Outline. Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC
|
|
- Allyson Tyler
- 5 years ago
- Views:
Transcription
1 Outline Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC Part 5: Completeness of Lifted Inference Part 6: Query Compilation Part 7: Symmetric Lifted Inference Complexity Part 8: Open-World Probabilistic Databases Part 9: Discussion & Conclusions
2 Question 2. Are lifted rules stronger than grounded? Alternative to lifting: 1. Ground the FO sentence 2. Do WMC on the propositional formula There is no reason why grounded inference should be weaker than lifted inference However, existing grounded algorithms are strictly weaker than lifted inference
3 Algorithms for Model Counting [Gomes 08] Based on full search DPLL: Shannon expansion. #F = #F[X=0] + #F[X=1] Caching. Store #F, look it up later Components. If Vars(F1) Vars(F2) = : #(F1 F2) = #F1 * #F2
4 Knowledge Compilation Definition (informal): represent the Boolean formula F in a circuit where WMC(F) is in PTIME in the size of the representation Why we care: The trace of any inference algorithm is a knowledge compilation Lower bounds on size(kc) give lower bounds on the algorithm s runtime
5 Knowledge Compilation Targets 0 0 Y X 1 0 U Z 0 1 FBDD: Decision-, sink-nodes OBDD: fixed variable order 1 0 X Y U 0 Z Decision-DNNF add: -nodes Children of have disjoint sets of variables 1 0 1
6 DPLL and Knowledge Compilation Fact: Trace of full-search DPLL KC: Basic DPLL decision trees DPLL + caching OBDD (fixed variable order) FBDD DPLL + caching + components decision-dnnf [Huang&Darwiche 2005]
7 Hard Queries H 0 = x y (R(x) S(x,y) T(y)) = non-hierarchical H k = hierarchical, has inversion, for k 1 Grounded Boolean formulas: F n (H 0 ) = i [n], j [n] (R i S ij T j ) Th. [Beame 14] Any FBDD for F n (H k ) has size 2 n-1 /n. Same holds for any non-hierarchical query. What about Decision-DNNFs?
8 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1
9 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1 Problem: 0 X 1
10 Decision-DNNF to FBDD Optimal [Razgon] Theorem If F has a Decision-DNNF with N nodes, then F has an FBDD with at most N 1+log(N) nodes. Proof idea No-op No-op 0 1 Problem: 0 X 1 Solution: copy the smaller child 0 X 1
11 Hard Queries Corollary Any Decision-DNNF for F n (H k ) has size 2 Ω( n) Same holds for any non-hierarchical query. Proof. N-node Decision-DNNF to N 1+log(N) nodes FBDD. N 1+log(N) > 2 n-1 /n, log(n) + log 2 (N) > n 1 log(n) log 2 (N) = Ω(n) log(n) = Ω( n)
12 Lifted v.s. Grounded Inference Lifted P(Q) Grounded P(F n (Q)) Non-hierarchical Q (e.g. H 0 ) #P-hard 2 Ω( n) What about hierarchical queries?
13 Inversion-Free Queries Definition An inversion in Q is a sequence of co-occurring vars: (x 0,y 0 ), (x 1,y 1 ),, (x k,y k ), such that: at(x 0 ) at(y 0 ), at(x 1 )=at(y 1 ),, at(x k-1 )=at(y k-1 ), at(x k ) at(y k ) For all i=1,..,k-1 there exists two atoms in Q of the form: S i (,x i-1,,y i-1, ) and S i (,x i,, y i, ) Inversion-free implies hierarchical, but converse fails Q=[R(x 0 ) S(x 0,y 0 )] [S(x 1,y 1 ) T(x 1 )] Inversion-free Inversion H 1 =[R(x 0 ) S(x 0,y 0 )] [S(x 1,y 1 ) T(y 1 )]
14 Easy Queries [Jha&S.11], [Beame 15] Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)
15 Easy Queries [Jha&S.11], [Beame 15] [Beame&Liew 15] Extended to SDD. Thus, over FO un, OBDD SDD Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)
16 Easy Queries [Bova 16] SDD more succint than OBDD (HWB) [Jha&S.11], [Beame 15] [Beame&Liew 15] Extended to SDD. Thus, over FO un, OBDD SDD Theorem Let Q in FO un 1. If Q has inversion then OBDD for F n (Q) has size 2 n-1 /n 2. Else, F n (Q) has OBDD of width 2 #atoms(q) (size O(n)) Proof (part 2 only next slide)
17 Q = [R(x) S(x,y)] [T(x ) S(x,y )]
18 Q = [R(x) S(x,y)] [T(x ) S(x,y )] n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2
19 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2
20 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 x = 1 x = 2
21 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 1 R 1 0 x = 1 x = 2 1 S 11 0 x = 1 S R S 22 S x =
22 C 1 = R(x) S(x,y) C 2 = T(x ) S(x,y ) = Q = [R(x) S(x,y)] [T(x ) S(x,y )] F 2 (C 1 ) = (R 1 S 11 ) (R 1 S 12 ) ( R 2 S 21 ) (R 2 S 22 ) n = 2 Π = R 1 T 1 S 11 S 12 R 2 T 2 S 21 S 22 1 R T 1 1 x = 1 x = 2 1 S 11 0 x = 1 0 S 11 1 R 2 S T 2 S Same variable order Π in both OBDDs! 22 1 S 22 S x = S 22 S OBDD for Q = C 1 C 2 has width = width1 width2
