AUTOMATED REASONING. Agostino Dovier. Udine, October 1, Università di Udine CLPLAB

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1 AUTOMATED REASONING Agostino Dovier Università di Udine CLPLAB Udine, October 1, 2018 AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

2 COURSE PLACEMENT International Master Degree in CS (mandatory course) This is why the course is in (a personal dialect of) English National Master Degree in Informatica (optional course for all, characterizing the path ARA, Algoritmi e ragionamento automatico ) Suggested to Mathematicians due to its roots in logic modeling 6CFUs Project + Oral exam Prerequisites? Some programming experience, some logical background. You have it! AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

3 COURSE PLACEMENT International Master Degree in CS (mandatory course) This is why the course is in (a personal dialect of) English National Master Degree in Informatica (optional course for all, characterizing the path ARA, Algoritmi e ragionamento automatico ) Suggested to Mathematicians due to its roots in logic modeling 6CFUs Project + Oral exam Prerequisites? Some programming experience, some logical background. You have it! AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

4 ACM-IEEE Reading Computer Science Curricula. Joint Task Force on Computing Curricula, ACM-IEEE, Dec we learn that there are Among them there is the area of: 18 Knowledge Areas (of C.S.) IS - Intelligent Systems AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

5 ACM-IEEE Reading Computer Science Curricula. Joint Task Force on Computing Curricula, ACM-IEEE, Dec we learn that there are Among them there is the area of: 18 Knowledge Areas (of C.S.) IS - Intelligent Systems AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

6 ACM-IEEE Reading Computer Science Curricula. Joint Task Force on Computing Curricula, ACM-IEEE, Dec we learn that there are Among them there is the area of: 18 Knowledge Areas (of C.S.) IS - Intelligent Systems AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

7 ACM-IEEE Reading Computer Science Curricula. Joint Task Force on Computing Curricula, ACM-IEEE, Dec we learn that there are Among them there is the area of: 18 Knowledge Areas (of C.S.) IS - Intelligent Systems AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

8 ACM-IEEE IS - INTELLIGENT SYSTEMS Artificial intelligence (AI) is the study of solutions for problems that are difficult or impractical to solve with traditional methods. [...] The solutions rely on a broad set of general and specialized knowledge representation schemes, problem solving mechanisms, and learning techniques. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

9 ACM-IEEE IS - INTELLIGENT SYSTEMS. 12 SUBAREAS IS/Fundamental Issues IS/Basic Knowledge Representation and Reasoning IS/Basic Search Strategies IS/Advanced Search IS/Advanced Representation and Reasoning IS/Agents IS/Basic Machine Learning IS/Advanced Machine Learning IS/Reasoning Under Uncertainty IS/Natural Language Processing IS/Robotics IS/Perception and Computer Vision The course of Automated Reasoning will deal with the 6 subareas in the left column. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

10 ACM-IEEE IS - INTELLIGENT SYSTEMS. 12 SUBAREAS IS/Fundamental Issues IS/Basic Knowledge Representation and Reasoning IS/Basic Search Strategies IS/Advanced Search IS/Advanced Representation and Reasoning IS/Agents IS/Basic Machine Learning IS/Advanced Machine Learning IS/Reasoning Under Uncertainty IS/Natural Language Processing IS/Robotics IS/Perception and Computer Vision The course of Automated Reasoning will deal with the 6 subareas in the left column. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

11 COURSE PROGRAM ACCORDING TO THE 6 SUBAREAS OF IS IS/Fundamental Issues Overview of AI problems, examples of successful recent AI applications IS/Basic Knowledge Representation and Reasoning Review of propositional and predicate logic Resolution and theorem proving IS/Advanced Representation and Reasoning Knowledge representation issues Non-monotonic reasoning Reasoning about action and change Planning Argumentation AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

12 COURSE PROGRAM ACCORDING TO THE 6 SUBAREAS OF IS IS/Basic Search Strategies Problem spaces (states, goals and operators), problem solving by search Factored representation (factoring state into variables) Uninformed search (breadth-first, depth-first, depth-first with iterative deepening) IS/Agents Heuristics and informed search (hill-climbing, generic best-first, A*) Space and time efficiency of search Constraint satisfaction (backtracking and local search methods) Definitions of agents Multi-agent systems (Collaborative agents) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

13 COURSE PROGRAM ACCORDING TO THE 6 SUBAREAS OF IS IS/Advanced Search Global constraints Large Neighborhood Search SAT and ASP solving with Learning Parallelism Students should encode and solve problems using Minizinc and Answer Set Programming (and will meet SAT, PDDL, and Picat) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

