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0306964d6ca220860c61c9ef772856808dc452ff | subsection | 19 | 41 | Polygonal approximation of the target object | For polygonal approximation of the target object we draw small line segments between two consecutive dominant points of the target object. Let us consider two dominant points D_1 and D_2 which are calculated using equation-(3). The path between this two points is a small line segments joining the said two points. There... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
"Kumar S. Ray"
] | [
"cs.CV",
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a5b6938973bd72904b167c8ad20543b53175fae0 | subsection | 20 | 41 | Fitness Function for PSO tracker | Every PSO model is based on some cost function. Each particle of the swarm computes that fitness function in each iteration to confirm whether it converges to the final solution or not. In this paper, our cost function is the perpendicular distance of the particle i to the small line segment which is a part of the appr... | {
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9104515cb1e04a8d7dc77671ef43d4de54d7028f | subsection | 21 | 41 | Formation of Multiswarms | Once the task of constructing the polygon of the target object is completed, for the first time,at frame 2 of the video sequence, we distribute particles over the entire image space. Note that the the population of the particles is a heuristic parameter which depends on the need of the problem and which has several opt... | {
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a428a118638c327627f7f2b328039767eac00327 | subsection | 22 | 41 | Body | When the dual trackers arrive at frame-3 of the video sequence, the shape of the polygon is automatically changed due to the movement of the target object which is in general non rigid in nature. In case of rigid object the shape of the polygon of the target object remains same. Once the shape of the polygon changes at... | {
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aa9e7b30063637b90c91cc94737eb1d56e927285 | subsection | 23 | 41 | Reinitialization of the particle of the individual swarm | At the time of updating the position the particles of the individual swarms, instead of converging over the small line segments of the changed polygon, the particle(particles) may be distracted from the said line segments to a far away distance even after several iteration of updation. In that case we need to reinitial... | {
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fd63505fc02aaa16e49e03e7c77a54f132a68f48 | subsection | 24 | 41 | Bounding Box formulation | To identify tracked target object usually a rectangular bounding box is utilized. There are some pre-defined algorithms exist for this purpose, but here we design our own bounding box based on PSO particle position which will best suite our object tracking algorithm.The main idea is whenever all particles in all swarms... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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0f289c22fe5a6ae3ae880e99df475a4f5f22cbb2 | subsection | 25 | 41 | Re-initialization of missing dominant points | Due to background clutter, occlusion, Illumination Variation, low resolution and scale variation of various video sequences, change of image background occurs frequently. So the optical flow method based Kanade–Lucas–Tomasi(KLT) tracker which is basically a point tracker is unable to track a single point throughout the... | {
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2b88b55c14dea04f7ccf9cefad78dbd5196dc918 | subsection | 26 | 41 | Pictorial Illustration of Reinitialization of missing dominant points | We explain reinitialization process of missing Dominant points using an example. Let’s consider a curvature C whose start (s) and end (e) points are dominant points. We track these dominant point using KLT method.In the following figure-(10) there are two dominant points (s and e) which are marked as RED. These two poi... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
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6366bcc8caf11e7f67e09e4c765aafa2fd4fe378 | subsection | 27 | 41 | Re initialization of the particles of the swarms | Reinitialization of particle is sometime required as it is inherent in nature of PSO that few particles are too diverged from their desired position and even after several updation may not bring them towards their goal. In our case it is also possible that some particles are too far away from curvature boundary and aft... | {
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144dffe4099510f4cc2b658cce4d44e1414c5117 | subsection | 28 | 41 | Re initialization of the particles of the swarms | So rather computing any complex mathematical function and performing extensive number of iteration we consider the most simple approach by placing the diverged particles positions directly on any of the two dominant points around which a swarm was already formed and from where particle(particles) were diverged.
Figure-... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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21e0e17ae84bea2051faac05c4d1a899ba8c60e9 | subsection | 29 | 41 | Some further illustration on the proposed dual tracking algorithm | Step 1: First we extract the first frame from input video and convert it into binary image.By trial and error we find a pixel point on the boundary of target object as shown in figure-(16). Here P_{i}(x_{i},y_{i}) is the boundary point. Note that for simplicity of illustration we consider the front view of an object. I... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
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] | [
"cs.CV",
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bc3811f47e2d9660b632c4d8587c2e8c840b8252 | subsection | 30 | 41 | Some further illustration on the proposed dual tracking algorithm | Blue arrows show the tracking of dominant points is performed by KLT.Yellow arrows show the tracking of the boundary(approximated by a straight line)of the curved object is performed by PSO particles.Step 7: Note that by process of dual tracking when the target object, which is polygonally approximated , reaches the 3r... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
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"Rajesh Misra",
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] | [
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355233ff6f430c77bb79ca407a4cb75a81e800ef | subsection | 31 | 41 | Algorithmic summary and Complexity analysis | The Dual Tracking Algorithm(DTA) is represented in pseuducode as follows and - | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
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971446aa4035db690de23433ded9515a5fbfb3a2 | subsection | 32 | 41 | Pseudocode | Procedure:DualTrackingAlgorithm(DTA)(videoSequence_with_traget_object)[Get Frames form the input video]Frames \leftarrow CALL Algorithm Frame_Extraction(videoSequence_with_traget_object)[see Appendix][Calculate Breakpoints of target objects and store those points in “brpts” variable]brpts \leftarrow CALL Algorithm BrPt... | {
"cite_spans": []
} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
"Kumar S. Ray"
] | [
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caa45b2b9a819e244d94a5ebb087d7f8d478cb17 | subsection | 33 | 41 | Complexity analysis | To calculate time complexity of Procedure DualTrackingAlgorithm(DTA)() we need to compute complexity of all the sub algorithms it called and summing up all those complexity will give us approximated time complexity of this algorithm.Complexity of Algorithm Frame_Extraction(Video_input)- reading a video file and extract... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
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] | [
"cs.CV",
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d91ada8dc0e7ec0573d0b3f6e5c26ce12357ff0d | subsection | 34 | 41 | Complexity analysis | And finially PSO will run for each frames required O(f), f is frame number.Total time complexity = [O(b_{2}) + f* O(n_{2}*N +n_{3}) + O(f)] where b – no. of breakpoints, f – no of frames and n – number of pso particle, and N – no. of pixels in KLT tracking. | {
"cite_spans": []
} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
"Kumar S. Ray"
] | [
"cs.CV",
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4f7012c6765c14dde5182c35d1d68be1ac1906b4 | subsection | 35 | 41 | Experimental Setting | The proposed dual tracking approach for variable background and static background under different challenges as stated earlier, is tested by MatLab 2015a on a 64 bit PC with Intel i5 processor with 3 GHz speed. The image size of the frame 180 X 144. Static video is 20 sec duration whereas variable background is 13 sec ... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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ec610824eec257128073d497369a42a8ab5e8cbb | subsection | 36 | 41 | Experimental Dataset-1 (Wu et.all) | All the experimental dataset has been taken from benchmark library created by Wu, Yi Wu, Jongwoo Lim and Ming-Hsuan Yang and which is available on http://pami.visual-tracking.net | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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2f40c3902bf3ce0c9346c2fd29044f5efc1083f5 | subsection | 37 | 41 | Tracking Results of the proposed method | Form the experimental test data set we pick up 3 video streams which have static background and point of interest is moving high to moderate rate. From TB-50 sequence Girl[SV,OCC,IPR,OPR],Walking2[SV,OCC,LR] and one Walking[LR,IV]. And 3 video stream from TB-100 sequence for dynamic background; jogging(1)(2)[OCC,DEF,OP... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
"Rajesh Misra",
"Kumar S. Ray"
] | [
"cs.CV",
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faf77a8395735c2c2711186253905f26cf78060a | subsection | 38 | 41 | Static background | The first experiment is on static background. We consider Walking [LR,IV], Girl[SV,OCC,IPR,OPR], Walking2[SV,OCC,LR] datasets in figure-(25) to figure-(30).
