The cultural evolution of socially situated cognition Because human cognition is creative and socially situated, knowledge accumulates, diffuses, and gets applied in new contexts, generating cultural analogs of phenomena observed in population genetics such as adaptation and drift. It is therefore commonly thought that elements of culture evolve through natural selection. However, natural selection was proposed to explain how change accumulates despite lack of inheritance of acquired traits, as occurs with templatemediated replication. It cannot accommodate a process with significant retention of acquired or horizontally (e.g. socially) transmitted traits. Moreover, elements of culture cannot be treated as discrete lineages because they constantly interact and influence one another. It is proposed that what evolves through culture is the mind; ideas and artifacts are merely reflections of its current evolved state. Interacting minds transform (in part) through through a non-Darwinian autopoietic process similar to that by which early life evolved, involving not survival of the fittest but actualization of their potential. Other papers in this issue explore how cognition is shaped by the social matrix in which it is embedded. In this paper the reader is invited to take a step back and consider the dynamics that emerge when cognitive agents don't just learn from one another, but 'put their own spin on' what they learn, make sense of it in their own terms, and then express or implement their 'take' on the ideas to others. Elements of culture start to create niches for one another. They become more complex with time, such that they might be thought of as constituting cultural lineages.. Some treat cultural change as merely a facet or dimension of biological evolution (Jablonka& Lamb, 2006) .1[1] Others, while not denying that culture has a dramatic impact on biological life, argue that culture constitutes a form of evolution in its own right, a second evolutionary process. On Earth the two are deeply intertwined (as illustrated by phenomena such as genetic assimilation, the Baldwin effect, and the fact that biological life had to come into being before culture could take hold). But in principle they need not be. For example, in a computer program in which artificial agents invent and imitate ideas for new gestures, but neither die nor give birth, there is still evolution (Gabora, 1995) . It is not the agents' physical form that is evolving, but their ideas. Fit gestures get imitated, and spread through the artificial society, while unfit gestures do not, such that over time the distribution of gestures implemented by agents becomes fitter. Indeed, cultural traits can be said to undergo descent with modification, and on the face of it cultural change is reminiscent of natural selection. It exhibits phenomena studied by population geneticists such as adaptation, punctuated equilibrium (Orsucci, in press) , and drift (Bentley et al., 2004; Durham, 1991; Gabora, 1995) , as well as features referred to by Mesoudi et al. (2004) as "key Darwinian properties", including variation, competition, and inheritance. Cultural change is also cumulative; humans have a propensity to not just generate novelty but build on it cumulatively, adapting old ideas to new circumstances (the Ratchet effect). One individual modifies the basic idea of a cup by giving it a flat enough bottom to stay put when not in use, another adds a handle, making it easier to grasp, and yet another adds a spout, making it easier to pour from. Moreover, this cumulative change is adaptive. With each instantiation, the basic idea remains the same but the details change to make it more useful with respect to the prevailing situation or need. It is also complex and open-ended; there is no limit to the cultural novelty that can be generated. This paper attempts to answer the question of whether the transmission and transformation of information across individuals occurs through a Darwinian process, and in what sense (if any) culture can rightly be said to 'evolve'. We begin by examining why organisms do not inherit acquired characteristics, and how this impacts their evolution. This is key because acquired change is inherited in culture2 [2] , though note that here 'inherited' merely means transmitted or 'passed on' without implying genetic mediation. To the extent that not inheriting acquired characteristics is central to how organisms evolve, answers from biology will not translate to culture. Noting that current origin of life theories suggest that the earliest life forms, referred to as protocells, also inherited acquired characteristics, we examine the hypothesis that the mechanisms by which culture evolves are more akin to those underlying the evolution of protocells than modern-day life. The human capacity to exchange and build on one another's ideas has completely transformed this planet.. It is tempting to describe the process of cultural change in Darwinian terms, in part because (1) the parallels between biology and culture are striking, and (2) Darwin's theory had a profoundly unifying effect on biology. Indeed Mesoudiet al. (2004) rightly assert that culture exhibits "key Darwinian properties". But having these properties does not imply that the mechanisms are similar, and they are not; indeed the paradox that necessitated the theory of natural selection in biology does not exist with respect to culture. Early biologists were motivated by the need to explain how change accumulates in a domain where acquired traits are not retained at the individual level. Selection theory was accordingly designed to describe descent with modification at the population level, in organisms that do not inherit acquired characteristics because they are von Neumann self-replicating automata. Self-replicating automata use a self-assembly code that is both actively transcribed to produce a new individual, and passively copied to ensure that the new individual can itself reproduce. Cultural traits and artifacts are not self-replicating automata, even those such as a village plan that contain a code, because the code consists of instructions for how to generate something else, not a copy of itself. They are allopoietic, not autopoietic. Cultural traits do inherit acquired traits-the effects of creative modification or contextual interaction with the environment are retained. So cultural theorists do not share the biologists' need to explain how change occurs in the face of periodic reversion to a previous state. Their challenge is to describe cognitive states that involve potentiality, and that change through interaction with an incompletely specified context, as occurs in (for example) dialogue or play.. It is sometimes said that culture evolves without any sort of self-replicating structure, or that since the kind of modification with descent exhibited in culture cannot be described by natural selection, there is no theory of evolution that can explain cultural change (e.g. Fracchia&Lewontin, 1999) . It does seem unlikely that culture involves von Neumann selfreplicating automata. But natural selection of self-replicating automata is not the only means by which evolution can occur. Indeed another sort of process was necessary to get life to the point where it replicated using a self-assembly code and was thereby able to evolve through natural selection. Current widely held theories about the origin of life hold that protocells, the earliest forms of life, evolved through a non-Darwinian process. They self-replicated without a selfassembly code through autopoiesis, i.e. mutual regeneration of parts to reconstitute a new whole. Because uncoded self-replicating structures can inherit traits acquired between replication events, they may either survive intact or go extinct, or do something in between these two extremes: they can transform through the horizontal exchange of innovation.. We found a candidate for this more primitive, uncoded kind of self-replicating structure in culture: the worldview. It is posited that humans achieved this kind of integrated internal model of the world in virtue of the large cranial capacity and resulting dense associative structure of human memory. The mind generates cultural novelty not by selecting amongst predefined randomly-generated alternatives but by thinking through how it could work, and its capacity to shift between analytic and associative thought processes makes it ingeniously suited to this kind of 'mulling over'. Associative thought often proceeds through interaction with a context: a fortuitous encounter with a triggering stimulus, talking something over, or simply re-assessing where one is going based on the effects of ones' actions. Natural selection cannot account for this kind of context-driven change because it cannot incorporate situations in which a product of culture could take many different forms depending on the worldviews of those creating it, and the circumstances under which they came together and interacted when they did. Consequently, it cannot account for the birth of new cultural traits except as the result of trial and error; moreover it cannot account for novelty that has new properties or that deviates in any substantial way from what has come before. It is proposed that worldviews evolve not through survival of the fittest but through transformation; they neither die nor survive intact but transform over generations as elements get incorporated and adapted to new circumstances. We cannot examine the evolution of worldviews directly, but we can do so indirectly, by examining how art, science, technology, media, social structures, and governing bodies are transforming the planet that sustains us.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


AN AUTOMATA-THEORETIC APPROACH TO THE STUDY OF THE INTERSECTION OF TWO SUBMONOIDS OF A FREE MONOID Abstract. We investigate the intersection of two finitely generated submonoids of the free monoid on a finite alphabet. To this end, we consider automata that recognize such submonoids and we study the product automata recognizing their intersection. We start by proving a result of Karhumaki on the characterization of the intersection of two submonoids of rank two, in the case of prefix (or suffix) generators. In a more general setting, for an arbitrary number of generators, we prove that if H and K are two finitely generated submonoids generated by prefix sets such that the product automaton associated to H ∩ K has a given special property then rk(where rk(L) = max(0, rk(L) − 1) for any free submonoid L.

