## Category: Monadic second

## Monadic second

By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. Maybe there is a solution. But, for that I assume there is an upper bound in the number of rounds needed, say n, and that the value b is fixed upfront. Then, there is the following EMSO formula. So if we can prove that if there is a solution then there is a solution of maximum n rounds which is dependent on the size of the graph then I think we have a solution.

A reference to a clear definition of Monadic Second-Order Logic and especially, some examples of what it can typically express would be helpful for answering the question. Maybe you should say precisely what language you want this expressed in? And what exactly you want expressed: that a given solution is feasible? Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 10 years, 3 months ago.

Active 7 years ago. Viewed times. Please help formulate this problem in MSO. Sergiy Kozerenko 1 1 gold badge 7 7 silver badges 16 16 bronze badges. Esha Esha 53 8 8 bronze badges. This problem is NP-hard on general graphs. But has polynomial time solution for some special classes. Active Oldest Votes. Abdullah Abdullah 56 3 3 bronze badges. Jun 29 '10 at They include references to the state of the art circa I think a lot of people who understand graph theory don't know monadic logic so it would help if you explained what other properties are typically expressible in MSO.

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### Monadic Sequential

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In a monadic sequential presentation scheme, each assessor will be exposed to at least two objects but only one at a timei. In statistics, monadic sequential designs are also known as crossover designs.

Monadic sequential designs can be contrasted with Monadic studies in which each assessor only evaluates one stimulus. They also differ from so-called simultaneous presentation schemes, often employed in discrimination or paired preference tests, in which the panelist receives multiple samples at the same time.

Of course, even in such cases, the assessor can only evaluate one sample at a time. The difference here is that all samples in the assessment set are evaluated prior to giving a response say, a discrete choice versus answering multiple questions about a given sample.

For example, panelists may only evaluate a subset of all of the samples in the study, in which case the design is referred to as an Incomplete Block. Also, all possible presentation orders need not be present in the design. In fact, a consumer study with six samples would require consumers in order to have every presentation order represented just once, which is far more consumers than is typically used in industry.

One variation of the monadic sequential approach is the Proto Monadic scheme in which, typically, the second sample is not evaluated using the same ballot that was used for the first sample, instead, a preference question is asked. The ballot portion would follow a typical monadic presentation scheme, whereas the preference question is similar to a standard comparative judgment in the sense that both samples are evaluated prior to making a choice.

Page Rating. Categories Vocabulary.In the mathematical fields of graph theory and finite model theorythe logic of graphs deals with formal specifications of graph properties using formulas of mathematical logic. There are several variations in the types of logical operation that can be used in these formulas.

The first order logic of graphs concerns formulas in which the variables and predicates concern individual vertices and edges of a graph, while monadic second order graph logic allows quantification over sets of vertices or edges.

Logics based on least fixed point operators allow more general predicates over tuples of vertices, but these predicates can only be constructed through fixed-point operators, restricting their power to an intermediate level between first order and monadic second order. The algorithmic problem of model checking concerns testing whether a given graph models a given sentence. The algorithmic problem of satisfiability concerns testing whether there exists a graph that models a given sentence.

Although both model checking and satisfiability are hard in general, several major algorithmic meta-theorems show that properties expressed in this way can be tested efficiently for important classes of graphs. Other topics of research in the logic of graphs include investigations of the probability that a random graph has a property specified within a particular type of logic, and methods for data compression based on finding logical formulae that are modeled by a unique graph.

In the first-order logic of graphs, a graph property is expressed as a quantified logical formula whose variables represent graph verticeswith predicates for equality and adjacency testing.

For instance, the condition that a graph does not have any isolated vertices may be expressed by the sentence.

**Monadic second-order logic on finite sequences**

The subgraph isomorphism problem for a fixed subgraph H asks whether H appears as a subgraph of a larger graph G. It may be expressed by a sentence that states the existence of vertices one for each vertex of H such that, for each edge of Hthe corresponding pair of vertices are adjacent; see picture. As a special case the clique problem for a fixed clique size may be expressed by a sentence that states the existence of a number of vertices equal to the clique size all of which are adjacent.

For simple undirected graphsthe first order theory of graphs includes the axioms. Other types of graphs, such as directed graphsmay involve different axioms, and logical formulations of multigraph properties require having separate variables for vertices and edges. That is, let S be a fixed first-order sentence, and choose a random n -vertex graph G n uniformly at random among all graphs on a set of n labeled vertices.

