Complete Proof System Research Articles

Logics for Actor Networks: A two-stage constrained-hybridisation approach

Actor Networks are a modelling 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 networks whose components are no longer limited to programs, but can also include humans and physical artefacts as actors. The main contribution of this paper is a logic for modelling and reasoning about such actor networks that results from a two-stage constrained-hybridisation process: the first stage corresponds to a logic that captures the structure of actor networks and the way knowledge or data flows across them; the second addresses their dynamic aspects, i.e., the way actor networks 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, and we illustrate how the framework can be used for modelling and analysing properties of cyber-physical-system protocols. This two-stage constrained-hybridisation process advances the theoretical and practical aspects of hybrid logics by providing new insights and results that go beyond the specific domain of actor networks. On the other hand, and in line with Milner's bigraph paradigm, the paper also makes a novel contribution to the development of formal methods for systems where connectivity and locality play a fundamental role.

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

Unification and generalization are operations on two terms computing respectively their greatest lower bound and least upper bound when the terms are quasi-ordered by subsumption up to variable renaming (i.e., t1⪯t2 iff t1=t2σ for some variable substitution σ). When term signatures are such that distinct functor symbols may be related with a fuzzy equivalence (called a similarity), these operations can be formally extended to tolerate mismatches on functor names and/or arity or argument order. We reformulate and extend previous work with a declarative approach defining unification and generalization as sets of axioms and rules forming a complete constraint-normalization proof system. These include the Reynolds-Plotkin term-generalization procedures, Maria Sessa's “weak” unification with partially fuzzy signatures and its corresponding generalization, as well as novel extensions of such operations to signatures with weaker functor similarities (i.e., with possibly different arities). One advantage of this approach is that it requires no modification of the conventional data structures for terms and substitutions. This and the fact that these declarative specifications are efficiently executable conditional Horn-clauses offers great practical potential for fuzzy information-handling applications.

Finite-trace linear temporal logic: coinductive completeness

Linear temporal logic (LTL) is suitable not only for infinite-trace systems, but also for finite-trace systems. Indeed, LTL is frequently used as a trace specification formalism in runtime verification. The completeness of LTL with only infinite or with both infinite and finite traces has been extensively studied, but similar direct results for LTL with only finite traces are missing. This paper proposes a sound and complete proof system for finite-trace LTL. The axioms and proof rules are natural and expected, except for one rule of coinductive nature, reminiscent of the Godel-Lob axiom. A direct decision procedure for finite-trace LTL satisfiability, a PSPACE-complete problem, is also obtained as a corollary.

Witnessing matrix identities and proof complexity

We use results from the theory of algebras with polynomial identities (PI-algebras) to study the witness complexity of matrix identities. A matrix identity of [Formula: see text] matrices over a field [Formula: see text]is a non-commutative polynomial (f(x1, …, xn)) over [Formula: see text], such that [Formula: see text] vanishes on every [Formula: see text] matrix assignment to its variables. For every field [Formula: see text]of characteristic 0, every [Formula: see text] and every finite basis of [Formula: see text] matrix identities over [Formula: see text], we show there exists a family of matrix identities [Formula: see text], such that each [Formula: see text] has [Formula: see text] variables and requires at least [Formula: see text] many generators to generate, where the generators are substitution instances of elements from the basis. The lower bound argument uses fundamental results from PI-algebras together with a generalization of the arguments in [P. Hrubeš, How much commutativity is needed to prove polynomial identities? Electronic colloquium on computational complexity, ECCC, Report No.: TR11-088, June 2011].We apply this result in algebraic proof complexity, focusing on proof systems for polynomial identities (PI proofs) which operate with algebraic circuits and whose axioms are the polynomial-ring axioms [P. Hrubeš and I. Tzameret, The proof complexity of polynomial identities, in Proc. 24th Annual IEEE Conf. Computational Complexity, CCC 2009, 15–18 July 2009, Paris, France (2009), pp. 41–51; Short proofs for the determinant identities, SIAM J. Comput. 44(2) (2015) 340–383], and their subsystems. We identify a decrease in strength hierarchy of subsystems of PI proofs, in which the [Formula: see text]th level is a sound and complete proof system for proving [Formula: see text] matrix identities (over a given field). For each level [Formula: see text] in the hierarchy, we establish an [Formula: see text] lower bound on the number of proof-steps needed to prove certain identities.Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of matrix identities, and beyond.