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Computing constructor forms with non terminating rewrite programs In the context of the study of rule-based programming, we focus in this paper on the property of C-reducibility, expressing that every term reduces to a constructor term on at least one of its rewriting derivations. This property implies completeness of function definitions, and enables to stop evaluations of a program on a constructor form, even if the program is not terminating. We propose an inductive procedure proving C-reducibility of rewriting. The rewriting relation on ground terms is simulated through an abstraction mechanism and narrowing. The induction hypothesis allows assuming that terms smaller than the starting terms rewrite into a constructor term. The existence of the induction ordering is checked during the proof process, by ensuring satisfiability of ordering constraints. The proof is constructive, in the sense that the branch leading to a constructor term can be computed from the proof trees establishing C-reducibility for every term. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. PPDP'06 July 10-12, 2006, Venice, Italy. Copyright c 2006 ACM 1-59593-388-3/06/0007. . . $5.00.Categories and Subject DescriptorsSufficient completeness plays an important role in algebraic specifications, as well as in rewriting-based programs. In both contexts, terms are built on operators, among them we can distinguish constructors. Constructors are basic symbols, allowing us to describe the values of computations. The other symbols, called defined symbols, represent functions defined on these values. One is naturally led to ask for a specification or a program, whether the functions are sufficiently defined to warrant that to every expression or term corresponds a value.From the point of view of specifications, where properties are described by equations, sufficient completeness ensures that every term is equivalent to a term built on constructors, called constructor term or constructor form. It allows inductive proofs, in particular by consistency methods [7] . Proof assistants like Coq or PVS include decision or semi-decision procedures based on rewrite rules and rely on complete definitions of functions.From the point of view of programming, sufficient completeness ensures that a program produces a completely computed form for every data. Studying the property in this context is pertinent since rule-based systems have recently gained considerable interest with the development of efficient compilers. Now, systems like ASF+SDF [25] Proving sufficient completeness is undecidable in general. It has already been widely studied, for example in [21, 30, 26, 23, 6, 22, 28, 2, 3] , but most of the time, the proposed approaches for proving the property need restrictions like termination and confluence.The property is strongly related to ground reducibility, which expresses that every ground instance of a term is reducible. Indeed, it is equivalent to ground reducibility of all patterns f (x1, . . . , xm) built on a defined symbol f , provided the rewrite system (RS in short) is terminating, confluent, and the normal form of a constructor term is again a constructor term [22] . Under these conditions, techniques developed for proving ground reducibility hold for sufficient completeness as well [31, 24, 22, 27, 4, 8] .In this paper, we address the problem of sufficient completeness from the programming point of view, and we go beyond the previous usual restrictions.We observe the case where a program or a RS can be neither confluent, nor terminating, and we study its 121 evaluations. We do not suppose other restrictions used in the domain as the constructor preserving property, or the absence of relation between constructors. The question is to know whether at least one evaluation of a given data gives a completely evaluated result; in other words, whether for every ground term, there exists a rewriting chain that eventually reaches a constructor term, even if the chains do not converge to a single term, and are infinite. We call this property C-reducibility.Since several years, we have been studying properties of rewriting in the context of programming, where a rewrite system interpreter or compiler uses a specific strategy for applying rewrite rules, most frequently an innermost strategy that consists in rewriting always at the lowest possible positions, and corresponds to a call-by-value evaluation of functions. We observe rewriting in considering its properties from an inductive point of view. We already have proposed proof procedures for the termination property, in the specific case of rewriting strategies, respectively for the innermost [20] , the outermost [14] and local strategies [13] . We also have given a constructive proof technique for weak innermost termination, enabling to reach the terminating branches [15] . Here, C-reducibility is tackled with a similar inductive approach.It is not based on analyzing the patterns issued from the RS, as do most of existing methods for sufficient completeness, but on observing the rewriting process starting from any term to see whether it is eventually transformed into a constructor term. For that, like for our previous procedures for termination, we reason by induction on the property itself, assuming that the terms t smaller than the starting terms for an induction ordering are C-reducible.This approach needs a noetherian ordering on terms, used in the induction principle. In contrast to classical induction proofs where the ordering is given, we do not need to define it a priori. We only have to check its existence by ensuring satisfiability of ordering constraints incrementally set along the proof.By alternatively applying an abstraction mechanism and well-covered narrowing steps, we generate proof trees representing at least one rewriting derivation for every ground term. The proof is constructive, in the sense that the rewriting branch leading to a constructor term can be computed from the proof trees establishing C-reducibility for every term, thus avoiding costly breadth-first computations to reach the constructor form. Although our method differs from previous approaches, we naturally encounter basic notions already used for proving sufficient completeness, as unification, used in [28] , or covering properties used in many works in the domain, for example in [26, 28] . In [3] , pattern trees are also developed, but under the assumption that the RS is terminating and ground confluent.Before we detail our procedure, we link C-reducibility to close properties like strong C-reducibility, expressing the existence of a constructor form on every rewriting chain, and their variants in the case of innermost rewriting. Sufficient completeness and ground reducibility are also considered in this comparison. In particular, C-reducibility directly implies sufficient completeness. In addition, we justify that ground reducibility just requires weak termination (expressing that every term has at least one terminating rewriting derivation) to imply sufficient completeness, thus weakening the condition of the well-known theorem of [22] . We also note that as the covering property we require is stronger than ground reducibility, this property directly implies sufficient completeness if we suppose weak termination. Thus, with respect to the sufficient completeness property, the proof technique we propose is interesting:• obviously when the RS is not even weakly terminating,• when it is terminating, or just weakly terminating, but we have no technique to prove it. The weak termination proof is difficult to handle in general, and to our knowledge, our proof procedure [15] is