14 COURSE PROGRAM Constraint Programming Introduction Constraint Propagation and Search Global constraints Modeling in Minizinc Logic Programming Definite programs Default Negation Answer Set Programming (ASP) Modeling in ASP SAT solving (learning) Advanced Topics AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

15 WHAT S GOING ON See International Conference on Constraint Programming (CP) International Conference on Logic Programming (ICLP) International Conference on Theory and Applications of Satisfiability Testing (SAT) (all together in Melbourne in Logic Programming and Non Monotonic Reasoning (LPNMR) International Joint Conference on Artificial Intelligence (IJCAI)... AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

16 CONSTRAINT PROGRAMMING INTRODUCTION Constraint Programming is a declarative programming paradigm rooted in AI, suited for modeling and solving (complex) combinatorial problems Problem modeling and solution searching are clearly separated Typically, the code is very readable and easy to modify You don t have restrictions on the kind of constraints Solution search is natural to parallelize Search heuristics are crucial and easy to be added AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

17 Sketchpad: Ivan Sutherland (PhD thesis, 1963) Notion of CSP (Ugo Montanari 1972) Domain filtering (Waltz 1975) Late Seventies/Early Eighties: constraint propagation (Freuder, Mackworth, others) Late Eighties: the first Constraint-Based Language (Constraint Logic Programming, or CLP) Ninenties: Global constraints and other languages New millennium: hybrid techniques, standardization 2010: Minizinc challenge (and many more: parallelism, learning,... ) Statement by Michael Trick Read aop-cambridge-core/content/view/ 60BF5956A25C6E531BD0C5B0438CD6D5/ S a.pdf/preface.pdf AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28 CONSTRAINT PROGRAMMING BRIEF HISTORY

18 INTRODUCTION ACP ACP Schools tw/ school2013/exercises.html Videos: Pascal Van Hentenryck Peter Stuckey Material (see the web page). However, this is a reference text for this part. Handbook final.pdf AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

19 EXAMPLE 4-QUEENS 4-Queens: put 4 queens on a chessboard 4 4 in such a way as they do not attack each other AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

20 EXAMPLE 4-QUEENS 4-Queens: put 4 queens on a chessboard 4 4 in such a way as they do not attack each other AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

21 EXAMPLE 4-QUEENS 4-Queens: put 4 queens on a chessboard 4 4 in such a way as they do not attack each other AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

22 EXAMPLE 4-QUEENS 4-Queens: put 4 queens on a chessboard 4 4 in such a way as they do not attack each other AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

23 EXAMPLE 4-QUEENS 4-Queens: put 4 queens on a chessboard 4 4 in such a way as they do not attack each other AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

24 EXAMPLE 4-QUEENS Variables Domains X 1,..., X 4 X i = j means a queen is in column i, row j D(X i ) = {1,..., 4}i = 1..4 Constraints (i, j = 1..4 e i < j) X i X j (horizontal attack) X j X i j i (diagonal attack) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

25 EXAMPLE 4-QUEENS X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 1 WRONG X i X j (horizontal) X j X i j i (diagonal) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

26 EXAMPLE 4-QUEENS X 1 = 2, X 2 = 4, X 3 = 1, X 4 = 3 CORRECT X i X j (horizontal) X j X i j i (diagonal) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

27 CONSTRAINT SATISFACTION PROBLEM CSP (CONSTRAINT SATISFACTION PROBLEM) Set of Variables V = {V 1,..., V n } Set of domains D = {D 1,..., D n } Set of Constraints (Vincoli) C on the variables V. We look for one (or all) the admissible solutions, namely an assignment σ : V D 1 D n that satisfies all domains and constraints. No restrictions on domains and kind of constraints AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

28 CONSTRAINT OPTIMIZATION PROBLEM COP (CONSTRAINT OPTIMIZATION PROBLEM) Set of Variables V = {V 1,..., V n } Set of domains D = {D 1,..., D n } Set of Constraints (Vincoli) C on the variables V. A function f : V A We look for one (or all) the assignments σ that are admissible solutions of the CSP that minimizes (equivalently, maximizes) the value of f. No restrictions on domains, kind of constraints, and on the function f AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

29 SEARCH SPACE Given a CSP X 1 D 1,..., X n D n ; C (C denotes the set of all the constraints) The search space is the set of the n-tuples consistent with the domains D 1 D n In this set we look for the points that satisfy the C This set is commonly represented as a tree, the search tree. Let us observe that, in general, the D i can be infinite. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

