Propositional dynamic logic with recursive programs We extend the propositional dynamic logic PDL of Fischer and Ladner with a restricted kind of recursive programs using the formalism of visibly pushdown automata (Alur, Madhusudan 2004) . We show that the satisfiability problem for this extension remains decidable, generalising known decidability results for extensions of PDL by non-regular programs. Propositional Dynamic Logic (PDL) is a modal logic that was introduced by Fischer and Ladner in [5] to capture the behaviour of programs. The models for PDL formulas are transition systems whose edges are labelled with atomic programs and whose states are labelled with atomic propositions. Formulas and programs are inductively (and mutually) defined from atomic propositions and programs. Formulas are closed by the standard Boolean operations, and for each program α and each formula ϕ, α ϕ is a formula meaning that there is an execution of program α that ends in a state where ϕ holds. A program is a regular language (represented by a regular expression of a finite automaton) over the set of atomic programs and tests (where tests correspond to formulas). As shown in [5] , satisfiability is decidable for PDL. Proofs for this result usually rely on model-theoretic properties of PDL, e.g., the small model property and the tree model property.. In order to capture more complex programs, several extensions of PDL have been considered. One can allow new programs operators as, e.g., converse [14] , or consider an intersection operator on programs to express concurrency properties. Recently, Lutz has shown that PDL with both intersection and converse is decidable [11] , extending a difficult result from Danecki [4] . One of the main difficulties when considering such extensions is that they do no longer have the tree model property.. Other extensions use non-regular programs to capture recursive behaviours [8, 9, 6] . Consider the following recursive program [7] : proc V { if p then {a; call V; b} else return }. The set of executions of this program can be described by the set {(p?a) i (¬p)?b i | i ≥ 0}, which is not regular and hence cannot be captured by standard PDL. To overcome this weakness various extensions of PDL by sets of non-regular programs have been considered.. For a class C of languages over the set of atomic programs as letters, one can distinguish the weak and the strong extension ( [6] ) of PDL by programs in C. The weak extension allows formulas α ϕ where α is (a placeholder for) some language from C, whereas in the strong extension the set of atomic programs is augmented by symbols for the languages from C (which can then be used in the regular expressions). For the weak extension of PDL by the class of context-free languages satisfiability is easily seen to be undecidable. Nevertheless, several strong extensions by single context-free languages are known to be decidable, e.g., the one by {a i b i | i ≥ 0}, even if they no longer have the finite model property. Surprisingly, the weak extension of PDL with {a i ba i | i ≥ 0} leads to undecidability. In order to establish the borderline between decidable and undecidable extensions of PDL by non-regular languages, restrictions on pushdown automata (and hence subclasses of context-free languages) have been considered: the largest class of languages, for which the strong extension of PDL is decidable, is the one of deterministic semi simple-minded languages considered in [6] . A pushdown automaton is semi simple-minded if the input letter determines whether to execute a pop, no stack operation, or a push according to a fixed partition of the input alphabet. In case of a push also the stack symbol to be pushed is fixed by the input letter.. A related but stronger subclass of context-free languages that has recently been defined in [3] is the class of visibly pushdown languages. The definition of visibly pushdown automata (Vpas) is the same as for semi simple-minded pushdown automata except that the stack symbol that is pushed may also depend on the current control state and not only on the input letter. It is not difficult to see that this additional freedom indeed allows to define more languages.. We define recursive PDL by replacing regular expressions as the formalism to define programs in PDL by Vpas. This extension of PDL with Vpas is more general in two senses than the strong extension of PDL with (semi) simpleminded pushdown automata as considered in [8] and [6] : the model of Vpa is more expressive than the model of (semi) simple-minded pushdown automaton, and we do not just augment the set of atomic programs by placeholders for Vpa languages (as in the strong extension) but use Vpas as formalism for defining programs. This second property allows not just to use Vpas over atomic programs but also to use tests inside the Vpas.. Our main result is that satisfiability for recursive PDL is decidable in doubly exponential time. To our knowledge, this captures all previous known results concerning decidable extensions of PDL using context-free languages. Futhermore, our proofs are simpler and less technical than the ones in [8, 6] because we can use general resutls and constructions from the theory of Vpas.. The remainder of the paper is organised as follows. In Section 2 we give the basic definitions and results concerning PDL and Vpas, and we define recursive PDL. In Section 3 we prove the decidability of the satisfiability problem for recursive PDL, and in Section 4 we extend the results to infinite computations. Using visibly pushdown automata we have defined recursive PDL as an extension of regular PDL that allows to capture the behaviour of recursive programs. The result on the satisfiability of this logic subsumes all known decidable extensions of PDL with context-free programs. Comparisons of recursive PDL with µ-calculus using relational fixed points and with visibly pushdown µ-calculus would be interesting. The first one [13] allows to capture the example from the introduction using the formula µR.((p?; a; R; b) ∪ (¬p)?) (for a binary relation symbol R), while the second one [1] embeds in the modal µ-calculus the formalism of visibly pushdown automata. Another possible direction for future research is to combine visibly pushdown automata with the game logic of Parikh [12] . that for all ψ ∈ cl(ϕ) and all states s ∈ S that ψ ∈ t(s) if and only if M, s |= ψ. The base case where ψ ∈ P comes from the definition of ν while the inductive step follows from the definition of Hintikka tree. ⊓ ⊔ Proposition 3. Let ϕ be a formula of recursive PDL. There is a Hintikka tree for ϕ if and only if there is a unique diamond path Hintikka tree for ϕ.. Proof. Consider a Hintikka tree t : [k] * → Σ ϕ for ϕ. Let k ′ = 2rk, where r = |cl ⋄ (ϕ)|, and fix an ordering ψ 0 , . . . , ψ r−1 on the diamond formulas from cl ⋄ (ϕ). We define a mapping η : [k ′ ] * → [k] * that associates to each element from [k ′ ] * an element of [k] * by induction on the length of words in [k ′ ] * as follows: η(ε) = ε and η(xd ′ ) = η(x)d where d = d ′ mod k. We finally define t ′ : [k ′ ] * → Σ ϕ by t ′ (x) = t(η(x)). It is not difficult to show that t ′ is a Hintikka tree.. In order to show that t ′ is a unique diamond path Hintikka tree, we have to define the mapping ρ : [k ′ ] * → (cl ⋄ (ϕ)×[k ′ ] * )∪{⊥}. The idea for this mapping is the following. Assume that ψ j is in t(x) for some x ∈ [k] * and that the last vertex in the witnessing path for ψ j is obtained from x by appending d 0 , d 1 , . . . , d m .. In t ′ we have ψ j ∈ t ′ (x ′ ) for x ′ with η(x ′ ) = x. The witnessing path in t ′ is then obtained by appending d ′ 0 , d ′ 1 , . . . , d ′ m to x with d ′ 0 = d 0 + (r + j)k and d ′ i = d i + jk for i ∈ {1, . . . , m}. Formally, we define ρ inductively on the length of the nodes. We set ρ(ε) = ⊥. Assume that ρ(x ′ ) is already defined for Using the idea sketched above on how to obtain the witnessing paths in t ′ from those in t, one can show by induction on the length of A-paths that ρ satisfies the conditions for t ′ being a unique diamond path Hintikka tree. ⊓ ⊔