the only one existing today. On the following small example, weakly terminating, and which is a classical definition of the booleans, enriched with a rule expressing the double application of not on and, and deliberately oriented in the divergent direction, it fails:• as an alternative of the termination proof, as on the following more realistic example of computation of quotient, which is innermost terminating:In addition, if the RS has no rule with a constructor lefthand side, our proof technique establishes weak termination at the same time. This is the case for the two examples above. In Section 2, the background is presented. In Section 3, we link known notions of completeness to C-reducibility. Section 4 introduces the basic concepts of the inductive proof mechanism. In Section 5, our procedure is defined with inference rules and a strategy to apply them. In Section 6, we explain how to extract a rewriting chain leading to a constructor term, from the proof of C-reducibility of terms.We assume that the reader is familiar with the basic definitions and notations of term rewriting given for instance in [11] . T (F, X ) is the set of terms built from a given set F of function symbols having an arity ar = n ∈ N, and a set X of variables denoted x, y . . .. In this paper, we only consider finite sets of symbols. T (F) is the set of ground terms (without variables). Symbols of arity 0 are called constants. Positions in a term are represented as sequences of integers. The empty sequence denotes the top position. The notation t|p stands for the subterm of t at position p. To emphasize that u contains subterms tj, j ∈ [1..p] respectively at positions {i1..ip}, we writeA substitution is an assignment from X to T (F, X ), written σ = (x → t) . . . (y → u). It uniquely extends to an endomorphism of T (F, X ). We identify a substitution σ = (x → t) . . . (y → u) with the finite conjunction of equations (x = t) ∧ . . . ∧ (y = u). The result of applying σ to a termA ground substitution or instantiation is an assignment from X to T (F). The composition of substitutions σ1 followed by σ2 is denoted σ2σ1. Given two substitutions σ1 and σ2, we write σ1 ≤ σ2 iff ∃θ such that σ2 = θσ1. We say that σ1 is a generalization of σ2.For F, we declare a set C of constructors and a set D of defined symbols. The terms of T (C) are called constructor terms, or more briefly C-terms (or C-forms). Given a set R of rewrite rules or term rewriting system on T (F, X ), the rewriting relation induced by R is denoted by →R (→ if there is no ambiguity on R). We denote s → A RS R (innermost) terminates iff every (innermost) rewriting derivation (or derivation chain) for R is finite. It weakly terminates if at least one derivation chain starting from every term is finite. It is confluent iff for every term t such that t → * u and t → * v, there exists t such that u → * t and v → * t . It is constructor preserving iff for each rule l → r of R, whenever l ∈ T (C), then r ∈ T (C).Before we compare them, let us now formally define the properties presented in the introduction.m]} be a RS on T (F, X ). The equational theory associated to R is the equational theory induced by the set of equationssuch that t =E u.Definition 3. Let R be a RS on T (F, X ). A term t ∈ T (F, X ) is ground reducible if every ground instance of t is reducible with R.Let us make some remarks on this property. If R does not have rules whose left-hand sides are C-terms, the reached Cforms are normal forms.If there are such rules in R, which is a case of relations between constructors, the C-forms may be reducible. Then, either R is constructor-preserving and the next terms in the rewriting derivations are still C-terms. Or R is not constructor-preserving and terms with defined symbols can be introduced after the C-terms in the derivations.In non terminating cases, C-reducibility allows us to introduce a weak pseudo-termination notion, expressing that the evaluation of every data stops on the first constructor encountered form, ensured to exist on at least one rewriting branch.Let us now observe the following diagram, summarizing the links between the different properties we consider. "Ground reducibility" stands for ground reducibility of all patterns f (x1, . . . , xm), f ∈ D; "C-reducibility" stands for Creducibility of all terms of T (F). Arrows represent implications between properties; they are labeled by the necessary conditions on the RS required by the implications. All links are quite obvious, except those between ground reducibility and sufficient completeness, given by Theorem 7 in [22] . From the diagram, we immediately set the following theorem, which weakens the terminating condition of the "if" part of Theorem 7 in [22] .Theorem 1. Let R be an innermost terminating RS onThis result can be verified in analyzing the proof of Theorem 7 in [22] , where the termination hypothesis can be indeed weakened into innermost termination, and even into weak termination.Note that sufficient completeness does not imply Creducibility in every case: for a given term, a C-form reachable by E can be unreachable by the rewriting relation, as illustrates the following terminating but not confluent example, where a, b, c are constants and c is the only constructor form:a → b a → c.For proving that a term t of T (F) is C-reducible, we proceed by induction on T (F) with a noetherian ordering , assuming the property for any t such that t t . To warrant non emptiness of T (F) and T (C), we assume that C contains at least a constant. The main intuition is to observe rewriting derivations starting from a ground term t ∈ T (F) which is any instance of a term g(x1, . . . , xm) ∈ T (F, X ), for some defined function symbol g ∈ D, and variables x1, . . . , xm. Proving the C-reducibility property on ground terms amounts to prove that at least one derivation leads to a constructor term.Rewriting derivations are simulated, using a lifting mechanism, by a proof tree developed from g(x1, . . . , xm) on T (F, X ), for every g ∈ D, by alternatively using two main concepts, namely narrowing and abstraction. More precisely, narrowing schematizes the rewriting possibilities of terms. Abstraction simulates the reduction of subterms in the derivations until these subterms become C-terms. It expresses the application of the induction hypothesis on these subterms.The schematization of ground rewriting derivations is achieved through constraints. The nodes of the developed proof trees are composed of a current term of T (F, X ), and a set of ground substitutions represented by a constraint progressively built along the successive abstraction and narrowing steps. Each node in an abstract tree schematizes a set of ground terms: all ground instances of the current term, that are solutions of the constraint.The constraint is in fact composed of two kinds of formulas: ordering constraints, set to warrant the validity of the inductive steps, and abstraction constraints combined to narrowing substitutions, which effectively define the relevant sets of ground terms.As said previously, we consider any term of T (F) as a ground instance of a term t of T (F, X ) occurring in a proof tree issued from a reference term t ref .