Reversible Regular Languages: Logical and Algebraic Characterisations

2021 ◽  
Vol 180 (4) ◽  
pp. 333-350
Author(s):  
Paul Gastin ◽  
Amaldev Manuel ◽  
R. Govind
Keyword(s):  
Normal Form ◽  
General Problem ◽  
Second Order ◽  
Regular Languages ◽  
First Order ◽  
Finite Semigroups ◽  
Order Case ◽  
Additional Equation

We present first-order (FO) and monadic second-order (MSO) logics with predicates ‘between’ and ‘neighbour’ that characterise the class of regular languages that are closed under the reverse operation and its subclasses. The ternary between predicate bet(x, y, z) is true if the position y is strictly between the positions x and z. The binary neighbour predicate N(x, y) is true when the the positions x and y are adjacent. It is shown that the class of reversible regular languages is precisely the class definable in the logics MSO(bet) and MSO(N). Moreover the class is definable by their existential fragments EMSO(bet) and EMSO(N), yielding a normal form for MSO formulas. In the first-order case, the logic FO(bet) corresponds precisely to the class of reversible languages definable in FO(<). Every formula in FO(bet) is equivalent to one that uses at most 3 variables. However the logic FO(N) defines only a strict subset of reversible languages definable in FO(+1). A language-theoretic characterisation of the class of languages definable in FO(N), called locally-reversible threshold-testable (LRTT), is given. In the second part of the paper we show that the standard connections that exist between MSO and FO logics with order and successor predicates and varieties of finite semigroups extend to the new setting with the semigroups extended with an involution operation on its elements. The case is different for FO(N) where we show that one needs an additional equation that uses the involution operator to characterise the class. While the general problem of characterising FO(N) is open, an equational characterisation is shown for the case of neutral letter languages.

FINITE STATE PROCESSES, Z-TEMPORAL LOGIC AND THE MONADIC THEORY OF THE INTEGERS

1992 ◽  
Vol 03 (03) ◽  
pp. 233-244 ◽  
Author(s):  
A. SAOUDI ◽  
D.E. MULLER ◽  
P.E. SCHUPP
Keyword(s):  
Temporal Logic ◽  
Linear Temporal Logic ◽  
Second Order ◽  
Regular Languages ◽  
Infinite Words ◽  
First Order ◽  
Monadic Theory ◽  
Finite State ◽  
Temporal Formula

We introduce four classes of Z-regular grammars for generating bi-infinite words (i.e. Z-words) and prove that they generate exactly Z-regular languages. We extend the second order monadic theory of one successor to the set of the integers (i.e. Z) and give some characterizations of this theory in terms of Z-regular grammars and Z-regular languages. We prove that this theory is decidable and equivalent to the weak theory. We also extend the linear temporal logic to Z-temporal logic and then prove that each Z-temporal formula is equivalent to a first order monadic formula. We prove that the correctness problem for finite state processes is decidable.

scholarly journals Consensus Indices of Two-Layered Multi-Star Networks: An Application of Laplacian Spectrum

2021 ◽  
Vol 9 ◽  
Author(s):  
Da Huang ◽  
Jicheng Bian ◽  
Haijun Jiang ◽  
Zhiyong Yu
Keyword(s):  
Asymptotic Properties ◽  
Second Order ◽  
Graph Spectra ◽  
Layered Systems ◽  
First Order ◽  
Layered Networks ◽  
Order Case ◽  
Coordination Systems ◽  
Multi Agent ◽  
Agent Networks

In this article, the convergence speed and robustness of the consensus for several dual-layered star-composed multi-agent networks are studied through the method of graph spectra. The consensus-related indices, which can measure the performance of the coordination systems, refer to the algebraic connectivity of the graph and the network coherence. In particular, graph operations are introduced to construct several novel two-layered networks, the methods of graph spectra are applied to derive the network coherence for the multi-agent networks, and we find that the adherence of star topologies will make the first-order coherence of the dual-layered systems increase some constants in the sense of limit computations. In the second-order case, asymptotic properties also exist when the index is divided by the number of leaf nodes. Finally, the consensus-related indices of the duplex networks with the same number of nodes but non-isomorphic structures have been compared and simulated, and it is found that both the first-order coherence and second-order coherence of the network D are between A and B, and C has the best first-order robustness, but it has the worst robustness in the second-order case.

