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A sound and complete fuzzy temporal constraint logic
Abstract
In this work, we define an extended fuzzy temporal constraint logic (EFTCL) based on possibilistic logic. EFTCL allows us to handle fuzzy temporal constraints between temporal variables and, therefore, enables us to express interrelated events through fuzzy temporal constraints. EFTCL is compatible with a theoretical temporal reasoning model: the fuzzy temporal constraint networks (FTCN). The syntax, the semantics and the deduction and refutation theorems for EFTCL are similar to those defined for the sound and noncomplete fuzzy temporal constraint logic (FTCL). In this paper, a resolution principle for performing inferences which take these constraints into account is proposed for EFTCL. Moreover, we prove the soundness and the completeness of the refutation by resolution in EFTCL.
... FTCLogic is inspired by an earlier logical formalism Extended Fuzzy Temporal Constraint Logic or EFTCL [13] [10] [12] [11]. Both are based on possibilistic logic [19] [24]. ...
... The syntax and the semantics of EFTCL is based on a first-order language. However, EFTCL is a reified logic [37] formalized in [10] [12] [13]. ...
... In the unification process, the formula below will be applied to calculate the τ c p 2 ∨p 3 i value associated to the resolvent for each variable x i . This formula is a simplification of EFTCL-resolution principle (see [13]). ...
In this paper we present FTCLogic, a formal first-order logic that can manage fuzzy temporal constraints between variables efficiently. In this logic, the use of explicit temporal axioms is unnecessary, and therefore the deduction mechanism doesn't slow down for this reason. FTCLogic has an immediate precedent: the Extended Fuzzy Temporal Constraint Logic or EFTCL. However, while EFTCL is based on Timed Possibilistic Logic, FTCLogic uses the Possibilistic Logic to formulate an original semantics according to its syntax. In fact, FTCLogic defines both syntax and semantics from a powerful combination of two formalisms: the Possibilistic Logic and the Fuzzy Temporal Constraints Networks. FTCLogic has provided the basis for the creation of a fuzzy temporal PROLOG: PROLogic, which is implemented through Haskell, and which is currently undergoing evaluation.
... More recently, FBTL has been chosen as the semantics of requirement specification language RELAX [40]. Other works on fuzzy temporal logics include [10,30,32,43]. ...
... More recently, FBTL has been chosen as the semantics of requirement specification language RELAX [40]. Other works on fuzzy temporal logics include [10,30,32,43]. ...
... Temporal fuzzy systems are novel approaches that are suitable for dynamic systems, when temporal information of the system is important [2]. Temporal fuzzy systems help to handle fuzzy temporal constraint between temporal variables and therefore are able to express interrelated events through fuzzy temporal constraints [3]. Temporal fuzzy systems are also used in data mining. ...
Temporal fuzzy systems are powerful systems that use temporal information in their reasoning process. Probabilistic fuzzy systems are the systems that combine the advantages of probabilistic and fuzzy uncertainty with each other. In this paper, we propose a novel approach, which combines the advantages of probabilistic fuzzy systems with temporal fuzzy systems. The proposed algorithm uses an evolutionary algorithm to find the parameters of the system. In order to examine the proposed method, a fuzzy controller for an overhead crane system is designed. Simulation results show that the proposed method has better performance than other methods.
... Proposals exist for clausal resolution with temporal logics such as PLT [14], PTL [15] or FNext Logic [12]. However, using logics the constraint propagation methods lay on the resolution principle, which is more inefficient than the constraint satisfaction techniques [5]. In the particular case of temporal point algebras [33], the advantages are: (1) the simplicity of temporal relations (before, equals, and after); and (2) the efficiency of their constraint propagation algorithms. ...