• first, some subterms t|j of the current term t of the proof tree are supposed to reduce into a C-term, by the induction hypothesis, if θt ref θt|j for the induction ordering and for every θ solution of the constraint associated to t. They are replaced in t by abstraction variables Xj representing respectively any of their constructor forms.Reasoning by induction allows us to only suppose the existence of the constructor forms without explicitly computing them. If the resulting term only contains constructor symbols and abstraction variables, the initial term is C-reducible. Else, ip} (where i1, . . . , ip are the abstraction positions in t) is narrowed in all possible ways into terms v, according to the possible instances of the Xj . This corresponds to rewriting ground instances of u (characterized by the constraint associated to u) with all possible rewrite rules.This technique is inspired from the one we have proposed for proving innermost termination. But in that case, as abstracted subterms were representing innermost normal forms, alternating abstraction and innermost narrowing steps allowed us to simulate all innermost rewriting derivations. Here, abstracted subterms do not generally represent normal forms, and we use standard narrowing, so the process only simulates some possible rewriting derivations, for the standard rewriting relation (without strategy) among which we try to prove the existence of a C-term.However, we have to ensure that every ground term represented by u is C-reducible, and so we have to prove the existence of at least one branch leading to a C-term for every ground instance of u.To exhibit such a branch for every ground instance of u containing at least one defined symbol, the narrowing steps have to cover (in a sense we will define later) all possible ground instances of u.Then C-reducibility of the ground instances of t is reduced to C-reducibility of the ground instances of the terms v. Now, if θt ref θv for every ground substitution θ that is a solution of the constraint associated to v, by the induction hypothesis, θv is supposed to be C-reducible. Else, the process is iterated on v, until we get a term t such that either θt ref θt , or θt is a C-term. We now introduce some concepts to formalize and automate this mechanism.The induction ordering is constrained along the proof by imposing constraints between terms that must be comparable, each time the induction hypothesis is used in the abstraction mechanism. As we are working with a lifting mechanism on the proof trees with terms of T (F, X ), we directly work with an ordering P on T (F, X ) such that t ref P u induces θt ref θu, for every θ solution of the constraint associated to u.So inequalities of the form t ref > u1, . . . , um are accumulated, which will be called ordering constraints. Any ordering P on T (F, X ) satisfying them and which is stable by substitution fulfills the previous requirements on ground terms. The ordering P , defined on T (F, X ), can then be seen as an extension of the induction ordering , defined on T (F). For convenience, the ordering P will also be written .It is important to note that, for establishing the inductive termination proof, it is sufficient to decide whether there exists such an ordering. Satisfiability of a constraint C of this form is undecidable. But a sufficient condition for an ordering to satisfy C is that is stable by substitution and t t for every constraint t > t of C.The inductive termination proof on ground terms by application of Theorem 2 requires that the noetherian induction ordering has the constructor subterm property. Simplification orderings fulfill such conditions. So in practice, it is sufficient to find a simplification ordering such that t t for any constraint t > t of C.Solving ordering constraints in finding simplification orderings is a well-known problem in rewriting. The simplest and automatable way to proceed is to test simple existing orderings like the subterm ordering, the Recursive Path Ordering, or the Lexicographic Path Ordering. This is often sufficient for the constraints considered here. If these simple orderings are not powerful enough, automatic solvers like Cime2 [9] can provide adequate polynomial orderings.Other constraints are introduced by the abstraction mechanism. As said previously, to abstract a term t at positions i1, . . . , ip, where the t|j, j ∈ {i1, . . . , ip} are supposed to have a reduced C-form t|j ↓ C , we replace the t|j by abstraction variables Xj representing respectively one of their possible C-forms. Let us define these special variables more formally.T (F, X ∪ N ) in the following way. Let X ∈ N; for any substitution σ (resp. instantiation θ) such that X ∈ Dom(σ), σX ∈ T (C, N ) (resp. θX ∈ T (C)) and then Var (σX) ⊆ N . C-reducibility on T (F) is proved by reasoning on terms with abstraction variables, i.e. on terms of T (F, X ∪ N ). Ordering constraints are extended to pairs of terms of T (F, X ∪ N ). When subterms ti are abstracted by Xi, we state constraints on abstraction variables, called abstraction constraints, to express that their instances can only be Cterms, rewritten from the corresponding instances of ti. Initially, they are of the form t↓ C = X where t ∈ T (F, X ∪ N ), and X ∈ N , but we will see later on how they will be combined with the substitutions used for the narrowing process.Definition 7 (abstractionAfter abstraction of the current term t into the term u, if u contains defined symbols, we test whether the ground instances of u are reducible, according to the possible values of the instances of the Xj . This is achieved by narrowing u in all possible ways. We recall the definition of narrowing.A term t ∈ T (F, X ∪ N ) narrows into a term t ∈ T (F, X ∪ N ) at the non-variable position p, using the rule l → r ∈ R with the substitution σ which is written t p,l→r,σ t iffwhere σ is the most general unifier of t at position p and l.A few remarks can be made on the choice of variables and on the domain of the substitutions generated during the proof process. It is always assumed that there is no variable in common between the rule and the term, i.e. that V ar(l) ∩ V ar(t) = ∅. This requirement of disjoint variables is easily fulfilled by an appropriate renaming of variables in the rules when narrowing is performed. Observe that for the most general unifier σ used in the above definition, Dom(σ) ⊆ V ar(l) ∪ V ar(t) and we can choose Ran(σ) ∩ (V ar(l) ∪ V ar(t)) = ∅, thus introducing in the range of σ only fresh variables. Moreover, narrowing is only performed on terms t of T

O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O


An Accept-Reject Algorithm For the Positive Multivariate Normal Distribution The need to simulate from a positive multivariate normal distribution arises in several settings, specifically in Bayesian analysis. A variety of algorithms can be used to sample from this distribution, but most of these algorithms involve Gibbs sampling. Since the sample is generated from a Markov chain, the user has to account for the fact that sequential draws in the sample depend on one another and that the sample generated only follows a positive multivariate normal distribution asymptotically. The user would not have to account for such issues if the sample generated was i.i.d. In this paper, an accept-reject algorithm is introduced in which variates from a positive multivariate normal distribution are proposed from a multivariate skew-normal distribution. This new algorithm generates an i.i.d. sample and is shown, under certain conditions, to be very efficient. A d−dimensional random vector, X = (X 1 , X 2 , . . . , X d ). T , is said to follow a truncated multivariate normal distribution if X has the density f X (x; µ, Σ, D), where terior density (or an approximation to the posterior density) of parameters that are forced to satisfy some constraint. In Chen and Deely (1996) , for example, the posterior density of ten regression coefficients β, where β = (β 1 , β 2 , . . . , β 10 ) T , is found to be proportional to the multivariate normal density, but only in that region where the coefficients are ordered and greater than 0. The posterior of β thus takes the form given in (1) , and in. Another example can be found