[Figure: It shows a sequence of frames of a single person moving towards a camera where background is static. Proposed dual tracking algorithm successfully tracks ... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
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"Rajesh Misra",
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f5e0d067d38d48c583e0138d31c84aa77a990f00 | subsection | 39 | 41 | Variable Background | Now we perform our experiment on a video where background is moving with object. Video is taken by a moving camera. Here we consider 3 video frames from TB-100 sequence namely jogging[1][2][OCC,DEF,OPR], Suv[OCC,IPR,OV], Walking[OCC,BC]. Both tracking results and Bounding Box representations are shown below from figure... | {
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} | 1808.08186 | Dual approach for object tracking based on optical flow and swarm
intelligence | [
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326818e4948e2ced28b797b7542c90085838f6cb | subsection | 40 | 41 | Analysis and Evaluation | We evaluate the proposed dual tracking algorithm(DTA) using three parameters: True Detection(TD), False Detection(FD), Missed Detection(MD). We consider the parameter Frames per Seconds(FPS) to denote the number of frames per second. There is substantial amount of impacts in tracking due to high speed (high FPS) video ... | {
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bcbfd17d83a6c40f2c3ea21da092243ab3598047 | abstract | 0 | 174 | Abstract | We generalize several recognizability theorems for free single-sorted
algebras to the field of many-sorted algebras and provide, in a uniform way and
without using neither regular tree grammars nor tree automata, purely algebraic
proofs of them based on the concept of congruence. | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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65a81a337b9c0e3ed9ef7469229575f4a58a5220 | subsection | 1 | 174 | Introduction | The definition of recognizable language set down by Rabin and Scott in , Definition 2, on p. 116, originated in the characterizations of regular languages given by Myhill , in terms of congruence relations of finite index, and Nerode , in terms of right invariant equivalence relations of finite index, and was formulate... | {
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{
"arxiv_id": "",
"doi": "10.1147/rd.32.0114",
"end": 411,
"openalex_id": "https://openalex.org/W2054801208",
"raw": "M. O. Rabin and D. Scott, Finite automata and their decision problems. IBM J. Res. Develop. 3 (1959) pp. 114–125.",
"source_ref_id": "9226adc3... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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b2e29e076c285469069569bd69dfab3ef75df91f | subsection | 2 | 174 | Introduction | For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions—they are not just repetitions of each other. [\ldots ] the more perspectives, the better!”, and (2) what Lang, referring to his own review of the historical development of class field theory, ... | {
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"doi": "10.1007/bfb0092624",
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"raw": "S. Lang, Topics in cohomology of groups. Translated from the 1967 French original by the author. Chapter X based on letters written by John Tate. Sp... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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65445bfe1517edf46e92fe6efb3e3e50e64bc8ec | subsection | 3 | 174 | Introduction | Then we show that each hyperderivor determines, in a canonical way, a tree homomorphism, which is, in fact, a homomorphism from a free many-sorted algebra obtained from the domain of the hyperderivor to another many-sorted algebra of the same many-sorted signature, itself derived from a many-sorted algebra obtained fro... | {
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"doi": "10.1007/978-1-4612-9839-7",
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"raw": "S. Mac Lane, Categories for the working mathematician. 2nd ed. Springer-Verlag, New York, 1998.",
"source_ref_id": "58dc025756fb0bff02... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
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] | 2,018 | en | Computer Science | [
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88894f9be2fffc8f1765bcdcca97ad5c6b48c3db | subsection | 4 | 174 | Introduction | If \Phi and \Psi are (binary) relations in a set A, then we will say that \Psi is a refinement of \Phi if \Psi \subseteq \Phi .
