bluuebunny/crossref_metadata_2025_split huggingface dataset:
A general theory of axiomatization for propositional temporal logic
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
There are many possible connectives and therefore many possible logics. Some connectives are specially distinguished either because they appear in applications or because they represent natural technical features of flows of time. We have mentioned before the connectives Fq, Pq (‘It will be the case that q’ and ‘It was the case that q’) as natural connectives to consider. In fact, they should be included in any temporal logic. Other natural connectives are S(p,q) and U(p, q), and the technically important Stavi connectives S´(p,q) and U´(p,q). These are also important from the point of view of expressive power of temporal logic.
Expressive power of one-dimensional temporal connectives: basic concepts
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
This chapter deals with the basic concepts and theorems involved in measuring the expressive power of a temporal system. We deal here with one-dimensional temporal logics, those in which the unit for truth value is a point of the flow of time. Below we define the notions and connectives involved. These definitions are a special case of the definition in chapter 4. We repeat them here for the sake of clarity and relative independence of chapters.
H-dimension
Authors: Gabbay Dov M , Hodkinson Ian , Reynolds Mark 
Published: November 2023

Abstract
We saw that expressive completeness of a set of connectives depends on the flow of time. Kamp’s Since and Until are expressively complete over the integers but not over the rationals. The natural question to ask is, given a flow of time, (T, &lt;, =), can one find a finite set of connectives which is expressively complete over (T, &lt;, =)?We shall see in section 13.2 below that a necessary condition for a positive answer to the question is that (T, &lt;, =) has a property known as finite Henkin or H-dimension. We can use this to show that because some flows of time have infinite H-dimension, they admit no finite expressively complete set of connectives. The property of finite H-dimension is important. In sections 13.4 and 13.5 we give a method of generating more and more flows of time with finite H-dimension. Some of these flows are non-linear. In section 13.6 we examine a notion related to finite H-dimension, the so-called k-variable property.
Canonical Models and Filtration
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
In this chapter we consider two best known methods of obtaining completeness results. One of them-themethod of canonical models-givena consistent logicLin Extlnt or NExtK, constructs a canonical Kripke model 𝔐L= (𝔉L, 𝔉L) characterizingL.Sometimes the frame 𝔉Lturns out to be a frame forL,and then we can say at once thatLis Kripke complete. Another method, known asfiltration,is intended for establishing the finite approximability by means of extracting from the canonical models finite refutation frames.
Kripke Completeness
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Perhaps the most desirable property of a logic is its decidability. However, the main tool for proving it (at least in the realm of modal and superintuitionistic logics) is Harrop’s theorem, according to which the decidability of a finitely axiomatizable logic follows from its finite approximability. Likewise, to prove many other properties of a logic, it is desirable first to establish its completeness with respect to some good class of frames, the simpler the better.
Finite Approximability
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Let us now go one step down the hierarchy of frame classes and consider a stronger form of completeness, viz., completeness with respect to the class of finite frames or, in other terms, finite approximability. We have already met with one way of proving this property—the filtration method, requiring a specialad hoctechnique in each particular case. Now we will show a few other methods which provide us in fact with general syntactical and semantic sufficient conditions of finite approximability.
Tabularity
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Now we consider tabular and locally tabular modal and superintuitionistic logics. The main question we try to answer here is how to determine whether a given logic is tabular or locally tabular. First we establish that every tabular modal (no matter normal or not) and silogic is finitely axiomatizable. This will be done with the help of a syntactic criterion of tabularity which uses the following formulas:
Interpolation
Authors: Chagrov Alexander , Zakharyaschev Michael 
Published: November 2023

Abstract
Recall that a logicLis said to have theCraig interpolation propertyif, for every implicationα鄤βinL,there exists a formula γ, called aninterpolantforα鄤βinL,such thatα鄤,γ∈L,γ鄤β ∈and Var γ ⊆,;Varaα ∩ Varβ. In this chapter we present the most important semantic methods of proving and disproving the interpolation property of modal and superintuitionistic logics.
Do it yourself
Authors: Clark Malcolm 
Published: November 2023

Abstract
The handful of commands which were introduced in the last chapter were introduced with the idea that we can now re-run the EXAMPLE. Tex file, changing the various parameters. We can generate more white space between lines by increasing the value associated with the baseline skip, put more white space between paragraphs, change the font we are using; but more useful at this stage would be to alter the page dimensions. Changing the page width in particular will most quickly generate interesting, educational, results.
Beginning mathematics
Authors: Clark Malcolm 
Published: November 2023

Abstract
In many respects this is where Tex gets to be lots more fun. There is something extraordinarily satisfying about being able to set mathematical formulae. Tex permits this both within the text ( text style) and displayed on lines by themselves (display style). The same expression can be given in text or display style, and there will be a few stylistic changes, which Tex takes care of automatically.
More words
Authors: Clark Malcolm 
Published: November 2023