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


Efficient feature tracking of time-varying surfaces using multi-scale motion flow propagation This paper presents a framework for efficient feature tracking of time-varying surfaces. The framework can not only capture the dynamic geometry features on the time-varying surfaces, but also can compute the accurate boundaries of the geometry features. The basic idea of the proposed approach is using the multi-scale motion flow and surface matching information to propagate the feature frame on the time-varying surfaces. We first define an effective multi-scale geometry motion flow for the time-varying surfaces, which efficiently propagates the geometry features along the time direction of the time-varying surfaces. By combining both the approximately invariant signature vectors and geometry motion flow vectors, we also incorporate the shape matching into the system to process feature tracking for time-varying surfaces in large deformation while with low frame sampling rate. Our approach does not depend on the topological connection of the underlying surfaces. Thus,it can process both mesh-based and point-based time-varying surfaces without vertex-to-vertex correspondence across the frames. Feature tracking results on different kinds of time-varying surfaces illustrate the efficiency and effectiveness of the proposed method. Time-varying surfaces, also called animated objects or dynamic shapes, are given as a sequence of surface frames, each of which is a static geometry model (mesh based surface or point based surface). Recently, with the progress of advanced scanning technologies, it is possible to reconstruct timevarying surfaces from the point clouds scanned for the deformed or animated objects [1] . Time-varying surfaces are more expressive compared with static geometry model, and they are now a popular data representation method with wide applications in various fields, such as virtual reality, computer games, 3D animated films, product design and scientific computation visualization. As a promising data representation, a large number of research work on time-varying surfaces have been proposed, such as segmentation [2, 3, 4] , geometry reconstruction [1, 5, 6] , geometry editing and modeling [7, 8] .. In this paper, we aim at tracking geometry features of time-varying surfaces, namely, tracking the features' motion paths of time-varying surfaces. Similar to video object tracking [9] , efficient feature tracking of time-varying surfaces is one of the critical tasks in many geometry processing applications such as geometry editing and modeling, geometry transmission, feature detection, and geometry description. However, unlike the video tracking, where there are pixel-to-pixel correspondences among the consequent image frames, time-varying surfaces such as point-based geometry usually do not hold the point-to-point correspondence among adjacent surfaces. Feature tracking of time-varying surfaces is a challenging job considering the following aspects:. • Capturing the dynamic geometry features with accurate boundaries over time-varying surfaces is difficult, especially when the boundaries of time-varying feature are fuzzy.. • Time-varying surfaces usually do not hold the point-to-point correspondence among adjacent surfaces, especially for point sampled geometry sequences, which increases the complexity of features tracking of timevarying surfaces.. • Data amount of time-varying surface is usually much larger than the single static object, thus the efficiency of feature tracking is one performance bottleneck which needs to be addressed.. To address aforementioned problems, we present a novel framework for efficient feature tracking of time-varying surfaces. The basic idea of the presented approach is to compute an efficient multi-scale geometry motion field for the time-varying surfaces, which propagates the current geometry feature to the next frame according to the motion field variation. For timevarying surfaces with large deformation while with low frame sampling rate, we first perform surface matching between the neighboring frames, and then combine both the surface matching results and geometry motion flow vectors for accurate feature tracking. We build localized classifiers based on GMMs (Gaussian Mixture Models) over initial boundaries of the tracked feature to compute accurate and smooth feature boundaries. Our approach does not depend on the topological connection of the underlying surfaces, thus, it can process both mesh-based and point-based time-varying surfaces.. This paper introduces the following contributions:. • Present a multi-scale geometry motion flow for efficient feature tracking of time-varying surfaces, which can process both mesh-based and pointbased time-varying surfaces without vertex-to-vertex correspondence across the frames.. • Integrate shape matching in the feature tracking system to process time-varying surfaces in large deformation with low frame sampling rate.. The rest of our paper is organized as follows. Section 2 reviews related work. Section 3 gives single-frame features and multi-frame propagation method. In section 4, we show experimental results and discussions. Finally, we conclude our paper in section 5. In this paper, we have presented a framework for efficient feature tracking of time-varying surfaces. We define an effective multi-scale geometry motion flow for the time-varying surfaces, which efficiently propagates the geometry features on the time-varying surfaces. By combining both the approximately invariant signature vectors and geometry motion flow vectors, we also corporate shape matching techniques into our system to process feature tracking for time-varying surfaces in large deformation with low frame sampling rate. Furthermore, multi-frame propagation system also has been developed to improve our feature tracking results.. In the future, we will report the following progress in latter publications. Firstly, optical flow computation has been widely investigated by computer vision researchers. When the objects in the input image sequences have the occlusion artifacts, these methods cannot correctly handle the flow estimation for the occluded region. Similarly, due to the occlusion and low surface albedo, scanned surface sequences usually exhibit large holes, such as the scanned data from Microsoft Kinect. In this case, motion flow computing for geometric features may not be accurate due to geometry incompletion in time-varying surfaces. We would like to develop a more stable motion flow computation to address this issue. Secondly, though SIFT matching of object features work well, the matching efficiency needs to be addressed when data amount of time-varying surfaces is much large. [10] Vlasic D, Baran I, Matusik W, Popović J. Articulated mesh animation from multi-view silhouettes. In: ACM Transactions on Graphics (TOG); vol. 27. ACM; 2008, p. 97. . where u, v and w are the averages of neighborhoods of velocities, we can receive:. We can solve for u , v and w as:. The above equations can be written as: We can use the Gauss-Seidel iteration method to solve above equations:. u k , v k and w k are averages of neighborhoods of velocities at iteration k, and typically set to 0.0. Many works on geometry processing of time-varying surfaces have been proposed, the comprehensive overview for these topics is beyond the scope of this paper. We only give the most related works such as feature tracking, shape matching, and surface reconstruction of the time-varying surfaces.. Several methods have been proposed for feature tracking of time-varying surfaces. Some approaches use strong shape prior knowledge to find features tracking of time-varying surface, such as parametric models [10, 11] , machine learning [12] or modal approaches [13] . Vlasic et al.. [10] and Gall et al. [11] optimized the pose of skeletal models before fitting the surface to visual data. In the case of data-driven time-varying face surfaces, Chai et al. [12] tracked a low number of facial features. By using machine learning techniques and a database of laser scanned faces, they mapped these features back onto a linear combination of previously acquired high resolution motion data to complete the feature tracking. Salzmann et al. [13] tracked a piece of 3D deforming cloth, and they applied Principal Component Analysis on a randomly generated set of possible configurations to constrain the space of acceptable deformations and to reduce the optimization complexity. One drawback of these approaches is that they depend on the strong shape prior knowledge.. Other approaches [14, 15] used little prior knowledge to perform geometry features tracking for time-varying surfaces. For example, segmentation methods [15] applied vertex positions optimization techniques and used prior knowledge as a regularization term to control vertex moving. However, these two methods operated at each vertex and would lead to difficult non-linear optimization for time-varying surfaces with strong noise. Patch based mesh tracking method [15] can receive satisfactory tracking optimization for timevarying surfaces with local deformation, however, this method assumed that the local deformations should be rigid transformations, and may not work well for tracking non-rigid features.. In fluid simulation community, many methods have been presented to track fluid surfaces. These method usually track topology-changing evolving surfaces incorporating with velocity fields or prescribed motions. Level set methods [16] and particle level set methods [17] are two preventative techniques for describing a dynamic implicit surface. These two methods consider the zero level set of a voxelized signed distance function, and integrate velocity information into the distance function to move the function, which can effectively track the surface evolution. Mesh-based surface tracking methods in surface evolvers also have been proposed, the idea behind these techniques is to evolve a triangle mesh according to a velocity field [18] . More recently, Hansen et al. [19] tracked temporally coherent, deforming triangle mesh with topology changing. They solved this problem using the surface geometry alone, without prior information like surface templates or velocity fields.. Shape matching is critical for geometry processing, a variety of methods have been proposed, refer to [20] for a survey. Shape matching between inconsistent surfaces is also critical for dynamic geometry processing and reconstruction, several methods also have been proposed [8, 21, 22] . Inspired by the SIFT method [23] , Li et al. [24] extracted feature points and proposed a multi-scale representation for the surface matching, in our paper, we will borrow some shape matching ideas from [24] for our feature tracking. Mitra et al. [5] computed a globally consistent motion spanning multiple frames for shape matching. They added a time coordinate to all the input points based on the ordering of the respective frames, and considered the problem of computing the motion of each frame as an estimation of certain kinematic properties of the resulting space-time surfaces. Wand et al. [6] estimated a globally consistent template matching model for the reconstruction of deforming geometry from unstructured noisy point cloud sequences, they later improved their method in [25] . More recently, Li et al. [1] proposed a temporally coherent completion method for scanned dynamic shapes using a non-rigid surface registration algorithm.. There are some other works that are closely related to ours, such as tracking and modeling of non-rigid objects [26] , surface reconstruction for time-varying surfaces [27, 6] , and performance capture [28, 29, 30, 31, 32] . For example, Torresani et al. [26] computed the deformable template and the motion directly from the image sequences. They recovered non-rigid structure from one single video assuming the shape lies within a linear subspace. This method is promising and work well from regular video streams. Sharf et al. [27] introduced a volumetric space-time technique for the reconstruction of deforming objects from point data, they described the motion of the object as an incompressible flow of material through time, and reconstructed coherent sequences of watertight models from incomplete scanner data. Bradley et al. [28] and De Aguiar et al. [29] took advantage of temporal information to deform or to track key models for topology-preserving reconstruction.