23 Lifted v.s. Grounded Inference Nonhierarchical Q (e.g. H 0 ) Inversion -free Q Lifted P(Q) #P-hard PTIME Grounded P(F n (Q)) 2 Ω( n) PTIME
24 Easy/Hard Queries Main result: a class of queries Q such that: Lifted inference: P((Q)) in PTIME Grounded inference: P(F n (Q)) exponential time Significance: limitation of DPLL-based algorithms for model counting
25 H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (x,y) S 2 (x,y) H k2 = x y S 2 (x,y) S 3 (x,y) H kk = x y S k (x,y) T(y) Clauses of H k
26 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (x,y) S 2 (x,y) H k2 = x y S 2 (x,y) S 3 (x,y) H kk = x y S k (x,y) T(y) f(z 0, Z 1,, Z k ) = a Boolean function
27 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (x,y) S 2 (x,y) H k2 = x y S 2 (x,y) S 3 (x,y) H kk = x y S k (x,y) T(y) f(z 0, Z 1,, Z k ) = a Boolean function Q = f(h k0, H k1,, H kk )
28 Clauses of H k H k0 = x y R(x) S 1 (x,y) H k1 = x y S 1 (x,y) S 2 (x,y) H k2 = x y S 2 (x,y) S 3 (x,y) H kk = x y S k (x,y) T(y) f(z 0, Z 1,, Z k ) = a Boolean function Q = f(h k0, H k1,, H kk ) Examples: f = Z 0 Z 1 Z k then f(h k0, H k1,, H kk ) = H k f = Z 0 Z 2 Z 0 Z 3 Z 1 Z 3 then f(h 30, H 31, H 31, H 33 ) = Q W
29 Easy/Hard Queries [Beame 14] Theorem For any Boolean function f(z 0, Z 1,, Z k ), denoting Q = f(h k0, H k1,, H kk ): Any FBDD for F n (Q) has size 2 Ω(n) Any Decision-DNNF has size 2 Ω( n). Consequence: Lifted inference computes compute P(Q W ) in PTIME Any DPLL-based algorithm takes time 2 Ω( n) Many other queries are like Q W
30 Lifted v.s. Grounded Inference Nonhierarchical Q (e.g. H 0 ) Inversion -free Q Q = f(h k0,,h kk ) Lifted P(Q) #P-hard PTIME PTIME or #P-hard Grounded P(F n (Q)) 2 Ω( n) PTIME 2 Ω( n)
31 Two Questions Question 1: Are the lifted rules complete? We know that they get stuck on some queries Should we add more rules? Complete for unate FO and for unate FO Question 2: Are lifted rules stronger than grounded? Lifted rules can also be grounded Any advantage over grounded inference? Strictly stronger than DPLL-based algorithms
32 #P-hard Möbius Über Alles FO un, FO un
33 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9
34 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Read Once Q U
35 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Poly-size OBDD,SDD = = inversion-free Q J Read Once Q U
36 #P-hard Möbius Über Alles FO un, FO un PTIME Q W Q 9 Poly-size FBDD, dec-dnnf Poly-size OBDD,SDD = = inversion-free Q J Q V Read Once Q U Open
37 Möbius Über Alles #P-hard Non-hierarchical hierarchical H 0 H 1 H 2 FO un, FO un PTIME Q W Q 9 H 3 Poly-size FBDD, dec-dnnf Poly-size OBDD,SDD = = inversion-free Q J Q V Read Once Q U Open
Brief Tutorial on Probabilistic Databases
Brief Tutorial on Probabilistic Databases Dan Suciu University of Washington Simons 206 About This Talk Probabilistic databases Tuple-independent Query evaluation Statistical relational models Representation,
More informationModel Counting of Query Expressions: Limitations of Propositional Methods
Model Counting of Query Expressions: Limitations of Propositional Methods Paul Beame University of Washington beame@cs.washington.edu Jerry Li MIT jerryzli@csail.mit.edu Dan Suciu University of Washington
More informationLower Bounds for Exact Model Counting and Applications in Probabilistic Databases
Lower Bounds for Exact Model Counting and Applications in Probabilistic Databases Paul Beame Jerry Li Sudeepa Roy Dan Suciu Computer Science and Engineering University of Washington Seattle, WA 98195 {beame,
More informationOutline. Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC
Outline Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC Part 5: Completeness of Lifted Inference Part 6: Query Compilation Part 7:
More informationTowards Deterministic Decomposable Circuits for Safe Queries
Towards Deterministic Decomposable Circuits for Safe Queries Mikaël Monet 1 and Dan Olteanu 2 1 LTCI, Télécom ParisTech, Université Paris-Saclay, France, Inria Paris; Paris, France mikael.monet@telecom-paristech.fr
More informationBasing Decisions on Sentences in Decision Diagrams
Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence Basing Decisions on Sentences in Decision Diagrams Yexiang Xue Department of Computer Science Cornell University yexiang@cs.cornell.edu
