Logic MTL Research Articles

Runtime Verification over Out-of-order Streams

We present an approach for verifying systems at runtime. Our approach targets distributed systems whose components communicate with monitors over unreliable channels, where messages can be delayed, reordered, or even lost. Furthermore, our approach handles an expressive specification language that extends the real-time logic MTL with freeze quantifiers for reasoning about data values. The logic’s main novelty is a new three-valued semantics that is well suited for runtime verification as it accounts for partial knowledge about a system’s behavior. Based on this semantics, we present online algorithms that reason soundly and completely about streams where events can occur out of order. We also evaluate our algorithms experimentally. Depending on the specification, our prototype implementation scales to out-of-order streams with hundreds to thousands of events per second.

준구조 퍼지 논리 MTL을 위한 집합 이론적 루트리-마이어형 의미론

준구조 퍼지 논리 MTL을 위한 집합 이론적 루트리-마이어형 의미론

Querying Log Data with Metric Temporal Logic


 
 
 We propose a novel framework for ontology-based access to temporal log data using a datalog extension datalogMTL of the Horn fragment of the metric temporal logic MTL. We show that datalogMTL is EXPSPACE-complete even with punctual intervals, in which case full MTL is known to be undecidable. We also prove that nonrecursive datalogMTL is PSPACE-complete for combined complexity and in AC0 for data complexity. We demonstrate by two real-world use cases that nonrecursive datalogMTL programs can express complex temporal concepts from typical user queries and thereby facilitate access to temporal log data. Our experiments with Siemens turbine data and MesoWest weather data show that datalogMTL ontology-mediated queries are efficient and scale on large datasets.
 
 

Logical foundation of the quintuple implication inference methods

The quintuple implication inference methods for fuzzy reasoning characterizes the solution B⁎(A⁎) for the fuzzy modus ponens (fuzzy modus tollens) as the formula that is best supported by A→B, A⁎→A and A⁎(A→B,B→B⁎ and B⁎). In this study, we provide a predicate formal representation of the solution for the quintuple implication inference methods based on the many-sorted first-order formal system Monoidal t-norm based logic MTL∀ms, including detailed logic proofs. We bring the quintuple implication inference methods within a logical framework and provide a sound logic foundation for the quintuple implication inference methods of fuzzy reasoning.

Axiomatizations of team logics

In a modular approach, we lift Hilbert-style proof systems for propositional, modal and first-order logic to generalized systems for their respective team-based extensions. We obtain sound and complete axiomatizations for the dependence-free fragment FO(∼) of Väänänen's first-order team logic TL, for propositional team logic PTL, quantified propositional team logic QPTL, modal team logic MTL, and for the corresponding logics of dependence, independence, inclusion and exclusion.As a crucial step in the completeness proof, we show that the above logics admit, in a particular sense, a semantics-preserving elimination of modalities and quantifiers from formulas.

Ontology-Based Data Access with a Horn Fragment of Metric Temporal Logic

We advocate datalogMTL, a datalog extension of a Horn fragment of the metric temporal logic MTL, as a language for ontology-based access to temporal log data. We show that datalogMTL is EXPSPACE-complete even with punctual intervals, in which case MTL is known to be undecidable. Nonrecursive datalogMTL turns out to be PSPACE-complete for combined complexity and in AC0 for data complexity. We demonstrate by two real-world use cases that nonrecursive datalogMTL programs can express complex temporal concepts from typical user queries and thereby facilitate access to log data. Our experiments with Siemens turbine data and MesoWest weather data show that datalogMTL ontology-mediated queries are efficient and scale on large datasets of up to 11GB.

On similarity in fuzzy description logics

This paper is a contribution to the study of similarity relations between objects represented as attribute-value pairs in Fuzzy Description Logics. For this purpose we use concrete domains in the fuzzy description logic IALCEF(D) associated either with a left-continuous or with a finite t-norm. We propose to expand this fuzzy description logic by adding a Similarity Box (SBox) including axioms expressing properties of fuzzy equalities. We also define a global similarity between objects from similarities between the values of each object attribute (local similarities) and we prove that the global similarity defined using a t-norm inherits the usual properties of the local similarities (reflexivity, symmetry or transitivity). We also prove a result relative to global similarities expressing that, in the context of the logic MTL∀, similar objects have similar properties, being these properties expressed by predicate formulas evaluated in these objects.

The logic of the strongest and the weakest t-norms

It is well known that the strongest t-norm, that is the largest with respect to the pointwise order, is the minimum. In 2001, the logic MTL was introduced as the base of a framework of many-valued logics, and in 2002 it was shown that it is the logic of all left-continuous t-norms and their residua. Within this family of logics, the many-valued logic associated with the minimum t-norm is the Gödel one, whilst there is no logic associated to the drastic product t-norm. Indeed the drastic product is not left-continuous, and hence it does not have a residuum. However, in a recent paper the logic DP has been studied, by showing that the monoidal operation of every DP-chain is like the drastic product t-norm. In this paper we present the logic EMTL, whose algebraic variety is the smallest to contain the ones of Gödel- and DP-algebras. We show that the chains in this algebraic variety are exactly all the Gödel- and DP-chains, we classify and axiomatize all the subvarieties, and we show some limitative results concerning the amalgamation property.