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


Using conditional random fields for result identification in biomedical abstracts Abstract. The abstracts of biomedical papers usually contain three sections: objective, methods, and results-conclusion. The results-conclusion section is the most important because it usually describes the main contribution of a paper. Unfortunately, not all biomedical journals follow this three-section format. In this paper, we propose a machine learning (ML) based approach to automatically identify the results-conclusion section. The results-conclusion section identification problem is formulated as a sequence labeling task. Four feature sets, including Position, Named Entity, Tense, and Word Frequency, are employed with Conditional Random Fields (CRFs) as the underlying ML model. The experiment results show that the proposed approach can achieve F-measure, precision, and recall of 97.08%, 96.63% and 97.53%, respectively. Integrated computer-aided tools are widely developed and demonstrated their effectiveness in improving the working efficiency in many domains [1, 3, 7, 20, 35, 44] . In the biomedical domain, the phenomenal growth in biomedical literature poses a major problem for biologists. In recent years, a range of text-mining applications have been developed to improve access to knowledge for biologists and database curators [6, 11, 21] . These applications perform functions like recognizing named entities (NEs) or identifying the relationships between them from an entire abstract without distinguishing the introduction from the methodology, the results or the conclusion. However, in the biomedical field, the results-conclusion section of an abstract usually describes the main contribution or new finding of a paper. For example, Fig. 1 shows a biomedical abstract (PubMed ID "15097232") comprised of four sections: objective, methods, results, and conclusion. The sentence in the results section, "Genetic variation in GRK4gamma was associated with HT in the subjects studied", summarizes the true contribution of the article. Therefore, distinguishing the results-conclusion section from the other parts of an abstract could provide detailed information on the article's key content. (For convenience, we refer to the results-conclusion section as the "results section" hereafter.). In this work, we use the Conditional Random Fields (CRFs) [27] machine learning (ML) algorithm to identify the result section from other paragraphs because CRF is one of the best known model in solving sequential labeling tasks such as named entity recognition (NER) [39] and part of speech tagging [33] , etc. Four feature sets, Position, Named Entity (NE), Tense, and Word Frequency (WF), are proposed. Because there are not any openly available section identification corpora, we generated our training and test data from a controlled source, hypertension-gene relation articles, In this paper, we proposed to use the CRFs machine learning model with four feature sets to deal with the result identification problem. Several experiments are conducted to analyze the characteristics of the individual and the combination of the proposed feature sets. Our experiment results show that 1) the NE information of the title can be interpreted as features to improve the precision, 2) carefully selected word unigrams and bigrams can be a useful information to enhance both the precision and recall, and 3) the tense of a sentence can be encoded as features to improve the overall performance.. In the future work, we plan to (1) find out stable feature sets which are general enough for cross domain abstracts (not just biomedical fields), and (2) improve our approach to recognize all sections in a given abstract including the "objective" and "methods" section. Several works [8, 23, 29, 43] have showed that the Position feature is profitable for the result identification task. However, in some cases, the Position feature alone cannot correctly disambiguate the result section. For example, in the training phase, given two abstracts with the same numbers of sentences in total, but the beginning boundary of their result sections are different; one starts from the sixth sentence and the other starts from the forth. In this case, the trained CRFs model will bias on the most frequent collocation of sentence tags and positions observed in the training data. In this section, we firstly discuss the effects of combining the position feature with the proposed three feature sets. Secondly, we explain our approach for threshold selection. Finally, we explain why we chose CRFs as our ML model. In the light of rhetorical structure theory [18] , clauses in text are relevant to one another through relations such as Background, Elaboration, and Contrast. These rhetorical relations when identified could be helpful for summarization, information retrieval, information extraction, and question answering. In natural language processing (NLP), researchers have undertaken to recognize rhetorical relations using manually crafted and statistical techniques [16, 32] . [19, 24, 34] have claimed that abstracts across scientific disciplines including the biomedical field follow consistent rhetorical roles or "argumentative moves" (e.g. Problem, Solution, Evaluation, and Conclusion). Teufel and Moens [36] have proposed a strategy for summarization by classifying sentences from scientific texts into seven rhetorical categories. Extracted sentences could be concatenated for automated usertailored summaries.. Since then, several result identification approaches have been proposed in recent years. In 2003, Shimbo et al. [29] present an experimental text retrieval system to facilitate search in the MEDLINE database. A unique feature of the system is that the user can search not only throughout the whole abstract text, but also from the limited "sections" in the text. For this purpose, they exploit the "structured" abstracts contained in the MEDLINE database in which sections are explicitly marked by the headings. These abstracts provide training data for constructing sentence classifiers that are used to section unstructured abstracts, in which explicit section headings are missing. They used support vector machine (SVM) [12] to classify sentences represented by bigrams, and contextual information and reported 91.9% accuracy.. Two years later, Yamamoto et al. [43] also used SVM classifiers with various novel features, such as subjectverb, verb tense, relative sentence location, and sentence score (i.e., the average TF-IDF score of constituent words) features and trained each of them for a different rhetorical status on structured abstracts. A structured abstract is one that has labels indicating rhetorical statuses of the sentences, while an unstructured abstract does not. The classifiers were tested on both structured and unstructured abstracts. The former were randomly obtained from the MEDLINE database and the latter were manually labeled by humans. Their method achieved F-measure of 87.2% and 89.8% for result and conclusion (structured abstract) and F-measure of 81.8% and 87.4% for result and conclusion (unstructured abstract).. Recently, Ruch et al. [23] reported on the construction of a categorizer, which classifies sentences of biomedical abstracts into a four-class argumentative model. The system is based on a set of Bayesian learners trained on automatically acquired corpora and augmented with distributional heuristics. Feature weighting was optimal with DF-thresholding. For the CON-CLUSION class, which has been reported to contain more highly informative contents than other sentences, they obtain an F-score of 85%.. Lin et al. [15] describes experiments with generative models for analyzing the discourse structure of medical abstracts, which generally follow the pattern of "introduction", "methods", "results", and "conclusions". They demonstrate that Hidden Markov Models [45] are capable of accurately capturing the structure of such texts, and can achieve classification accuracy comparable to that of discriminative techniques. Kneser-Ney discounting and Katz backoff are utilized to generate bigram language models for each section and achieved an F-measure of 89.8% for the result section and 89.7% for the conclusion section.. Wu et al. [42] introduces a method for computational analysis of move structures in abstracts of research articles. They proposed a six-step learning process to train a collocation classifier and exploited the HMM to tag sentences. Their system achieved 80.54% precision.

