Markov's Principle Research Articles

Prediction of building carbon emission based on grey Markov theory

The increase in carbon emissions in the construction industry is one of the factors that lead to environmental problems and hinder the sustainable development of society and the construction industry. Therefore, there is an urgent need to rationally control carbon emissions by formulating scientific and effective energy-saving and emission reduction policies so as to safeguard the development of the industry. However, because of the uncertainty and complexity of CO2 emissions, more reliable prediction and assessment tools are needed to comprehensively analyze and predict CO2 emissions. This study combines the gray system theory and Markov principle to give full play to the advantages of the two methods, using Markov theory to determine the state transition probability, find out the characteristics of carbon emissions in the construction industry, and form a suitable gray Markov prediction model. The feasibility of the prediction model is demonstrated by calculating and examining the prediction model using data related to carbon emissions from the U.S. buildings from 2002 to 2022, which greatly improves the accuracy of CO2 emissions prediction.

Dialectica principles via Gödel doctrines

Gödel's Dialectica interpretation was conceived as a tool to obtain the consistency of Heyting arithmetic in the 40s. In recent years, several proof theoretic transformations, based on Gödel's Dialectica interpretation, have been used systematically to extract new content from classical proofs, following a suggestion of Kreisel. Thus, the interpretation has found new relevant applications in several areas of mathematics and computer science. Several authors have explained the Dialectica interpretation in categorical terms. In particular, de Paiva introduced the notion of a Dialectica category as an internal version of Gödel's Dialectica Interpretation in her doctoral work. This was generalised by Hyland and Hofstra, who considered the interpretation in terms of fibrations. In our previous work, we introduced an intrinsic presentation of the Dialectica construction via a generalisation of Hofstra's work, using the notion of Gödel fibration and its proof-irrelevant version, a Gödel doctrine. The key idea is that Gödel fibrations (and doctrines) can be thought of as fibrations generated by some basic predicates playing the role of quantifier-free predicates. This categorification of quantifier-free predicates is crucial not only to show that our notion of Gödel fibration is equivalent to Hofstra's Dialectica fibration in the appropriate way, but also to show how Gödel doctrines embody the main logical features of the Dialectica Interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded by Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely (suitable versions of) the Markov Principle (MP) and the Independence of Premise (IP) principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight correspondence between the logical system and the categorical framework. This tight correspondence should come in handy not only when discussing the traditional applications of Dialectica, but also when dealing with some newer uses of the interpretation, as in modelling games or concurrency theory. Finally, to complete our analysis, we characterise categories obtained as a result of the Hyland, Johnstone and Pitts tripos-to-topos construction when applied to Gödel doctrines.

Dialectica logical principles: not only rules

Abstract Gödel’s Dialectica interpretation was designed to obtain the consistency of Peano arithmetic via a proof of consistency of Heyting arithmetic and double negation. In recent years, proof theoretic transformations (the so-called proof interpretations) based on Gödel’s Dialectica interpretation have been used systematically to extract new content from proofs and so the interpretation has found relevant applications in several areas of mathematics and computer science. Following our previous work on ‘Gödel fibrations’, we present a (hyper)doctrine characterization of the Dialectica, which corresponds exactly to the logical description of the interpretation. To show that, we derive the soundness of the interpretation of the implication connective, as expounded on by Spector and Troelstra, in the categorical model. This requires extra logical principles, going beyond intuitionistic logic, namely Markov Principle and the Independence of Premise principle, as well as some choice. We show how these principles are satisfied in the categorical setting, establishing a tight (internal language) correspondence between the logical system and the categorical framework. We make sure that this tight correspondence extends to the use of the principles above, instead of the weaker rules we had proved earlier on. This tight correspondence should come handy not only when discussing the traditional applications of the Dialectica but also when dealing with newer uses in modelling games or concurrency theory.

Instance reducibility and Weihrauch degrees

We identify a notion of reducibility between predicates, called instance reducibility, which commonly appears in reverse constructive mathematics. The notion can be generally used to compare and classify various principles studied in reverse constructive mathematics (formal Church's thesis, Brouwer's Continuity principle and Fan theorem, Excluded middle, Limited principle, Function choice, Markov's principle, etc.). We show that the instance degrees form a frame, i.e., a complete lattice in which finite infima distribute over set-indexed suprema. They turn out to be equivalent to the frame of upper sets of truth values, ordered by the reverse Smyth partial order. We study the overall structure of the lattice: the subobject classifier embeds into the lattice in two different ways, one monotone and the other antimonotone, and the $\lnot\lnot$-dense degrees coincide with those that are reducible to the degree of Excluded middle. We give an explicit formulation of instance degrees in a relative realizability topos, and call these extended Weihrauch degrees, because in Kleene-Vesley realizability the $\lnot\lnot$-dense modest instance degrees correspond precisely to Weihrauch degrees. The extended degrees improve the structure of Weihrauch degrees by equipping them with computable infima and suprema, an implication, the ability to control access to parameters and computation of results, and by generally widening the scope of Weihrauch reducibility.