Then in the limit as n tends to infinity the probability that G n models S will tend either to zero or to one:. Moreover, there is a specific infinite graph, the Rado graph Rsuch that the sentences modeled by the Rado graph are exactly the ones for which the probability of being modeled by a random finite graph tends to one:.

For random graphs in which each edge is included independently of the others with a fixed probability, the same result is true, with the same sentences having probabilities tending to zero or to one. The computational complexity of determining whether a given sentence has probability tending to zero or to one is high: the problem is PSPACE-complete. A similar analysis can be performed for non-uniform random graphs, where the probability of including an edge is a function of the number of vertices, and where the decision to include or exclude an edge is made independently with equal probability for all edges.

However, for these graphs the situation is more complicated. In this case, a first-order property may have one or more thresholds, such that when the edge inclusion probability is bounded away from the threshold then the probability of having the given property tends to zero or one. These thresholds can never be an irrational power of nso random graphs where the edge inclusion probability is an irrational power obey a zero-one law analogous to the one for uniformly random graphs. If a first-order sentence includes k distinct variables, then the property it describes can be tested in graphs of n vertices by examining all k -tuples of vertices; however, this brute force search algorithm is not particularly efficient, taking time O n k.In mathematical logicmonadic second-order logic MSO is the fragment of second-order logic where the second-order quantification is limited to quantification over sets.

Second-order logic allows quantification over predicates. However, MSO is the fragment in which second-order quantification is limited to monadic predicates predicates having a single argument. This is often described as quantification over "sets" because monadic predicates are equivalent in expressive power to sets the set of elements for which the predicate is true. Existential monadic second-order logic EMSO is the fragment of MSO in which all quantifiers over sets must be existential quantifiersoutside of any other part of the formula.

The first-order quantifiers are not restricted. By analogy to Fagin's theoremaccording to which existential non-monadic second-order logic captures precisely the descriptive complexity of the complexity class NPthe class of problems that may be expressed in existential monadic second-order logic has been called monadic NP.

The restriction to monadic logic makes it possible to prove separations in this logic that remain unproven for non-monadic second-order logic. For instance, in the logic of graphstesting whether a graph is disconnected belongs to monadic NP, as the test can be represented by a formula that describes the existence of a proper subset of vertices with no edges connecting them to the rest of the graph; however, the complementary problem, testing whether a graph is connected, does not belong to monadic NP.

By contrast, when we wish to check whether a Boolean MSO formula is satisfied by an input finite treethis problem can be solved in linear time in the tree, by translating the Boolean MSO formula to a tree automaton [4] and evaluating the automaton on the tree.

In terms of the query, however, the complexity of this process is generally nonelementary. For MSO formulas that have free variableswhen the input data is a tree or has bounded treewidth, there are efficient enumeration algorithms to produce the set of all solutions [6]ensuring that the input data is preprocessed in linear time and that each solution is then produced in a delay linear in the size of each solution, i.

There are also efficient algorithms for counting the number of solutions of the MSO formula in that case. The monadic second order theory of the infinite complete binary treecalled S2S, is decidable. As a consequence of this result, the following theories are decidable:.

From Wikipedia, the free encyclopedia. Computational complexity of evaluation [ edit ] Existential monadic second-order logic EMSO is the fragment of MSO in which all quantifiers over sets must be existential quantifiersoutside of any other part of the formula.

As a consequence of this result, the following theories are decidable: The monadic second-order theory of trees. Note that for binary numbers represented by subsetsaddition is definable even in wS1S. Cambridge University Press. Retrieved Mathematical Systems Theory. Parikh, Rohit ed.Philosophy An indivisible, impenetrable unit of substance viewed as the basic constituent element of physical reality in the metaphysics of Leibniz. Biology A single-celled microorganism, especially a flagellate protozoan formerly classified in the taxonomic group Monadina.

All rights reserved. Logic logic maths of an operator, predicate, etc having only a single argument place. Mathematics logic maths of an operator, predicate, etc having only a single argument place. Mentioned in? Cells dyadic monad monadic operation operation polyadic singulary unary unary operation.

References in periodicals archive? The death of a strong, great, bad man: an ethnography of soul incorporation. Beyond description, classification, and analysis--the purview of the "natural" sciences and of most "social" sciences--humanistic disciplines come together in the ethical sphere, the place where human beings transcend their monadic singularity and act as members of the species.

Humility and Method. The trouble with the virtual. The bias towards understating willingness to prescribe can be attenuated to some extent by administering the task in a monadic fashion; that is, the sample is split into two or more cells, with each cell answering the willingness to prescribe question with a different price in each cell.

The role of pricing research in assessing the commercial potential of new drugs in development. Languages and genes: Can they be built up through random change and natural selection?