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


Isolation Concepts for Clique Enumeration: Comparison and Computational Experiments We do computational studies concerning the enumeration of isolated cliques in graphs. Isolation, as recently introduced, measures the degree of connectedness of the cliques to the rest of the graph. Isolation helps both in getting faster algorithms than for the enumeration of maximal general cliques and in filtering out cliques with special semantics. We compare three isolation concepts and their combination with two enumeration modi for maximal cliques ("isolated maximal" vs "maximal isolated"). All studied concepts exhibit the fixed-parameter tractability of the enumeration task with respect to the parameter "degree of isolation". We provide a first systematic experimental study of the corresponding enumeration algorithms, using synthetic graphs (in the G n,m,p model), financial networks, and a music artist similarity network (proposing the enumeration of isolated cliques as a useful instrument in analyzing financial and social networks). We study the enumeration of maximal cliques of an undirected graph G = (V, E), that is, the enumeration of all vertex subsets V ′ ⊆ V such that the induced subgraph G[V ′ ] is complete and there is no V ′′ V ′ such that the induced subgraph G[V ′′ ] is also complete. Unfortunately, already finding one maximum-cardinality clique is a notoriously hard computational problem, being NP-hard [10] as well as W [1] -hard [8] and hard to approximate [11] . By way of contrast, finding cliques is very important in many practical applications and has been subject of the second DIMACS implementation challenge in 1996. Recent papers describe applications in computational finance [2, 3] and computational biochemistry and genomics [5, 7] . Moreover, clique finding also plays a role in classical computer science fields such as the analysis of web graphs for instance to identify web communities [9] or "link farms" (for the purpose of spam deletion and analysis) [20] .. Enumerating all maximal cliques needs exponential time. For instance, a recent paper by Tomita et al. [21] proved a worst-case time complexity of Θ(3 n/3 ) for an n-vertex graph, arguing for its optimality due to the fact that there are example graphs having 3 n/3 maximal cliques [18] . Ito et al. [14] (also see the journal version [13] ) proposed to restrict the search to certain types of cliques, that is, specifically isolated cliques. A clique V ′ of k vertices is called c-isolated in a graph G if there are less than c·k edges leaving the induced subgraph G[V ′ ] in G. This concept is interesting for two reasons. First, since one does not ⋆ A preliminary version of this paper appeared in the proceedings of the 2nd Annual International Conference on Combinatorial Optimization and Applications (CO-COA '08), held in St. John's, Newfoundland, Canada, August 21-24 [12] . Note that in our conference version [12] we state that the work by Ito et al. [14] , which is the conference version of Ito and Iwama [13] , is flawed. A reason for this statement was a misunderstanding of Ito et al.'s [14] enumeration concept: they enumerate all isolated maximal cliques, and not all maximal isolated cliques, as we were assuming.. Email addresses: hueffner@tau.ac.il (Falk Hüffner), c.komus@uni-jena.de (Christian Komusiewicz), hannes.moser@uni-jena.de (Hannes Moser), rolf.niedermeier@uni-jena.de (Rolf Niedermeier). 1 Supported by the Deutsche Forschungsgemeinschaft, Emmy Noether research group PIAF (fixed-parameter algorithms), NI 369/4, and the Edmond J. Safra Foundation. Major parts of the work done while the author was with the FriedrichSchiller-Universität Jena. 2 Partially supported by the Deutsche Forschungsgemeinschaft, project OPAL (optimal solutions for hard problems in computational biology), NI 369/2, and a PhD fellowship of the Carl-Zeiss-Stiftung. 3 Supported by the Deutsche Forschungsgemeinschaft, project ITKO (iterative compression for solving hard network problems), NI 369/5, and project AREG (algorithms for generating quasi-regular structures in graphs), NI 369/9. search for all maximal cliques anymore, faster enumeration algorithms are possible. Second, isolated cliques may be an intrinsically relevant concept, because these cliques can represent structures with particularly interesting properties that are detected in this way.. In a companion paper [16] , we extended Ito et al.'s [14, 13] studies by distinguishing three natural isolation concepts instead of only one. Moreover, whereas Ito et al. considered the task to enumerate all maximal cliques that are isolated ("isolated maximal"), in our follow-up work [16] we studied the task to enumerate cliques that are maximal additionally with respect to the property of being isolated ("maximal isolated"). For the first time, here we systematically compare all different combination possibilities, after all leading to five different fixed-parameter algorithms (see [19] for more on fixed-parameter algorithms) with respect to the parameter c for enumerating c-isolated cliques. A typical running time of these algorithms looks like O(2 c · c 5 · |E|).. The main focus of our work is on computational experiments to explore the practical utility of the new clique enumeration algorithms based on isolation.. Before doing that, we provide a systematic comparison of all five enumeration algorithms for isolated cliques. In particular, we can improve a previous running time of O(4 c ·c 4 ·|E|) [13] to O(2 c ·c 5 ·|E|) for one of the mentioned enumeration algorithms. Our experiments are based on synthetic (G n,m,p graphs) as well as real-world data (financial networks, music artist similarity network). With the help of the corresponding empirical investigations we can spot subtle but important (practical) differences between the various isolation concepts and enumeration tasks. The main conclusion substantiated by our findings is that the consideration of isolated cliques pays off because one may • achieve faster algorithms for relevant special cases of clique enumeration in comparison with the famous Bron-Kerbosch algorithm and its variants [4, 15, 21, 6] , and • isolation concepts help filtering out semantically particularly interesting maximal cliques that may have remained undiscovered when enumerating all maximal cliques.. In conclusion, we believe that with our study (and the corresponding, freely available open source code) we contribute a practically useful tool for cliquebased network analysis in all fields of applications.. Our work is organized as follows. In the next section, we overview and compare known theoretical results and present a small improvement concerning the running time of one of the studied fixed-parameter algorithms. Then, in Section 3, we discuss relevant issues concerning the implementation of the algorithms experimented with in Section 4. There, we first present results for synthetic data-emphasizing efficiency issues-and then we present results for real-world data-emphasizing semantic issues. In Section 5, we draw some conclusions for future work and briefly summarize our findings and their potential impact on clique-based network analysis.