Abstract
Even the most erudite and involved mathematical treatise needs the glue of words to hold the pieces together. Despite the incredible ease and simplicity with which ‘!EX can handle equations, as we have already discovered, it also handles non-technical material in a straightforward and direct way. Unfortunately, the rules for ‘normal’ text are less rigid. Why unfortunately? The greater the amount of choice, the greater the amount of disagreement. We can generally agree when equations ‘look wrong’, but decid ing when text ‘looks right’ is rather more tricky. The general approach adopted here is to concentrate on the plainer aspects, assuming that the content is the important component, and that the presentation should be as bland and anonymous as possible. At the extreme, we hope the presentation will be invisible. If we are very fortunate, it may even contribute subtly to the message. But, before the heavyweight material, something carefree.
Commands
Authors: Clark Malcolm 
Published: November 2023

Abstract
A great many of ‘IE;X’s mysteries are bound up in commands. Alternative terms are ‘macros’, ‘control sequences’, or ‘definitions’: although Knuth refers to them as macros, unless you are familiar with computer science jargon, there is no advantage in this usage. The 300 or so fundamental ‘Tex commands are the building blocks which can be assembled quite easily (and all too readily) into rather arcane powerful commands. It is easy to be overwhelmed by the apparent simplicity and sheer power of commands. It is this feature of ‘Tex which gives it the appearance of a programming language, rather than a simple set of commands. One of the first places we might look to see working commands could be plain. tex which is the record of all that plain ‘Tex.
Boxing
Authors: Clark Malcolm 
Published: November 2023

Abstract
To explain how Tex handles boxes we can first return briefly to some of the ideas introduced very briefly in Chapter 3. There the boxes considered were those which were associated with individual letters which were themselves assembled into a larger box - the line. The horizontal lines are then stacked together into a vertical list of boxes. Let’s go back to the simplest cases. There are two fundamental sorts of boxes: horizontal and vertical. When Tex is creating a line of text, it is creating a horizontal box. When it assembles several paragraphs into a page, it is creating a vertical box.
Coininands#1
Authors: Clark Malcolm 
Published: November 2023

Abstract
Commands (definition or macro) start to get much more interesting when they have parameters. The basic form of the definition of a command with parameters is something like where the first #1 says that there is to be one single parameter. The repetition of #1 within the braces merely tells ‘It;X how you want the parameter (or argument) to be treated. Having defined\one, we can use it by saying \one{argument). All this does is to place argument into the text at this point. In this instance the command is dull and rather pointless, so let’s have a look at a more interesting single-parameter command.
Matrix Manipulations
Authors: Clark Malcolm 
Published: November 2023

Abstract
So far we have merely looked at fairly simple equations which can conveniently be written either in the text, or in a one-line display. We have not considered more ‘two-dimensional’ structures, like matrices. It is probably fair to say that large equations, which go over more than one line, as well as matrices (or tables) are unlikely to be found in text rather than display. Almost all of the following discussion therefore assumes implicitly that we are talking about displayed equations and formulae.
Pages
Authors: Clark Malcolm 
Published: November 2023

Abstract
Although a great deal of Tex has been presented already, only a very small amount has been directed towards the handling of fairly ordinary text. Even the most esoteric mathematics usually requires the ‘glue’ of words to hold it together. In this chapter, some of the tools and techniques available will be discussed. Text looks deceptively simple. It is probably considerably more difficult to format text than mathematics. The range of possibilities for text is so much greater. Examination of a couple of non-technical books will probably yield more variation in typesetting than com parable technical (especially mathematical) books.
Tables again
Authors: Clark Malcolm 
Published: November 2023

Abstract
There is quite another way entirely to tackle tables. Tex has two commands, \halign and \ valign, which provide an enormously flexible way of handling tabular material. With the exception of the two ‘programming’ examples from the last chapter, all the tabular material presented so far would (in my opinion) be more easily handled through this alternative technique. I confess that I very rarely use \settabs. The\halign command in particular seems to lend itself to a variety of problems. To begin with, its use gives the illusion of awkwardness. This is more a function of unfamiliarity, rather than intrinsic difficulty.
Rules
Authors: Clark Malcolm 
Published: November 2023

Abstract
But there is something fairly important missing. What about the rules? We normally expect tables to be boxed in. This we can do, but the alignments tend to get even more hidden in the welter of extra detail. The use of the horizontal rule to divide some rows is still quite common, but vertical rules are now infrequently found. A curious inversion of reality has occurred. The latest edition of the Chicago Manual of Style says One style that has changed since the last edition of this manual was published is the use of vertical rules in tabular matter. In line with a nearly universal trend among scholarly and commercial publishers, the University of Chicago Press has given up vertical rules as a standard feature of tables in the books and journals that it publishes. The handwork necessitated by including vertical rules is costly no matter what mode of com position is used, and in the Press’s view the expense of it can no longer be justified by the additional refinement it brings.
Fonts
Authors: Clark Malcolm 
Published: November 2023