in Albert and Chib (1993) . In the analysis of some binary data y, where y = (y 1 , y 2 , . . . , y N ) T , the posterior density of N latent variables Z, where In conclusion, it is clear the proposed accept-reject algorithm can significantly increase the acceptance probability from p SAR (µ, Σ). Areas of further research include how the parameters of the alternative skew-normal distribution are selected for this new accept-reject algorithm. In this paper, we select values of β, ξ, Ω, R, A, and l that are easy to compute, correspond to an alternative skew normal density that is easy to sample from, and result in an acceptance probability that is much larger than p SAR (µ, Σ). The author is still in search of values of β, ξ, Ω, R, A, and l that are just as easy to calculate, correspond to an alternative skew normal density that is just as easy to sample from, and result in an even larger acceptance probability than the one reported in this paper. Identifying such parameters, though, still appears to be a challenge.. issue involves the location of x. Observe that at y = µ, m (y; β, Ω) = −α T Ω −1 α which is negative for all α ̸ = 0 since Ω −1 is positive definite. Since. , α has to equal 0, which implies β = µ

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Managing irrelevant knowledge in CBR models for unsolicited e-mail classification The problem of unsolicited e-mail has been increasing during recent years. Fortunately, some advanced technologies have been successfully applied to spam filtering, achieving promising results. Recently, we have introduced SPAMHUNTING, a successful spam filter able to address the concept drift problem by combining a relevant term identification technique with an evolving sliding window strategy.Several successful spam filtering techniques use continuous learning strategies to achieve better adaptation capabilities and address concept drift issues. Nevertheless, due to the presence of concept drift and hidden changes in the environment, the presence of obsolete and irrelevant knowledge becomes a serious drawback. Soon after the launch of the filter, many decisions are made based on irrelevant and/or obsolete knowledge. Therefore, in such a situation, the use of forgetting strategies is as important as the implementation of continuous learning approaches.In this paper we introduce a novel technique designed for identifying and removing the obsolete and irrelevant knowledge that has accumulated over to the passage of time. We have carried out several experiments to test for the suitability of our proposal showing the results obtained and its applicability. This initiative shares an extensive corpus containing more than nine thousands messages in EML format (RFC-822 (Crocker, 1982) ). Another excellent alternative is the usage of the Ling-Spam corpus 2 containing more than two thousands messages shared as a collection of plain text in tokenized files. Although these corpuses do not contain message headers, they can be used for the execution and testing of anti-spam filters.. The main limitation of these public datasets is the low number of messages. A great amount of available e-mails 1 Available at http://apache.spamassassin.org/. 2 Available at http://www.iit.demokritos.gr/ facilitates the observation of the effects caused by the presence of irrelevant knowledge. Moreover, we should bear in mind the source of the Ling-Spam dataset, which has been extracted from a public linguistics e-mail distribution list. As a consequence, the subject matter of all legitimate messages belonging to this corpus is relative to Linguist issues.. To carry out the optimization of the cut-off threshold a, we will use a set of messages compiled since the year 2004 from several students and teachers belonging to our University (the SING corpus). This dataset contains a great amount of spam and legitimate e-mails. Finally, due to its large amount of messages, we found SpamAssassin as a good choice for the comparison of different spam filtering models. Table 1 summarizes the main characteristics of the above mentioned corpuses.. As seen in Table 1 , the SpamAssassin corpus is an old dataset while our corpus has been mainly constructed using recent e-mails. In addition to this, the SING corpus contains a large amount of messages. The current disadvantage of our corpus is the presence of private messages that should be carefully handled to preserve the identity of their senders/receivers. Therefore, we think that SING corpus is a good choice only for optimizing purposes while SpamAssassin should be used for model analysis.. Seven well-known measurements have been used for comparison purposes including: (i) percentage of correct classifications, false positive and false negative error rates, (ii) batting average (Graham-Cumming, 2004) , (iii) precision and recall, (iv) F-score (Rijsbergen, 1979) , (v) balanced F-score (Shaw, Burgin, & Howell, 1997; Yang, 1999) , (vi) Total Cost Ratio (TCR) (Androutsopoulos, Koustias, Chandrinos, Paliouras, & Spyropoulos, 2000) and (vii) Receiver Operating Characteristic (ROC) curves (Egan, 1975) .. To guarantee the quality of our experimental results, we have used a 10-fold stratified cross-validation for all experiments (Kohavi, 1995) . The results reported are the means of the 10-folds. With the goal of demonstrating the suitability of our SPAMHUNTING system working with the proposed technique in dynamic environments, we have compared its performance level with the results achieved by the ECUE model , a well-known lazy learning spam filter.. Finally, talking about the SPAMHUNTING configuration, the percentage of AI selected from each message has been set to 60%, as it has proved to be a reasonable value (Mén-dez et al., 2006) . In addition to this, Section 4.2contains information about the benchmark protocol used for optimizing a parameter as well as the results obtained.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The Capacity of Channels With Feedback In this paper, we introduce a general framework for treating channels with memory and feedback. First, we prove a general feedback channel coding theorem based on Massey's concept of directed information. Second, we present coding results for Markov channels. This requires determining appropriate sufficient statistics at the encoder and decoder. We give a recursive characterization of these sufficient statistics. Third, a dynamic programming framework for computing the capacity of Markov channels is presented. Fourth, it is shown that the average cost optimality equation (ACOE) can be viewed as an implicit single-letter characterization of the capacity. Fifth, scenarios with simple sufficient statistics are described. Sixth, error exponents for channels with feedback are presented. T HIS paper presents a general framework for proving coding theorems for channels with memory and feedback. Because of increased demand for wireless communication and networked systems there is a renewed interest in this problem. Feedback can increase the capacity of a noisy channel, decrease the complexity of the encoder and decoder, and reduce latency.. Recently, Verdú and Han presented a very general formulation of the channel coding problem without feedback [34] . Specifically, they provided a coding theorem for finite-alphabet channels with arbitrary memory. They worked directly with the information density and provided a Feinstein-like lemma for the converse result. Here we generalize that formulation to the case of channels with feedback. In this case, we require the use of code functions as opposed to codewords. A code function maps a message and the channel feedback information into a channel input symbol. Shannon introduced the use of