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


Probabilistic abduction without priors This paper considers the simple problem of abduction in the framework of Bayes theorem, when the prior probability of the hypothesis is not available, either because there are no statistical data to rely on, or simply because a human expert is reluctant to provide a subjective assessment of this prior probability. This abduction problem remains an open issue since a simple sensitivity analysis on the value of the unknown prior yields empty results. This paper tries to propose some criteria a solution to this problem should satisfy. It then surveys and comments on various existing or new solutions to this problem: the use of likelihood functions (as in classical statistics), the use of information principles like maximum entropy, Shapley value, maximum likelihood. The formal setting includes de Finetti's coherence approach, which does not exclude conditioning on contingent events with zero probability. We show that the ad hoc likelihood function method, that can be reinterpreted in terms of possibility theory, is consistent with most other formal approaches. However, the maximum entropy solution is significantly different, despite some formal analogies. Consider the basic problem of Bayesian abduction: Let H be a Boolean proposition interpreted as a hypothesis, a disease, a fault, a cause, etc. pertaining to the state of a system. Let E be another proposition representing a hypothetically observed (that is, observable) fact, a symptom, an alarm, an effect, etc. Numerical assessments of positive conditional probability values P (E|H) = a and P (E|H c ) = b ≤ a are supplied by an agent, who either uses available statistical data or proposes purely subjective assessments. The problem is to evaluate the relative plausibility of the hypothesis and its negation after observing event E. If a prior probability P (H) is assigned and b > 0, the question is solved by Bayes theorem. But, due to sheer ignorance, suppose no prior probability P (H) is assigned and observation E is made, or that probabilities a or b are set to zero. What can be said about the support given to hypotheses H vs. H c upon observing E?. In this work, we review past proposals for dealing with this problem and propose either new solutions or rigorous Copyright c 2006, American Association for Artificial Intelligence (www.aaai.org). All rights reserved.. formalization of previously proposed solutions, in connection with various approaches to probability theory, and to imprecise probabilities, such as maximum entropy, Shapley value, conditional events, de Finetti's coherence setting, possibility theory and the like.. Here, by definition, we do not take for granted the Bayesian credo according to which whatever their state of knowledge, rational agents should produce a prior probability. Indeed the idea that point probability functions should be in one to one correspondence with belief states means that a probability degree IS equated to a degree of belief. Then, in case of total ignorance about H agents should assign equal probabilities to H and its complement, due to symmetry arguments. This claim can be challenged, and was challenged by many scholars (e.g., (Shafer 1976; Dubois & Prade 1990; Smets & Kennes 1994; Walley 1991) ): Indeed agents must assign equal probabilities to H and its complement, when they know that the occurrence of H is driven by a genuine random process, and when they know nothing. The two epistemic states are different but result in the same probability assessment. Here, we take ignorance about H for granted, assuming P (H) is unspecified (in other words the agent refuses to bet on a value of P (H)), and see what was done in the past, what can legitimately be done to cope with ignorance, and how to formally justify various solutions to this problem.. Investigating the problem of abduction in a formal Bayesian framework here allows us to deal both with consistency based diagnosis (i.e. when evidence contradicts some hypotheses made about a system) and purely abductive reasoning (i.e. finding a minimal set of faults that explain the observations) in a common framework. This paper is organized as follows: We put the problem into formal terms in the next section, and some criteria are laid bare for a solution to the problem to be acceptable. We continue by recalling three classical approaches for its solution. Afterwards, various information principles are applied to solve the problem, and compared with each other. As a main contribution of this paper, we present a novel maximum likelihood approach by taking the conditional probabilities as kind of midvalues. We conclude the paper with a summary and an outlook on further work. Several approaches to the problem of probabilistic abduction have been reviewed, and some novel solutions have been proposed, based on maximum entropy, Shapley value and maximum likelihood reasoning. This study suggests that the key issue is a suitable representation of the available probabilistic knowledge, and a suitable choice of a reasoning principle. Table 1 summarizes the performance of the considered approaches with respect to the criteria 1, 2, 4, 5 laid bare in the methodology section. The agreement between the use of Shapley value and the classical Bayesian assumption of uniform priors under ignorance is noticeable. The maximum entropy approach is shown to significantly differ from the Bayesian tradition of uniform priors and the non-Bayesian approach based on likelihoods. Only the maximum likelihood bluntly applied when likelihoods are known leads to a contradiction (criterion 3). Our new maximum likelihood approach under a relaxed interpretation of the causal knowledge provides an original solution to the probabilistic abduction problem that bridges the gap between the straightforward use of likelihood functions and the assumption of a uniform prior, being more informative than the pure sensitivity analysis approach but less precise than the Bayesian, Shapley and maxent solutions when the likelihoods are too high to enable any hypothesis rejection.. More work is needed to fully interpret the obtained results. In particular, a systematic comparative study of first principles underlying the Shapley value and the maximum entropy approach is certainly in order. We should also compare our results with what the imprecise probability school has to say about this problem in a more careful way. Finally, another point to study is the influence of irrelevant information on the results of the various approaches.. It could be interesting to develop the work made in this paper by applying the relaxed maximum likelihood approach to more general knowledge bases, and also for notions of coherence other than coherent inference. In particular, the case of multiple-valued universes for hypotheses and pieces of evidence is worth investigating.. Moreover, in order to evaluate the cognitive plausibility (see Requirement 5) of the different approaches more thoroughly, psychological testing could be carried out with experts. . where s 1 , . . . , s n are n arbitrary real numbers. Let g h be the value of G when C h is true. Of course g 0 = 0 (notice that g 0 will play no role in the definition of coherence). We denote by G|H n the restriction of G to H n ; hence G|H n ∈ {g 1 , . . . , g m } , min G|H n = min {g 1 , . . . , g m } , max G|H n = max {g 1 , . . . , g m } .. Then, the function P defined on K is said coherent if and only if, for every integer n, for every finite sub-family F ⊆ K and for every s 1 , . . . , s n , one has. or equivalently. Coherence with penalty criterion: De Finetti (De Finetti 1974) has proposed another operational definition of probabilities, which can be extended to conditional events (Gilio 1990) . With the pair (F, p) we associate the loss. we denote by L h the value of L if C h is true. If You specify the assessment p on F as representing your belief's degrees, You are required to pay a penalty L h when C h is true. Then, the function P is said coherent if and only if do not exist an integer n, a finite subfamily F ⊆ K, and an assessment p * = (p * 1 , . . . , p * n ) on F such that, for the loss. h ≤ L h , h = 1, . . . , m , with L * h < L h in at least one case. Notice that the betting scheme and the penalty criterion are equivalent (Gilio 1990) ; this means that a probability assessment p is coherent under the betting scheme if and only if it is coherent under the penalty criterion. If P is coherent, then it is called a conditional probability on K. Notice that, if P is coherent, then P satisfies all the well known properties of conditional probabilities (while the converse is not true; see (Gilio 1995) , Example 8; or (Coletti & Scozzafava 2002 ), Example 13).. We can develop a geometrical approach to coherence by associating, with each constituent C h contained in H n , a point Q h = (q h1 , . . . , q hn ) , where. Denoting by I the convex hull of the points Q 1 , . . . , Q m , based on the penalty criterion, the following result can be proved ( (Gilio 1990) ) Theorem 3 The function P is coherent if and only if, for every finite sub-family F ⊆ K, one has p ∈ I. Notice that, if F = {E | H}, p = (P (E | H)) = (p), we have. Then, by Theorem 3, it immediately follows Corollary 4 A probability assignment P (E | H) = p is coherent iff it holds that p = 0, EH = ∅, p = 1, EH = H, p ∈ [0, 1], ∅ ⊂ EH ⊂ H.. The betting scheme and the penalty criterion are equivalent, as it follows by following reasoning: (i) the condition p ∈ I amounts to solvability of the following system (S) in the unknowns λ 1 , . . . , λ m m h=1 q hj λ h = p j , j = 1, . . . , n ; m h=1 λ h = 1 , λ h ≥ 0 , h = 1, . . . , m.. (ii) let x = (x 1 , . . . , x m ), y = (y 1 , . . . , y n ) t and A = (a ij ) be, respectively, a row m−vector, a column n−vector and a m × n−matrix. The vector x is said semipositive if x i ≥ 0, ∀ i, and x 1 + · · · + x m > 0 . Then, we have (cf. (Gale 1960) , Theorem 2.9) Theorem 5 Exactly one of the following alternatives holds. (i) the equation xA = 0 has a semipositive solution; (ii) the inequality Ay > 0 has a solution. We observe that, choosing a ij = q ij − p j , ∀ i, j, the solvability of xA = 0 means that p ∈ I, while the solvability of Ay > 0 means that, choosing s i = y i , ∀ i, one has min G|H n > 0 (and hence p would be incoherent). Therefore, applying Theorem 5 with A = (q ij − p j ), we obtain max G|H n ≥ 0 iff (S) is solvable, that is, max G|H n ≥ 0 iff p ∈ I.. Checking coherence. It could seem that, in order to verify coherence, we should check the condition p ∈ I for every F ⊆ K (which tends to become intractable). We show that this is not the case, by restricting the attention to the checking of coherence of the assessment p on F. Let S be the set of solutions Λ = (λ 1 , . . . , λ m ) of the system (S). Then, define Φ j (Λ) = Φ(λ 1 , . . . , λ m ) = r:Cr⊆Hj λ r , j = 1, . . . , n ; M j = max Φ j (Λ) , j = 1, . . . , n ; I 0 = {j ∈ J : M j = 0} . Notice that I 0 is a strict subset of {1, . . . , n}. We denote by (F 0 , p 0 ) the pair associated with I 0 . Given the pair (F, p) and a subset J ⊂ {1, . . . , n}, we define H J = j∈J H j . Moreover, we denote by (F J , p J ) the pair associated with J and by (S J ) the corresponding system. We observe that (S J ) is solvable if and only if p J ∈ I J , where I J is the convex hull associated with the pair (F J , p J ). Then, it can be proved: Theorem 6 Given the assessment p on F, assume that (S) is solvable, i.e. p ∈ I, and let J be a subset of {1, . . . , n}. If there exists a solution (λ 1 , . . . , λ m ) of (S) such that r:Cr⊆H J λ r > 0 , then (S J ) is solvable, i.e. p J ∈ I J .. Theorem 7 Given the assessment p on F, assume that (S) is solvable, i.e. p ∈ I. Then, for every J ⊂ {1, . . . , n} such that J \ I 0 = ∅, one has p J ∈ I J .. By the previous results, we obtain: p coherent ⇐⇒ p ∈ I ; if I 0 = ∅, then p 0 is coherent.. Then, we can check coherence by the following procedure: Algorithm 1 Let be given the pair (F, p ) In this paper, the following notations are adopted: Ω is the sure event, AB is short for A ∧ B (conjunction), and the complement of an event A is denoted A c . Moreover, we use the same symbol to denote an event and its indicator.