Abstract
In this chapter we will look at some of the aspects of dealing with fonts. To some extent the basics have already been covered. As already related, a simple \font\tenss=cmss10 loads the Computer Modern Sans Serif font at 10 pt. There are potential problems since plain is only guaranteed to set up the basic 16 fonts. These are the eight Computer Modern fonts in Figure 18.1. All are at 10 pt, but four of them are also loaded at some other sizes: A number of others are \preloaded in plain. All that \preloaded means is that the font information is read in so that should we need those fonts sometime there is no overhead.
Making pages
Authors: Clark Malcolm 
Published: November 2023

Abstract
Although we have looked at some aspects of building pages, we have not tackled this in a systematic way. In this chapter we will look at the default arrangements that Tex has for page building, and the ways this may be modified for more specialist requirements. In particular, we shall examine some of the techniques which may be employed in order to achieve double-column output. This seems a perennial problem, not least since the most common size of paper available, generally of the order of just over 8 inches, or about 750 points, requires some alternative to the default strategy when faced with 10 point fonts.
Breaking up
Authors: Clark Malcolm 
Published: November 2023

Abstract
Some topics, although essential, have been treated in a rather diffuse manner. In particular the problem of line, paragraph, and page breaking requires closer examination. It reveals some fascinating aspects of Tex- In order to be able to tackle this, we need to have some knowledge about three kinds of costs associated with this activity. In a nutshell, Tex seeks to minimize some function of the costs incurred in dividing a paragraph into lines. The three costs are termed badness, penalties, and merits (or demerits). Penalties have at least been introduced in a rather loose and informal way. The function of a penalty is to modify Tex’s ‘normal’ or default behaviour. All of these behaviour characteristics can be modified either globally or locally by either modifying global penalty values, or by locally introducing specific penalties. Only penalties may be specified or altered. The other costs are determined by Tex itself. ‘Badness’ is associated with individual lines (or potential lines) within a paragraph, while ‘merit’ is calculated over the whole paragraph. Tex can report their values, but you have no mechanism for altering them directly. The merit figure may be altered indirectly through manipulation of \penalty.
Delays and deferments
Authors: Clark Malcolm 
Published: November 2023

Abstract
The action of page creation has some interesting consequences. There are certain matters which Tex deliberately delays until a page has been created. Clearly, the \output routine makes explicit certain actions, but there are a few implicit actions too. To be specific, the \mark command has several compatriots, namely the triad \topmark, \firstmark, and \botmark whose meaning is determined through page breaking. The contents of these three will change automatically, without any ‘human’ (or even explicit command) intervention. The basic strategy is this: somewhere in the text, there will be some \marked text. The first time that \mark is used, all of the \mark triad take the same value.
Collections
Authors: Clark Malcolm 
Published: November 2023

Abstract
The purpose of this chapter is to draw together some of the commands which have been uncovered in the evolution of the primer. From time to time it has been necessary to introduce members of various classes of commands in a somewhat unstructured way. They were needed to solve particular problems, but elaborating upon them and introducing similar commands would confuse the flow of the text. The account of 1E)C’s primitive conditionals has been rather scattered throughout the preceding chapters. It is useful to draw all the variations into a more ordered exposition, augmenting them with examples where appropriate. Some of the conditionals are used infrequently, although it may be comforting to know that they are available. Naturally these tests would not occur in ‘normal’ text: they would be located within more elaborate commands.
Random Walk On A Lattice
Authors: Hughes Barry d. 
Published: November 2023

Abstract
The problem of wandering at random in a network of streets, or as we would now describe it, of random walk on a lattice, was first raised by Polya (1919, 1921), though its one-dimensional antecedents may be found in the theory of games of chance, especially the problem of gambler’s ruin (Feller 1970); in one dimension, Polya’s problem is also equivalent to the problem of random flight in one dimension, discussed in §2.2.3. Polya addressed in particular the following problem*. A wandering point (or in modern parlance, a random walker) moves at random between the sites of a simple hypercubic lattice in d dimensions, so that the allowed positions of the random walker are points indexed by a d-dimensional vector with integer components. The motion consists of a sequence of steps between nearest-neighbour sites-one of the coordinates of the position changes by ±1 in any step. It is assumed that there is unbiased random motion, so that all of the 2d possible next steps away from the current position are equally likely, each having probability 1/(2d).
Continuous-Time Random Walks
Authors: Hughes Barry d. 
Published: November 2023