On some questions concerning the axiomatisation of WNM-algebras and their subvarieties

In a seminal paper Esteva and Godo introduced monoidal t-norm-based logic MTL and some of its prominent extensions such as NM and WNM. We notice that NM is axiomatisable from IMTL, and hence MTL, with one-variable axioms, by instantiating the WNM axiom over one variable. This observation leads us here to study the logic axiomatised by extending MTL by this one-variable axiom. We shall refer to its equivalent algebraic semantics as the variety of GHP-algebras, for those algebras will be shown to form the largest variety of MTL-algebras such that the falsum-free reducts of the positive cones of their chains are the most general totally ordered Gödel hoops. Among other results we obtain a general description of GHP standard algebras, and use the latter to characterise those extensions of WNM that can be obtained from GHP via the same set of extending axioms.

Logics of formal inconsistency arising from systems of fuzzy logic

This paper proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this paper we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs.

Uninorm logic with the [formula omitted]-potency axiom

This paper investigates a class of extensions CnUL of uninorm logic UL, which is obtained by adding the n-potency axiom to UL. The main result of this paper is that by generalizing Jenei and Montagna-style approach for proving standard completeness for monoidal t-norm based logic MTL, we obtain the standard completeness for CnUL. In addition, we present the suitable analytic hypersequent calculi for CnUL.

Graded properties of unary and binary fuzzy connectives

The paper studies basic graded properties of unary and binary fuzzy connectives, i.e., unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic MTL of left-continuous t-norms. The properties studied in this paper are graded generalizations of monotony, Lipschitz continuity, null and unit elements, idempotence, commutativity, and associativity. The paper elaborates the initial study presented in previous papers and focuses mainly on parameterization of graded properties by conjunction-multiplicities of subformulae in the defining formulae, preservation of graded properties under compositions and slight variations of fuzzy connectives, the values of graded properties for basic connectives of the ground logic, and the dependence of the values on the ground logic. The results are proved in the formal framework of higher-order fuzzy logic MTL, also known as Fuzzy Class Theory (FCT). General theorems provable in FCT are illustrated on several semantic examples.

On the independence of axioms in BL and MTL

We prove that the axiom expressing that the multiplicative conjunction of two formulae implies the first one of them is redundant in the standard Hilbert-style calculi of Hájek's basic logic BL and Esteva and Godo's monoidal t-norm based logic MTL. This proof does not use the axiom expressing that multiplicative conjunction is commutative, which is already known to be redundant. Therefore both of these axioms are simultaneously redundant. We also show that all the other axioms are independent of each other.

Dual tableau for monoidal triangular norm logic MTL

Monoidal triangular norm logic MTL is the logic of left-continuous triangular norms. In the paper we present a relational formalization of the logic MTL and then we introduce relational dual tableau that can be used for verification of validity of MTL-formulas. We prove soundness and completeness of the system.

Algebras for Galois-style connections and their discrete duality

Bounded distributive lattices with pairs of operators forming Galois, dual Galois, residuation and dual residuation connections, respectively, are considered. A discrete duality for these classes of algebras and the corresponding classes of relational systems is established. These results are extended to the fuzzy logic MTL.

An early approach toward graded identity and graded membership in set theory

The paper considers an early approach toward a (fuzzy) set theory with a graded membership predicate and a graded equality relation which had been developed by the German mathematician Klaua in 1965. In the context of the mathematical fuzzy logic MTL of left-continuous t-norms we discuss some properties of these graded relations. We compare the simultaneous recursive definitions of these relations with the very similar approach toward Boolean algebra valued interpretations of membership and equality, presented in 1967 by Scott and R. Solovay in the context of independence proofs for ZF set theory. Finally we speculate about possible reasons why Klaua soon abandoned this approach.

Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions

Extensions of monoidal t-norm logic MTL and related fuzzy logics with truth stresser modalities such as globalization and “very true” are presented here both algebraically in the framework of residuated lattices and proof-theoretically as hypersequent calculi. Completeness with respect to standard algebras based on t-norms, embeddings between logics, decidability, and the finite embedding property are then investigated for these logics.

On the expressiveness of MTL in the pointwise and continuous semantics

We show that the pointwise version of the logic MTL is strictly less expressive than the continuous version, over finitewords. The proof is constructive in that we exhibit a timed language, which is definable in the continuous semantics but is not definable in the pointwise semantics.

Short note: on the redundancy of axiom (A3) in BL and MTL

In this short note we prove the redundance of axiom (A3) of Hajek's Basic Fuzzy Logic. We also extend this result to Esteva and Godo's logic MTL.

Simplification and independence of axioms of fuzzy logic systems IMTL and NM

Monoidal t-norm-based logic MTL is the common formalization of all fuzzy logic of left continuous t-norms and their residua. This system has two important schematic extensions: involutive monoidal t-norm-based logic IMTL and nilpotent minimum logic NM. In this paper, we discuss simplification and independence of axioms of IMTL and NM, the shorter axioms of both logics, and an independent axioms of the system IMTL are given.