We will denote by \mathrm {Fnc}(A,B) the set of all functions from A to B. We recall that a function from A to B is a subset F of A\times B satisfying the functional condition, i.e., such that... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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4dae532dc1c1432bd91feb8e7f4b75fc68d3e6ef | subsection | 5 | 174 | Preliminaries | In this section we collect the basic facts, mostly without proofs, about many-sorted sets, many-sorted algebras, and recognizability for arbitrary many-sorted algebras, that we will need.Assumption From now on S stands for a set of sorts in {\mathcal {U}}, fixed once and for all.Definition 2.1 An S-sorted set is a mapp... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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bd5663ff249bf662c64d7b9876fbfee67ba9d932 | subsection | 6 | 174 | Preliminaries | The ordered pair (\prod _{i\in I}A^{i},(\mathrm {pr}^{i})_{i\in I}) has the following universal property: For every S-sorted set B and every I-indexed family of S-sorted mappings (f^{i})_{i\in I}, where, for every i\in I, f^{i} is an S-sorted mapping from B to A^{i}, there exists a unique S-sorted mapping \left<f^{i}\r... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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3f12df7cd4aab93a3a9851946c03aae93f52bc71 | subsection | 7 | 174 | Preliminaries | The ordered pair (\coprod _{i\in I}A^{i},(\mathrm {in}^{i})_{i\in I}) has the following universal property: For every S-sorted set B and every I-indexed family of S-sorted mappings (f^{i})_{i\in I}, where, for every i\in I, f^{i} is an S-sorted mapping from A^{i} to B, there exists a unique S-sorted mapping [f^{i}]_{i\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f8a0f9ff1bfa4a558d9b4be9391b228e9221b4dd | subsection | 8 | 174 | Preliminaries | If X = \lbrace x\rbrace , then, for simplicity of notation, we will write \delta ^{t,x} instead of \delta ^{t,\lbrace x\rbrace }. Moreover, for a sort t in S, \delta ^{t,1}, the delta of Kronecker associated to (t,1), will be denoted by \delta ^{t} and called delta of Kronecker.Remark For a sort t\in S and a set X, the... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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7fd79c7e95f4ede7d3196d1a8b829f2719e5dad0 | subsection | 9 | 174 | Preliminaries | To this we add the following facts: (1) \lbrace \,\delta ^{s}\mid s\in S\,\rbrace is the set of atoms of the Boolean algebra \mathbf {Sub}(1^{S}), of subobjects of 1^{S}; (2) the Boolean algebras \mathbf {Sub}(1^{S}) and \mathbf {Sub}(S) are isomorphic; (3) for every s\in S, \delta ^{s} is a projective object; and (4) ... | {
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"doi": "10.1145/947864.947865",
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"raw": "J. Goguen and J. Meseguer, Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(1985), pp. 307–334.",
"source_... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2cf79a4378895079ec88043c2279ae510ebed7eb | subsection | 10 | 174 | Preliminaries | The kernel of f, denoted by \mathrm {Ker}(f), is (\mathrm {Ker}(f_{s}))_{s\in S} and the image of f, denoted by \mathrm {Im}(f), is f[A]. Moreover, if X\subseteq A, then the restriction of f to X, denoted by f\!\!\upharpoonright _{X}, is f\circ \mathrm {in}_{X,A}, where \mathrm {in}_{X,A} = (\mathrm {in}_{X_{s},A_{s}})... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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df3bd51da185f05c4af587abbcf5aee717b927bd | subsection | 11 | 174 | Preliminaries | Moreover, f^{\mathfrak {p}} is clearly the unique S-sorted mapping from X^{\wp } to Y^{\wp } satisfying the aforementioned conditions.Corollary 2.8 Let \mathbf {Set}^{S}_{\wp ,\mathrm {ca}} be the category whose objects are the S-sorted sets X^{\wp }, where X\in {\mathcal {U}}^{S}, and in which the set of morphisms fro... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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22ef82802e5e9be81c4518233827a45879a04a83 | subsection | 12 | 174 | Preliminaries | One should be careful not to confuse f[\cdot ], which is a mapping from the set \mathrm {Sub}(X) to the set \mathrm {Sub}(Y), and f^{\wp }, which is an S-sorted mapping from the S-sorted set X^{\wp } = (\mathrm {Sub}(X_{s}))_{s\in S} to the S-sorted set Y^{\wp } = (\mathrm {Sub}(Y_{s}))_{s\in S}. On the other hand, it ... | {
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"raw": "H. Herrlich and G. E. Strecker, Category theory: an introduction. Allyn and Bacon Series in Advanced Mathematics. Allyn and Bacon Inc., Boston, Mass., 1973.",
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many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0d347e0c1fa57d19405aba919422a731f746dc7f | subsection | 13 | 174 | Preliminaries | Then the support of A, denoted by \mathrm {supp}_{S}(A), is the set \lbrace \,s\in S\mid A_{s}\ne \varnothing \,\rbrace .Remark An S-sorted set A is finite if and only if \mathrm {supp}_{S}(A) is finite and, for every s\in \mathrm {supp}_{S}(A), A_{s} is finite.We next recall, after fixing some notation with regard to ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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48ad6af1d59c78cda4a72a52fc9199f4ca82f6c0 | subsection | 14 | 174 | Preliminaries | We will denote by \mathrm {Eqv}(A) the set of all S-sorted equivalences on A (which is an algebraic closure system on A\times A), by \mathbf {Eqv}(A) the algebraic lattice (\mathrm {Eqv}(A),\subseteq ), by \nabla _{A} the greatest element of \mathbf {Eqv}(A), and by \Delta _{A} the least element of \mathbf {Eqv}(A).For... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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beb594039f9b4dd65ff676effe8c531b1b35e223 | subsection | 15 | 174 | Preliminaries | In particular, if \Psi is an S-sorted equivalence relation on A such that \Phi \subseteq \Psi , then we will denote by \mathrm {p}_{\Phi ,\Psi } the unique S-sorted mapping from A/\Phi to A/\Psi such that \mathrm {p}_{\Phi ,\Psi }\circ \mathrm {pr}_{\Phi } = \mathrm {pr}_\Psi .Remark Let \mathbf {ClfdSet}^{S} be the ca... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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271befd6bbae8f15d587533fe3b8da6eb69bce70 | subsection | 16 | 174 | Preliminaries | Moreover, if \Psi is another equivalence relation on A, \Psi is a refinement of \Phi , i.e., \Psi \subseteq \Phi , and X^{\Phi } is a transversal of A/\Phi in A, then, for every s\in S and every a\in A_{s}, there exists a unique x\in X^{\Phi }_{s} such that [a]_{\Psi _{s}}\subseteq [x]_{\Phi _{s}}.Definition 2.14 Let A... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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5fd58bb4e6b3d664033342770ba2acaa7cef117f | subsection | 17 | 174 | Preliminaries | Then\Phi \subseteq \Psi \, \text{if and only if }\, \forall X\subseteq A\;([[X]^{\Psi }]^{\Phi }=[X]^{\Psi }).Moreover, for a sort s\in S, we have that\Phi _{s}\subseteq \Psi _{s}\, \text{if and only if }\, \forall X\subseteq A_{s}\;([[X]^{\Psi _{s}}]^{\Phi _{s}}=[X]^{\Psi _{s}}).We restrict ourselves to proving the fi... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2c4faf10ed7e7dc7881e4fec3ec3f2bb2e7b58b8 | subsection | 18 | 174 | Preliminaries | Moreover, for s\in S and L\subseteq A_{s}, if \Phi _{s}\subseteq \Psi _{s} and L = [L]^{\Psi _{s}}, then L = [L]^{\Phi _{s}}.Remark If, for an S-sorted set A, we denote by (\cdot )\text{-}\mathrm {Sat}(A) the mapping from \mathrm {Eqv}(A) to \mathrm {Sub}(\mathrm {Sub}(A)) which sends \Phi to \Phi \text{-}\mathrm {Sat}... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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3a0d327583a17e349a4c49f334269f4e4965f0b8 | subsection | 19 | 174 | Preliminaries | Then, by definition, there exists an a\in X_{s} such that (a,b)\in (\Phi \cap \Psi )_{s} = \Phi _{s}\cap \Psi _{s}. Hence, (a,b)\in \Phi _{s} and (a,b)\in \Psi _{s}. Therefore b\in [X]^{\Phi }_{s} and b\in [X]^{\Psi }_{s}. Consequently, b\in ([X]^{\Phi }\cap [X]^{\Psi })_{s}. Thus [X]^{\Phi \cap \Psi }\subseteq [X]^{\P... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0dd3e121f93d0d833d3ebe379b76bdba7cbe5288 | subsection | 20 | 174 | Preliminaries | And \Phi \text{-}\mathrm {Sat}(A) = \mathrm {Fix}([\cdot ]^{\Phi }), where \mathrm {Fix}([\cdot ]^{\Phi }) is the set of all fixed point of the operator [\cdot ]^{\Phi }.Proposition 2.21 Let A be an S-sorted set and \Phi \in \mathrm {Eqv}(A). Then the ordered pair
\Phi \text{-}\mathbf {Sat}(A) = (\Phi \text{-}\mathrm {... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2fa665b97a254d00038bbffa47d95f9f1b473d75 | subsection | 21 | 174 | Preliminaries | A word w\in A^{\star } is usually denoted as a sequence (a_{i})_{i\in \vert w \vert }, where, for i\in \vert w \vert , a_{i} is the letter in A satisfying w(i)=a_{i}. We will denote by \eta _{A} the mapping from A to A^{\star } that sends a\in A to (a)\in A^{\star }, i.e., to the mapping (a)\colon 1\usebox {
}A that se... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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3ba8ae322e320ad2c280c0688aaf4e9817a58104 | subsection | 22 | 174 | Preliminaries | We will say that a occurs in w if there are words u, v in A^{\star } such that w = u\operatorname{\curlywedge }(a)\operatorname{\curlywedge }v. Note that a occurs in w if and only if there exists an i\in \vert w \vert such that w(i) = a. We will denote by \vert w \vert _{a} the natural number \mathrm {card}(\lbrace i\i... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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e0145ae99edfd57e313f70c9939ae55d5a9d9d1a | subsection | 23 | 174 | Preliminaries | Therefore, since, for every w\in A^{\star }, the A-indexed family (\vert w \vert _{a})_{a\in A} in \mathbb {N} is such that \vert w \vert _{a} = 0 for all but a finite number of elements a in A, i.e., is such that \mathrm {card}(\lbrace a\in A\mid \vert w \vert _{a}\ne 0\rbrace )<\aleph _{0}, we have that \sum _{a\in A... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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09cd4040bad8f355108563a36f8416b8c8e0557d | subsection | 24 | 174 | Preliminaries | Then we will denote by \left(\!\begin{}c\\ \cdot \end{}\!\right)(w) the mapping from (A^{\star })^{\vert w \vert _{c}} to A^{\star } that assigns to (q^{c}_{\alpha })_{\alpha \in \vert w \vert _{c}}\in (A^{\star })^{\vert w \vert _{c}} the term \left(\!\begin{}c\\(q^{c}_{\alpha })_{\alpha \in \vert w \vert _{c}}\end{}\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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423f5162051367ec0c0379d3c4587a077061093f | subsection | 25 | 174 | Preliminaries | S-sorted signatures and morphisms between S-sorted signatures form a category which we will denote henceforth by \mathbf {Sig}(S).Remark For every set of sorts S, the category \mathbf {Sig}(S) is \mathbf {Set}^{S^{\star }\times S}.Assumption From now on \Sigma stands for an S-sorted signature, fixed once and for all.We... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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b0ee6e861d9995fba57ade2696e822117303c7cb | subsection | 26 | 174 | Preliminaries | A \Sigma -homomorphism from \mathbf {A} to \mathbf {B}, where \mathbf {B} = (B,G), is a triple (\mathbf {A},f,\mathbf {B}), abbreviated to f\colon \mathbf {A}\usebox {
}\mathbf {B}, where f is an S-sorted mapping from A to B such that, for every (w,s)\in S^{\star }\times S, every \sigma \in \Sigma _{w,s}, and every (a_... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.... |
f7a28c9ec3ff8d828ebf4f0cba985b9968504826 | subsection | 27 | 174 | Preliminaries | We will say that \mathbf {A} is finite if A, the underlying S-sorted set of \mathbf {A}, is finite.Remark In \mathbf {Alg}(\Sigma ), as was the case with \mathbf {Set}^{S}, there is another notion of finiteness: A \Sigma -algebra \mathbf {A} is called S-finite or locally finite, abbreviated as \mathrm {lf}, if and only... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/947864.947865",
"end": 765,
"openalex_id": "https://openalex.org/W2062341323",
"raw": "J. Goguen and J. Meseguer, Completeness of many-sorted equational logic. Houston Journal of Mathematics, 11(1985), pp. 307–334.",
"source_r... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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8cfb08d076015aa2eb80d5fad82d9d6a4d9fe2dc | subsection | 28 | 174 | Preliminaries | The \Sigma -algebra \mathbf {A}^{\wp } is called the subset algebra associated to \mathbf {A} (this notion, but for single-sorted algebras, is due to Mezei and Wright, cf. , Definition 2.2). Moreover, there exists a pointwise monomorphic natural transformation \lbrace \cdot \rbrace ^{\Sigma } from \mathrm {Id}_{\mathbf... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0019-9958(67)90353-1",
"end": 190,
"openalex_id": "https://openalex.org/W2050407768",
"raw": "J. Mezei and J. Wright, Algebraic automata and context-free sets. Information and Control, 11 (1967), pp. 3–29.",
"source_ref_id": ... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