More informationKnowledge Compilation Meets Database Theory : Compiling Queries to Decision Diagrams
Knowledge Compilation Meets Database Theory : Compiling Queries to Decision Diagrams Abhay Jha Computer Science and Engineering University of Washington, WA, USA abhaykj@cs.washington.edu Dan Suciu Computer
More informationOutline. Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC
Outline Part 1: Motivation Part 2: Probabilistic Databases Part 3: Weighted Model Counting Part 4: Lifted Inference for WFOMC Part 5: Completeness of Lifted Inference Part 6: Quer Compilation Part 7: Smmetric
More informationCV-width: A New Complexity Parameter for CNFs
CV-width: A New Complexity Parameter for CNFs Umut Oztok and Adnan Darwiche 1 Abstract. We present new complexity results on the compilation of CNFs into DNNFs and OBDDs. In particular, we introduce a
More informationTractable Lineages on Treelike Instances: Limits and Extensions
Tractable Lineages on Treelike Instances: Limits and Extensions Antoine Amarilli 1, Pierre Bourhis 2, Pierre enellart 1,3 June 29th, 2016 1 Télécom ParisTech 2 CNR CRItAL 3 National University of ingapore
More informationReduced Ordered Binary Decision Diagram with Implied Literals: A New knowledge Compilation Approach
Reduced Ordered Binary Decision Diagram with Implied Literals: A New knowledge Compilation Approach Yong Lai, Dayou Liu, Shengsheng Wang College of Computer Science and Technology Jilin University, Changchun
More informationOn the Relative Efficiency of DPLL and OBDDs with Axiom and Join
On the Relative Efficiency of DPLL and OBDDs with Axiom and Join Matti Järvisalo University of Helsinki, Finland September 16, 2011 @ CP M. Järvisalo (U. Helsinki) DPLL and OBDDs September 16, 2011 @ CP
More informationA Dichotomy. in in Probabilistic Databases. Joint work with Robert Fink. for Non-Repeating Queries with Negation Queries with Negation
Dichotomy for Non-Repeating Queries with Negation Queries with Negation in in Probabilistic Databases Robert Dan Olteanu Fink and Dan Olteanu Joint work with Robert Fink Uncertainty in Computation Simons
More informationFirst-Order Knowledge Compilation for Probabilistic Reasoning
First-Order Knowledge Compilation for Probabilistic Reasoning Guy Van den Broeck based on joint work with Adnan Darwiche, Dan Suciu, and many others MOTIVATION 1 A Simple Reasoning Problem...? Probability
More informationOn the Role of Canonicity in Bottom-up Knowledge Compilation
On the Role of Canonicity in Bottom-up Knowledge Compilation Guy Van den Broeck and Adnan Darwiche Computer Science Department University of California, Los Angeles {guyvdb,darwiche}@cs.ucla.edu Abstract
More informationAlgebraic Model Counting
Algebraic Model Counting Angelika Kimmig a,, Guy Van den Broeck b, Luc De Raedt a a Department of Computer Science, KU Leuven, Celestijnenlaan 200a box 2402, 3001 Heverlee, Belgium. b UCLA Computer Science
More informationPropositional Fragments for Knowledge Compilation and Quantified Boolean Formulae
1/15 Propositional Fragments for Knowledge Compilation and Quantified Boolean Formulae Sylvie Coste-Marquis Daniel Le Berre Florian Letombe Pierre Marquis CRIL, CNRS FRE 2499 Lens, Université d Artois,
More informationProbabilistic Theorem Proving. Vibhav Gogate (Joint work with Pedro Domingos)
Probabilistic Theorem Proving Vibhav Gogate (Joint work with Pedro Domingos) Progress to Date First-order variable elimination [Poole, 2003; de Salvo Braz, 2007; etc.] Lifted belief propagation [Singla
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 31. Propositional Logic: DPLL Algorithm Malte Helmert and Gabriele Röger University of Basel April 24, 2017 Propositional Logic: Overview Chapter overview: propositional
More informationQuery Evaluation on Probabilistic Databases. CSE 544: Wednesday, May 24, 2006
Query Evaluation on Probabilistic Databases CSE 544: Wednesday, May 24, 2006 Problem Setting Queries: Tables: Review A(x,y) :- Review(x,y), Movie(x,z), z > 1991 name rating p Movie Monkey Love good.5 title
More informationOn First-Order Knowledge Compilation. Guy Van den Broeck
On First-Order Knowledge Compilation Guy Van den Broeck Beyond NP Workshop Feb 12, 2016 Overview 1. Why first-order model counting? 2. Why first-order model counters? 3. What first-order circuit languages?
