Let $\xi(s)=\frac12 s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$ be the completed Riemann xi-function, and let $F(x)=\frac{\xi'}{\xi}\!\left(\frac{1}{1-x}\right)=\sum_{n\ge 0} f_nx^n$ initially denote its germ at the origin. We introduce the real symmetric Toeplitz--Hankel matrix $c_{ij}=f_{|i-j|}-f_{i+j+1}+\delta_{ij}f_0, \qquad i,j\ge 0.$ We prove that the Riemann hypothesis is equivalent to positive semidefiniteness of every finite leading principal block of this matrix. More strongly, the full matrix condition is equivalent, for the purpose of testing the Riemann hypothesis, to only the adjacent local inequalities $2f_0-f_{2n+1}\ge 0,$ $(2f_0-f_{2n+1})(2f_0-f_{2n+3})\ge (f_1-f_{2n+2})^2 \qquad(n\ge 0).$ The converse implication uses a Pringsheim bootstrap: these local inequalities force the germ of $F$ to have Taylor radius at least one, hence exclude zeros of $\xi$ from the half-plane $\Re s>1/2$. If $\lambda_n$ are the Keiper--Li coefficients, then $f_n=\lambda_{n+1}-2\lambda_n+\lambda_{n-1}$, so the criterion is a local quadratic condition on their second finite differences. This is an equivalent reformulation, not a proof of the Riemann hypothesis. To the best of our knowledge, the exact Toeplitz--Hankel matrix and its reduction to adjacent $2\times2$ conditions have not appeared previously.
The Connes–van Suijlekom truncated Weil quadratic form, indexed by a cutoff parameter c that controls the primes p ≤ c entering the operator, produces a ground state whose Fourier–Mellin zeros provably lie on the critical line; whether they converge to the Riemann zeros as c → ∞ is open (Connes 2026; Connes–Consani–Moscovici 2025). We present, to our knowledge, the first independent public implementation of the Connes–van Suijlekom Galerkin matrix at sixteen cutoffs (c = 13 through 67, plus c = 100). Across the in-sample window c = 13 through c = 67 at N = 100, the first-zero absolute error |γ1 − γ1Riemann| shrinks monotonically from ∼2×10−55 to ∼1.5×10−168 — a 113-OOM convergence across fifteen cutoffs. The smallest-positive even-sector eigenvalue λmineven separately reaches ∼10−334 at c = 100, N = 250 (275-OOM span from c = 13). Out-of-sample test at c = 100. On the four-point N-sweep N ∈ {100, 150, 200, 250} at dps = 500, consecutive first-difference ratios 0.837 and 0.836 match to two decimal places. Aitken-Δ2 on the two overlapping triples yields log10|λ∞even| ≈ −536.8 and ≈ −533.7, approaching the Connes 2026 §6.4 heuristic prediction (≈ −530.4) monotonically with N (6.4 and 3.3 OOM gaps out of |x∞| ∼ 530). The same eigenvector recovers γ1, …, γ10 to 307–329 matching digits at N = 250, dps = 500. Under the unitary equivalence with Connes–Consani–Moscovici Lemma 5.1, this is the deepest such Galerkin-truncation recovery in the public Connes–van Suijlekom / Connes–Consani–Moscovici literature, subject to a hypothesis-status caveat: the raw finite-N matrix carries a small block of dps-stable negative-sign eigenvalues, so we report the smallest-positive branch (continuum positivity of QWλ is RH-equivalent and is not assumed at λ = √100). The fit |log10 λmin| ≈ 13.24 c0.634 on c ≤ 67 at N = 100 is shown to be a finite-N rate, falsified at c = 100, N = 200 by 49 OOM in the direction of faster decay. Structural observations include approximate eigenvector c-invariance (overlap ≥ 0.9498 on all 105 cutoff pairs despite eigenvalues differing by 113 OOM), multi-zero convergence universality (all ten detectable zeros within 3.8% of each other), an empirical Galerkin-convergence exponent s(c) ≈ 55 log c − 128, un-rescaled Galerkin bulk-spectrum Poisson statistics (β < 0.05; this is a structural diagnostic of the truncated operator, not a test of Montgomery's conjecture, which applies to locally-rescaled zero spacings), and tight bulk invariants log|det Qc| ≈ −65.6 c + 542 (R2 = 0.997). We make no claim of proof; the contribution is reproducible numerical data and its careful interpretation under the existing CvS / CCM framework. All code, data, and ancillary files are publicly available.