We use the lower of the two monadic values in a dyad to measure the exposure of a dyad to capital investments. If one tried to imagine a universe in which this poststructuralist notion really applied, it's a very strange, what Leibnitz might call a " monadic ," existence, where one is self-enclosed entirely and impinged upon by objects, by reality, rather than engaging with it. The ideal to aim at is not a monadic subject, but a person who expresses his or her authentic experiences all-inclusively to others.

Mundane social policies in the context of the fragmentation of welfare-state-oriented alcohol policy in Finland. Bacon thinks that an individual is not uniquely characterized by the conjunction of all its monadic properties the principle of the identity of indiscernibles being false.

Existential second-order logic ESO and monadic second-order logic MSO have attracted much interest in logic and computer science. Dictionary browser? Full browser?This book studies the relationship between automata and monadic second-order logic, focusing on classes of automata that describe the concurrent behavior of distributed systems.

It provides a unifying theory of communicating automata and their logical properties. Based on Hanf's Theorem and Thomas's graph acceptors, it develops a result that allows characterization of many popular models of distributed computation in terms of the existential fragment of monadic second-order logic. Enter your mobile number or email address below and we'll send you a link to download the free Kindle App.

Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. To get the free app, enter your mobile phone number. Read more Read less. Kindle Cloud Reader Read instantly in your browser. Register a free business account. Review From the reviews: "The book deals with one essential problem of communication systems, namely formal description of protocols and systems and verification of correctness of their operation.

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Amazon Advertising Find, attract, and engage customers. Amazon Drive Cloud storage from Amazon. Alexa Actionable Analytics for the Web.Answers to these critical questions will steer the course of product development and determine the success or failure of the product in the marketplace. The best approach to obtain maximum learning from your product testing is to design a product testing system. Three benefits of a system are: 1 consistency of approach in terms of sample size, confidence range, methodology options, questionnaire design, analysis, and report format; 2 the ability to generate a normative database on key measures repeated from study to study; and 3 the opportunity to conduct meta analysis across product tests to understand the product elements that drive ratings or preference.

## Logic of graphs

The design of a product test must be handled with care. The methodology employed depends on the objectives of the study i. A key consideration is whether the test will be monadic, paired comparison, proto-monadic, sequential monadic, or multi-product three or more products. Another important factor is whether respondents will be exposed to blinded test product or identified test product.

After decades of conducting product tests for Fortune consumer packaged goods companies, the Consumer Products Division of Ipsos Insight has established guidelines to ensure that product tests are designed correctly.

There are two general types of product tests: monadic and paired, which includes paired comparison, proto-monadic, and sequential monadic tests. Another type of product test is multi-product three or more products. These tests are very useful when the goal is to optimize a product by determining the product elements that drive performance.

Multi-product tests require product formulations based on an experimental design of product elements. As such, the design of multi-product tests is quite different from monadic or paired tests. Our subsequent discussion focuses only on monadic and paired tests, which are described below. Each respondent tries two products and then answers one survey, which includes questions asking the respondent to compare the two products. Each respondent tries two products, and answers two surveys.

Specifically, each respondent tests the first product and answers a monadic survey and then tests the second product and answers a comparative survey. The order of products tested is rotated among the respondents. By comparing monadic results between cells, you will learn whether the differences between the products are meaningful when tested separately; by pooling comparative results across cells, you will learn which product is preferred.

Each respondent tries two products and answers two surveys, which are consecutive monadic surveys. Specifically, each respondent tests one product and then answers a monadic survey. Next, each respondent tests a second product and then answers a second monadic survey. Preference questions are asked at the end of the second survey.

The first product is evaluated without knowledge that a second product will follow. Sequential monadic tests are used to obtain both monadic and comparative ratings.

However, once the first product is tested, the ratings of the second product are no longer monadic and can be difficult to interpret. For this reason, we recommend proto-monadic tests over sequential monadic tests whenever possible. The monadic and paired methods differ from one another largely based on two issues: validity and sensitivity.

Monadic testing offers greater validity as consumers use only one product at a time as they would in the real world. Paired testing offers greater sensitivity, as consumers are exposed to two separate stimuli, and using products one after the other magnifies differences.

So, how does one decide which design to apply? Generally, monadic product tests should be used when there are readily discernible product differences.

This scenario typically occurs during product development when the issue is whether consumers like the product or hate it. Monadic tests are also very suitable for testing innovative products for which no benchmark or competition exists and for testing products having long purchase cycles which require a long usage period making it hard to compare products.

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