Non-Classical Circular Definitions

Circular denitions have primarily been studied in revision theory in the classical scheme. I present systems of circular denitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of denitions, the intrinsic denitions, naturally arises in both schemes. I survey some of the features of this class of denitions.

A logic of goal-directed knowing how

In this paper, we propose a decidable single-agent modal logic for reasoning about goal-directed “knowing how”, based on ideas from linguistics, philosophy, modal logic, and automated planning in AI. We first define a modal language to express “I know how to guarantee \(\varphi \) given \(\psi \)” with a semantics based not on standard epistemic models but on labeled transition systems that represent the agent’s knowledge of his own abilities. The semantics is inspired by conformant planning in AI. A sound and complete proof system is given to capture valid reasoning patterns, which highlights the compositional nature of “knowing how”. The logical language is further extended to handle knowing how to achieve a goal while maintaining other conditions.

Parametrized verification diagrams: temporal verification of symmetric parametrized concurrent systems

This paper studies the problem of verifying temporal properties (including liveness properties) of parametrized concurrent systems executed by an unbounded number of threads. To solve this problem we introduce parametrized verification diagrams (PVDs), that extend the so-called generalized verification diagrams (GVDs) adding support for parametrized verification. Even though GVDs are known to be a sound and complete proof system for non-parametrized systems, the application of GVDs to parametrized systems requires using quantification or finding a potentially different diagram for each instantiation of the parameter (number of threads). As a consequence, the use of GVDs in parametrized verification requires discharging and proving either quantified formulas or an unbounded collection of verification conditions. Parametrized verification diagrams enable the use of asinglediagram to represent the proof that all possible instances of the parametrized concurrent system satisfy the given temporal specification. Checking the proof represented by a PVD requires proving only a finite collection of quantifier-free verification conditions. The PVDs we present here assume that the parametrized systems are symmetric, which covers a large class of concurrent and distributed systems, including concurrent data types. Our second contribution is an implementation of PVDs and its integration into Leap, our prototype theorem prover. Finally, we illustrate empirically, using Leap, the practical applicability of PVDs by building and checking proofs of liveness properties of mutual exclusion protocols and concurrent data structures. To the best of our knowledge, these are the first machine-checkable proofs of liveness properties of these concurrent data types.

Semantics-based program verifiers for all languages

We present a language-independent verification framework that can be instantiated with an operational semantics to automatically generate a program verifier. The framework treats both the operational semantics and the program correctness specifications as reachability rules between matching logic patterns, and uses the sound and relatively complete reachability logic proof system to prove the specifications using the semantics. We instantiate the framework with the semantics of one academic language, KernelC, as well as with three recent semantics of real-world languages, C, Java, and JavaScript, developed independently of our verification infrastructure. We evaluate our approach empirically and show that the generated program verifiers can check automatically the full functional correctness of challenging heap-manipulating programs implementing operations on list and tree data structures, like AVL trees. This is the first approach that can turn the operational semantics of real-world languages into correct-by-construction automatic verifiers.

Reasoning about actions with loops via Hoare logic

Plans with loops are more general and compact than classical sequential plans, and gaining increasing attentions in artificial intelligence (AI). While many existing approaches mainly focus on algorithmic issues, few work has been devoted to the semantic foundations on planning with loops. In this paper, we first develop a tailored action language AL K, together with two semantics for handling domains with non-deterministic actions and loops. Then we propose a sound and (relative) complete Hoare-style proof system for efficient plan generation and verification under 0-approximation semantics, which uses the so-called idea offline planning and on-line querying strategy in knowledge compilation, i.e., the agent could generate and store short proofs as many as possible in the spare time, and then perform quick query by constructing a long proof from the stored shorter proofs using compositional rule. We argue that both our semantics and proof system could serve as logical foundations for reasoning about actions with loops.