9d9ac9d2c2ee750f3cc26e9c4fa898f62ea58e86 | subsection | 29 | 174 | Preliminaries | It is easily seen that, for every \Sigma -algebra \mathbf {A}, \lbrace \cdot \rbrace ^{\Sigma }_{\mathbf {A}} is an injective homomorphism from \mathbf {A} to \mathbf {A}^{\wp } and that \lbrace \cdot \rbrace ^{\Sigma } = (\lbrace \cdot \rbrace ^{\Sigma }_{\mathbf {A}})_{\mathbf {A}\in \mathrm {Alg}(\Sigma )} is, in fa... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.... |
e99490cf1fc4da2d72dfa2a64dc432997afbc482 | subsection | 30 | 174 | Preliminaries | Besides, \mathbf {Sg}_{\mathbf {A}}(X) denotes the algebra determined by \mathrm {Sg}_{\mathbf {A}}(X).Remark Let \mathbf {A} be a \Sigma -algebra. Then the algebraic closure operator \mathrm {Sg}_{\mathbf {A}} is uniform, i.e., for every X, Y\subseteq A, if \mathrm {supp}_{S}(X) = \mathrm {supp}_{S}(Y), then we have t... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0202476922422647... |
d43a987dfbe2fb83db4dc86248129762e65b3f06 | subsection | 31 | 174 | Preliminaries | The algebra of \Sigma -rows in X, denoted by \mathbf {W}_{\Sigma }(X), is defined as follows:The underlying S-sorted set of \mathbf {W}_{\Sigma }(X), written as \mathrm {W}_{\Sigma }(X), is precisely the S-sorted set ((\coprod \Sigma \amalg \coprod X)^{\star })_{s\in S}, i.e., the mapping from S to {\mathcal {U}} const... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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43f0a136bfb2b3de7ae527a22a9d5f6201393b63 | subsection | 32 | 174 | Preliminaries | For every (w,s)\in S^{\star }\times S, and every \sigma \in \Sigma _{w,s}, the structural operation F_{\sigma } associated to \sigma is the mapping from \mathrm {W}_{\Sigma }(X)_{w} to {\mathrm {W}_{\Sigma }(X)}_{s} which sends (P_{i})_{i\in \vert w \vert } \in \mathrm {W}_{\Sigma }(X)_{w} to (\sigma )\curlywedge \text... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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030d70cdccbdd1ce1c9a3e2aa27a8d4cd925a4b5 | subsection | 33 | 174 | Preliminaries | We will denote by \mathrm {T}_{\Sigma }(X) the underlying S-sorted of \mathbf {T}_{\Sigma }(X) and, for s\in S, we will call the elements of \mathrm {T}_{\Sigma }(X)_{s} terms of type s with variables in X or (X,s)-terms.Remark Since (\lbrace (x)\mid x\in X_{s}\rbrace )_{s\in S} is a generating subset of \mathbf {T}_{\... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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1f8d62f125b0c58dbb65c82ece071998aeba1326 | subsection | 34 | 174 | Preliminaries | Moreover, the three possibilities are mutually exclusive.From now on, for simplicity of notation, we will write x, \sigma , and \sigma (P_{0},\ldots ,P_{\vert w \vert -1}) or \sigma ((P_{i})_{i\in \vert w \vert }) instead of (x), (\sigma ), and (\sigma )\curlywedge \text{\Large $\curlywedge $}(P_{i})_{i\in \vert w \ver... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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d12bb75e5c5c819f55cc2bcf855badc5ab671a63 | subsection | 35 | 174 | Preliminaries | Moreover, for the case of free algebras, such a preorder is, in fact, an order and this allows us to define the notion of subterm of a given term.Definition 2.38 Let \mathbf {A}=(A,F) be a \Sigma -algebra. Then <_{\mathbf {A}} denotes the binary relation on \coprod A consisting of the ordered pairs ((a,s),(b,t))\in (\c... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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4e67a5f15b3a89bc2b286b15b1a86a5f76653f23 | subsection | 36 | 174 | Preliminaries | Moreover, \mathrm {Subt}(P) can also be characterized as the smallest subset \mathcal {L} of \mathrm {T}_{\Sigma }(X) which satisfies the following conditions: (1) P\in \mathcal {L}_{t} and (2) for every (w,s)\in S^{\star }\times S, every \xi \in \Sigma _{w,s}, and every (Q_{i})_{i\in \vert w \vert }\in \mathrm {T}_{\S... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.00... |
d476075e1802d11e9813e6716d0b74fd12680645 | subsection | 37 | 174 | Preliminaries | Let
\delta ^{X}=(\delta ^{X}_{s})_{s\in S} be the S-sorted mapping defined, for every s\in S, as\delta ^{X}_{s}
\left\lbrace
\begin{array}{@{\:}c@{\:}c@{\:}l}
X_{s} &\usebox {
}& \mathrm {Sub}_{\mathrm {f}}(X) \\
x &\longmapsto & \delta ^{s,x}
\end{array}
\right.Then we will denote by \mathrm {Var}^{X} the unique ho... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
7c2d5d70bb9d259ff67cd30dec35b301ddf4df1c | subsection | 38 | 174 | Preliminaries | We will say that \Phi is an
S-sorted congruence on (or, to abbreviate, a congruence on) \mathbf {A} if, for every (w,s)\in (S^{\star }-\lbrace \lambda \rbrace )\times S, every \sigma \in \Sigma _{w,s},
and every (a_{i})_{i\in \vert w \vert },(b_{i})_{i\in \vert w \vert }\in A_{w}, if, for every i\in \vert w\vert , (a_{... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f7237d7da811bdef962d09e843f74f6d557e2888 | subsection | 39 | 174 | Preliminaries | The ordered pair (\mathbf {A}/\Phi ,\mathrm {pr}_{\Phi }) has the following universal property: \mathrm {Ker}(\mathrm {pr}_{\Phi }) is \Phi and, for every \Sigma -algebra \mathbf {B} and every homomorphism f from \mathbf {A} to \mathbf {B}, if \mathrm {Ker}(f)\supseteq \Phi , then there exists a unique homomorphism h f... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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ace7f89fb74287164e7757631c799cfd6d5a9cfb | subsection | 40 | 174 | Preliminaries | Then, for every classified \Sigma -algebra (\mathbf {A},\Phi ), there exists a universal mapping from (\mathbf {A},\Phi ) to G, which is precisely the ordered pair (\mathbf {A}/\Phi ,\mathrm {pr}_{\Phi }) with \mathrm {pr}_{\Phi }\colon (\mathbf {A},\Phi )\usebox {
}(\mathbf {A}/\Phi ,\Delta _{\mathbf {A}/\Phi }).We ne... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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c6e83a22d4ef31392c339bd7711bee4c4a92349c | subsection | 41 | 174 | Preliminaries | Then we will denote by \mathrm {Tl}_{t}(\mathbf {A}) the subset
(\mathrm {Tl}_{t}(\mathbf {A})_{s})_{s\in S} of (\mathrm {Hom}(A_{t},A_{s}))_{s\in S} defined, for every s\in S, as follows: For every mapping T\in \mathrm {Hom}(A_{t},A_{s}), T\in \mathrm {Tl}_{t}(\mathbf {A})_{s} if and only if there is an n\in \mathbb {... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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d304e2eca22f07ddb228e1cf631e6edcdd54cea9 | subsection | 42 | 174 | Preliminaries | Therefore, for every t\in S, \mathrm {End}_{\mathbf {Tl}(\mathbf {A})}(t) is equipped with a structure of monoid.Given a \Sigma -algebra \mathbf {A} and a translation T\in \mathrm {Tl}_{t}(\mathbf {A})_{s} we next define, associated to the mappings T[\cdot ]\colon \mathrm {Sub}(A_{t})\usebox {