More informationBinary Decision Diagrams Boolean Functions
Binary Decision Diagrams Representation of Boolean Functions BDDs, OBDDs, ROBDDs Operations Model-Checking over BDDs 72 Boolean functions:b = {0,1}, f :B B B Boolean Functions Boolean expressions: t ::=
More informationSums of Products. Pasi Rastas November 15, 2005
Sums of Products Pasi Rastas November 15, 2005 1 Introduction This presentation is mainly based on 1. Bacchus, Dalmao and Pitassi : Algorithms and Complexity results for #SAT and Bayesian inference 2.
More informationBDD Based Upon Shannon Expansion
Boolean Function Manipulation OBDD and more BDD Based Upon Shannon Expansion Notations f(x, x 2,, x n ) - n-input function, x i = or f xi=b (x,, x n ) = f(x,,x i-,b,x i+,,x n ), b= or Shannon Expansion
More informationEECS 219C: Computer-Aided Verification Boolean Satisfiability Solving III & Binary Decision Diagrams. Sanjit A. Seshia EECS, UC Berkeley
EECS 219C: Computer-Aided Verification Boolean Satisfiability Solving III & Binary Decision Diagrams Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: Lintao Zhang Announcement Project proposals due
More informationarxiv: v1 [cs.ai] 21 Sep 2014
Oblivious Bounds on the Probability of Boolean Functions WOLFGANG GATTERBAUER, Carnegie Mellon University DAN SUCIU, University of Washington arxiv:409.6052v [cs.ai] 2 Sep 204 This paper develops upper
More informationTutorial 1: Modern SMT Solvers and Verification
University of Illinois at Urbana-Champaign Tutorial 1: Modern SMT Solvers and Verification Sayan Mitra Electrical & Computer Engineering Coordinated Science Laboratory University of Illinois at Urbana
More informationSpace complexity of cutting planes refutations
Space complexity of cutting planes refutations Nicola Galesi, Pavel Pudlák, Neil Thapen Nicola Galesi Sapienza - University of Rome June 19, 2015 () June 19, 2015 1 / 32 Cutting planes proofs A refutational
More informationEnumeration: logical and algebraic approach
Enumeration: logical and algebraic approach Yann Strozecki Université Paris Sud - Paris 11 Novembre 2011, séminaire ALGO/LIX Introduction to Enumeration Enumeration and logic Enumeration and polynomials
More informationLifted Probabilistic Inference: A Guide for the Database Researcher
Lifted Probabilistic Inference: A Guide for the Database Researcher Eric Gribkoff University of Washington eagribko@cs.uw.edu Dan Suciu University of Washington suciu@cs.uw.edu Guy Van den Broeck UCLA,
More informationLimitations of Algorithm Power
Limitations of Algorithm Power Objectives We now move into the third and final major theme for this course. 1. Tools for analyzing algorithms. 2. Design strategies for designing algorithms. 3. Identifying
More informationSymmetric Weighted First-Order Model Counting
Symmetric Weighted First-Order Model Counting ABSTRACT Paul Beame University of Washington beame@cs.washington.edu Eric Gribkoff University of Washington eagribko@cs.washington.edu The FO Model Counting
More informationCompiling Knowledge into Decomposable Negation Normal Form
Compiling Knowledge into Decomposable Negation Normal Form Adnan Darwiche Cognitive Systems Laboratory Department of Computer Science University of California Los Angeles, CA 90024 darwiche@cs. ucla. edu
More informationPropositional Logic. Methods & Tools for Software Engineering (MTSE) Fall Prof. Arie Gurfinkel
Propositional Logic Methods & Tools for Software Engineering (MTSE) Fall 2017 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
More informationKnowledge Compilation in the Modal Logic S5
Knowledge Compilation in the Modal Logic S5 Meghyn Bienvenu Universität Bremen Bremen, Germany meghyn@informatik.uni-bremen.de Hélène Fargier IRIT-CNRS, Université Paul Sabatier, Toulouse, France fargier@irit.fr
More informationThe Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
More informationExercises - Solutions
Chapter 1 Exercises - Solutions Exercise 1.1. The first two definitions are equivalent, since we follow in both cases the unique path leading from v to a sink and using only a i -edges leaving x i -nodes.