A common aim of epidemiological research is to estimate the association between a particular exposure and a particular outcome, with the aim to better understand potential causal mechanisms behind this association. In 2020, Li et al1 proposed a method they called ‘Inference about causation from examination of familial confounding’ (abbreviated ICE FALCON), which uses twin data (or, more generally, data on related individuals) to assess whether an association is due to a causal effect of the exposure, reverse causation, familial confounding, or a mix of the three. In brief, this method requires the analyst to first regress an individual’s outcome on her own exposure, then on her twin’s exposure, and then finally on her own exposure and her twin’s exposure simultaneously. Li et al1 argued that by examining the pattern of the estimated regression coefficients the analyst could infer the true mechanism that generated the data, and claimed the method was analogous to Mendelian randomization (MR), but with certain additional benefits. Since this methodological paper was published it has been cited 31 times according to Google Scholar, and in a later paper, Li et al2 claimed that the ICE FALCON method has ‘provided causal evidence for numerous exposure and outcomes’, citing 14 publications for this claim. In this cautionary note, we argue that the ICE FALCON method is based on highly unrealistic assumptions that were not clearly stated by Li et al,1 and several statistical errors. The note is organized as follows. We first review the key elements of the ICE FALCON method. Next, we lay out our critique against the method. We finally consider the real data example given by Li et al,1 and argue that far less can be said about underlying mechanisms in this example than promised by the ICE FALCON method. Throughout, we ignore sampling variability and focus on the large sample behaviour of estimated regression coefficients. We adopt the notation by Li et al.1 Let Xself and Yself be the exposure and outcome for a particular individual, and Xco-twin and Yco-twin be the exposure and outcome for that individual’s co-twin. Li et al1 considered the possible mechanisms illustrated by the causal diagrams in Figure 1 (identical to their Figure 1). In these diagrams, SXY is the set of family-constant confounders for X and Y, SX is the set of family-constant factors that influence X but not Y, and SY is the set of family-constant factors that influence Y but not X. Finally, U is the set of unmeasured within-individual confounders, i.e. the confounders that may have different values for the two twins in a pair. Apart from unmeasured within-individual confounding, in Figure 1a the statistical association between X and Y is due to familial confounding, in Figure 1b it is due to causation, and in Figure 1c it is due to reverse causation. Figure 1d illustrates a mixture of familial confounding and causation, and Figure 1e illustrates a mixture of familial confounding and reverse causation. Like Li et al,1 we first restrict attention to the three simpler scenarios in Figure 1a-c, and later comment on mixtures of these. Possible mechanisms underlying a statistical association between X and Y, as proposed by Li et al1 Note that, in Models 1–3, the intercepts α, α′ and α″ are generally different, as well as the coefficients βself and βself′, and the coefficients βco-twin and βco-twin′. Note also that, since X and Y are available for both twins in each pair, each twin contributes as both ‘self’ and ‘co-twin’ to the fitted models. Li et al1 argued that the mechanisms in Figure 1a-c imply the coefficient patterns summarized in Table 1 (identical to their Table 1), where ρX and ρY are the within-pair correlations in X and Y, respectively. Since each pattern (e.g. column in the table) is unique, they concluded that the estimated regression coefficients could be used to distinguish between these three mechanisms. For instance, if we observe that βself′=βself and βco-twin′=0, then, according to Li et al,1 we can rule out both familial confounding and reverse causation, since this coefficient pattern is unique for causation. Unique regression coefficient patterns implied by the three mechanisms in Figure 1, according to Li et al1 Unique regression coefficient patterns implied by the three mechanisms in Figure 1, according to Li et al1 Li et al1 provided both graphical arguments (in their main text) and mathematical proofs (in their Supplementary material) for the coefficient patterns in Table 1. However, there are two important problems with these arguments and proofs: (1) they ignore the influence of unmeasured within-individual confounders, and (2) they conflate statistical conditioning with accounting for correlated observations. Below, we discuss these problems in more detail. In the Supplementary material we show that, when these issues