Similarity is an essential concept in case-based reasoning (CBR). In domains in which time plays a relevant role, CBR systems require good temporal similarity measures to compare cases. Temporal cases are traditionally represented by a set of temporal features, defining time series and temporal event sequences. In the particular situation where these features are not homogeneous (i.e. combination of qualitative and quantitative information), systems find difficulties in performing the CBR cycle. Furthermore, temporal similarity measures cannot directly apply the efficient time series techniques, requiring new approaches to deal with these heterogeneous sequences. To this end, recent proposals are focused on direct matching between pairs of features within sequences, mainly based on classical distances. However, three limitations to the traditional approaches have been identified: (1) they do not consider the implicit temporal relations amongst all features of the sequence (ignoring a large amount of temporal information); (2) they ignore the uncertainty produced in any process of analogy; (3) they are designed to compare pairs of sequences, limiting their use to basic aspects of the Retrieval step of CBR (no benefits on other CBR steps). Temporal constraint networks have proved to be useful tools for temporal representation and reasoning, and can be easily extended to manage imprecision and uncertainty. An approach to solve similarity problems could be the transformation of these heterogeneous sequences into uncertain temporal relations, obtaining a temporal constraint network. The overall uncertainty of this network can be considered as an effective indicator of the sequences similarity. Therefore, this paper proposes a non-classical approach to measure temporal similarity of cases which are heterogeneous temporal event sequences. Given two or more sequences, the temporal similarity is measured by describing a unique temporal scenario of possibilistic temporal relations and calculating the uncertainty produced.
This paper shows that the fuzzy temporal logic can model figures of thought to describe decision-making behaviors. In order to exemplify, some economic behaviors observed experimentally were modeled from problems of choice containing time, uncertainty and fuzziness. Related to time preference, it is noted that the subadditive discounting is mandatory in positive rewards situations and, consequently, results in the magnitude effect and time effect, where the last has a stronger discounting for earlier delay periods (as in, one hour, one day), but a weaker discounting for longer delay periods (for instance, six months, one year, ten years). In addition, it is possible to explain the preference reversal (change of preference when two rewards proposed on different dates are shifted in the time). Related to the Prospect Theory, it is shown that the risk seeking and the risk aversion are magnitude dependents, where the risk seeking may disappear when the values to be lost are very high.
https://arxiv.org/abs/1901.01970
The aim of this project (MEDICI: Inpatient/Outpatient Monitoring for Diagnosis and Medical Research in Ischaemic Cardiopathy: Tools for Acquisition, Visualization, Integra- tion and Knowledge Discovery) is the design, development and implementation of a set of tools for inpatient and outpatient monitoring of patient sufiering Ischaemic Cardiopathy. These tools will provide support for both medical care and clinical research tasks. The work is organized around three development lines: a) Subsystems for data collection in difierent scenarios, including clinical history data, monitored electric and hemodynamic signals and data related to remotely monitored signals outside the hospital. b) Techni- ques for integrating the collected information and presenting it as a unique virtual clinical history associated to the patient; and c) tools for the extraction and validation of medical knowledge.
High-level descriptions of real-time systems often use fuzzy notions of time that are left open to domain specific interpretations. In order to verify that a given implementation conforms to such loosely defined specifications, the typical approach is to verify the implementation to be correct within well defined limits of time tolerance. This approach determines whether the real-time requirements are met, but does not reflect how well it is met.
Our goal in this paper is to prescribe the development of timed specifications using fuzzy notions of time, and to present a methodology for computing the quality of satisfaction of the specification on a given implementation using domain specific fuzzy membership functions. With this objective, we combine the notions of real-time interval temporal logic (like Metric Interval Temporal Logic) and fuzzy logic to derive FRTL, a fuzzy real-time temporal logic. The novelty of the proposed logic is in introducing the notion of fuzzy time intervals into the core fabric of conventional metric temporal logic. We present a method for evaluating the fuzzy truth of FRTL properties on finite traces. We discuss the motivation of computing the fuzzy truth towards evaluating the quality of control in time critical embedded control system applications. We also show that two important related problems from the domain of mixed-signal design verification, are subsumed by the proposed framework of analysis.