More informationCompute the Fourier transform on the first register to get x {0,1} n x 0.
CS 94 Recursive Fourier Sampling, Simon s Algorithm /5/009 Spring 009 Lecture 3 1 Review Recall that we can write any classical circuit x f(x) as a reversible circuit R f. We can view R f as a unitary
More informationQueries and Materialized Views on Probabilistic Databases
Queries and Materialized Views on Probabilistic Databases Nilesh Dalvi Christopher Ré Dan Suciu September 11, 2008 Abstract We review in this paper some recent yet fundamental results on evaluating queries
More informationReordering Rule Makes OBDD Proof Systems Stronger
Reordering Rule Makes OBDD Proof Systems Stronger Sam Buss University of California, San Diego, La Jolla, CA, USA sbuss@ucsd.edu Dmitry Itsykson St. Petersburg Department of V.A. Steklov Institute of Mathematics
More informationLecture 5: Splay Trees
15-750: Graduate Algorithms February 3, 2016 Lecture 5: Splay Trees Lecturer: Gary Miller Scribe: Jiayi Li, Tianyi Yang 1 Introduction Recall from previous lecture that balanced binary search trees (BST)
More informationResolution for Predicate Logic
Logic and Proof Hilary 2016 James Worrell Resolution for Predicate Logic A serious drawback of the ground resolution procedure is that it requires looking ahead to predict which ground instances of clauses
More informationIntroduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010
Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010 Today Representation in Propositional Logic Semantics & Deduction
More informationA Lower Bound Technique for Nondeterministic Graph-Driven Read-Once-Branching Programs and its Applications
A Lower Bound Technique for Nondeterministic Graph-Driven Read-Once-Branching Programs and its Applications Beate Bollig and Philipp Woelfel FB Informatik, LS2, Univ. Dortmund, 44221 Dortmund, Germany
More informationChapter 5 The Witness Reduction Technique
Outline Chapter 5 The Technique Luke Dalessandro Rahul Krishna December 6, 2006 Outline Part I: Background Material Part II: Chapter 5 Outline of Part I 1 Notes On Our NP Computation Model NP Machines
More informationLifted Inference: Exact Search Based Algorithms
Lifted Inference: Exact Search Based Algorithms Vibhav Gogate The University of Texas at Dallas Overview Background and Notation Probabilistic Knowledge Bases Exact Inference in Propositional Models First-order
More informationarxiv: v3 [cs.cc] 19 Feb 2015
A Strongly Exponential Separation of DNNFs from CNF Formulas Simone Bova Florent Capelli Stefan Mengel Friedrich Slivovsky arxiv:1411.1995v3 [cs.cc] 19 Feb 2015 Abstract Decomposable Negation Normal Forms
More informationLecture 10 Algorithmic version of the local lemma
Lecture 10 Algorithmic version of the local lemma Uriel Feige Department of Computer Science and Applied Mathematics The Weizman Institute Rehovot 76100, Israel uriel.feige@weizmann.ac.il June 9, 2014
More informationCSC 2429 Approaches to the P vs. NP Question and Related Complexity Questions Lecture 2: Switching Lemma, AC 0 Circuit Lower Bounds
CSC 2429 Approaches to the P vs. NP Question and Related Complexity Questions Lecture 2: Switching Lemma, AC 0 Circuit Lower Bounds Lecturer: Toniann Pitassi Scribe: Robert Robere Winter 2014 1 Switching
More informationSymbolic Model Checking with ROBDDs
Symbolic Model Checking with ROBDDs Lecture #13 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de December 14, 2016 c JPK Symbolic
More informationCS Introduction to Complexity Theory. Lecture #11: Dec 8th, 2015
CS 2401 - Introduction to Complexity Theory Lecture #11: Dec 8th, 2015 Lecturer: Toniann Pitassi Scribe Notes by: Xu Zhao 1 Communication Complexity Applications Communication Complexity (CC) has many
More informationA Knowledge Compilation Map
Journal of Artificial Intelligence Research 17 (2002) 229-264 Submitted 12/01; published 9/02 A Knowledge Compilation Map Adnan Darwiche Computer Science Department University of California, Los Angeles