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


Aggregative quantification for regression The problem of estimating the class distribution (or prevalence) for a new unlabelled dataset (from a possibly different distribution) is a very common problem which has been addressed in one way or another in the past decades. This problem has been recently reconsidered as a new task in data mining, renamed quantification when the estimation is performed as an aggregation (and possible adjustment) of a single-instance supervised model (e.g., a classifier). However, the study of quantification has been limited to classification, while it is clear that this problem also appears, perhaps even more frequently, with other predictive problems, such as regression. In this case, the goal is to determine a distribution or an aggregated indicator of the output variable for a new unlabelled dataset. In this paper, we introduce a comprehensive new taxonomy of quantification tasks, distinguishing between the estimation of the whole distribution and the estimation of some indicators (summary statistics), for both classification and regression. This distinction is especially useful for regression, since predictions are numerical values that can be aggregated in many different ways, as in multi-dimensional hierarchical data warehouses. We focus on aggregative quantification for regression and see that the approaches borrowed from classification do not work. We present several techniques based on segmentation which are able to produce accurate estimations of the expected value and the distribution of the output variable. We show experimentally that these methods especially excel for the relevant scenarios where training and test distributions dramatically differ. Quantification can be seen as a group of closely related tasks where the goal is to determine some global indicator or the whole distribution from a set of (individual) unlabelled examples. Aggregative quantification is the view of quantification as a problem which is solved by aggregating the predictions of an underlying predictive model: a classification model if the output value is categorical, or a regression model if the output value is numerical. Where many proposals have been recently proposed for classification quantification, to our knowledge this is the first paper addressing the problem of regression quantification, as shaped here.. Quantification is a common problem in many data mining applications:. -Multidimensional hierarchical data. Data warehouses are frequently the source of minable views. A predictive model learnt at a fine-grained level may be required to be applied to a different distribution (region, period or category) or at a different level of aggregation. Quantification techniques are crucial for this problem. -Overall estimation. Problems are frequently presented in the form of a batch, where we need to anticipate (e.g., assign resources) an assessment of some indicators or the whole distribution for a batch of customers, patients, complaints, errors, diagnoses, etc., before making any individual decision. -Distribution shift detection. Clearly a quantifier can be used to determine when (and of course how much) a distribution is shifting. This may be used to trigger some revision or re-training of the models, instead of triggering this by the degradation of the models (which can be originated by other factors, such as obsolescence, but not necessarily a distribution shift). -Calibration and cost applications. An interesting application would be to use quantification methods to calibrate supervised models or to use them in cost applications, only incipiently explored by Forman (2008) . However, the methods introduced in this paper modify all the single predictions, so the model resulting from adjustment, segmentation and spreading can still be used as a single-instance supervised model (this is not the case, for instance, with methods such as T50 or MS in classification).. As discussed throughout the paper, the problem becomes really meaningful when we have an important distribution shift between the training data and the deployment data. This distribution shift entails a change in the categorical distribution (class frequencies) in classification problems. In regression, the distribution shift may involve a change in location, dispersion and shape of the data distribution. This is in contrast to the classical problem of estimating the distribution from a small sample, even if there is no distribution shift. All the above issues have triggered a comprehensive analysis of quantification as a group of data mining tasks on its own, which is consolidated by the new taxonomy we have introduced in this paper. From here, we can see how much similar the quantification tasks are, what metrics are required for each of them and the desired detail of the quantification result, as a single indicator or a whole distribution. Given this taxonomy we have focussed on the two tasks which, to our knowledge, have not received proper attention in the literature (in the terms set here). These families of tasks are regression quantification obtaining the whole distribution and the task focussed on a particular indicator. One of the principles we have followed when looking for solutions for these problems is that the solutions should be general to any underlying regression technique.. We have seen that the direct adaptation of ideas for classification quantification to regression does not work. Moreover, the methods based on segmentation introduced here, especially when each bin is used to do local adjustments, have shown a very significant improvement. We have seen good results for both the mean and the median estimation and the whole distribution estimation (Q RI and Q RD tasks).. There are also some limitations of the approach presented in this paper. Aggregative regression quantification still relies on good regression models to get good quantification. While it is possible to get good quantification from some poor models, there is still much to be done in order to understand which regression models (and techniques) are more suitable for quantification. While some quantification methods excel on average, they can perform worse than the trivial RS method for some datasets and techniques. Detecting when this is the case, and using method combinations with this information could lead to improved results. Also, the results for the whole distribution case Q RD can be improved in terms of shape, since asymmetries and multimodalities may not be properly identified in many cases.. In the end, this is just a first attempt for regression quantification (as considered here) and many things can be improved. For instance, we have set the same value for the parameter α for all techniques and methods in order to ease the comparison, but we may choose a particular α for each problem, regression technique and quantification method, or link it to some parameters of the model on the training set, such as correlation or the variance. Also, new smoothing methods could be devised as well by reusing the information from the segmentation. Segmentation could also be done in other ways, by the use of other clustering methods, sliding windows or kernels, with possibly overlapping bins.. As a side possibility, we might wonder what learning parameters and metrics should be used for training supervised models on purpose for a quantification problem. For instance, we have used regression techniques with default parameters usually aiming at minimising MSE, but we could also use other techniques with the purpose of being median-unbiased estimators by the use of the absolute error instead. The approaches using classification may also be considered. Some of the analysis of existing performance metrics (Ferri et al. 2009; Hernández-Orallo et al. 2012 ) could be overhauled having quantification in mind. The use of different loss functions and metrics could also lead to extensions of the taxonomy, by accommodating ordinal regression, hierarchical classification, etc.. As mentioned above, quantification is a very common problem, and there are also many possible scenarios and applications for the regression case. In fact, the interaction of regression quantification with common hierarchical data organisations deserves to be explored. In fact, quantification could be applied in an iterative way. For instance, we may need to calculate the expected total benefits for a city, then aggregate for regions, then for countries, etc. This stepwise aggregation may give better results than aggregating the predictions from the lowest level to the highest one. This, of course, suggests the use of (hierarchical) clustering for both the input and the output values. In fact, many different segmentation methods could be used instead of the E F, E W and k M methods. All this could also be explored as future work.. All in all, regression quantification may have been approached in many different (and ad-hoc) ways in the literature and specific applications, but there has not been a systematic and generalised account of this data mining task on its own. Given the number of alternatives and the ideas that could be used for different task variants of the regression quantification problem, we think this paper is a resolute step in the recognition of this family of data mining tasks and the everyday application of better suited solutions. In this section, we will introduce some notation to express what quantification is precisely. The understanding of this problem will be helped by two examples and a proper account of related work, including a short description of the methods which have been previously introduced for (classification) quantification. In this section, we present the experimental evaluation of the methods for regression quantification introduced in the previous section: those based on Regress & Sum or Splice, namely RS and a RS, and the six methods based on data segmentation. We will examine the results for the quantification problems Q RI (mean and median) and Q RD in Table 1 and we will use their corresponding metrics (V SE and Cramér-von Mises u metric, cvmu). Finally, we will also include some results of the approach to regression quantification using classification quantification.