are acknowledged, there is no guarantee that the mechanisms in Figure 1a-c will produce distinct coefficient patterns. In particular, we give numerical examples where all three mechanisms result in identical coefficient patterns. In virtually all real family studies, unmeasured within-individual confounders U will be present. Or stated differently: there are likely to exist confounders which are not identically shared by the twins, and which we cannot perfectly adjust for in our regression models. Depending on the context, these may have larger or smaller magnitude than the familial confounders SXY, and may affect X and Y in the same or opposite direction. We have previously shown that unmeasured within-individual confounders have important consequences for the interpretation of sibling comparison studies, since their effect will be numerically different in the full cohort and when conditioning on the sibling pair.3 However, Li et al1 effectively ignored unmeasured within-individual confounders when deriving Table 1. For instance, they wrote (page 1261, second column) ‘If there is familial confounding only … (Figure 1a), there will be associations between… Yself and Xco-twin (βco-twin, Model 2). Adjusting for Xself (Model 3), there will still be a conditional association between Yself and Xco-twin (βco-twin′), but it will be attenuated towards the null compared with βco-twin’. This conclusion is not generally valid when Uself is present, since adjusting for Xself then opens the path Yself←Uself→Xself←SX→Xco-twin at which Xself is a collider, which may inflate the association between Yself and Xco-twin. As another example, they wrote: ‘If there is a causal effect from X to Y only (Figure 1b), … the conditional [on Xself] association between Yself and Xco-twin (βco-twin′) will be null…’ This conclusion is also invalid when Uself is present, since adjusting for Xself then again opens the aforementioned path, which will induce a statistical association between Yself and Xco-twin. Similar issues apply to several of the other coefficient patterns in Table 1. In their Supplementary material, Li et al1 explicitly articulated the assumption of no unmeasured within-individual confounders. However, they did this by writing ‘For simplicity we assume that … there was no within-individual confounding’ (italics added), without commenting on the implausibility of this assumption in real studies or acknowledging that the ICE FALCON method is invalid when the assumption is violated. This gives the reader the misleading impression that the assumption was made purely for mathematical convenience, and that it is not required by the ICE FALCON method. This impression is further enforced by the fact that, in the main text of the paper, the assumption is not mentioned at all; on the contrary, all the causal diagrams in the paper include within-individual confounders, and the text repeatedly refer to these as if they are not assumed absent. For instance, when comparing the ICE FALCON method to MR, they state (in their Table 4), as a relative advantage of the former, that ‘Xco-twin is theoretically unrelated to unmeasured confounders specific to an individual only [italics added]’. We have not scrutinized all 14 papers that Li et al2 cite for using the ICE FALCON method, but several of them appear to have entirely missed the crucial role of unmeasured within-individual confounders for the method. For instance, Zheng et al4 studied the association between socio-economic status and obesity among adult monozygotic and dizygotic twins, which may clearly suffer from unmeasured within-individual confounding by various life-style and (for dizygotic twins) genetic factors. The authors indeed included within-individual confounders in their causal diagrams (identical to Figure 1a-c); however, they then proceeded by analysing data with the ICE FALCON method as if unmeasured within-individual confounders were absent altogether. Then, Model 2* would still produce incorrect standard errors if fitted with ordinary linear regression instead of a GEE, since each twin contributes as both ‘self’ and ‘co-twin’ to the model. This conceptual mistake led Li et al1 to several incorrect conclusions about the coefficient patterns in Table 1. For instance, they wrote ‘If there is a causal effect from Y to X only (Figure 1c), … [then in] Model 2, there is no open path between Yself and Xco-twin—the path through SY is closed due to the fact that Yco-twin is conditioned on’. Presumably, they here referred to the path Yself←SY→Yco-twin→Xco-twin, which would indeed be closed if Yco-twin were conditioned on. However, since fitting Model 2 with a GEE does not condition on Yco-twin the path remains open. Ironically, if Yco-twin were truly conditioned on, then the path Yself←SY→Yco-twin←Uco-twin→Xco-twin at which Yco-twin is a collider would be open