More informationComplexity, P and NP
Complexity, P and NP EECS 477 Lecture 21, 11/26/2002 Last week Lower bound arguments Information theoretic (12.2) Decision trees (sorting) Adversary arguments (12.3) Maximum of an array Graph connectivity
More informationBinary Decision Diagrams
Binary Decision Diagrams Beate Bollig, Martin Sauerhoff, Detlef Sieling, and Ingo Wegener FB Informatik, LS2, Univ. Dortmund, 44221 Dortmund, Germany lastname@ls2.cs.uni-dortmund.de Abstract Decision diagrams
More informationLifted Probabilistic Inference in Relational Models
Lifted Probabilistic Inference in Relational Models Guy Van den Broeck Dan Suciu KU Leuven U. of Washington Tutorial UAI 2014 1 About the Tutorial Slides available online. Bibliography is at the end. Your
More informationReduced Ordered Binary Decision Diagrams
Reduced Ordered Binary Decision Diagrams Lecture #13 of Advanced Model Checking Joost-Pieter Katoen Lehrstuhl 2: Software Modeling & Verification E-mail: katoen@cs.rwth-aachen.de June 5, 2012 c JPK Switching
More informationOn the correlation of parity and small-depth circuits
Electronic Colloquium on Computational Complexity, Report No. 137 (2012) On the correlation of parity and small-depth circuits Johan Håstad KTH - Royal Institute of Technology November 1, 2012 Abstract
More informationA Relaxed Tseitin Transformation for Weighted Model Counting
A Relaxed Tseitin Transformation for Weighted Model Counting Wannes Meert, Jonas Vlasselaer Computer Science Department KU Leuven, Belgium Guy Van den Broeck Computer Science Department University of California,
More informationPolynomial time Prediction Strategy with almost Optimal Mistake Probability
Polynomial time Prediction Strategy with almost Optimal Mistake Probability Nader H. Bshouty Department of Computer Science Technion, 32000 Haifa, Israel bshouty@cs.technion.ac.il Abstract We give the
More informationOn Unit-Refutation Complete Formulae with Existentially Quantified Variables
On Unit-Refutation Complete Formulae with Existentially Quantified Variables Lucas Bordeaux 1 Mikoláš Janota 2 Joao Marques-Silva 3 Pierre Marquis 4 1 Microsoft Research, Cambridge 2 INESC-ID, Lisboa 3
More informationComputing Query Probability with Incidence Algebras
Computing Query Probability with Incidence Algebras Nilesh Dalvi Yahoo Research Santa Clara, CA, USA ndalvi@yahoo-inc.com Karl Schnaitter UC Santa Cruz Santa Cruz, CA, USA karlsch@soe.ucsc.edu Dan Suciu
More informationOrganization. Informal introduction and Overview Informal introductions to P,NP,co-NP and themes from and relationships with Proof complexity
15-16/08/2009 Nicola Galesi 1 Organization Informal introduction and Overview Informal introductions to P,NP,co-NP and themes from and relationships with Proof complexity First Steps in Proof Complexity
More informationLecture 1: Introduction to Sublinear Algorithms
CSE 522: Sublinear (and Streaming) Algorithms Spring 2014 Lecture 1: Introduction to Sublinear Algorithms March 31, 2014 Lecturer: Paul Beame Scribe: Paul Beame Too much data, too little time, space for
More informationTopics in Model-Based Reasoning
Towards Integration of Proving and Solving Dipartimento di Informatica Università degli Studi di Verona Verona, Italy March, 2014 Automated reasoning Artificial Intelligence Automated Reasoning Computational
More informationAlgorithms for Satisfiability beyond Resolution.
Algorithms for Satisfiability beyond Resolution. Maria Luisa Bonet UPC, Barcelona, Spain Oaxaca, August, 2018 Co-Authors: Sam Buss, Alexey Ignatiev, Joao Marques-Silva, Antonio Morgado. Motivation. Satisfiability
More informationA Lower Bound Technique for Restricted Branching Programs and Applications
A Lower Bound Technique for Restricted Branching Programs and Applications (Extended Abstract) Philipp Woelfel FB Informatik, LS2, Univ. Dortmund, 44221 Dortmund, Germany woelfel@ls2.cs.uni-dortmund.de
More informationCS6840: Advanced Complexity Theory Mar 29, Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K.