Complete Proof Systems for Amortised Probabilistic Bisimulations

The notion of amortisation has been integrated in quantitative bisimulations to make long-term behavioral comparisons between nondeterministic systems. In this paper, we present sound and complete proof systems for amortised strong probabilistic bisimulation and its observational congruence on a process algebra with probability and nondeterminism, and prove their soundness and completeness. Our results make it possible to reason about long-term (observable) probabilistic behaviors by syntactic manipulations.

Dual Erotetic Calculi and the Minimal $${\mathsf{LFI}}$$ LFI

An erotetic calculus for a given logic constitutes a sequent-style proof-theoretical formalization of the logic grounded in Inferential Erotetic Logic (\({\mathsf{IEL}}\)). In this paper, a new erotetic calculus for Classical Propositional Logic (\({\mathsf{CPL}}\)), dual with respect to the existing ones, is given. We modify the calculus to obtain complete proof systems for the propositional part of paraconsistent logic \({\mathsf{CLuN}}\) and its extensions \({\mathsf{CLuNs}}\) and \({\mathsf{mbC}}\). The method is based on dual resolution. Moreover, the resolution rule is non-clausal. According to the authors knowledge, this is the first account of resolution for \({\mathsf{mbC}}\). Last but not least, as the method is grounded in \({\mathsf{IEL}}\), it constitutes an important tool for the so-called question-processing.

A Conceptual Framework for Secrecy-preserving Reasoning in Knowledge Bases

In many applications, Knowledge Bases (KBs) contain confidential or private information (secrets). The KB should be able to use this secret information in its reasoning process but in answering user queries care must be exercised so that secrets are not revealed to unauthorized users. We consider this problem under the Open World Assumption (OWA) in a setting with multiple querying agents M 1 ,…, M m that can pose queries against the KB K and selectively share answers that they receive from K with one or more other querying agents. We assume that for each M i , the KB has a prespecified set of secrets S i that need to be protected from M i . Communication between querying agents is modeled by a communication graph, a directed graph with self-loops. We introduce a general framework and propose an approach to secrecy-preserving query answering based on sound and complete proof systems. The idea is to hide the truthful answer from a querying agent M i by feigning ignorance without lying (i.e., to provide the answer ‘Unknown’ to a query q if it needs to be protected. Under the OWA, a querying agent cannot distinguish between the case that q is being protected (for reasons of secrecy) and the case that it cannot be inferred from K . In the pre-query stage we compute a set of envelopes E 1 , …, E m (restricted to a finite subset of the set of formulae that are entailed by K ) so that S i ⊆ E i , and a query α posed by agent M i can be answered truthfully whenever α ∉ E i and ¬ α ∉ E i . After the pre-query stage, the envelope is updated as needed. We illustrate this approach with two simple cases: the Propositional Horn KBs and the Description Logic AL KBs.

Complete proof systems for weighted modal logic

The weighted transition systems (WTS) considered in this paper are transition systems having both states and transitions labeled with real numbers: the state labels denote quantitative resources, while the transition labels denote costs of transitions in terms of resources. Weighted Modal Logic (WML) is a multi-modal logic that expresses qualitative and quantitative properties of WTSs. While WML has been studied in various contexts and for various application domains, no proof system has been developed for it. In this paper we solve this open problem and propose both weak-complete and strong-complete axiomatizations for WML against WTSs. We prove a series of metatheorems including the finite model property and the existence of canonical models. We show how the proof system can be used in the context of a process-algebra semantics for WML to convert a model-checking problem into a theorem-proving problem. This work emphasizes a series of similarities between WML and the probabilistic/stochastic modal logics for Markov processes and Harsanyi type spaces, such as the use of particular infinitary rules to guarantee the strong-completeness.

On the complexity of stream equality

AbstractWe study the complexity of deciding the equality of streams specified by systems of equations. There are several notions of stream models in the literature, each generating a different semantics of stream equality. We pinpoint the complexity of each of these notions in the arithmetical or analytical hierarchy. Their complexity ranges from low levels of the arithmetical hierarchy such as Π02for the most relaxed stream models, to levels of the analytical hierarchy such as Π11and up to subsuming the entire analytical hierarchy for more restrictive but natural stream models. Since all these classes properly include both the semi-decidable and co-semi-decidable classes, it follows that regardless of the stream semantics employed, there is no complete proof system or algorithm for determining equality or inequality of streams. We also discuss several related problems, such as the existence and uniqueness of stream solutions for systems of equations, as well as the equality of such solutions.