30 SEARCH SPACE 4-QUEENS AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

31 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 1 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

32 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 1 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

33 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 2 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

34 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 2 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

35 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 3 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

36 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 3 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

37 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 4 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

38 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 1, X 4 = 4 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

39 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 2, X 4 = 1 not a solution AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

40 CSP/COP NAIVE SEARCH: GENERATE AND TEST Non deterministic technique in which the potential solutions are generated sequentially and verified (akin to Guess & Verify of NP). X 1 = 1, X 2 = 1, X 3 = 2, X 4 = 1 not a solution... We are visiting all the tree. The risk is of doing 4 4 = 256 attempts AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

41 CSP/COP NAIVE SEARCH: PARTIAL ASSIGNMENTS, PARTIAL CHECK, BACKTRACKING 19 attempts. Assignments with X 1 = 3 and X 1 = 4 are symmetrical wrt those already visited. Symmetry breaking will be a crucial issue in modeling. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

42 CSP/COP NAIVE SEARCH: PARTIAL ASSIGNMENTS, PARTIAL CHECK, BACKTRACKING 19 attempts. Assignments with X 1 = 3 and X 1 = 4 are symmetrical wrt those already visited. Symmetry breaking will be a crucial issue in modeling. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

43 CSP/COP MAIN NON NAIVE TECHNIQUES USED FOR SOLUTION S SEARCH Local Search (Integer) Linear Programming Translation to SAT and use of a SAT solver Constraint Programming Answer Set Programming (born for KR purposes, but very effective) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

44 CSP OR COP? COP CSP Assume to have a (black box) COP solver Given a CSP P = X 1 D 1,..., X n D n ; C Define f as a constant (e.g. f (X 1,..., X n ) = 0) call the solver of P, f. It is clear that the solution of the COP is also a solution of the CSP. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

45 CSP OR COP? CSP COP Suppose now to have a solver (think to it as to a black box) capable of solving a CSP P = X 1 D 1,..., X n D n ; C, f (X 1,..., X n ) k Let M a (even rough) upper bound of the expected value of the function. Given P is usually easy to find an upper bound. Assume f ranges on N. Using bisection (with a number of calls limited by log 2 M) we can solve the COP riesco a risolvere il COP Eg, suppose the minimum is 15 (but we don t know it yet) and M = 205. Let s solve P with k = 205, 102, 51, 25, 12. With 12 we have the first no. Thus the minimum is between 13 and 25. Let s try (yes), (yes), (no). The doubt is now between 14 or 15. Calling the solver on P with k = 14 we discover it. If f ranges on R it is the same with some approximation considerations. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

46 CSP OR COP? COP CSP In principle solving CSP and COP is not too much different. However, finding a solution to a even big CSP can be easy (if we are lucky) finding the minimum in very big COPs imply the visit of the whole (or, at least of a big protion of the) search space Sometimes we are happy with good solutions, not necessarily the minimum. AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

47 NP-HARDNESS OF CSP Let us recall the 3-coloring problem: does a coloring of the nodes of a graph using three colors in such a way that adjacent nodes have different colors exist? D 1 = = D 6 = {red, green, black}, X 1 X 2, X 1 X 3, X 1 X 5,..., X 5 X 6 AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

48 NP-HARDNESS OF CSP Let us recall the 3-coloring problem: does a coloring of the nodes of a graph using three colors in such a way that adjacent nodes have different colors exist? D 1 = = D 6 = {red, green, black}, X 1 X 2, X 1 X 3, X 1 X 5,..., X 5 X 6 AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

49 NP-HARDNESS OF CSP Let us recall the 3-coloring problem: does a coloring of the nodes of a graph using three colors in such a way that adjacent nodes have different colors exist? D 1 = = D 6 = {red, green, black}, X 1 X 2, X 1 X 3, X 1 X 5,..., X 5 X 6 AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28

50 A FAMOUS CSP domain([x 1,1,..., X 9,9 ], 1, 9) X 1,3 = 1, X 2,3 = 2, X 2,5 = 3, X 2,9 = 4,..., X 9,7 = 5 alldifferent(x 1,1,..., X 1,9 )... alldifferent(x 9,1,..., X 9,9 ) alldifferent(x 1,1,..., X 9,1 )... alldifferent(x 1,9,..., X 9,9 ) alldifferent(x 1,1,..., X 3,3 )... alldifferent(x 7,7,..., X 9,9 ) AGOSTINO DOVIER (CLPLAB) AUTOMATED REASONING UDINE, OCTOBER 1, / 28