CS684: Advanced Complexity Theory Mar 29, 22 Lecture 46 : Size lower bounds for AC circuits computing Parity Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K. Theme: Circuit Complexity Lecture Plan: Proof
More informationINAPPROX APPROX PTAS. FPTAS Knapsack P
CMPSCI 61: Recall From Last Time Lecture 22 Clique TSP INAPPROX exists P approx alg for no ε < 1 VertexCover MAX SAT APPROX TSP some but not all ε< 1 PTAS all ε < 1 ETSP FPTAS Knapsack P poly in n, 1/ε
More informationLogical Compilation of Bayesian Networks
Logical Compilation of Bayesian Networks Michael Wachter & Rolf Haenni iam-06-006 August 2006 Logical Compilation of Bayesian Networks Michael Wachter & Rolf Haenni University of Bern Institute of Computer
More informationBinary Decision Diagrams. Graphs. Boolean Functions
Binary Decision Diagrams Graphs Binary Decision Diagrams (BDDs) are a class of graphs that can be used as data structure for compactly representing boolean functions. BDDs were introduced by R. Bryant
More informationBinary Decision Diagrams
Binary Decision Diagrams An Introduction and Some Applications Manas Thakur PACE Lab, IIT Madras Manas Thakur (IIT Madras) BDDs 1 / 25 Motivating Example Binary decision tree for a truth table Manas Thakur
More informationComputing Query Probability with Incidence Algebras Technical Report UW-CSE University of Washington
Computing Query Probability with Incidence Algebras Technical Report UW-CSE-10-03-02 University of Washington Nilesh Dalvi, Karl Schnaitter and Dan Suciu Revised: August 24, 2010 Abstract We describe an
More informationWitnessing Matrix Identities and Proof Complexity
Witnessing Matrix Identities and Proof Complexity Fu Li Iddo Tzameret August 4, 2016 Abstract We use results from the theory of algebras with polynomial identities PI algebras to study the witness complexity
More informationBounds on the OBDD-Size of Integer Multiplication via Universal Hashing
Bounds on the OBDD-Size of Integer Multiplication via Universal Hashing Philipp Woelfel Dept. of Computer Science University Dortmund D-44221 Dortmund Germany phone: +49 231 755-2120 fax: +49 231 755-2047
More informationQuery answering using views
Query answering using views General setting: database relations R 1,...,R n. Several views V 1,...,V k are defined as results of queries over the R i s. We have a query Q over R 1,...,R n. Question: Can
More informationSum-Product Networks: A New Deep Architecture
Sum-Product Networks: A New Deep Architecture Pedro Domingos Dept. Computer Science & Eng. University of Washington Joint work with Hoifung Poon 1 Graphical Models: Challenges Bayesian Network Markov Network
More informationPrinciples of Program Analysis: Control Flow Analysis
Principles of Program Analysis: Control Flow Analysis Transparencies based on Chapter 3 of the book: Flemming Nielson, Hanne Riis Nielson and Chris Hankin: Principles of Program Analysis. Springer Verlag
More information4.8 Huffman Codes. These lecture slides are supplied by Mathijs de Weerd
4.8 Huffman Codes These lecture slides are supplied by Mathijs de Weerd Data Compression Q. Given a text that uses 32 symbols (26 different letters, space, and some punctuation characters), how can we
More informationLogical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
More informationALGEBRAIC PROOFS OVER NONCOMMUTATIVE FORMULAS
Electronic Colloquium on Computational Complexity, Report No. 97 (2010) ALGEBRAIC PROOFS OVER NONCOMMUTATIVE FORMULAS IDDO TZAMERET Abstract. We study possible formulations of algebraic propositional proof
More informationGAV-sound with conjunctive queries
GAV-sound with conjunctive queries Source and global schema as before: source R 1 (A, B),R 2 (B,C) Global schema: T 1 (A, C), T 2 (B,C) GAV mappings become sound: T 1 {x, y, z R 1 (x,y) R 2 (y,z)} T 2
More informationHeuristics for Efficient SAT Solving. As implemented in GRASP, Chaff and GSAT.
