scholarly journals First-Order Logic with Two Variables and Unary Temporal Logic

2002 ◽  
Vol 179 (2) ◽  
pp. 279-295 ◽  
Author(s):  
Kousha Etessami ◽  
Moshe Y. Vardi ◽  
Thomas Wilke
Keyword(s):  
Temporal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

A review of temporal logics

1989 ◽  
Vol 4 (2) ◽  
pp. 141-162 ◽  
Author(s):  
Derek Long
Keyword(s):  
Artificial Intelligence ◽  
Temporal Logic ◽  
Temporal Reasoning ◽  
Order Logic ◽  
First Order Logic ◽  
Temporal Logics ◽  
Detailed Review ◽  
First Order ◽  
Reasoning Tasks

AbstractA series of temporal reasoning tasks are identified which motivate the consideration and application of temporal logics in artificial intelligence. There follows a discussion of the broad issues involved in modelling time and constructing a temporal logic. The paper then presents a detailed review of the major approaches to temporal logics: first-order logic approaches, modal temporal logics and reified temporal logics. The review considers the most significant exemplars within the various approaches, including logics due to Russell, Hayes and McCarthy, Prior, McDermott, Allen, Kowalski and Sergot. The logics are compared and contrasted, particularly in their treatments of change and action, the roles they seek to fulfil and the underlying models of time on which they rest. The paper concludes with a brief consideration of the problem of granularity—a problem of considerable significance in temporal reasoning, which has yet to be satisfactorily treated in a temporal logic.

scholarly journals “Most of” leads to undecidability: Failure of adding frequencies to LTL

2021 ◽  
pp. 82-101
Author(s):  
Bartosz Bednarczyk ◽  
Jakub Michaliszyn
Keyword(s):  
Model Checking ◽  
Formal Verification ◽  
Temporal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Statistical Reasoning ◽  
First Order ◽  
High Interest ◽  
The Past ◽  
Influence Networks

AbstractLinear Temporal Logic (LTL) interpreted on finite traces is a robust specification framework popular in formal verification. However, despite the high interest in the logic in recent years, the topic of their quantitative extensions is not yet fully explored. The main goal of this work is to study the effect of adding weak forms of percentage constraints (e.g. that most of the positions in the past satisfy a given condition, or that $$\sigma $$ σ is the most-frequent letter occurring in the past) to fragments of LTL. Such extensions could potentially be used for the verification of influence networks or statistical reasoning. Unfortunately, as we prove in the paper, it turns out that percentage extensions of even tiny fragments of LTL have undecidable satisfiability and model-checking problems. Our undecidability proofs not only sharpen most of the undecidability results on logics with arithmetics interpreted on words known from the literature, but also are fairly simple. We also show that the undecidability can be avoided by restricting the allowed usage of the negation, and discuss how the undecidability results transfer to first-order logic on words.

scholarly journals A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

2017 ◽  
Vol 28 (4) ◽  
pp. 508-532 ◽  
Author(s):  
PAOLO BOTTONI ◽  
DANIELE GORLA ◽  
STEFANO KASANGIAN ◽  
ANNA LABELLA
Keyword(s):  
Temporal Logic ◽  
Computational Models ◽  
Logical Structure ◽  
Order Logic ◽  
First Order Logic ◽  
Equivalence Relations ◽  
First Order ◽  
Concurrent Processes ◽  
The Difference ◽  
Algebraic Modelling

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB, obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational models.

scholarly journals Temporal Logic with Cyclic Counting and the Degree of Aperiodicity of Finite Automata

2001 ◽  
Vol 8 (53) ◽  
Author(s):  
Zoltán Ésik ◽  
Masami Ito
Keyword(s):  
Temporal Logic ◽  
Regular Language ◽  
Finite Automata ◽  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Direct Products ◽  
First Order ◽  
Closure Conditions ◽  
Positive Integers

We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QA_M of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings. These closure conditions define q-varieties of finite automata. We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages. We also characterize QA_M as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata. We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli. It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M.

scholarly journals First-Order Logic with Two Variables and Unary Temporal Logic

1997 ◽  
Vol 4 (5) ◽  
Author(s):  
Kousha Etessami ◽  
Moshe Y. Vardi ◽  
Thomas Wilke
Keyword(s):  
Computational Complexity ◽  
Temporal Logic ◽  
Linear Temporal Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Model Property ◽  
First Order ◽  
The Past ◽  
Small Model ◽  
Input Formula

We investigate the power of first-order logic with only two variables over<br />omega-words and finite words, a logic denoted by FO2. We prove that FO2 can<br />express precisely the same properties as linear temporal logic with only the unary temporal operators: “next”, “previously”, “sometime in the future”, and “sometime in the past”, a logic we denote by unary-TL. Moreover, our translation from FO2 to unary-TL converts every FO2 formula to an equivalent unary-TL formula that is at most exponentially larger, and whose operator depth is at most twice the quantifier depth of the first-order formula. We show that this translation is optimal.<br />While satisfiability for full linear temporal logic, as well as for<br />unary-TL, is known to be PSPACE-complete, we prove that satisfiability<br />for FO2 is NEXP-complete, in sharp contrast to the fact that satisfiability<br />for FO3 has non-elementary computational complexity. Our NEXP time<br />upper bound for FO2 satisfiability has the advantage of being in terms of<br />the quantifier depth of the input formula. It is obtained using a small model property for FO2 of independent interest, namely: a satisfiable FO2 formula has a model whose “size” is at most exponential in the quantifier depth of the formula. Using our translation from FO2 to unary-TL we derive this small model property from a corresponding small model property for unary-TL. Our proof of the small model property for unary-TL is based on an analysis of unary-TL types.

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.

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.

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