Algorithm Design of Port Cargo Throughput Forecast Based on the ES‐Markov Model

At present, the existing prediction algorithm of a port cargo throughput neglects the correction of the initial value of the cargo series data model, which leads to a large error in a port cargo throughput prediction. Therefore, a prediction algorithm of a port cargo throughput based on the ES‐Markov model is designed. A decompose function is used to decompose the time series of a port cargo throughput, and the trend elements of a port cargo throughput are divided into long‐term trend, seasonal trend, fluctuation trend, and irregular trend. In this study, the ES‐Markov model is introduced, and the initial prediction is obtained by using the cubic exponential smoothing method, and the state transition matrix is obtained by the Markov principle. Based on the results of the time‐series analysis and the ES‐Markov model, the prediction algorithm of a port cargo throughput is designed. In the experimental design, the Elman neural network is used to construct an experimental sample data model. The monthly cargo throughput data of a certain port for eight months from May 2020 to December 2020 are collected and sorted according to the time series. The experimental results show that the prediction results of the proposed algorithm are closer to the actual value and the fluctuation of the prediction results is less than that of the reference.

Not all Kripke models of [formula omitted] are locally [formula omitted

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA. For every node k in K, we can view the classical structure of k, Mk as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T, iff for every k in K, Mk⊨T. One of the most important problems in the model theory of HA is the following question: Is every Kripke model ofHAlocallyPA? We answer this question negatively. We introduce two new Kripke model constructions to this end. The first construction actually characterizes the arithmetical structures that can be the root of a Kripke model K⊩HA+ECT0 (ECT0 stands for Extended Church Thesis). The characterization says that for every arithmetical structure M, there exists a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr=M iff M⊨ThΠ2(PA). One of the consequences of this characterization is that there is a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr⊭IΔ1 and hence K is not even locally IΔ1. The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficient condition from this construction that describes when for an arithmetical structure M, there exists a rooted Kripke model K⊩T with the root r such that Mr=M. As applications of this sufficient condition, we construct two new Kripke models. The first one is a Kripke model K⊩HA+¬θ+MP (θ is an instance of ECT0 and MP is Markov's principle) which is not locally IΔ1. The second one is a Kripke model K⊩HA such that K forces exactly the sentences that are provable from HA, but it is not locally IΔ1. Also, we will prove that every countable Kripke model of intuitionistic first-order logic can be transformed into another Kripke model with the full infinite binary tree as the Kripke frame such that both Kripke models force the same sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally IΔ1.

Certain Partial Conservativeness Properties of Intuitionistic Set Theory with the Principle of Double Complement of Sets

The Zermelo–Fraenkel set theory with the underlying intuitionistic logic (for brevity, we refer to it as the intuitionistic Zermelo–Fraenkel set theory) in a two-sorted language (where the sort 0 is assigned to numbers and the sort 1, to sets) with the collection scheme used as the replacement scheme of axioms (the ZFI2C theory) is considered. Some partial conservativeness properties of the intuitionistic Zermelo–Fraenkel set theory with the principle of double complement of sets (DCS) with respect to a certain class of arithmetic formulas (the class all so-called AEN formulas) are proved. Namely, let T be one of the theories ZFI2C and ZFI2C + DCS. Then (1) the theory T+ECT is conservative over T with respect to the class of AEN formulas; (2) the theory T+ECT+M is conservative over T+M{su−} with respect to the class of AEN formulas. Here ECT stands for the extended Church’s thesis, Mis the strong Markov principle, and M{su−} is the weak Markov principle. The following partial conservativeness properties are also proved: (3) T+ECT+M is conservative over T with respect to the class of negative arithmetic formulas; (4) the classical theory ZF2 is conservative over ZFI2C with respect to the class of negative arithmetic formulas.