Parametric completeness for separation theories

In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic , and validity in an intended class of separation models , as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory . Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binder-free extension captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of "pure" axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature.

A complete proof system for propositional projection temporal logic

The paper presents a proof system for Propositional Projection Temporal Logic (PPTL) with projection-plus. The syntax, semantics, and logical laws of PPTL are introduced together with an axiom system consisting of axioms and inference rules. To facilitate proofs, some of the frequently used theorems are proved. A normal form of PPTL formulas is presented, and the soundness and completeness of the proof system are demonstrated. To show how the axiom system works, a full omega regular property for the mutual exclusion problem is specified by a PPTL formula and then a deductive proof of the property is performed.

A bialgebraic approach to automata and formal language theory

A bialgebra is a structure which is simultaneously an algebra and a coalgebra, such that the algebraic and coalgebraic parts are compatible. Bialgebras are usually studied over a commutative ring. In this paper, we apply the defining diagrams of algebras, coalgebras, and bialgebras to categories of semimodules and semimodule homomorphisms over a commutative semiring. We then treat automata as certain representation objects of algebras and formal languages as elements of dual algebras of coalgebras. Using this perspective, we demonstrate many analogies between the two theories. Finally, we show that there is an adjunction between the category of “algebraic” automata and the category of deterministic automata. Using this adjunction, we show that K-linear automaton morphisms can be used as the sole rule of inference in a complete proof system for automaton equivalence.

A Logic for Choreographies

We explore logical reasoning for the global calculus, a coordination model based on the notion of choreography, with the aim to provide a methodology for specification and verification of structured communications. Starting with an extension of Hennessy-Milner logic, we present the global logic (GL), a modal logic describing possible interactions among participants in a choreography. We illustrate its use by giving examples of properties on service specifications. Finally, we show that, despite GL is undecidable, there is a significant decidable fragment which we provide with a sound and complete proof system for checking validity of formulae.

Gödel logics with monotone operators

We consider the extension of Gödel logic by a unary operator interpreted by functions on the unit interval with certain monotonicity properties. We prove that validity of propositional formulas is decidable by giving a sound and complete proof system with finitely many axioms. We show also how to transfer the deduction theorem, the lifting lemma and the agreement of entailment and 1-entailment from Gödel logic to the propositional fragment of our extension. Finally, we prove an enumerability result for a ring-normal prenex fragment.

Dealing with Satisfiability and n-ary CSPs in a Logical Framework

We investigate in this work a generalization of the known CNF representation that we call General Normal Form (GNF) and extend the resolution rule of Robinson to design a new sound and complete resolution proof system for the GNF. We show that by using a cardinality operator in the GNF we obtain the new representation CGNF that allows a natural and efficient Boolean encoding for n-ary finite discrete extensional CSPs. We prove that the space complexity of a CSP encoded in the CGNF is identical to its space complexity when it is expressed in the classical CSP representation. We prove that the generalized resolution rule applies for the CGNF encoding and introduce a new inference rule whose application until saturation achieves arc-consistency in a linear time complexity for n-ary CSP expressed in the CGNF encoding. Two enumerative methods for solving CSP instances encoded in this Boolean framework are studied: the first one (equivalent to MAC in CSPs) maintains full arc-consistency at each node of the search tree while the second (equivalent to FC in CSPs) performs partial arc-consistency on each node. Both methods are experimented and compared on some instances of the Ramsey problem, randomly generated 3/4-ary CSPs and promising results are obtained. We also adapted the notion of clause learning defined in SAT for the CGNF encoding. Finally, we show how the proposed inference rule can be used to achieve Path-consistency in the case of binary CSPs encoded in CGNF, and how it can be used to enforce strong path consistency when it is combined with the generalized resolution rule.