code functions, which he called strategies, in his work on transmitter side information [28] . Code functions are also sometimes called codetrees [21] .. We convert the channel coding problem with feedback into a new channel coding problem without feedback. The channel inputs in this new channel are code functions. Unfortunately, the space of code functions can be quite complicated to work with. We show that we can work directly with the original space Manuscript of channel inputs by making explicit the relationship between code-function distributions and channel input distributions. This relationship allows us to convert a mutual information optimization problem over code-function distributions into a directed information optimization problem over channel input distributions. Directed information was introduced by Massey [23] who attributes it to Marko [22] . Directed information can be viewed as a causal version of mutual information. Kramer [20] , [21] used directed information to prove capacity theorems for general discrete memoryless networks. These networks include the memoryless two-way channel and the memoryless multiple-access channel. In this paper, we examine single-user channels with memory and feedback. One of the main difficulties in this problem has to do with the fact that the transmitter and the receiver may have different information about the state of the channel. We show how to choose appropriate sufficient statistics at both the transmitter and the receiver.. The problem of optimal channel coding goes back to the original work of Shannon [26] . The channel coding problem with feedback goes back to early work by Shannon, Dobrushin, Wolfowitz, and others [27] , [12] , [38] . In particular, Shannon introduced the feedback problem. Both Shannon and Dobrushin examined the case of memoryless channels with feedback. Wolfowitz, in his book, describes a variety of finite-state channels with state calculable by the sender or the receiver. We generalize these results to general Markov channels with output feedback. We do not assume that the state is known to either the transmitter or the receiver.. There is a long history of work regarding Markov channels and feedback. Here we describe a few connections to that literature. Mushkin and Bar-David [24] determined the capacity of the Gilbert-Elliot channel without feedback. Blackwell, Breiman, and Thomasian [4] examine finite-state indecomposable channels without feedback. Goldsmith and Varaiya [16] examine nonintersymbol interference (ISI) Markov channels without feedback. For the case of independent and identically distributed (i.i.d.) inputs and symmetric channels, they introduce sufficient statistics that lead to a single-letter formula. In this paper, we identify the appropriate statistics when feedback is available. In a certain sense, to be explained in this paper, the Markov channel with feedback problem is easier than the Markov channel without feedback problem. This is because in the full channel output feedback case the decoder's information pattern is nested in the encoder's information pattern [36] . In this paper, we do not treat noisy feedback.. Viswanathan [35] , Caire and Shamai [6] , and Das and Narayan [10] all examine different classes of channels with memory and side information at the transmitter and the receiver. Chen and Berger [7] examine Markov channels when the state is known to both the transmitter and the receiver. In this paper, 0018-9448/$25.00 © 2009 IEEE we present a general framework for treating Markov channels with ISI and feedback.. Many authors consider conditions that ensure the Markov channel is information stable [25] . For example, Cover and Pombra [8] show that Gaussian channels with feedback are always information stable. Shannon [29] introduced the notion of "recoverable state property" by which a channel can be reset to a known state by using a fixed finite sequence of channel inputs. In addition, some authors consider conditions that ensure the Markov channel is indecomposable [15] , [4] . In our work, it is shown that solutions to the associated average cost optimality equation (ACOE) imply information stability. In addition, the sufficient condition provided here for the existence of a solution to the ACOE implies a strong mixing property of the underlying Markov channel in the same way that indecomposability does. The ACOE can be viewed as an implicit single-letter characterization of the channel capacity.. We consider Markov channels with finite-state, channel input, and channel output alphabets. But with the introduction of appropriate sufficient statistics, we quickly find ourselves working with Markov channels over general alphabets and states. As shown by Csiszár [9] , for example, treating general alphabets involve many technical issues that do not arise in the finite-alphabet case.. Tatikonda first introduced the dynamic programming approach to the directed information optimization problem [31] . Yang, Kavcic, and Tatikonda have examined the case of finite-state machine Markov channels [39] . Here we present a stochastic control framework that treats many Markov channels including finite-state machine Markov channels.. In general, it is difficult to solve the ACOE. This is because the sufficient statistics can be quite complicated. Hence, it will be difficult to get an explicit formula for the feedback channel capacity. There are, though, many scenarios when the sufficient statistics become much simpler and hence the ACOE becomes simpler. We discuss these scenarios in Section VIII. In these cases, one can apply exact or approximate dynamic programming techniques to solve the ACOE.. In summary, the main contributions of this paper are as follows. 1) We prove a general feedback channel coding theorem based on Massey's concept of directed information [23] . 2) We present coding results for Markov channels. This requires determining appropriate sufficient statistics at the encoder and decoder. We give a recursive characterization of these sufficient statistics. 3) A dynamic programming framework for computing the capacity of Markov channels is presented. 4) It is shown that the ACOE can be viewed as an implicit single-letter characterization of the capacity. 5) Scenarios with simple sufficient statistics are described. 6) Error exponents for channels with feedback are presented. Preliminary versions of this work have appeared in [31] - [33] . We have presented a general framework for treating channels with memory and feedback. We first proved a general coding theorem based on Massey's concept of directed information and Dobrushin's program of communication as interconnection. We then specialized this result to the case of Markov channels. To compute the capacity of these Markov channels, we converted the directed information optimization problem into a partially observed MDP. This required identifying appropriate sufficient statistics at the encoder and the decoder. The ACOE verification theorem was presented and sufficient conditions for the existence of a solution were provided. The complexity of many feedback problems can now be understood by examining the complexity of the associated ACOE. Error exponents were presented.. The framework developed herein allows one to apply approximate dynamic programming techniques, such as value and policy iteration and reinforcement learning, for computing the capacity. Such dynamic programming techniques are presented in [39] for the case of finite-state machine Markov channels. Finally, the framework presented here allows one to compute the capacity under restricted classes of policies. This is useful if one is willing to sacrifice capacity for the benefit of a simpler policy.