instead. Curiously, whereas Li et al1 claimed that Model 2 becomes conditioned on Yco-twin when accounting for correlated observations, they did not make such claims for Models 1 and 3, whereas, in fact, the issue of correlated observations is equally present for these models. Thus, by their (incorrect) logic, Models 1 and 3 would also be conditioned on Yco-twin when fitted with a GEE. In real scenarios, the true mechanism behind an observed association will likely be a mixture of confounding and causation, or between confounding and reverse causation, as in Figure 1d and e, respectively. For such scenarios, Li et al1 wrote (page 1262, first column): ‘…the result will be a mixture [of the patterns in Table 1]… The changes in the pair of regression coefficients from comparing Model 3 with Models 1 and 2 will apply, allowing assessment of evidence for causality still to be made’. To illustrate, Li et al1 provided two real data examples. In one of their examples, the exposure was body mass index (BMI) measured at baseline, and the outcome was BMI measured at a later time point during follow-up. Data were collected on adult twins from 250 monozygotic twin pairs. From the temporal order of the BMI measures, we can rule out reverse causation a priori. However, there is clearly a high potential for both confounding (familial as well as within-individual) and causation. In particular, the members of an adult twin pair may have very different eating and exercise habits from each other, which could then be strong unmeasured within-individual confounders for BMI measures over time. Li et al1 presented the following estimated regression coefficients: βself=0.81, βco-twin=0.73, βself′=0.73 and βco-twin′=0.15. The pattern of these coefficients is a mixture of ‘Familial confounding’ and ‘X causes Y’ in Table 1, with the attenuation from βco-twin to βco-twin′ being stronger than the attenuation from βself to βself′. Li et al1 concluded that these results are ‘… consistent with a longitudinal causation, as well as a small amount of familial confounding’. We do not disagree with this conclusion, which sounds intuitively appealing given the nature of BMI and its stability over time, even before doing any study on the topic. However, since the pattern of regression coefficients in Table 1 are derived under the unrealistic assumption of no unmeasured within-individual confounding, we fail to see that the presented regression coefficients give any strong further support for this hypothesis. If we are open to the presence of unmeasured within-individual confounding, the results would also be consistent with other possible mechanisms, such as a large amount of familial confounding together with a small amount of causation, or even a total absence of causation. We demonstrate this with numerical examples in the Supplementary material where the mechanisms in Figure 1a-c all give virtually the same coefficients as above for the BMI data. In the presence of unmeasured within-individual confounding, the regression coefficients alone simply cannot discriminate between these three mechanisms. This can also be seen directly from the causal diagrams. Consider the causal diagram in Figure 1a, where causation is entirely absent. In this diagram, Xself and Yself are associated through two paths: Xself←SXY→Yself and Xself←Uself→Yself, whereas Xco-twin and Yself are only associated though the first path, which could explain why βself> βco-twin. When conditioning on Xco-twin, the path Xself←SX→Xco-twin←SXY→Yself becomes open, which could explain why βself>β′self. Similarly, when conditioning on Xself, the path Xco-twin←SX→Xself←Uself →Yself becomes open, which could explain why βco-twin>β′co-twin. In this note, we have demonstrated some inaccuracies in the derivation of the ICE FALCON method, and highlighted that the method fails to distinguish between competing causal hypotheses in the presence of within-individual confounding. Even in their title, Li et al1 proposed that ICE FALCON was ‘analogous’ to MR, and further claimed in their key messages that a benefit over MR would be that ICE FALCON ‘does not make strong assumptions’. We think the comparison to MR is overstated and misleading, as the ICE FALCON cannot be described as an instrumental variable method, uses completely different types of data, and does not lend itself to an estimation of the causal effect even when the (indeed) very strong assumptions underlying it are fulfilled. Whereas we have focused on the most important problems with the paper by Li et al,1 there are other, more technical issues with the paper as well, which we discuss in the Supplementary material. In brief, their mathematical derivations have an error that potentially invalidates their results, even in the absence of unmeasured within-individual confounding, and they used GEEs in an inappropriate way that is almost