O B O O O O O O O O B 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 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 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 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 B 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 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 B 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 B 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 B O O O O O O O O O O B 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 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 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 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 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 O O O B 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 O O O O B 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 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 B 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 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 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 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 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 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 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 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 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 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 B 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 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 O O O 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 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 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 B 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 B O O B


Reliable communication for highly mobile agents The provision of a reliable communication infrastructure for mobile agents is still an open research issue. The challenge to reliability we address in this work does not come from the possibility of faults, but rather from the mere p r esence of mobility, which slightly complicates the problem of ensuring the delivery of information even in a fault-free network. For instance, the asynchronous nature of message passing and agent migration may cause situations where messages forever chase a mobile agent that moves frequently from one host to another. Current solutions rely on conventional technologies that either do not provide a solution for the aforementioned p r oblem, because they were not designed with mobility in mind, or enforce continuous connectivity with the message source, which in many cases defeats the very purpose of using mobile agents.In this paper, we propose an algorithm that guarantees delivery to highly mobile agents using a technique similar to a distributed snapshot. A number of enhancements to this basic idea are discussed, which limit the scope of message delivery by allowing dynamic creation of the connectivity graph. Notably, the very structure of our algorithm makes it amenable not only to guarantee message delivery to a given mobile agent, but also to provide multicast communication to a group of agents|another open problem in research on mobile agents. After presenting our algorithm and its properties, we discuss its implementability by analyzing the requirements on the underlying mobile agent platform, and argue about its applicability. Mobile agent systems currently provide an increasing degree of sophistication in the abstractions and mechanisms they support, well beyond the purpose of achieving agent migration. However, it is questionable whether the features that are being added on top of plain agent migration are really focused on the needs of application developers, or they really address the problems that are peculiar to mobility.. A good example of the gap between what is provided and what is needed is the problem of providing a communication infrastructure for mobile agents. This aspect is often overlooked or misunderstood in the context of mobile agent research. For instance, significant e orts are being devoted to the problem of enabling communication among mobile agents by de ning a common semantic layer for the exchange of information, as in KQML 6 . Despite their relevance, the questions posed by researchers in this area are not particularly a ected by the presence of mobility, and focus on the problem of communication at a completely di erent, and much higher, abstraction level than the one we are concerned with in this paper. Even if we assume that the problem of ensuring a proper semantic level for agent communication is somehow solved, we are still left with the problem of reliably delivering a message to a mobile agent whose patterns of mobility are potentially unknown a priori. This is the problem we address in this paper.. The challenge to reliable communication persists even under the assumption of an ideal transport mechanism that guarantees a correct delivery of information in the presence of faults in the underlying communication link or in the communicating nodes. It is the sheer presence of mobility, and not the possibility of faults, that undermines the notion of reliability. If mobile agents are allowed to move freely from one host to another according to some a priori unknown migration pattern, the challenge in delivering information properly is caused by the di culty in determining where the mobile agent is, and in ensuring that the information reaches the mobile agent before it moves again.. By and large, currently available mobile agent sys-tems rely either on conventional communication facilities like sockets and remote procedure or method call 1, 8, 13 , or implement their own message passing facility 10 . To our knowledge, none of them satisfactorily addresses the aforementioned problem. They require knowledge about the location of the mobile agent, which is obtained either by overly restricting the freedom of mobility or by assuming continuous connectivity|assumptions that in many cases defeat the whole purpose of using mobile agents.. In this paper, we propose an algorithm that guarantees message delivery to highly mobile agents in a fault-free network. We focus on message passing as the communication mechanism that we want to adapt to mobility, because it is a fundamental and well understood form of communication in a distributed system. This incurs no loss of generality because more complex mechanisms like remote procedure call and method invocation are easily built on top of message passing. Our algorithm does not assume knowledge about the location of agents, and constrains the movement of agents only in its most enhanced form and only for a limited amount of time. Furthermore, its structure makes it inherently amenable to an extension that provides multicast communication to a group of agents dispersed in the network, another problem for which satisfactory solutions do not yet exist.. The paper is structured as follows. Section 2 discusses the motivation for this work, and the current state of the art in the eld. Section 3 presents our algorithm, starting with the underlying assumptions and illustrating subsequent re nements of the original key idea. Section 4 discusses the applicability and implementability of a communication mechanism embodying our algorithm in a mobile agent platform. Finally, Section 5 provides some concluding remarks. In this work we point out how the sheer presence of mobility makes the problem of guaranteeing the delivery of a message to a mobile agent inherently di cult, even in absence of faults in the network. To our knowledge, this problem has not been addressed by the research community. Currently available mobile agent systems employ techniques that either do not provide guarantees, or overly constrain the movement or connectivity of mobile agents, thus to some extent reducing their usefulness. In this work, we propose a solution based on the concept of a distributed snapshot. Several extensions of the basic idea allow us to cope with di erent levels of dynamicity and, along the way, provide a straightforward way to implement group communication for mobile agents. Our communication mechanism is meant to complement those currently provided by mobile agent systems, thus allowing the programmer to trade reliability for bandwith consumption. Further work will address fault tolerance and exploit an implementation of our mechanism to evaluate its tradeo s against those of conventional mechanisms.