Presents an environment for the exploration of low-power memory configurations. The environment uses accurate models of memory energy, area and delay to evaluate different configurations. Taking the constraints specified by the user and memory profiling information from the application, the environment can optimize the memory organization for a specific application. The environment also places and routes the organizations to provide more accurate energy, area and delay estimates. Our overall memory system estimation is highly accurate, with average absolute percentage errors within 7%. Within the environment, the user can explore powerful energy/area/delay trade-offs. We present results which show energy reductions within the memory system of up to 63%
. This paper is an attempt to cast both uncertainty and time in a logical framework. It generalizes possibilistic logic, previously developed by the authors, where each classical formula is associated with a weight which obeys the laws of possibility theory. In the possibilistic temporal logic we present here, each formula is associated with a time set (a fuzzy set in the more general case) which represents the set of instants where the formula is certainly true (more or less certainly true in the general case). When a particular instant is fixed we recover possibilistic logic. Timed possibilistic logic generalizes possibilistic logic also in the sense that we substitute the lattice structure of the set of the (fuzzy) subsets of the temporal scale to the lattice structure underlying the certainty weights in possibilistic logic. Thus many results from possibilistic logic can be straightforwardly generalized to timed possibilistic logic. Illustrative examples are given. 1. Introduction ...
We expand the applicability of the clausal resolution technique to the branching-time temporal logic ECTL+. ECTL+ is strictly more expressive than the basic computation tree logic CTL and its extension, ECTL, as it allows Boolean combinations of fairness and single temporal operators. We show that any ECTL+ formula can be translated to a normal form the structure of which was initially defined for CTL and then applied to ECTL. This enables us to apply to ECTL+ a resolution technique defined over the set of clauses. Our correctness argument also bridges the gap in the correctness proof for ECTL: we show that the transformation procedure for ECTL preserves unsatisfiability.
We introduce hybrid Ockhamist temporal logic, which combines the
mechanisms of hybrid logic with Ockhamist semantics by employing
nominals, satisfaction operators, binders, and quantifiers over
branches. We provide a complete (with respect to bundled trees
semantics) axiomatic system for the basic hybrid Ockhamist temporal
logic (HOT) and for some of its extensions including the full hybrid
Ockhamist temporal logic. The fill system is expressively equivalent to
the first-order logic over trees extended with branch quantifiers which
was proved decidable previously
An approach to automated deduction under uncertainty, based on
possibilistic logic, is described; for that purpose we deal with clauses
weighted by a degree that is a lower bound of a necessity or a
possibility measure, according to the nature of the uncertainty. Two
resolution rules are used for coping with the different situations, and
the classical refutation method can be generalized with these rules.
Also, the lower bounds are allowed to be functions of variables involved
in the clauses, which results in hypothetical reasoning capabilities. In
cases where only lower bounds of necessity measures are involved, a
semantics is proposed in which the completeness of the extended
resolution principle is proved. The relation between our approach and
the idea of minimizing abnormality is briefly discussed. Moreover,
deduction from a partially inconsistent knowledge base can be managed in
this approach and captures a form of nonmonotonicity
In this paper we propose a propositional temporal language based on fuzzy temporal constraints which turns out to be expressive enough for domains like many coming from medicine where knowledge is of propositional nature and an explicit handling of time, imprecision and uncertainty are required. The language is provided with a natural possibilistic semantics to account for the uncertainty issued by the fuzziness of temporal constraints. We also present an inference system based on specific rules dealing with the temporal constraints and a general fuzzy modus ponens rule whereby behaviour is shown to be sound. The analysis of the different choices as fuzzy operators leads us to identify the well-known Lukasiewicz implication as very appropriate to define the notion of possibilistic entailment, an essential element of our inference system.
In this article, we examine how clausal resolution can be applied to a specific, but widely used, non-classical logic, namely discrete linear temporal logic. Thus, we first define a normal form for temporal formulae and show how arbitrary temporal formulae can be translated into the normal form, while preserving satisfiability. We then introduce novel resolution rules that can be applied to formulae in this normal form, provide a range of examples and examine the correctness and complexity of this approach is examined and. This clausal resolution approach. Finally, we describe related work and future developments concerning this work.