guaranteed to give bias for twin data. The latter issue does not invalidate the ICE FALCON method per se, but it does potentially invalidate their real data analysis results. Yet another issue worth mentioning is the extension of the ICE FALCON method to non-linear models. Li et al1 appear to claim that this extension is trivial, by stating (page 1266) that ‘ICE FALCON is based on regression, so the method can be applied to continuous and binary outcomes using ordinary and logistic regression, respectively, and potentially to survival data using Cox regression. There are no restrictions on the measurement scale of exposures’. This seems overly optimistic, given that their Supplementary material proofs use analytic results for linear regression coefficients that are not easily transferred to non-linear models. In particular, due to the non-collapsibility of odds ratios and hazard ratios,5 we conjecture that it is very hard to derive universal patterns of the regression coefficients of logistic regression models and Cox regression models under the mechanisms in Figure 1, even in the absence of unmeasured within-individual confounders. The idea that underlying mechanisms can be inferred for family data by comparing (changes in) regression coefficients is not new. Hudson et al6 also considered the causal diagrams in Figure 1, and discussed what can be inferred about familial confounding from Models 1–3. However, they were substantially more modest in their conclusions than Li et al.1 They concluded that familial confounding can be ruled in by a non-zero coefficient βco-twin, provided that one a priori rules out both causation and reverse causation. This fact follows immediately by noting that, in Figure 1a, the only explanation for an association between Xco-twin and Yself (e.g. non-zero βco-twin) is the presence of familial confounders SXY. They also carried out a simulation, which indicated that the degree of familial confounding may often lie between βco-twin and β′co-twin, when causation is present. However, they cautioned the reader that a simulation does not provide definitive evidence, and unlike Li et al, acknowledged that their particular simulation made several strong assumptions (e.g. normally distributed errors, no statistical interactions, all effects being positive, etc) that will not hold in all real studies. We have repeatedly argued that the complete absence of unmeasured within-individual confounding is very unlikely in practice, but some might counter that the absence of unaccounted for confounding is necessary to draw causal conclusions from all observational studies, and thus that this is not a unique weakness of the ICE FALCON method. To this we would reply that there is a large difference between the common practice in epidemiology where we remain open to the idea that the reported association remains partly confounded (discussing the risk and magnitude of residual confounding, often in relation to differently adjusted models), and the ICE FALCON where we on the one hand must assume that there is confounding by some factors shared identically by relatives (else the method is meaningless), and on the other hand assume that there is no confounding by anything which is correlated less than 1 between twins. Since most potential confounders are somewhere in between (genetic markers would, e.g. be correlated 0.5 in first degree relatives), we can only imagine this to be a plausible assumption when we already have so much knowledge about the research question that it is no longer meaningful to rule out alternative hypotheses for the causation. In particular, if we are interested in estimating a causal effect when we suspect there may be familial confounding but no other bias, we should use a standard sibling comparison design,7 which requires fewer models and gives us a direct estimate of the effect size. We also note that, if the ICE FALCON method is used for other types of relatives than twins (e.g. ordinary siblings or half-siblings), then one would expect the sets of within-individual confounders to be larger for less related individuals. Finally, we note that the ICE FALCON method can only, at best, indicate which underlying mechanisms are at play, but does not provide estimates of their relative importance. Such estimates can be obtained with variance decomposition through structural equation modelling; see Maes et al8 and the references therein. Since there are no new data associated with this article, there was no need for ethics approval. There are no new data associated with this article. Supplementary data are available at IJE online. All work for this paper was carried out by Arvid Sjölander and Thomas Frisell jointly. This work was supported by the Swedish Research Council [2020-01188 to A.S.]. None declared. Artificial intelligence (AI) was not used for any parts of this paper.