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


Kernel bounds for disjoint cycles and disjoint paths Abstract. In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic assumptions. We build upon recent results by Bodlaender et al. [3] and Fortnow and Santhanam [13] , that show that NPcomplete problems that are or-compositional do not have polynomial kernels, unless NP ⊆ coN P/poly. To this machinery, we add a notion of transformation, and thus obtain that Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coN P/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(k log k). In many practical settings, exact solutions to NP-hard problems are needed. A common approach in such cases is to start with a preprocessing or data reduction algorithm: before employing a slow exact algorithm (e.g., ILP, branch and bound), we try to transform the input to an equivalent, smaller input.. Currently, the theory of fixed parameter complexity gives us tools to make a theoretical analysis of such data reduction or preprocessing algorithms. A kernelisation algorithm is an algorithm, that uses polynomial time, and transforms an input for a specific problem to an equivalent input whose size is bounded by some function of a parameter. The resulting instance is also called a kernel. Questions of both theoretical and practical interests are for a specific problem: does it have a kernel, and if so, how large can this kernel be? An excellent overview of much recent work on kernelisation was made by Guo and Niedermeier [15] .. For the question, whether a specific (parameterised) problem has a kernel, the fixed parameter tractability theory introduced by Downey and Fellows gives good tools to answer these. We say a problem is fixed parameter tractable (in FPT), if it has an algorithm that runs in time O(n c f (k)), with n the input size, k the parameter, c a constant, and f any function. It is well known that a decidable problem is in FPT, if and only if it has a kernel. (See [7] .). Recently, Bodlaender et al. [3] gave a framework to give evidence that problems (in FPT) do not have a kernel of polynomial size. The framework is based upon the notion of compositionality. There are actually two forms: andcompositionality, and or-compositionality. We have a parameterised problem, whose variant as a decision problem is NP-complete, and it is and-compositional, then it does not have a kernel whose size is bounded by a polynomial, unless the and-distillation conjecture does not hold. Similarly for or-compositionality, and the or-distillation conjecture, but in this case, one can use a result by Fortnow and Santhanam, and strengthen the conjecture to N P ⊆ coN P/poly [13] . In this paper, we extend the framework by introducing a notion of transformation. While the main idea parallels classic notions of transformation, we think that our contribution is a new important tool for the theory of data reduction/kernelisation.We use our framework to show for the following problems that they do not have a kernel of polynomial size unless N P ⊆ coN P/poly: Disjoint Cycles, Disjoint Paths. The result for Disjoint Paths is not so surprising, given the very complicated FPT algorithms for this problem [19] . The result for Disjoint Cycles came unexpected to us, also because polynomial kernels are known for the closely related problems: Feedback Vertex Set (see [21] ) and Disjoint Cycle Packing (see Section 4) .. Concerning the size of a kernel of a parameterised problem, we can summarise the situation in Table 1 . Assuming that the problem is decidable, the second and third column give the main available positive or negative, respectively, evidence that the problem has a kernel of the size given in the first column. E.g., W [1]-hardness indicates that a problem is not in FPT; a problem is in FPT, if and only if it has a kernel of any size (i.e., bounded by a function of k.) In this paper, we showed that the Disjoint Cycles and Disjoint Paths problems do not have polynomial kernels, assuming that N P ⊆ coN P/poly. The result for Disjoint Cycles came unexpected to us, given the similarity of the problem to Feedback Vertex Set and Disjoint Cycle Packing, which do have polynomial kernels. Our initial expectation that techniques for these problems would carry over to Disjoint Cycles proved to be false. Thus, the result in our paper for e.g. Disjoint Cycles plays the role that it can direct further research efforts, i.e., it appears not to be useful to aim at finding a polynomial kernel for the problem; this is somewhat comparable to stating that an NPcompleteness proof directs our research efforts away from finding a polynomial time algorithm for a problem.. Transformations are a powerful mechanism to derive no-polynomial-kernel results. In a longer and earlier version of this paper [6] , we showed that Hamiltonian Path and Hamiltonian Circuit parameterised by treewidth has no polynomial kernel, unless the AND-distillation conjecture does not hold. Fernau et al. [12] show that k-Leaf-Out-Branching has no polynomial kernel, unless N P ⊆ coN P/poly. A large collection of no-polynomial-kernel results were obtained by Dom et al. [8] , using intricate and clever reductions.. The further development of the theory of kernel sizes is an interesting topic for further research. An important topic with many recent results (see e.g., the overview paper by Guo and Niedermeier [15] ) is to find kernels of sizes as small as possible for concrete combinatorial problems. Some questions we want to add to this are: The theory so far allows to distinguish between constant size, polynomial size, and any size kernels: can we refine this? Are the classes N P K and and N P K or the same? (There are some problems that belong to both classes.) Are there complete or "hardest" problems for these classes? Another still open problem is whether there exists a result for and-distillation that is similar to the result of Fortnow and Santhanam for the or-distillation conjecture, i.e., can we relate the and-distillation conjecture to more widely known and believed complexity theoretic conjectures?. Finally, we mention a few concrete open problems: does Disjoint Paths have a polynomial kernel when restricted to planar graphs? Is there a polynomial kernel for Feedback Arc Set or Directed Feedback Vertex Set? Or, if not, do these problems have a polynomial kernel when restricted to directed planar graphs?

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