Heuristics for Efficient SAT Solving As implemented in GRASP, Chaff and GSAT. Formulation of famous problems as SAT: k-coloring (1/2) The K-Coloring problem: Given an undirected graph G(V,E) and a natural
More informationComputational Complexity of Inference
Computational Complexity of Inference Sargur srihari@cedar.buffalo.edu 1 Topics 1. What is Inference? 2. Complexity Classes 3. Exact Inference 1. Variable Elimination Sum-Product Algorithm 2. Factor Graphs
More informationIntelligent Agents. Pınar Yolum Utrecht University
Intelligent Agents Pınar Yolum p.yolum@uu.nl Utrecht University Logical Agents (Based mostly on the course slides from http://aima.cs.berkeley.edu/) Outline Knowledge-based agents Wumpus world Logic in
More informationLimits to Approximability: When Algorithms Won't Help You. Note: Contents of today s lecture won t be on the exam
Limits to Approximability: When Algorithms Won't Help You Note: Contents of today s lecture won t be on the exam Outline Limits to Approximability: basic results Detour: Provers, verifiers, and NP Graph
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (previous lecture, Part I) Chapter 7.5 (this lecture, Part II) (optional: 7.6-7.8)
More informationChebyshev Polynomials, Approximate Degree, and Their Applications
Chebyshev Polynomials, Approximate Degree, and Their Applications Justin Thaler 1 Georgetown University Boolean Functions Boolean function f : { 1, 1} n { 1, 1} AND n (x) = { 1 (TRUE) if x = ( 1) n 1 (FALSE)
More information173 Logic and Prolog 2013 With Answers and FFQ
173 Logic and Prolog 2013 With Answers and FFQ Please write your name on the bluebook. You may have two sides of handwritten notes. There are 75 points (minutes) total. Stay cool and please write neatly.
More informationStanford University CS254: Computational Complexity Handout 8 Luca Trevisan 4/21/2010
Stanford University CS254: Computational Complexity Handout 8 Luca Trevisan 4/2/200 Counting Problems Today we describe counting problems and the class #P that they define, and we show that every counting
More informationEMPORIUM H O W I T W O R K S F I R S T T H I N G S F I R S T, Y O U N E E D T O R E G I S T E R.
H O W I T W O R K S F I R S T T H I N G S F I R S T, Y O U N E E D T O R E G I S T E R I n o r d e r t o b u y a n y i t e m s, y o u w i l l n e e d t o r e g i s t e r o n t h e s i t e. T h i s i s
More informationParallelism and Machine Models
Parallelism and Machine Models Andrew D Smith University of New Brunswick, Fredericton Faculty of Computer Science Overview Part 1: The Parallel Computation Thesis Part 2: Parallelism of Arithmetic RAMs
More informationSatisfiability Modulo Theories (SMT)
CS510 Software Engineering Satisfiability Modulo Theories (SMT) Slides modified from those by Aarti Gupta Textbook: The Calculus of Computation by A. Bradley and Z. Manna 1 Satisfiability Modulo Theory
More informationLecture 3 (Notes) 1. The book Computational Complexity: A Modern Approach by Sanjeev Arora and Boaz Barak;
Topics in Theoretical Computer Science March 7, 2016 Lecturer: Ola Svensson Lecture 3 (Notes) Scribes: Ola Svensson Disclaimer: These notes were written for the lecturer only and may contain inconsistent
More informationBinary Decision Diagrams
Binary Decision Diagrams Logic Circuits Design Seminars WS2010/2011, Lecture 2 Ing. Petr Fišer, Ph.D. Department of Digital Design Faculty of Information Technology Czech Technical University in Prague
More informationOn the Optimal Approximation of Queries Using Tractable Propositional Languages
On the Optimal Approximation of Queries Using Tractable Propositional Languages Robert Fink and Dan Olteanu Oxford University Computing Laboratory {robert.fink,dan.olteanu}@comlab.ox.ac.uk ABSTRACT This
More informationPropositional Logic: Methods of Proof (Part II)
Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short
More informationMath 262A Lecture Notes - Nechiporuk s Theorem
Math 6A Lecture Notes - Nechiporuk s Theore Lecturer: Sa Buss Scribe: Stefan Schneider October, 013 Nechiporuk [1] gives a ethod to derive lower bounds on forula size over the full binary basis B The lower
More informationDatabase Theory VU , SS Complexity of Query Evaluation. Reinhard Pichler
Database Theory Database Theory VU 181.140, SS 2018 5. Complexity of Query Evaluation Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 17 April, 2018 Pichler
More informationHierarchical Matrices. Jon Cockayne April 18, 2017
Hierarchical Matrices Jon Cockayne April 18, 2017 1 Sources Introduction to Hierarchical Matrices with Applications [Börm et al., 2003] 2 Sources Introduction to Hierarchical Matrices with Applications
More information