正定矩阵是一类特殊的矩阵，作为对称矩阵的子类，在数学分析中判别多元函数极值、判断函数单调性等具有广泛的应用。他不仅具备对称矩阵可对角化的性质，而且具有对称矩阵不具备的更高的性质。但是要想运用正定矩阵的这些性质，我们就要首先会判断一个矩阵是正定矩阵。可以看到的是在一些课本中或辅导教材中给出了很多正定矩阵的证明方法。一般都是求解特征值，然后证明其特征值大于零。但在一些比较复杂的题目中往往掺杂大量的中间结论的证明，学者在思考过程中如果忽略其中的隐含结论的证明，就会导致整个题目都做不出来。因此做这种题目往往需要我们有大量的知识储备。在本文中我们将要一起讨论针对一类特殊正定矩阵的证明方法，并给出了一种简单的证明方法。 Positive definite matrix is a special kind of matrix. As a subclass of symmetric matrix, positive definite matrix has a wide range of applications in mathematical analysis, such as discriminating the extreme value of multivariate functions and judging the monotonicity of functions. It possesses not only the diagonalization property of symmetric matrices, but also the higher property that symmetric matrices do not possess. But in order to use these properties of positive definite matrices, we first have to know that a matrix is positive definite. And you can see that there are a lot of proofs for positive definite matrices that are given in some books or in some tutorial books. You usually solve for the eigenvalues, and then you prove that the eigenvalues are greater than zero. However, in some more complex topics, a large number of proofs of intermediate conclusions are often mixed in. If scholars ignore the proofs of implicit conclusions in the process of thinking, they will fail to complete the whole topic. Therefore, to do this kind of problem often requires us to have a large knowledge reserve. Today we are going to discuss the proof for a special class of matrices, and give a simple proof method.
Asynchronous iterative algorithms are parallel iterative algorithms in which communications and iterations are not synchronized among processors. Thus, as soon as a processing unit finishes its own calculations, it starts the next cycle with the latest data received during a previous cycle, without waiting for any other processing unit to complete its own calculation. These algorithms increase the number of updates in some processors (as compared to the synchronous case) but suppress most idle times. This usually results in a reduction of the (execution) time to achieve convergence. Optimized Schwarz methods (OSM) are domain decomposition methods in which the transmission conditions between subdomains contain operators of the form \linebreak $\partial/\partial \nu +\Lambda$, where $\partial/\partial \nu$ is the outward normal derivative and $\Lambda$ is an optimized local approximation of the global Steklov-Poincar\'e operator. There is more than one family of transmission conditions that can be used for a given partial differential equation (e.g., the $OO0$ and $OO2$ families), each of these families containing a particular approximation of the Steklov-Poincar\'e operator. These transmission conditions have some parameters that are tuned to obtain a fast convergence rate. Optimized Schwarz methods are fast in terms of iteration count and can be implemented asynchronously. In this thesis we analyze the convergence behavior of the synchronous and asynchronous implementation of OSM applied to solve partial differential equations with a shifted Laplacian operator in bounded rectangular domains. We analyze two cases. In the first case we have a shift that can be either positive, negative or zero, a one-way domain decomposition and transmission conditions of the $OO2$ family. In the second case we have Poisson's equation, a domain decomposition with cross-points and $OO0$ transmission conditions. In both cases we reformulate the equations defining the problem into a fixed point iteration that is suitable for our analysis, then derive convergence proofs and analyze how the convergence rate varies with the number of subdomains, the amount of overlap, and the values of the parameters introduced in the transmission conditions. Additionally, we find the optimal values of the parameters and present some numerical experiments for the second case illustrating our theoretical results. To our knowledge this is the first time that a convergence analysis of optimized Schwarz is presented for bounded subdomains with multiple subdomains and arbitrary overlap. The analysis presented in this thesis also applies to problems with more general domains which can be decomposed as a union of rectangles.