}\mathrm {Sub}(A_{s}) and... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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dfc02a808844f15a7ca4e65a20cc53ca67607adb | subsection | 43 | 174 | Preliminaries | Had the operators T[\cdot ] and T^{-1}[\cdot ] been defined (for u\in S-\lbrace s,t\rbrace ) differently, then the suitably modified counterparts of the just stated connections would not be met.As announced above, we next provide, by using the notions of elementary translation and of translation, two characterizations ... | {
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{
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"raw": "G. Matthiessen, Theorie der Heterogenen Algebren, Mathematik-Arbeitspapiere, Nr. 3, Universität Bremen Teil A, Mathematische Forchungspapiere, 1976.",
"source_ref_id": "c48080e969b1b8336200a2d... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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5369dc02f28432c6fa5dd52d9a1a956e397a9d30 | subsection | 44 | 174 | Preliminaries | Since, for every j\in i, (a_{j},a_{j})\in \Phi _{w_{j}}, for every k\in \vert w \vert -(i+1), (a_{k},a_{k})\in \Phi _{w_{k}}, and, in addition, (x,y)\in \Phi _{t} = \Phi _{w_{i}}, then (T(x),T(y))\in \Phi _{s}.Reciprocally, let us suppose that, for every t, s\in S, every x, y\in A_{t}, and every T\in \mathrm {Etl}_{t}(... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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84471335ff00c263a7c05e86037d7a714026401a | subsection | 45 | 174 | Preliminaries | Therefore (F_{\sigma }(a), F_{\sigma }(b))\in \Phi _{u}.We shall now proceed to verify that (2) and (3) are equivalent.Since every elementary translations on \mathbf {A} is a translation on \mathbf {A}, it is obvious that if \Phi is closed under the translations on \mathbf {A}, then \Phi is closed under the elementary ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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926fed0cc05d33c171588e437ba428ea994f02df | subsection | 46 | 174 | Preliminaries | For every S-sorted set A, the mapping from \mathrm {Sub}(A) to \mathrm {Hom}(A,(2)_{s\in S}) that assigns to L\in \mathrm {Sub}(A) precisely \mathrm {ch}_{L}, the character of L, i.e., the S-sorted mapping from A to (2)_{s\in S} whose s-th coordinate, for s\in S, is \mathrm {ch}_{L_{s}}, the characteristic mapping of L... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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4f92e3df422a2d7ebc12327e4b7c9d9aeed7fdce | subsection | 47 | 174 | Preliminaries | For every congruence \Phi on \mathbf {A}, if \Phi \subseteq \mathrm {Ker}(\mathrm {ch}_{L}), then \Phi \subseteq \Omega ^{\mathbf {A}}(L).In other words, \Omega ^{\mathbf {A}}(L) is the greatest congruence on \mathbf {A} which saturates L.To prove (1) it suffices to take into account Proposition REF . To prove (2), giv... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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b4b72f3c22d10f0adc3e5a52356e0d9544e74b6a | subsection | 48 | 174 | Preliminaries | We will call \Omega ^{\mathbf {A}} the congruence cogenerating operator for \mathbf {A} (with regard to subsets of A).With regard to the congruence cogenerated by an equivalence it is worthwhile to quote what Büchi, in , on p. 113, wrote: “The notion of induced [\equiv cogenerated, we add] congruence therefore is clear... | {
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"raw": "G. Lallement, Semigroups and combinatorial applications. Pure and Applied Mathematics. A Wiley-Interscience Publication. John Wiley \\!\\!\\!\\! Sons, New York-Chich... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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dbe265a8a66d34a30e308a8e4900f9531f442892 | subsection | 49 | 174 | Preliminaries | We will call \widetilde{\Omega }^{\mathbf {A}}(\Phi ) the congruence on \mathbf {A} cogenerated by \Phi .Since: (1) the join in \mathbf {Eqv}(A) of a family (\Psi ^{i})_{i\in I} of congruences on \mathbf {A} is the S-sorted equivalence \Psi on A whose s-th coordinate, \Psi _{s}, for s\in S, is:\Psi _{s} =
\biggl \lbrac... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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53ec8ab4c3e0558ef473f63984d50143d9412104 | subsection | 50 | 174 | Preliminaries | We will call \widetilde{\Omega }^{\mathbf {A}} the congruence cogenerating operator for \mathbf {A} (with regard to S-sorted equivalences on A).Remark For every subset L of the underlying S-sorted set of a \Sigma -algebra \mathbf {A}, \Omega ^{\mathbf {A}}(L) = \widetilde{\Omega }^{\mathbf {A}}(\mathrm {Ker}(\mathrm {c... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2512e231e70b96daf5745c9e4d8cb43c1cb12b8b | subsection | 51 | 174 | Preliminaries | Moreover, \widetilde{\Omega }^{\mathbf {A}}(\Delta _{\mathbf {A}}) = \Delta _{\mathbf {A}}, \widetilde{\Omega }^{\mathbf {A}}(\nabla _{\mathbf {A}}) = \nabla _{\mathbf {A}} and, for every nonempty set I in {\mathcal {U}} and every (\Phi ^{i})_{i\in I}\in \mathrm {Eqv}(A)^{I}, \widetilde{\Omega }^{\mathbf {A}}(\bigcap _... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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cf4cf07b74f45542b36c5920a429083bf2f51a7a | subsection | 52 | 174 | Preliminaries | Then we have, on the one hand, the functor \mathrm {Eqv} from \mathbf {Alg}(\Sigma )^{\mathrm {op}}_{\mathrm {epi}}, the dual of \mathbf {Alg}(\Sigma )_{\mathrm {epi}}, to \mathbf {Set} which assigns to a \Sigma -algebra \mathbf {A} the set \mathrm {Eqv}(A), and to an epimorphism f\colon \mathbf {A}\usebox {
}\mathbf {... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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23693db4d473b120adf71852a76fd1d4d618b1f5 | subsection | 53 | 174 | Preliminaries | Then L\in \Phi \text{-}\mathrm {Sat}(A), i.e., L = [L]^{\Phi }, if and only if \Phi \subseteq \Omega ^{\mathbf {A}}(L). Moreover, for s\in S and L\subseteq A_{s}, we have that L = [L]^{\Phi _{s}} if and only if \Phi _{s}\subseteq \Omega ^{\mathbf {A}}(\delta ^{s,L})_{s}.Let us suppose that L = [L]^{\Phi }. Then, since ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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7e56939be3aae2a4fc3616df323ba62a04cdf308 | subsection | 54 | 174 | Preliminaries | But \bigcap _{i\in I}\Omega ^{\mathbf {A}}(L^{i}) \subseteq \bigcap _{i\in I}\mathrm {Ker}(\mathrm {ch}_{L^{i}}) and, in addition, we have that \bigcap _{i\in I}\mathrm {Ker}(\mathrm {ch}_{L^{i}}) \subseteq \mathrm {Ker}(\mathrm {ch}_{\bigcap _{i\in I}L^{i}}). Therefore \bigcap _{i\in I}\Omega ^{\mathbf {A}}(L^{i}) \su... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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3114beb345bfb4580603eace007f1f7898e990fd | subsection | 55 | 174 | Preliminaries | Moreover, if f is an epimorphism, then