O B O O O B O O O O O O O O O O O O O O B O O O O O B O O O O O O O O B I O O O O O O B I O O O O B I O O O O O O O O O O O O B I O O B I O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O B I O B O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O B O O O O O O O O B I O B O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B I O O O O O O O O B I O O O O B I O O O O O O O O O O O O B I O O B I O O O O B O B I O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O B O O B I O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B O B I O O O O O O O O O O O B I O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O


Understanding the Value of Software Engineering Technologies Abstract-When AI search methods are applied to software process models, then appropriate technologies can be discovered for a software project. We show that those recommendations are greatly affected by the business context of its use. For example, the automatic defect reduction tools explored by the ASE community are only relevant to a subset of software projects, and only according to certain value criteria. Therefore, when arguing for the value of a particular technology, that argument should include a description of the value function of the target user community. There are many software engineering (SE) technologies that a manager might apply in the hope of improving their software development project. Some of these technologies are paper-based methods like the checklists proposed by orthogonal defect classification [1] . Other technologies are tool-based such as using the new generation of functional programming languages or execution and testing tools [2] or automated formal analysis [3] . Yet other technologies are more process-based including process improvement initiatives, changing an organization's hiring practices, or a continual renegotiation of the requirements as part of an agile software development cycle [4] . SE technologies can be inappropriately applied to projects if a project manager do not assess the technology's benefits against its associated drawbacks. For example, using lessskilled developers is tempting (since there are so many of them), but the resulting product may be more defect-prone. As a result, products might get to market faster but contain too many defects.. In theory, software process models can be used to model the trade-offs associated with different technologies. However, such models can suffer from tuning instability. Large instabilities make it hard to recognize important influences on a project. For example, consider the following simplified COCOMO [5] model,. While simplified, the equation presents the core assumption of COCOMO; i.e. that software development effort is exponential on the size of the program. In this equation, (a, b) control the linear and exponential effects (respectively) on model estimates; while pmat (process maturity) and acap (analyst capability) are project choices adjusted by managers. Equation 1 contains two features (acap, pmat) and a full COCOMO-II model contains 22 [5] .. Baker [6] reports a study that learned values of (a, b) for a full COCOMO model using Boehm's local calibration method [7] from 30 randomly selected samples of 90% of the available project data. The ranges varied widely: . Such large variations make it possible to misunderstand the effects of project options. Suppose some proposed technology doubles productivity, but a moves from 9 to 4.5. The improvement resulting from that change would be obscured by the tuning variance. Previously we have reported some success with stochastic AI tools that search through the space of possible tunings within process models [8] - [12] . This approach returned conclusions about the project that were stable across the entire space of possibilities. For example, in two studies [8] , [9] we found that automated defect removal tools (such as those discussed at the ASE conference) were often required to achieve minimum defects/ development effort/ development time 1 . Elsewhere [10] , [11] , we have used this tool to comparatively assess different proposed changes to a project.. Our prior reports did not compare the recommendations found by different AI search methods. Nor did we explore the effect of changing the value function that models user goals. Here, we compare half a dozen AI methods and apply the best one to four case studies. Next, we repeat that analysis using a different value function.. The results were very surprising. Prior to this research, we had a pre-experimental intuition that concepts of value might change the organization of a project. However, we suspected that some things would remain constant (e.g., condoning the use of execution testing tools).. This turned out not to be the case. For the two value functions explored here, if one function approves of X then the other usually approves of not X. This result that value can change everything should motivate much future work on the business context of our tools. 1 Time refers to calendar months required to complete a project while effort refers to the number of staff hours within those months. For example, 4 people working for one year takes time = 12 months and ef f ort = 48 months. This work was inspired by Barry Boehm's 2004 ASE keynote [13] in which he advocated assessing SE tools by their value to a stake-holder, rather than just via their functionality. More specifically, one of the value functions used in this report was developed by Boehm and Huang [14] .. In his description of value-based SE, Boehm favors continuous optimization methods to conduct cost/benefit tradeoff studies. Tuning instability can confuse such continuous optimization methods. For example, the results of gradient descent methods can be misleading if the coefficients on the partial functions are very uncertain.. In our view, uncertainty in software process models is a very under-explored area. For example, the variance of Equation 2 remained undiscovered for 26 years until Baker, at our suggestion, looked for it. Much of the related work on uncertainty in software engineering uses a Bayesian analysis. For example, Pendharkar et al. [15] demonstrate the utility of Bayes networks in effort estimation while Fenton and Neil explore Bayes nets and defect prediction [16] (but unlike this paper, neither of these teams links defect models to effort models). We elect to take a non-Bayesian approach since most of the industrial and government contractors we work with use parametric models like COCOMO.. Other related work is the search-based SE approach advocated by Harman [17] . Search-Based Software Engineering (SBSE) uses optimization techniques from operations research and meta-heuristic search (e.g., simulated annealing and genetic algorithms) to hunt for near-optimal solutions to complex and over-constrained software engineering problems. Harman takes care to distinguish AI search-based methods from those seen in standard numeric optimizations. Such optimizers usually offer settings to all controllables. This may result in needlessly complex recommendations since a repeated empirical observation is that many model inputs are noisy or correlated in similar ways to model outputs [18] . Such noisy or correlated variables can be pruned away to generate simpler solutions that are easier and quicker to understand. In continuous domains, there is much work on feature selection [19] and techniques like principal component analysis [20] to reduce the number of dimensions reported by an analysis. Comparative studies report that discrete AI-based methods can do better at reducing the size of the reported theory [18] .. The SBSE approach can and has been applied to many problems in software engineering (e.g., requirements engineering [21] ) but most often in the field of software testing [2] . Harman's writing inspired us to try simulated annealing to search the what-ifs in untuned COCOMO models [9] . SA is a widely-used algorithm, perhaps due to the simplicity of its implementation and its very low memory requirements. Our results, shown below, indicate that several other algorithms out-perform SA (at least, on our models).. This result strongly suggests that proponents of SA should try a broader range of search engines.. The paper compares SEESAW to five AI search algorithms. These half a dozen algorithms hardly represent an exhaustive list of possibilities. For example, Gu et al. [22] list hundreds of optimization algorithms and no single conference paper can experiment with them all. However, we make one comment as to why the above list does not include integer programming methods. Coarfa et al. [23] found that integer programming-based approaches ran an order of magnitude slower than discrete methods like SEESAW. Similar results were reported by Gu et al. where discrete methods ran 100 times faster than integer programming [22] .. Our research grew out of a frustration with standard methods to reduce tuning variance. Previously, we have tried reducing that variance in various ways:. • Feature selection to prune spurious details [24] ;. • Instance selection to prune irrelevancies [25] ;. • Extended data collection. Despite all that work, the variance observed in our models remains very large. Even the application of techniques such as instance-based learning have failed to reduce variance in our effort predictions [26] . Feature subset selection has also been disappointing: while it reduces the median performance variance somewhat (in our experiments, from 150% to 53% [25] ), the residual error rates are large enough that it is hard to use the predictions of these models as evidence for the value of some proposed approach. Lastly, further data collection has not proven useful. Certainly, there is an increase in the availability of historical data on prior projects 2 . However, Kitchenham [27] cautions that the literature is contradictory regarding the value of using data from other companies to learn local models.. Having failed to tame tuning variance, despite years of research, we turned to alternate methods.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Connectivity of mobile ad hoc sensor networks for vehicles Reasonable network models are abstracted from the mobile ad hoc sensor network for the real road traffic system, and the connectivity of one-dimension (1-D) and two-dimension (2-D) networks are discussed separately by percolation theory. In the twodimension (2-D) network, a novel connectivity strategy, the pointbased coverage strategy, is introduced. Then through simulation experiments reasonable transmission radiuses of wireless nodes in real traffic network are obtained. Intelligent Transportation Systems (ITS) combine many advanced technologies including information, communication and sensor to facilitate the transportation management. Sensors have become the "nerve ending" of vehicles, and users can obtain more information about roads through the Inter-Vehicles Communication (IVC), such as projects of CarTALK and FleetNet in EU, PATH in USA and DEMO2000 in Japan. In our ad hoc sensor network for ITS, there is only a vague idea of the number of nodes and the shape/size of the deployment region, but the expected node density is approximately known. Under the situation, percolation is the best research model. Percolation theory was introduced by BroadBent et al. as a mathematical model to study the flow of fluid in random media. Dousse et al. [1] [2] analyzed the connectivity of ad hoc/sensor networks, and found the critical value of λ . In this paper, we use dynamic percolation theory to research the connectivity of our networks. The paper is structured as follows. Models are presented in Section 2. In Section 3, we study the characteristic of connecting in 1-D and 2-D dynamic random networks, and introduce a novel coverage strategy, the pointbased coverage strategy. Section 4 presents simulations and concludes of the connectivity of road vehicular network for Shanghai on our simulation platform.

