denotational semantics based on least fixed points to allow for the use of inductive verification methods. We capture the semantics of communication using finite traces similar to the denotations of CSP. In addition, we examine properties of this semantics and give an example that demonstrates reasoning about communication and jumps. With this semantics, we lay the foundations for a proof calculus that captures both, the semantics of unstructured low-level code and communication.

A multi-focused proof system isomorphic to expansion proofs

The sequent calculus is often criticized for requiring proofs to contain large amounts of low-level syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps---such as instantiating a block of quantifiers---by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically non-interfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proof-nets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical first-order logic. The essential element of our approach is the use of a multi-focused sequent calculus as the means for abstracting away low-level details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique appears to be a systematic way to recover the "essence of proof" from within sequent calculus proofs.

Extracting BB′IW Inhabitants of Simple Types From Proofs in the Sequent Calculus $${LT_\to^{t}}$$ L T → t for Implicational Ticket Entailment

The decidability of the logic of pure ticket entailment means that the problem of inhabitation of simple types by combinators over the base { B, B′, I, W } is decidable too. Type-assignment systems are often formulated as natural deduction systems. However, our decision procedure for this logic, which we presented in earlier papers, relies on two sequent calculi and it does not yield directly a combinator for a theorem of \({T_\to}\) . Here we describe an algorithm to extract an inhabitant from a sequent calculus proof—without translating the proof into another proof system.

A Hilbert-Style Axiomatisation for Equational Hybrid Logic

This paper introduces an axiomatisation for equational hybrid logic based on previous axiomatizations and natural deduction systems for propositional and first-order hybrid logic. Its soundness and completeness is discussed. This work is part of a broader research project on the development a general proof calculus for hybrid logics.

Lifting lower bounds for tree-like proofs

It is known that constant-depth Frege proofs of some tautologies require exponential size. No such lower bound result is known for more general proof systems. We consider tree-like sequent calculus proofs in which formulas can contain modular connectives and only the cut formulas are restricted to be of constant depth. Under a plausible hardness assumption concerning small-depth Boolean circuits, we prove exponential lower bounds for such proofs. We prove these lower bounds directly from the computational hardness assumption. We start with a lower bound for cut-free proofs and "lift" it so it applies to proofs with constant-depth cuts. By using the same approach, we obtain the following additional results. We provide a much simpler proof of a known unconditional lower bound in the case where modular connectives are not used. We establish a conditional exponential separation between the power of constant-depth proofs that use different modular connectives. We show that these tree-like proofs with constant-depth cuts cannot polynomially simulate similar dag-like proofs, even when the dag-like proofs are cut-free. We present a new proof of the non-finite axiomatizability of the theory of bounded arithmetic I Δ0(R). Finally, under a plausible hardness assumption concerning the polynomial-time hierarchy, we show that the hierarchy $${G_i^*}$$ of quantified propositional proof systems does not collapse.

On Mathematicians' Different Standards When Evaluating Elementary Proofs

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians.

Proof Nets for Herbrand’s Theorem

This article explores Herbrand’s theorem as the source of a natural notion of abstract proof object for classical logic, embodying the “essence” of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (Herbrand nets) as an analytic proof system with syntactic cut-elimination. Herbrand nets can also be seen as a natural generalization of Miller’s expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LK H ; each net corresponds to an equivalence class of LK H proofs under natural proof transformations. A surprising property of our cut-reduction algorithm is that it is non-confluent despite not supporting the usual examples of non-confluent reduction in classical logic.

A formal framework for specifying sequent calculus proof systems

Intuitionistic logic and intuitionistic type systems are commonly used as frameworks for the specification of natural deduction proof systems. In this paper we show how to use classical linear logic as a logical framework to specify sequent calculus proof systems and to establish some simple consequences of the specified sequent calculus proof systems. In particular, derivability of an inference rule from a set of inference rules can be decided by bounded (linear) logic programming search on the specified rules. We also present two simple and decidable conditions that guarantee that the cut rule and non-atomic initial rules can be eliminated.

A Complete Axiomatization of Quantified Differential Dynamic Logic for Distributed Hybrid Systems

We address a fundamental mismatch between the combinations of dynamics that occur in cyber-physical systems and the limited kinds of dynamics supported in analysis. Modern applications combine communication, computation, and control. They may even form dynamic distributed networks, where neither structure nor dimension stay the same while the system follows hybrid dynamics, i.e., mixed discrete and continuous dynamics. We provide the logical foundations for closing this analytic gap. We develop a formal model for distributed hybrid systems. It combines quantified differential equations with quantified assignments and dynamic dimensionality-changes. We introduce a dynamic logic for verifying distributed hybrid systems and present a proof calculus for this logic. This is the first formal verification approach for distributed hybrid systems. We prove that our calculus is a sound and complete axiomatization of the behavior of distributed hybrid systems relative to quantified differential equations. In our calculus we have proven collision freedom in distributed car control even when an unbounded number of new cars may appear dynamically on the road.

Bunched Logics Displayed

We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cut-elimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of display-equivalence in our calculi in such a way that an exhaustive proof search need be only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist.

A logic with imprecise probabilities and an application to automated reasoning using rewriting techniques

A new probabilistic logic for handling imprecise probabilities is introduced, implemented in a rewriting system, and used to carry out an experiment. Each well-formed formula of the probabilistic logic is labeled with two values that represent possible minimum and maximum probabilities associated with the event related to the unlabeled formula. The aim of this logic is to facilitate application of rules to obtain an approximation of the probability interval associated with an event, without the necessity of knowing the precise probability of other events. The logic is described as a formal theory by means of its language, semantics and a proof theory. The soundness of the proof theory has been proven. Rewriting techniques are a powerful method for testing the behavior of a formal proof calculus through translation of calculus inference rules into rewrite rules. Implementation of the logic rules in a rewriting language such as Maude allows an automated reasoning system to be easily obtained, which can be consulted by applications. We developed a small game application and an experiment to test the application.