Open access
Advanced Numerical Methods in Computational Mathematics
The order of the pole singularities encountered in spectral method of moments formulations for source-free periodic problems is investigated. The solution of the source-free problem is often obtained by searching for the zeros of the Z matrix determinant using an iterative algorithm. During this process, pole singularities of the determinant are encountered and may cause numerical instability. In order to cancel the poles, their order must be known. A rigorous proof of the pole singularity order in the Z matrix determinant is given. The proof is general and holds for any problem which is periodic in at least one of the spatial directions. This knowledge enables to cancel the poles by an appropriate fixed factor with a simple routine.
Ruy de Queiroz, Luiz Carlos Pereira, Edward Hermann Hæusler
This volume contains the Proceedings of the 10th Workshop on Logic, Language, Information and Computation (WoLLIC'2003). The Workshop was held in Ouro Preto, Minas Gerais, Brazil from July 29 to August 1, 2003, in the Escola de Minas of the Universidade Federal de Ouro Preto ( UFOP ). WoLLIC is a series of workshops which started in 1994 with the aim of fostering interdisciplinary research in pure and applied logic . The idea is to provide a forum which is large enough in the number of possible interactions between logic and the sciences related to information and computation, and yet is small enough to allow for concrete and useful interaction among participants. Previous versions were held at: Recife (Pernambuco, Brazil) in 1994 and 1995; Salvador (Bahia, Brazil) in 1996; Fortaleza (Ceará, Brazil) in 1997; São Paulo (Brazil) in 1998; Itatiaia (Rio de Janeiro, Brazil) in 1999; Natal (Rio Grande do Norte) in 2000; Brasília (Distrito Federal, Brazil) in 2001; Rio de Janeiro (Brazil) in 2002. Scientific sponsorship comes from the Interest Group in Pure and Applied Logics ( IGPL ), the European Association for Logic, Language and Information ( FoLLI ), the Association for Symbolic Logic ( ASL ), European Association for Theoretical Computer Science ( EATCS ), the Sociedade Brasileira de Computação ( SBC ), and the Sociedade Brasileira de Lógica ( SBL ). Funding was kindly given by:(i) CNPq ( Conselho Nacional de Desenvolvimento Científico e Tecnológico , the scientific and technological development council of the Brazilian Ministério da Ciência e Tecnologia ) (grant 450709/2003-5);(ii) CAPES ( Fundação Coordenação de Apoio ao Aperfeiçoamento de Pessoal de Nível Superior , a Foundation for the Development of Higher-Education under the Brazilian Ministério da Educação e do Desporto ) (grant PAEP0565/03);(iii) FAPEMIG ( Fundação de Amparo à Pesquisa do Estado de Minas Gerais , the Minas Gerais state foundation for the support of scientific research);(iv) Escola de Minas da UFOP ( Universidade Federal de Ouro Preto ). Contributions were received in the form of short papers in all areas related to logic, language, information and computation, including:pure logical systems, proof theory, model theory, algebraic logic, type theory, category theory, constructive mathematics, lambda and combinatorial calculi, program logic and program semantics, logics and models of concurrency, logic and complexity theory, proof complexity, foundations of cryptography (zero-knowledge proofs), descriptive complexity, nonclassical logics, nonmonotonic logic, logic and language, discourse representation, logic and artificial intelligence, automated