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Answering queries in hybrid Bayesian networks using importance sampling In this paper we propose an algorithm for answering queries in hybrid Bayesian networks where the underlying probability distribution is of class MTE (mixture of truncated exponentials). The algorithm is based on importance sampling simulation. We show how, like existing importance sampling algorithms for discrete networks, it is able to provide answers to multiple queries simultaneously using a single sample. The behaviour of the new algorithm is experimentally tested and compared with previous methods existing in the literature. Bayesian networks [17, 36] have become a popular tool for representing uncertainty in decision support systems. A review of recent literature shows the variety of applications in which they have been successfully used [1, 10, 24, 35, 45] . One of the main reasons for using them as the inference engine in a decision support system is that efficient reasoning algorithms can be designed, taking advantage of their structure [2, 3, 16, 44, 43, 30, 29] .. Most of the methodological development around Bayesian networks has concentrated on the case in which all the variables involved are qualitative or discrete. However, decision support systems usually have to operate in domains described in terms of both discrete and continuous variables simultaneously. In such scenarios, there is always the possibility of discretising the continuous variables [20, 34] , in order to be able to use methods designed for discrete variables. But such a solution in general conveys a loss of information.. Continuous and discrete variables can be handled simultaneously, with no need to discretise, in the so-called hybrid Bayesian networks. The first advances in this field came along with the definition of the Conditional Gaussian (CG) model [25, 26, 28] . The limitations of this approach are the assumption of normality over the continuous variables, and also the fact that dependencies of discrete variables conditional on continuous ones, are not allowed. This structural restriction is overcome in the augmented Conditional Linear Gaussian (CLG) networks, where discrete nodes are allowed to have continuous parents, by representing their conditional distributions as softmax functions [27] . However this model also relies on the normality assumption. Furthermore, exact inference is not possible in augmented CLG networks, and the solution proposed in [27] is based on a Gaussian approximation of the product of the Gaussian and softmax functions, which provides exact marginals for the discrete variables and also is able to obtain exact values only for the first and second order moments of the distribution of the continuous variables.. A more general proposal is based on the use of mixtures of truncated exponentials (MTEs), which do not impose any restriction and also do not rely on the normality assumption [31] . This model has been successfully applied to decision problems [6] . An important feature of MTEs is that they are compatible with efficient exact inference algorithms like, for instance, the Shenoy-Shafer architecture [44] and the variable elimination scheme [49] . As MTEs are able to approximate a wide variety of probability distributions [7] , they can be used as a general framework for carrying out inference in hybrid Bayesian networks, just by approximating each conditional distribution in the network by an MTE and then using an exact inference algorithm. This approach has been analysed in [22] , by solving a network involving Logistic and Gaussian distributions using MTEs, variational approximations [18] , discretisation [33] and Markov Chain Monte Carlo [12] .. A recent approach, similar in essence to MTEs, is based on representing the distribution in a hybrid Bayesian network as a Mixture of Polynomials (MOPs) [42] . Both MTEs and MOPs have been generalised in a global framework for representing hybrid Bayesian networks, called Mixtures of Truncated Basis Functions (MoTBFs) [23] . However, even though MOPs have some advantages over MTEs, specially the ability of dealing with a wider class of deterministic relationships, so far they lack of an algorithm for learning the models from data, while this issue has been solved for MTEs [38] . Hence, MTEs can be used as an exact model and not only as an approximation of other distributions. In that sense, MTEs behave as a nonparametric model, where no assumption is made about the underlying distribution.. Even though Bayesian networks allow efficient inference algorithms to operate over them, it is known that exact probabilistic inference is an NPhard problem [8] . Furthermore, approximate probabilistic inference is also an NP-hard problem if a given precision is required [9] . For that reason, approximate algorithms that tradeoff complexity for accuracy have been developed for discrete Bayesian networks. An important class of such approximate algorithms are based on the importance sampling technique, that provides a flexible approach to construct anytime reasoning algorithms [4, 13, 32, 46, 47, 48] .. Inference in hybrid Bayesian networks with MTEs does not escape from the above mentioned complexity. If the model is learnt from a database using the algorithm in [38] , it can be too complex if the number of variables is high. But even using the approximations in [7] , inference may become unfeasible if the model is complex enough.. With this motivation, in this paper we propose an approximate algorithm for computing fast and accurate answers to precise queries in hybrid Bayesian networks with MTEs. The algorithm is based on importance sampling, and therefore it is an anytime algorithm [37] in the sense that the accuracy of its results is proportional to the time it is allowed to use for computing the answer. We show how our proposal outperforms the previous state-of-the-art method for approximate inference with MTEs, introduced in [40] .. The rest of the paper is organised as follows. We establish the notation and define some preliminary concepts in Sec. 2. The problem addressed here is formally posted in Sec. 3. The core of the methodological contributions is in Sec. 4, and the details of the algorithm can be found in Sec. 5. The experimental analysis carried out to test the performance of the algorithm is reported in Sec. 6. The concluding remarks are given in Sec. 7. We have introduced a method for solving multiples queries in hybrid. Bayesian networks with MTEs. The method is based on importance sampling, which makes it an anytime algorithm. The algorithm is able to compute answers to multiple questions using a unique sample. We have shown that the variance remains bounded if the same sample is also used to compute the numerator and denominator in each query.. The experiments conducted illustrate the behaviour of the proposed algorithm, and they support the idea that the IS algorithm outperforms the two algorithms previously used for carrying out probabilistic reasoning in hybrid Bayesian networks with MTEs. Therefore, the methodology introduced here expands the class of problems that can be handled using hybrid. Bayesian networks, and more precisely, it provides versatility to the MTE model, by increasing the efficiency in solving probabilistic inference tasks.. We expect to continue this research line by developing methods for answering more complex queries. For instance, a query consisting on finding the most probable explanation to an observed fact in terms of a set of target variables, which is called abductive inference [11] . We also plan to study the application of the proposed algorithm to MOPs [42] . The main difference would be in Alg. 1, as in the case of MOPs, each term may oscillate within an interval, while MTEs are smoother.

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