scholarly journals Universal Invariant Checking of Parametric Systems with Quantifier-free SMT Reasoning

2021 ◽  
pp. 131-147
Author(s):  
Alessandro Cimatti ◽  
Alberto Griggio ◽  
Gianluca Redondi
Keyword(s):  
Communication Protocols ◽  
Order Logic ◽  
Automatic Verification ◽  
First Order Logic ◽  
Transition Systems ◽  
Fully Automatic ◽  
And Control ◽  
Inductive Invariants ◽  
Automatic Technique ◽  
Parametric Systems

AbstractThe problem of invariant checking in parametric systems – which are required to operate correctly regardless of the number and connections of their components – is gaining increasing importance in various sectors, such as communication protocols and control software. Such systems are typically modeled using quantified formulae, describing the behaviour of an unbounded number of (identical) components, and their automatic verification often relies on the use of decidable fragments of first-order logic in order to effectively deal with the challenges of quantified reasoning.In this paper, we propose a fully automatic technique for invariant checking of parametric systems which does not rely on quantified reasoning. Parametric systems are modeled with array-based transition systems, and our method iteratively constructs a quantifier-free abstraction by analyzing, with SMT-based invariant checking algorithms for non-parametric systems, increasingly-larger finite instances of the parametric system. Depending on the verification result in the concrete instance, the abstraction is automatically refined by leveraging canditate lemmas from inductive invariants, or by discarding previously computed lemmas.We implemented the method using a quantifier-free SMT-based IC3 as underlying verification engine. Our experimental evaluation demonstrates that the approach is competitive with the state of the art, solving several benchmarks that are out of reach for other tools.

scholarly journals A Quantified Coalgebraic van Benthem Theorem

2021 ◽  
pp. 551-571
Author(s):  
Paul Wild ◽  
Lutz Schröder
Keyword(s):  
Modal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Transition Systems ◽  
First Order ◽  
Wide Range ◽  
Symmetry Assumption ◽  
Invariant Properties ◽  
Bounded Rank

AbstractThe classical van Benthem theorem characterizes modal logic as the bisimulation-invariant fragment of first-order logic; put differently, modal logic is as expressive as full first-order logic on bisimulation-invariant properties. This result has recently been extended to two flavours of quantitative modal logic, viz. fuzzy modal logic and probabilistic modal logic. In both cases, the quantitative van Benthem theorem states that every formula in the respective quantitative variant of first-order logic that is bisimulation-invariant, in the sense of being nonexpansive w.r.t. behavioural distance, can be approximated by quantitative modal formulae of bounded rank. In the present paper, we unify and generalize these results in three directions: We lift them to full coalgebraic generality, thus covering a wide range of system types including, besides fuzzy and probabilistic transition systems as in the existing examples, e.g. also metric transition systems; and we generalize from real-valued to quantale-valued behavioural distances, e.g. nondeterministic behavioural distances on metric transition systems; and we remove the symmetry assumption on behavioural distances, thus covering also quantitative notions of simulation.

scholarly journals Automatic white-box testing of first-order logic ontologies

2019 ◽  
Vol 29 (5) ◽  
pp. 723-751
Author(s):  
Javier Álvez ◽  
Montserrat Hermo ◽  
Paqui Lucio ◽  
German Rigau
Keyword(s):  
Automated Reasoning ◽  
Computational Cost ◽  
Order Logic ◽  
First Order Logic ◽  
Automated Generation ◽  
First Order ◽  
Formal Ontologies ◽  
Expected Information ◽  
Fully Automatic ◽  
White Box Testing

AbstractFormal ontologies are axiomatizations in a logic-based formalism. The development of formal ontologies is generating considerable research on the use of automated reasoning techniques and tools that help in ontology engineering. One of the main aims is to refine and to improve axiomatizations for enabling automated reasoning tools to efficiently infer reliable information. Defects in the axiomatization cannot only cause wrong inferences, but can also hinder the inference of expected information, either by increasing the computational cost of or even preventing the inference. In this paper, we introduce a novel, fully automatic white-box testing framework for first-order logic (FOL) ontologies. Our methodology is based on the detection of inference-based redundancies in the given axiomatization. The application of the proposed testing method is fully automatic since (i) the automated generation of tests is guided only by the syntax of axioms and (ii) the evaluation of tests is performed by automated theorem provers (ATPs). Our proposal enables the detection of defects and serves to certify the grade of suitability—for reasoning purposes—of every axiom. We formally define the set of tests that are (automatically) generated from any axiom and prove that every test is logically related to redundancies in the axiom from which the test has been generated. We have implemented our method and used this implementation to automatically detect several non-trivial defects that were hidden in various FOL ontologies. Throughout the paper we provide illustrative examples of these defects, explain how they were found and how each proof—given by an ATP—provides useful hints on the nature of each defect. Additionally, by correcting all the detected defects, we have obtained an improved version of one of the tested ontologies: Adimen-SUMO.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

2021 ◽  
Vol 178 (1-2) ◽  
pp. 1-30
Author(s):  
Florian Bruse ◽  
Martin Lange ◽  
Etienne Lozes
Keyword(s):  
Model Checking ◽  
Lower Bounds ◽  
Expressive Power ◽  
Higher Order ◽  
Order Logic ◽  
High Complexity ◽  
Transition Systems ◽  
Recursive Formulas ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

scholarly journals GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

2021 ◽  
pp. 1-28
Author(s):  
IVANO CIARDELLI ◽  
GIANLUCA GRILLETTI
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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