On fixed-point theorems in synthetic computability

Lawvere's fixed point theorem captures the essence of diagonalization arguments. Cantor's theorem, Gödel's incompleteness theorem, and Tarski's undefinability of truth are all instances of the contrapositive form of the theorem. It is harder to apply the theorem directly because non-trivial examples are not easily found, in fact, none exist if excluded middle holds. We study Lawvere's fixed-point theorem in synthetic computability, which is higher-order intuitionistic logic augmented with the Axiom of Countable Choice, Markov's principle, and the Enumeration axiom, which states that there are countably many countable subsets of $\mathbb{N}$. These extra-logical principles are valid in the effective topos, as well as in any realizability topos built over Turing machines with an oracle, and suffice for an abstract axiomatic development of a computability theory. We show that every countably generated $\omega$-chain complete pointed partial order ($\omega$cppo) is countable, and that countably generated $\omega$cppos are closed under countable products. Therefore, Lawvere's fixed-point theorem applies and we obtain fixed points of all endomaps on countably generated $\omega$cppos. Similarly, the Knaster-Tarski theorem guarantees existence of least fixed points of continuous endomaps. To get the best of both theorems, namely that all endomaps on domains ($\omega$cppos generated by a countable set of compact elements) have least fixed points, we prove a synthetic version of the Myhill-Shepherdson theorem: every map from an $\omega$cpo to a domain is continuous. The proof relies on a new fixed-point theorem, the synthetic Recursion Theorem, so called because it subsumes the classic Kleene-Rogers Recursion Theorem. The Recursion Theorem takes the form of Lawvere's fixed point theorem for multi-valued endomaps.

A separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle

We prove constructively that every uniformly continuous convex function f:X→R+ has positive infimum, where X is the convex hull of finitely many vectors. Using this result, we prove that a separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle are constructively equivalent. This is the first time that important theorems are classified into Markov's principle within constructive reverse mathematics.

Some principles weaker than Markov’s principle

We systematically study several principles and give a principle which is weaker than disjunctive Markov's principle (MP?). We also show that the principle is underivable and strictly weaker than MP? in certain extensions of the system EL of elementary analysis.

On the Disjunctive Markov Principle

In this note we show that over a strong (semi-)intuitionistic base theory, the recursive comprehension principle $${\Delta^0_1}$$Δ10 -CA does not imply the disjunctive Markov principle MP$${^{\vee}}$$?.

Independence of the uniformity principle from Church's thesis in intuitionistic set theory

We prove the independence of the strong uniformity principle from Church's thesis with choice in intuitionistic set theory with the axiom of extensionality extended by Markov's principle and the double complement for sets.

Predictability of soil succession patterns under different agricultural land use practices: Continual conventional cultivation versus transformation to organic cultivation or fallow periods

Abstract Changes in agricultural management practices create conditions that directly or indirectly promote the onset of secondary succession processes in the soil. Previous research has demonstrated that the succession of temperate forests follows the Markov principle, which states that the next state of a process depends only on the preceding state; hence, we set out to test this principle for soils subjected to different land uses over a 2 year period. Specifically, (1) continued conventional cultivation (i.e. the monoculture of shell beans, Phaseolus vulgaris L., with chemical fertilisers); (2) transformation of land-use from conventional cultivation to organic cultivation (shell beans fertilised with manure); and (3) transformation of land-use to a fallow state. The soil was sampled three times at specific shell bean growth stages (June, the seedling stage; August, mid-growth; October, one month after harvest) in both years, with various chemical and biochemical soil profiles being conducted (including carbon, nitrogen and phosphorus levels, and enzymatic activity). Correlations among successive soil chemical and biochemical profiles were stronger (two successive events: June and August, August and October) in the first year compared to the second year (one successive event: August and October) for the organic plots. For soils in which conventional land management was continued, significant correlation was only detected for one successive profile (August–October) in both years, whereas no significant correlation was obtained for any successive profiles in the fallow plots. Therefore, the Markov principle only holds for the entire cultivation period under certain conditions for bean crops; specifically, initial conventional cultivation followed by transformation to organic cultivation. In the case of continued conventional monoculture, the Markov principle is partially supported. In contrast, fallow plots did not follow any predictable pattern of changes in soil chemical and biochemical profiles. In conclusion, agricultural management causes soil succession processes to be predictable, due to the environmental homogeneity induced by manure and/or the limited plant effect, rather than natural processes.