In this paper we present FTCLogic, a formal first-order logic that can manage fuzzy temporal constraints between variables efficiently. In this logic, the use of explicit temporal axioms is unnecessary, and therefore the deduction mechanism doesn't slow down for this reason. FTCLogic has an immediate precedent: the Extended Fuzzy Temporal Constraint Logic or EFTCL. However, while EFTCL is based on Timed Possibilistic Logic, FTCLogic uses the Possibilistic Logic to formulate an original semantics according to its syntax. In fact, FTCLogic defines both syntax and semantics from a powerful combination of two formalisms: the Possibilistic Logic and the Fuzzy Temporal Constraints Networks. FTCLogic has provided the basis for the creation of a fuzzy temporal PROLOG: PROLogic, which is implemented through Haskell, and which is currently undergoing evaluation.
In this work we present a model for the representation of imprecise temporal information. The model permits the representation of absolute and relative temporal references and includes qualitative and quantitative relationships. We also present an efficient algorithm that combines the information represented, minimizes imprecisions, and detects inconsistencies. We discuss the practical use of the model through examples in different domains.
This chapter discusses temporal and modal logic. The chapter describes a multiaxis classification of systems of temporal logic. The chapter describes the framework of linear temporal logic. In both its propositional and first-order forms, linear temporal logic has been widely employed in the specification and verification of programs. The chapter describes the competing framework of branching temporal logic, which has seen wide use. It also explains how temporal logic structures can be used to model concurrent programs using non-determinism and fairness. The chapter also discusses other modal and temporal logics in computer science. The chapter describes the formal syntax and semantics of Propositional Linear Temporal Logic (PLTL). The chapter also describes the formal syntax and semantics for two representative systems of propositional branching-time temporal logics.
In this work we present a new Automated Theorem Prover, called TAS-FNext, applied to temporal logic. This is part of a broader project developed by our research group GIMAC. It is an extension of works [4], [5] and [6] concerns classical logic and [9] Minimal Temporal Logic.
TAS-FNext is strongly based on formula structures and, specifically, on the structure of the syntactic tree of each formula. It works by making transformations on these syntactic trees (TAS stands for Transformaciones de rboles Sintcticos, Spanish rendering of Syntactic Tree Transformations).
The power of TAS-FNext is mainly based on its capacity to extract efficiently any potentially useful information contained in the syntactic trees with two aims: to detect and classify any subformulas found, whether or not they are valid, unsatisfiable, equivalent or equal, and to transform the formula in question into a simultaneous unsatisfiable, but with less size, formula.
TAS-FNext is sound and complete, and, moreover, it generates counter-models in a natural way [8].
In this paper we establish the relationship between the syntax and semantics of a fuzzy temporal constraint logic (FTCL) proposed by Crdenas et al. FTCL enables us to express interrelated events by means of fuzzy temporal constraints. Moreover, it provides a resolution principle for performing inferences which take these constraints into account. FTCL is compatible with the theoretical temporal reasoning model proposed by Marn et al. – the Fuzzy Temporal Constraint Networks (FTCN). The main contributions of this paper are, on the one hand, the proofs of the FTCL-deduction and the FTCL-refutation theorems, and, on the other, the proof of the soundness of the refutation by resolution in this formal system, together with an exhaustive study of its completeness.
A computer program capable of acting intelligently in the world must have a general representation of the world in terms of which its inputs are interpreted. Designing such a program requires commitments about what knowledge is and how it is obtained. Thus, some of the major traditional problems of philosophy arise in artificial intelligence.
In this work we propose a fuzzy temporal constraint logic. First of all, we provide the formal language which will allow the expression of well-formed formulas related to the temporal events by means of temporal constraints. Secondly, we introduce a valid resolution principle in order to solve the queries in this logic. Finally, we will show that this resolution principle is a generalization of the resolution principle proposed for a possibilistic logic with fuzzy predicates (Dubois and Prade, Internat. J. Approx. Reason. 4 (1990) 1–21). All this will serve to reason within a context of the theoretical model of the temporal reasoning proposed by Marı́n and Barro (Fuzzy Temporal Constraint Network, FTCN) (Marı́n et al., Cybernet. Systems 25(2) (1994) 207–215). This model underlies a module for the resolution of temporal queries. This module belongs to a diagnostic and intelligent monitoring system of patients, based on temporal reasoning. The system is applied to the patients admitted in the Intensive Care Units with severe ischemic cardiopathy, submitted to continuous monitoring of the electrical and mechanical signals of the heart. However, what is exposed here in this document is not limited to a field of application in particular, but instead, it is completely general.