NOTES ONω-INCONSISTENT THEORIES OF TRUTH IN SECOND-ORDER LANGUAGES

2013 ◽  
Vol 6 (4) ◽  
pp. 733-741 ◽  
Author(s):  
EDUARDO BARRIO ◽  
LAVINIA PICOLLO
Keyword(s):  
Classical Theory ◽  
Second Order ◽  
Revision Theory ◽  
Theories Of Truth ◽  
First Order ◽  
Order Case ◽  
Desirable Feature ◽  
Conceptual Problems ◽  
Inconsistent Theories

It is widely accepted that a theory of truth for arithmetic should be consistent, butω-consistency is less frequently required. This paper argues thatω-consistency is a highly desirable feature for such theories. The point has already been made for first-order languages, though the evidence is not entirely conclusive. We show that in the second-order case the consequence of adoptingω-inconsistent truth theories for arithmetic is unsatisfiability. In order to bring out this point, well knownω-inconsistent theories of truth are considered: the revision theory of nearly stable truthT#and the classical theory of symmetric truthFS. Briefly, we present some conceptual problems withω-inconsistent theories, and demonstrate some technical results that support our criticisms of such theories.

scholarly journals Applying the method of normal forms to second-order nonlinear vibration problems

2010 ◽  
Vol 467 (2128) ◽  
pp. 1141-1163 ◽  
Author(s):  
Simon A. Neild ◽  
David J. Wagg
Keyword(s):  
Normal Form ◽  
Nonlinear Vibration ◽  
Normal Forms ◽  
Equations Of Motion ◽  
Second Order ◽  
Form Analysis ◽  
First Order ◽  
Invariance Properties ◽  
Vibration Problems ◽  
Second Order Equations

Vibration problems are naturally formulated with second-order equations of motion. When the vibration problem is nonlinear in nature, using normal form analysis currently requires that the second-order equations of motion be put into first-order form. In this paper, we demonstrate that normal form analysis can be carried out on the second-order equations of motion. In addition, for forced, damped, nonlinear vibration problems, we show that the invariance properties of the first- and second-order transforms differ. As a result, using the second-order approach leads to a simplified formulation for forced, damped, nonlinear vibration problems.

scholarly journals Categoricity by convention

2021 ◽  
Author(s):  
Julien Murzi ◽  
Brett Topey
Keyword(s):  
Basic Problem ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Language ◽  
Standard Interpretation ◽  
Permutation Invariance ◽  
First Order ◽  
Order Case ◽  
Intended Interpretation ◽  
Second Order Logic

AbstractOn a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order and Zermelo’s quasi-categoricity theorem for second-order —these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic.

Two interpolation theorems for a predicate calculus

1971 ◽  
Vol 36 (2) ◽  
pp. 262-270
Author(s):  
Shoji Maehara ◽  
Gaisi Takeuti
Keyword(s):  
Normal Form ◽  
Completeness Theorem ◽  
Interpolation Theorem ◽  
Second Order ◽  
Predicate Calculus ◽  
Cut Elimination ◽  
First Order ◽  
Image Position ◽  
Order Predicate Calculus

A second order formula is called Π1 if, in its prenex normal form, all second order quantifiers are universal. A sequent F1, … Fm → G1 …, Gn is called Π1 if a formulais Π1If we consider only Π1 sequents, then we can easily generalize the completeness theorem for the cut-free first order predicate calculus to a cut-free Π1 predicate calculus.In this paper, we shall prove two interpolation theorems on the Π1 sequent, and show that Chang's theorem in [2] is a corollary of our theorem. This further supports our belief that any form of the interpolation theorem is a corollary of a cut-elimination theorem. We shall also show how to generalize our results for an infinitary language. Our method is proof-theoretic and an extension of a method introduced in Maehara [5]. The latter has been used frequently to prove the several forms of the interpolation theorem.