(f\times f)^{-1}[\Omega ^{\mathbf {B}}(M)] = \Omega ^{\mathbf {A}}(f^{-1}[M]).Remark Let \mathrm {P}^{-} be the functor from \mathbf {Alg}(\Sigma )^{\mathrm {op}}_{\mathrm {epi}} to \mathbf {Set} which assigns to a \Sigma -algebra \mathbf {A} the set \mathrm {Sub}(A), and to an ep... | {
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"raw": "F. Gécseg and M. Steinby, Tree automata. Akadémiai Kiadó, Budapest, 1984.",
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"start": 1169
... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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021ecd14558b2ccbf9f1cd8cddbe190345b41a32 | subsection | 56 | 174 | Preliminaries | We will say that \Phi is of finite index, abbreviated as \mathrm {fi}, if A/{\Phi }\in \mathrm {Sub}_{\mathrm {f}}(A^{\wp }), i.e., if \mathrm {card}(\mathrm {supp}_{S}(A/\Phi )) is finite and, for every s\in \mathrm {supp}_{S}(A/\Phi ), A_{s}/\Phi _{s} is finite. We will denote by \mathrm {Cgr}_{\mathrm {fi}}(\mathbf ... | {
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"raw": "B. Courcelle, On recognizable sets and tree automata. In H. Aït-Kaci and M. Nivat, editors. Algebraic Techniques: Resolution of Equ... | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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fa02ae172b4ac2459859556cf69654f6af819745 | subsection | 57 | 174 | Preliminaries | Therefore, the following conditions are equivalent: (1) for every \Sigma -algebra \mathbf {A}, \mathrm {supp}_{S}(\mathbf {A}) is finite, (2) for every \Sigma -algebra \mathbf {A}, \mathrm {Cgr}_{\mathrm {fi}}(\mathbf {A})\ne \varnothing , and (3) S is finite.Since the first assertion is straightforward, we restrict ou... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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cc20745681a362f4ab57ed5cce0e90b677993f03 | subsection | 58 | 174 | Preliminaries | Thenfor every n\in \mathbb {N}-\lbrace 0\rbrace and every (\Phi ^{i})_{i\in n}\in \mathrm {Cgr}_{\mathrm {fi}}(\mathbf {A})^{n}, \bigcap _{i\in n}\Phi ^{i}\in \mathrm {Cgr}_{\mathrm {fi}}(\mathbf {A});
\mathrm {Cgr}_{\mathrm {fi}}(\mathbf {A}) is an upward closed set of the lattice \mathbf {Cgr}\mathbf {(A)}, i.e., fo... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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9963baf5e6a973daf186d9dc8452b5b709f92843 | subsection | 59 | 174 | Preliminaries | Let us note that \mathrm {p}^{(\Phi ^{i})_{i\in n}} is \left<\mathrm {p}_{\bigcap _{i\in n}\Phi ^{i},\Phi ^{i}}\right>_{i\in n} (the unique homomorphism from \mathbf {A}/\bigcap _{i\in n}\Phi ^{i} to \prod _{i\in n}\mathbf {A}/\Phi ^{i} such that, for every i\in n, \mathrm {pr}_{\Phi ^{i}}\circ \left<\mathrm {p}_{\bigc... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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e35998da800948b75c759556909cb16e55a1d135 | subsection | 60 | 174 | Preliminaries | Then \mathrm {Cgr}_{\mathrm {lfi}}(\mathbf {A}) is a filter of the lattice \mathbf {Cgr}\mathbf {(A)}.As for congruences of finite index on a many-sorted algebra, for many-sorted languages we have, on the one hand, those whose definition is based on the categorial notion of finiteness and which will be called recogniza... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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cd97133c481dd634c3e5dfe7ff3970486651bfe8 | subsection | 61 | 174 | Preliminaries | We will call the elements of \mathrm {Rec}(\mathbf {A}) and \mathrm {Rec}_{\mathrm {lf}}(\mathbf {A}) recognizable and \mathrm {lf}-recognizable, respectively.For s\in S we will denote by \mathrm {Rec}_{s}(\mathbf {A}) and \mathrm {Rec}_{\mathrm {lf},s}(\mathbf {A}) the sets \mathrm {Rec}_{\lbrace s\rbrace }(\mathbf {A... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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81266865357368a2c891e3bcc92dd65391140177 | subsection | 62 | 174 | Preliminaries | In particular, we investigate such a relationship for the case of the congruence cogenerated by a many-sorted language.Proposition 2.64 Let \mathbf {A} be a \Sigma -algebra, T\subseteq S, and L\subseteq A\!\!\upharpoonright _{T}. Then the following assertions are equivalent:L is T-recognizable.
There exists a congruen... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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df4be23039196953b9c6ce8afc38d2eae2481b2a | subsection | 63 | 174 | Preliminaries | Then, since L = f^{-1}[M], we have that L_{s} = f^{-1}_{s}[M_{s}], thus f_{s}(x)\in M_{s}, but f_{s}(a) = f_{s}(x), therefore f_{s}(a)\in M_{s}, consequently a\in L_{s}. From this it follows that [L]^{\mathrm {Ker}(f)}\subseteq L.We next prove that (2)\Rightarrow (3).Let us suppose that there exists a congruence \Phi o... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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dbd045b2978a9ff816c411860afc36693846a098 | subsection | 64 | 174 | Preliminaries | But, by Proposition REF , L is \Omega ^{\mathbf {A}}(L)-saturated and, since a\in L_{s}, we have that b\in L_{s}. Therefore (\mathrm {pr}_{\Omega ^{\mathbf {A}}(L)})^{-1}[\mathcal {M}]\subseteq L.Proposition 2.66 Let \mathbf {A} be a \Sigma -algebra, s\in S, and L\subseteq A_{s}. Then the following assertions are equiv... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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... |
2550e8f5c0dd4d9e84b6c0618999807823f37fe9 | subsection | 65 | 174 | Preliminaries | Thus there exists an embedding from \mathrm {Rec}(\mathbf {A}) into \prod _{s\in S}\mathrm {Rec}_{s}(\mathbf {A}) and \mathrm {Rec}(\mathbf {A}) is a subdirect product of (\mathrm {Rec}_{s}(\mathbf {A}))_{s\in S}.If L\in \mathrm {Rec}(\mathbf {A}), then, from the definitions, it follows that, for every s\in S, L_{s}\in... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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2b979821d247fce43e2aed80c8d9e7487a59c803 | subsection | 66 | 174 | Preliminaries | Therefore, by part (2) of Proposition REF and since, by hypothesis, S is finite, \bigcup _{s\in S}\delta ^{s,L_{s}} = L\in \mathrm {Rec}(\mathbf {A}).The mapping from \mathrm {Rec}(\mathbf {A}) to \prod _{s\in S}\mathrm {Rec}_{s}(\mathbf {A}) that sends L in \mathrm {Rec}(\mathbf {A}) to L in \prod _{s\in S}\mathrm {Re... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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f2c3abfa337ac59d31b92f8a4e43f721949b98ee | subsection | 67 | 174 | Preliminaries | There exists a congruence \Phi on \mathbf {A} of S-finite index such that L = [L]^{\Phi \!\upharpoonright _{T}}, where \Phi \!\!\upharpoonright _{T} = (\Phi _{t})_{t\in T} and, for every t\in T, [L]^{\Phi \!\upharpoonright _{T}}_{t} = [L_{t}]^{\Phi _{t}}.