In this paper we present a model for the representation and handling of fuzzy temporal references. We define the concepts of date, time extent, and interval, according to the formalism of possibility theory. We introduce relations between the temporal entities dates and intervals, interpreted as constraints on the distance between dates and projected onto Fuzzy Temporal Constraint Satisfaction Networks. We introduce a language for the representation and manipulation of temporal entities and relations, which captures some of the terms we use in our expressions in the natural language and therefore, it is a flexible and powerful interface for those systems in which the representation of fuzzy temporal information is necessary. Our approach permits a common interpretation of qualitative and quantitative temporal relations, facilitating the relativization of the meaning of the temporal relations to each specific application context and the verification of relations between temporal entities.
We propose three methods for obtaining solutions in fuzzy constraint networks and study their application to the problem of ordering fuzzy numbers. The techniques proposed may be classified as defizziification functions which are applicable to any set of mutually dependent fuzzy numbers in which the dependence relationships are represented by means of metric constraints. We suggest the use of these techniques for ordering linked variables in an efficient manner, and discuss their behavior regarding several quality criteria. The first application realm of these techniques is temporal reasoning.
Clausal temporal resolution is characterised by a translation of the formulae whose satisfiability is to be established to a normal form, step resolution (similar to classical resolution) on formulae occurring at the same states and temporal resolution between formulae describing properties over a longer period. The most complex part of the method occurs in searching for candidates for the temporal resolution operation, something that may need to be carried out several times.
In this paper we consider a new technique for finding the candidates for the temporal resolution operation. Although related to the previously developed external search procedure, this new approach not only allows the temporal resolution operation to be carried out at any moment, but also simplifies any subsequent search required for similar temporal formulae.
Finally, in contrast with previous approaches, this search can be seen as an inherent part of the resolution process, rather than an external procedure that is only called in certain situations.
In this paper we propose a propositional tem poral language based on fuzzy temporal con straints which turns out to be expressive enough for domains -like many coming from medicine- where knowledge is of propositional nature and an explicit handling of time, impre cision and uncertainty are required. The lan guage is provided with a natural possibilistic semantics to account for the uncertainty issued by the fuzziness of temporal constraints. We also present an inference system based on spe cific rules dealing with the temporal constraints and a general fuzzy modus ponens rule whereby behaviour is shown to be sound. The analysis of the different choices as fuzzy operators leads us to identify the well-known Lukasiewicz impli cation as very appropriate to define the notion of possibilistic entailment, an essential element of our inference system.
In this work we propose a fuzzy temporal constraint logic. First of all, we provide the formal language which will allow the expression of well-formed formulas related to the temporal events by means of temporal constraints. Secondly, we introduce a valid resolution principle in order to solve the queries in this logic. Finally, we will show that this resolution principle is a generalization of the resolution principle proposed for a possibilistic logic with fuzzy predicates (Dubois and Prade, Internat. J. Approx. Reason. 4 (1990) 1{21). All this will serve to reason within a context of the theoretical model of the temporal reasoning proposed by Marn and Barro (Fuzzy Temporal Constraint Network, FTCN) (Marn et al., Cybernet. Systems 25(2) (1994) 207{215). This model underlies a module for the resolution of temporal queries. This module belongs to a diagnostic and intelligent monitoring system of patients, based on temporal reasoning. The system is applied to the patients admitted in the Intensive Care Units with severe ischemic cardiopathy, submitted to continuous monitoring of the electrical and mechanical signals of the heart. However, what is exposed here in this document is not limited to a eld of application in particular, but instead, it is completely general. c 2001 Elsevier Science B.V. All rights reserved.