scholarly journals On the Union Closed Fragment of Existential Second-Order Logic and Logics with Team Semantics

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Matthias Hoelzel ◽  
Richard Wilke
Keyword(s):  
Normal Form ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Team Semantics ◽  
Key Factor ◽  
Undecidable Property ◽  
Open Question ◽  
Second Order Logic

We present syntactic characterisations for the union closed fragments of existential second-order logic and of logics with team semantics. Since union closure is a semantical and undecidable property, the normal form we introduce enables the handling and provides a better understanding of this fragment. We also introduce inclusion-exclusion games that turn out to be precisely the corresponding model-checking games. These games are not only interesting in their own right, but they also are a key factor towards building a bridge between the semantic and syntactic fragments. On the level of logics with team semantics we additionally present restrictions of inclusion-exclusion logic to capture the union closed fragment. Moreover, we define a team based atom that when adding it to first-order logic also precisely captures the union closed fragment of existential second-order logic which answers an open question by Galliani and Hella.

Second-order languages and mathematical practice

1985 ◽  
Vol 50 (3) ◽  
pp. 714-742 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Infinite Cardinal ◽  
First Order ◽  
Order Case ◽  
Skolem Theorem ◽  
Infinite Model ◽  
Second Order Logic

There are well-known theorems in mathematical logic that indicate rather profound differences between the logic of first-order languages and the logic of second-order languages. In the first-order case, for example, there is Gödel's completeness theorem: every consistent set of sentences (vis-à-vis a standard axiomatization) has a model. As a corollary, first-order logic is compact: if a set of formulas is not satisfiable, then it has a finite subset which also is not satisfiable. The downward Löwenheim-Skolem theorem is that every set of satisfiable first-order sentences has a model whose cardinality is at most countable (or the cardinality of the set of sentences, whichever is greater), and the upward Löwenheim-Skolem theorem is that if a set of first-order sentences has, for each natural number n, a model whose cardinality is at least n, then it has, for each infinite cardinal κ (greater than or equal to the cardinality of the set of sentences), a model of cardinality κ. It follows, of course, that no set of first-order sentences that has an infinite model can be categorical. Second-order logic, on the other hand, is inherently incomplete in the sense that no recursive, sound axiomatization of it is complete. It is not compact, and there are many well-known categorical sets of second-order sentences (with infinite models). Thus, there are no straightforward analogues to the Löwenheim-Skolem theorems for second-order languages and logic.There has been some controversy in recent years as to whether “second-order logic” should be considered a part of logic, but this boundary issue does not concern me directly, at least not here.

A High-Order Solution for the Added Stability Lobes in Intermittent Machining

2000 ◽  
Author(s):  
William T. Corpus ◽  
William J. Endres
Keyword(s):  
Closed Form ◽  
High Speed ◽  
Higher Order ◽  
Second Order ◽  
Machining Processes ◽  
Analytical Technique ◽  
First Order ◽  
Order Solution ◽  
Order Case ◽  
Speed Stability

Abstract An earlier work by the authors presented a solution for the added ultrahigh-speed stability lobe that has been shown to exist for intermittent and other periodically time varying machining processes. That earlier first-order solution was not clearly extendible to a higher order. A more general analytical technique presented here does permit higher-order results. The solution is developed first for the case of zero damping for which a final closed-form symbolic result can be realized up to second order. More important than improved accuracy, the higher-order nature of the result confirms that there exist multiple added lobes and permits a mathematical description of their locations along the spindle-speed axis. A solution is then derived for the structurally damped case, where the first-order case permits a final closed-form symbolic result while the second-order case requires computational evaluation. The first-order result matches perfectly the previously published one, as expected. The second-order result improves accuracy, measured relative to numerical simulation results, and, more important, permits a second added lobe to be predicted. The second added lobe tends to cut into the region of the high-speed stability peak that is predicted under traditional zero-frequency (time-averaged) analyses. The damped solutions also indicate that structural damping of the dominant mode becomes virtually unimportant at ultrahigh speeds.

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