deduction, foundations of logic programming, logic and computation, and logic engineering. Apart from the contributed papers (15), and the invited talks (5), the programme includes 5 tutorial lectures: 1. Algorithmic Randomness and Derandomization by Eric Allender (Department of Computer Science, Rutgers, the State University of New Jersey, USA) 2. Generalized Quantifiers by Lauri Hella (Department of Mathematics, Statistics and Philosophy, University of Tampere, Finland) 3. Implicit computational complexity by Jean-Baptiste Joinet (Preuves-Programmes-Systèmes, Université Paris 7, France) 4. Proof search foundations for logic programming by Dale Miller (INRIA/Futurs/Saclay, and Laboratoire d'Informatique, École Polytechnique, France) 5. Iterated theory change by Hans Rott (Institut für Philosophie, Universität Regensburg, Germany) All papers in the volume were reviewed by the program committee consisting of Mauricio Ayala-Rinóon ( Departamento de Matemática, Universidade de Brasília, Brazil ) Argimiro Arratia ( Depto. Matematicas, Universidad Simon Bolivar, Venezuela ) Alessandra Carbone ( Institut des Hautes Études Scientifiques, and Université de Paris XII, France ) Marcelo Coniglio ( Centro de Lógica e Epistemologia, Universidade Estadual de Campinas, Brazil ) Gilles Dowek ( INRIA, France ) Arnaud Fleury ( Facoltà di Scienze, Università di Verona, Italy ) Dexter Kozen ( Cornell University, USA ) Maarten Marx ( ILLC, Faculty of Science, Universiteit Amsterdam, The Netherlands ) Anto˚nio Carlos da Rocha Costa ( Escola de Informática, Universidade Católica de Pelotas, Brazil ) Dieter Spreen ( Fachbereich Mathematik, Theoretische Informatik, Universität Siegen, Germany ) Luiz Carlos Pereira ( Departamento de Filosofia, PUC-Rio and UFRJ, Brazil ) Jouko Väänänen ( Department of Mathematics, University of Helsinki, Finland ) Renata Wassermann ( Departamento de Cie˚ncia da Computação, Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil ) The organising committee consisted of Lucília Figueiredo ( Departamento de Computação, Universidade Federal de Ouro Preto, Brazil ) Fred Ulisses Maranhão ( Centro de Informática, Universidade Federal de Pernambuco, Brazil ) Anjolina Grisi de Oliveira ( Center of Informatics, Universidade Federal de Pernambuco, Brazil ) Elaine Pimentel ( Departamento de Matemática, Universidade Federal de Minas Gerais, Brazil ) (Co-Chair) Ruy de Queiroz ( Center of Informatics, Universidade Federal de Pernambuco, Brazil ) (Co-Chair) Maria Angela Weiss ( Departamento de Matemática, Universidade de São Paulo, Brazil ) The volume will be published as volume 84 in the series Electronic Notes in Theoretical Computer Science ( ENTCS ). This series is published electronically through the facilities of Elsevier B.V. and its auspices. The volumes in the ENTCS series can be accessed at the URL http://www.elsevier.nl/locate/entcs A printed version of the current volume has been distributed to the participants at the workshop in Ouro Preto. We are very grateful to the following persons, whose help has been crucial for the success of WoLLIC'2003: Mike Mislove, one of the Managing Editors of the ENTCS series, for his assistance with the use of the ENTCS style files; Thanks are also due to the Department of Mathematics of Universidade Federal de Minas Gerais and the Department of Computing of the Universidade Federal de Ouro Preto, which has provided the logistic support to the organising committee. August 2, 2003 Ruy de Queiroz, Elaine Pimentel, Lucilia Figueiredo