\Omega ^{\mathbf {A}}([L,\varnothing ^{S-T}]) is of S-finite in... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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4e1d1b8df9ab0f489947ebd6eaa662d2e229fdc1 | subsection | 68 | 174 | Recognizable subsets of a free many-sorted algebra | This section is devoted to provide congruence based proofs of the recognizability theorems for free many-sorted algebras.
Actually, in order to deal with the different cases of recognizability, classified according to the type of operator under consideration, we have divided this section into several subsections. In Su... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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8a975ed46a092fa52c067f27d373bd22cc1b70ec | subsection | 69 | 174 | Basic terms | In this subsection we prove some basic recognizability results relative to a free algebra on an S-sorted set. Concretely, we prove that the final sets consisting, respectively, of a variable, a constant, and the action of an operator symbol on a family of variables, are recognizable.Assumption In this subsection we wil... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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3092ec23936a6359fc2621e5100734d2f73bf1f5 | subsection | 70 | 174 | Basic terms | Then the language \lbrace \sigma \rbrace \subseteq \mathrm {T}_{\Sigma }(X)_{s} is recognizable.Let \mathbf {2}=(2,H^{\mathbf {2}}) be the finite \Sigma -algebra defined as follows: H^{\mathbf {2}}_{\sigma } is the mapping from 2_{\lambda } to 2 that sends the unique member of 2_{\lambda } to 1, and, for every (w,t)\in... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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c204307d71a1bdeefcd67b6d60de251815e8ca84 | subsection | 71 | 174 | Basic terms | If x=(x_{i})_{i\in \vert w \vert }\in X_{w}, i.e., if x is a mapping from \vert w \vert to \bigcup _{i\in \vert w \vert }X_{w_{i}} such that, for every i\in \vert w \vert , x_{i}\in X_{w_{i}}, then we will denote by \overline{x} the S-sorted mapping from \downarrow \!\!w to X which is associated, in virtue of the natur... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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1f957754978e0861f58a3d743410f427f90db88b | subsection | 72 | 174 | Basic terms | Then let \mathbf {K}=(K,I^{\mathbf {K}}) be the \Sigma -algebra defined as follows: For (u,t)\in S^{\star }\times S and \tau \in \Sigma _{u,t}, if (u,t)\ne (w,s) or \tau \ne \sigma , then I^{\mathbf {K}}\colon K_{u}\usebox {
}K_{t} is the constant mapping with value k_{t}, and for \tau =\sigma , I^{\mathbf {K}} is the ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0860e860702af7ae9a0a068186a682784e82c20e | subsection | 73 | 174 | Substitutions | In this subsection we introduce several substitution operators associated to a free algebra with the aim of proving that, if the languages under consideration are recognizable, then the language that results from the substitution operator applied to these languages is also recognizable.Definition 3.4
Let X be an S-sor... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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0.0... |
77982aac342934fd9f02db181ac22903a70fb615 | subsection | 74 | 174 | Substitutions | Then we will denote by \left(\!\begin{}z\\ \cdot \end{}\!\right)(P) the mapping from \mathrm {T}_{\Sigma }(X)_{u}^{\vert P \vert _{z}} to \mathrm {T}_{\Sigma }(X)_{s} that assigns to (Q^{z}_{\alpha })_{\alpha \in \vert P \vert _{z}} in \mathrm {T}_{\Sigma }(X)_{u}^{\vert P \vert _{z}} the term \left(\!\begin{}z\\(Q^{z}... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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9f351aa61ce452f20c2f96260a27021587aa53db | subsection | 75 | 174 | Substitutions | Then we will denote by \left(\!\begin{}x\\ \cdot \end{}\!\right)_{x\in X_{s}}(P) the mapping from \prod _{x\in X_{s}}\mathrm {T}_{\Sigma }(X)_{s}^{\vert P \vert _{x}} to \mathrm {T}_{\Sigma }(X)_{s} that assigns to \left((Q^{x}_{\alpha })_{\alpha \in \vert P \vert _{x}}\right)_{x\in X_{s}} in \prod _{x\in X_{s}}\mathrm... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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66dd4f8cd1a72211b838de14a0373c8ce8a093d0 | subsection | 76 | 174 | Substitutions | We will call \left(\!\begin{}x\\(Q^{x}_{\alpha })_{\alpha \in \vert P \vert _{x}}\end{}\!\right)_{x\in X_{s}}(P) the substitution of (Q^{x}_{\alpha })_{\alpha \in \vert P \vert _{x}} for x in P for every x in X_{s}, and \left(\!\begin{}x\\ \cdot \end{}\!\right)_{x\in X_{s}}(P) the substitution operator for P.Since (1) ... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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-0.... |
34ea0812b0bdc1fd0305742f8ad4c4a9bf0ad2ba | subsection | 77 | 174 | Substitutions | Similarly \left(\!\begin{}z\\ \cdot \end{}\!\right)(P) is, essentially, the restriction of \left(\!\begin{}(x,t)\\ \cdot \end{}\!\right)_{(x,t)\in \coprod X}(P) to \mathrm {T}_{\Sigma }(X)_{u}^{\vert P \vert _{z}}.We next prove that, for every s\in S and every P\in \mathrm {T}_{\Sigma }(X), \left(\!\begin{}(x,t)\\ \cdo... | {
"cite_spans": []
} | 10.1093/logcom/exz032 | 1808.08217 | Congruence based proofs of the recognizability theorems for free
many-sorted algebras | [
"Juan Climent Vidal",
"Enric Cosme Llópez"
] | [
"cs.FL"
] | 2,018 | en | Computer Science | [
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-0... |
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