This paper presents a reified temporal logic for representing and reasoning about temporal and non-temporal relationships between non-temporal assertions, A clear syntax and semantics for the logic is formally provided. Three types of predicates, temporal predicates, non-temporal predicates and meta-predicates, are introduced. Terms of the proposed language are partitioned into three types, temporal terms, non-temporal terms and propositional terms. Reified propositions consist of formulae with each predicate being either a temporal predicate or a meta-predicate. Meta-predicates may take both temporal terms and propositional terms together as arguments or take propositional terms alone. A standard formula of the classical first-order language with each predicate being a non-temporal predicate taking only non-temporal terms as arguments is reified as just a propositional term. A general time ontology has been provided which can be specialized to a variety of existing temporal systems, The new logic allows one to predicate and quantify over propositional terms while according a special status of time; for example, assertions such as 'effects cannot precede their causes' is ensured in the logic, and some problematic temporal aspects including the delay time between events and their effects can be conveniently expressed. Applications of the logic are presented including the characterization of the negation of properties and their contextual sentences, and the expression of temporal relations between actions and effects.
The notion of time is uniquitous in any activity that requires intelligence. In particular, several important notions like change, causality, action are described in terms of time. Therefore, the representation of time and reasoning about time is of crucial importance for many Artificial Intelligence systems. Specifically during the last 10 years, it has been attracting the attention of many AI researchers. In this survey, the results of this work are analysed. Firstly, Temporal Reasoning is defined. Then, the most important representational issues which determine a Temporal Reasoning approach are introduced: the logical form on which the approach is based, the ontology (the units taken as primitives, the temporal relations, the algorithms that have been developed,...) and the concepts related with reasoning about action (the representation of change, causality, action,...). For each issue the different choices in the literature are discussed.
First-order temporal logic is a concise and powerful notation, with many potential applications in both Computer Science and Artificial Intelligence. While the full logic is highly complex, recent work on monodic first-order temporal logics has identified important enumerable and even decidable fragments. In this paper, we develop a clausal resolution method for the monodic fragment of first-order temporal logic over expanding domains. We first define a normal form for monodic formulae and then introduce novel resolution calculi that can be applied to formulae in this normal form. We state correctness and completeness results for the method. We illustrate the method on a comprehensive example. The method is based on classical first-order resolution and can, thus, be efficiently implemented.
In this paper we describe a module, the temporal solving module
(TSM), for answering queries sent to a data blackboard which must take
into account fuzzy temporal constraints among blackboard variables. The
TSM receives the queries written in an interaction language that is
translated into a language of a reified many-sorted first-order logic,
called fuzzy temporal constraint logic (FTCL). This logic has been
formalized in previous works, together with a sound resolution principle
for it. We justify the search strategy used in the implementation of the
unit resolution. The main results achieved in this work are the
simplification criteria and the heuristic evaluation function given for
improving the search process to achieve the refutation. First, we use
the fuzzy number ordering functions proposed by Marin et al. (1997) for
obtaining a simplification of queries that imply an exhaustive search in
the data blackboard. Secondly, in order to prune the tree, we use a
function to check the possibility of obtaining a fuzzy temporal instant
different to the empty set, in the unification of two imprecise temporal
variables
Intelligent systems require a systematic way to represent and handle temporal information containing uncertainty. In particular, a logical framework is needed that can represent uncertain temporal information and its relationships with logical formulae. Fuzzy linear temporal logic (FLTL), a generalization of propositional linear temporal logic (PLTL) with fuzzy temporal events and fuzzy temporal states defined on a linear time model, was previously proposed for this purpose. However, many systems are best represented by branching time models in which each state can have more than one possible future path. In this paper, fuzzy branching temporal logic (FBTL) is proposed to address this problem. FBTL adopts and generalizes concurrent tree logic (CTL*), which is a classical branching temporal logic. The temporal model of FBTL is capable of representing fuzzy temporal events and fuzzy temporal states, and the order relation among them is represented as a directed graph. The utility of FBTL is demonstrated using a fuzzy job shop scheduling problem as an example.
Efficient resolution mechanism for fuzzy temporal constraint logic
- S Goodwin
- A Trudel
Reified temporal logics: Semantical and ontological considerations
- B Du Boulay
- D Hogg
- L Steels