Kripke Sheaf Completeness of some Superintuitionistic Predicate Logics with a Weakened Constant Domains Principle

The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [8] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: (Q-H + D*), (Q-H + D*&K), (Q-H + D*&K&J). Here Q-H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda's axiom, and D* (cf. [12]) is a weakened version of the well-known constant domains principle D. Namely, the formula D states that any individual has ancestors in earlier worlds, and D* states that any individual has $${\neg\neg}$$ -ancestors (i.e., ancestors up to $${\neg\neg}$$ -equality) in earlier worlds. In particular, the logic (Q-H + D*&K&J) is the Kripke sheaf completion of (Q-H + E&K&J), where E is a version of Markov's principle (cf. [12]). On the other hand, we show that the logic (Q-H + D*&J) is incomplete w.r.t. Kripke sheaves.

Study on the Model of Construction Accident Forecast Based on Grey-Markov Theory

According to the grey system theory and markov principle, the fitting accuracy index is stochastic and fluctuant when time series datadevelopment trend is fitted through the grey theory; but the markov principle is suitable for processing the system with great stochastic fluctuation; therefore, a forecast model is put forward which is in accordance with the characteristics of construction accidents’ statistical data and is based on the grey system theory and markov principle. The model is suitable for the dynamic forecasting process featured by short forecasting time, small data quality and great stochastic fluctuation. The conclusion about the development trend of construction accidents with the model based on the grey-markov theory is more ideal compared with the grey model. The application example shows that the forecasting conclusion based on grey-markov theory is more accurate and more reliable. So the model is available enough to forecast the construction accidents.

Design of Experiments for Categorical Repeated Measurements in Packet Communication Networks

We study the optimal measurement of packet loss and delay in packet networks by treating all measurements as numerical experiments to which we apply the theory of the design of experiments. Specifically we seek to find the optimal times at which to inject survey (probe) packets. Our approach is to model the target node in the packet communication network, an access buffer, as a discrete-time Markov chain. Given that we may only make a limited number of observations, we present a method for optimally designing the observation times for the chain, and derive both exact and continuous optimal designs. Our results show that, for common optimality criteria, measuring at a uniform rate may not be optimal. This has significance for influencing commercial practice as uniform probing is standard. We show how our method may be generalized to Markov systems with larger state space, and describe computational methods to find optimal designs on any system which evolves according to the Markov principle.

Unavoidable sequences in constructive analysis

Five recursively axiomatizable theories extending Kleene's intuitionistic theory FIM of numbers and numbertheoretic (choice) sequences are introduced and shown to be consistent, by a modified relative realizability interpretation which verifies that every sequence classically defined by a Π11 formula is unavoidable (cannot fail to exist) and that no sequence can fail to be classically Δ11. The analytical form of Markov's Principle fails under the interpretation. The notion of strongly inadmissible rule of inference is introduced, with examples (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

On Two Attempts of Describing Propositional Realizability Logic

The study of propositional realizability logic was initiated in the 50th of the last century. Unfortunately, no description of the class of realizable propositional formulas is found up to now. Nevertheless, some attempts of such a description were made. In 1974, the author proved that every known realizable propositional formula has the property that every one of its closed arithmetical instances is deducible in the system obtained by adding Extended Church's Thesis and Markov Principle as axiom schemes to Intuitionistic Arithmetic. Visser calls this system Markov's Arithmetic. In 1990, another attempt of describing the class of realizable propositional formulas was made by Varpakhovskii who proposed a calculus in an extended propositional language and proved that all known realizable propositional formulas are deducible in this calculus. In this article we prove that every propositional formula deducible in Varpakhovskii's calculus has the property that each of its closed arithmetical instances is deducible in Markov's Arithmetic.

Completeness and incompleteness for intuitionistic logic

AbstractWe call a logicregularfor a semantics when the satisfaction predicate for at least one of its nontheorems is closed under double negation. Such intuitionistic theories as second-order Heyting arithmeticHASand the intuitionistic set theoryIZFprove completeness for no regular logics, no matter how simple or complicated. Any extensions of those theories proving completeness for regular logics are classical, i.e., they derive thetertium non datur. When an intuitionistic metatheory features anticlassical principles or recognizes that a logic regular for a semantics is nonclassical, it proves explicitly that the logic is incomplete with respect to that semantics. Logics regular relative to Tarski, Beth and Kripke semantics form a large collection that includes propositional and predicate intuitionistic, intermediate and classical logics. These results are corollaries of a single theorem. A variant of its proof yields a generalization of the Gödel-Kreisel Theorem linking weak completeness for intuitionistic predicate logic to Markov's Principle.

Closed Fragments of Provability Logics of Constructive Theories

AbstractIn this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.