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.

Small substructures and decidability issues for first-order logic with two variables

2012 ◽  
Vol 77 (3) ◽  
pp. 729-765 ◽  
Author(s):  
Emanuel Kieroński ◽  
Martin Otto
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Equivalence Relations ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
Surrounding Structure ◽  
Small Model ◽  
Exponential Size

AbstractWe study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

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 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

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.

Propositionalism

2021 ◽  
Vol 118 (8) ◽  
pp. 430-449
Author(s):  
A. C. Paseau ◽  
Keyword(s):  
Propositional Logic ◽  
Logical Structure ◽  
Order Logic ◽  
First Order Logic ◽  
Grammatical Form ◽  
The Real ◽  
First Order ◽  
Clear Sense ◽  
The Rich

Propositionalism is the claim that all logical relations can be captured by propositional logic. It is usually regarded as obviously false, because propositional logic seems too weak to capture the rich logical structure of language. I show that there is a clear sense in which propositional logic can match first-order logic, by producing formalizations that (i) are valid iff their first-order counterparts are, and (ii) also respect grammatical form as the propositionalist construes it. I explain the real reason propositionalism fails, which is more subtle and more interesting.

scholarly journals Paraconsistent games and the limits of rational self-interest

2015 ◽  
Vol 12 (1) ◽  
Author(s):  
Arief Daynes ◽  
Panagiotis Andrikopoulos ◽  
Paraskevas Pagas ◽  
David Latimer
Keyword(s):  
Paraconsistent Logic ◽  
Logical Structure ◽  
Social Contexts ◽  
Order Logic ◽  
First Order Logic ◽  
Material Objects ◽  
Moral Choice ◽  
First Order ◽  
Moral Framework ◽  
Self Interest

It is shown that logical contradictions are derivable from natural translations into first order logic of the description and background assumptions of the Soros Game, and of other games and social contexts that exhibit conflict and reflexivity. The logical structure of these contexts is analysed using proof-theoretic and model-theoretic techniques of first order paraconsistent logic. It is shown that all the contradictions that arise contain the knowledge operator K. Thus, the contradictions do not refer purely to material objects, and do not imply the existence of inconsistent, concrete, physical objects, or the inconsistency of direct sensory experience. However, the decision-making of rational self-interested agents is stymied by the appearance of such intensional contradictions. Replacing the rational self-interest axioms with axioms for an appropriate moral framework removes the inconsistencies. Rational moral choice in conflict-reflexive social contexts then becomes possible.

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