Abstract
In Chapter 4, we obtained the governing equations of continuous-time random processes by taking a suitable joint limit Δ -→0 and τ -→0 of a lattice random walk, where Δ is the lattice spacing and τ the constant time between successive steps. The continuous-time processes so obtained are governed by stable densities (§4.3), with the Gaussian density representing the most typical case. In taking the continuum limit the lattice structure and the identity of individual steps are lost, and the analysis of many properties of the processes so obtained requires rather deep and subtle aspects of advanced probability theory, in contrast to the elegant conceptual simplicity of the original random walk. There is a less drastic way to obtain a continuous time process related to the ordinary random walk, without losing either the lattice structure or the notion of well-defined individual steps. This approach, the continuous time random walk model, owes its popularity in contemporary physics to Montroll and Weiss (1965). From a mathematical standpoint it is a particularly useful special case of the semi-Markov process (Levy 1954, Smith 1955, Pyke 1961, Feller 1964, Qinlar 1969, 1975).
Exploration And Trapping
Authors: Hughes Barry d. 
Published: November 2023

Abstract
In Chapter 3 we addressed at some length for lattice random walks questions arising from P6lya’s problem, which may be succinctly described as ‘Will I come back, and if so, when?’ and ‘Where am I likely to be now?’. In the resolution of these questions, we were neither obliged to record or study the time-history of the walk, nor to take note of the portion of the lattice explored by the walker. We did remark in §3.2.6 that for walks which are not strongly transient, very tortuous paths back to the starting site are statistically significant. It may be recalled from the corollary to P6lya’s Theorem in §3.3.3 that an unbiased walk with finite mean-square displacement per step on a periodic lattice of dimension d is strongly transient if and only if d ≥ 4. In Chapter 4 we observed that the set of sites visited may, in some statistical sense, possess a fractal dimension, influenced by the range of the allowed displacements. The geometrical properties of the set of sites visited are important in some contexts. For example, if one places the random flight model of a polymer chain (§2.6) on a lattice (cf. Chapter 7), the geometry of the set of sites visited in a specific walk determines the shape of individual polymer chains.
The Self-Avoiding Walk
Authors: Hughes Barry d. 
Published: November 2023

Abstract
Therandom walk problems discussed in Chapters 2-5 are loosely described by physicists as Markovian, by which they mean the absence of dependence of future evolution on the entire past history of the process, that is, the absence of memory. For example, in P6lya’s problem of a random walk on a lattice, to determine the probability laws governing future motion of the walker one needs to know only the current position; where the walker was three steps ago is of no significance. In the context of continuous-time random walks, we examined in Chapter 5 some walks with a ‘convolution-type memory’, exhibited explicitly in the generalized master equation formalism. Allowing the walker to have such a memory does not destroy the elegant simplicity of the random walk problem. What kind of memory effects have to be introduced to present real difficulties in the analysis? It is evident that a finite-range memory, in which the transition probability distribution at the nth step depends on the k most recent steps (with k &lt; ∞), leads to nothing new. Such a walk may always be re-defined in terms of a walk with internal states (§3.5). The situation is entirely different, however, if one allows the walker to remember where he has been ever since his walk began, with a memory which does not fade, as the following canonical example shows.
Bernoulli Site Percolation
Authors: Hughes Barry D 
Published: November 2023

Abstract
This chapter contains the rigorous proofs of a number of important results of percolation theory. The results are established here for the specific case of Bernoulli site percolation on a lattice for which the coordination number of a site is bounded. Since the Bernoulli bond problem can always be converted to a Bernoulli site problem (on the covering lattice), the discussion effectively covers the Bernoulli bond problem also. Many of the results of this section have analogues for a more general class of models, allowing for directed bonds and for long-range connections between sites, some of which are summarized in §1.5.7 and §1.5.8.
Critical Exponents In Percolation Theory
Authors: Hughes Barry D 
Published: November 2023

Abstract
It is believed that the behaviour of the key statistics of percolation theory close to the percolation threshold can be described in terms of ‘critical exponents’. Some of the critical exponents are defined solely in terms of the cluster-size distribution Pn(p) and functions related to it, such as the mean cluster size x(p) and the percolation probability P00(p). These are discussed for arbitrary homogeneous lattices in §4.1. A number of relations between the critical exponents defined via Pn(P) are believed to hold (§4.2). These relations are closely analogous to scaling relations in statistical mechanics {cf. Volume 1, §7.6) and can be derived by heuristic scaling arguments (§4.2.2 and §4.2.3). One of these relations can be established rigorously as an inequality by a simple argument (§4.2.4), while a number of other rigorous inequalities can be derived via lattice animal expansions (§4.3).