Reasoning and search

1. Reasoning and search

2. An overview • Representation and reasoning system • Rule-based reasoning • Forward reasoning • Backward reasoning • Conflict resolution • Advantages and disadvantages • Solving problem by search – search strategies • hill-climbing search • backtracking search • graph search strategies Reasoning and search

3. Representation and reasoning system (RRS) Reasoning and search

4. Introduction to RRS • question: how can computer solve • difficult, non-trivial, complex, unusually large problems • in a non-trivial human-like (intelligent) way • application examples: • toy problems (15-puzzle, n-queens, chess playing) • route-finding (automatic travel advisory system) • diagnostic problems (medical diagnosis) • natural language translation systems • expert systems • RRSs • tools for the automation of problem solving tasks • problem  representation  computation Reasoning and search

5. Main components of RRS • An RRS consists of: • language to communicate with the computer: formal language • a way to assign meaning to the symbol: semantics • a procedure to compute answer or solve problem: reasoning • An implementation of RRS consists of: • language parser: maps sentences of the language into data structures • reasoning procedure: implementation of reasoning + search strategy Note: the semantics aren’t reflected in the implementation! Reasoning and search

6. Using an RRS (example) • begin with a task domain that you want to characterize • distinguish those things you want to talk about in domain (ontology) • coffee-machine, valves, heater, valve_in is open, flow water through valve, heating the water • choose symbols in the computer to represent objects and relations in the domain • v1, v2, k, valve(V, open), flow(V), heating(W) • tell the system knowledge about the domain • switch(on), valve(v1,open), switch(on)  heating(water) • ask the system questions • ?valve(v1,X), ?heating(Y) Reasoning and search

7. Questions and answers in RRS • determining what follows from the formal description • whether the question is implied by the defined clauses (logical consequence of the knowledge base – set of clauses claimed to be true) • construction of logical consequence/proof procedure  reasoning theory – a possible nondeterministic specification of how an answer can be derived from the knowledge base • nondeterministic specification: • have to make decision before knowing enough to make the right choice • make a choice and keep track of the alternatives (in case of fail) • can think as a search through the space of possible choices • an implementation of a reasoning theory together with a search strategy  reasoning procedure Reasoning and search

8. Simplifying assumptions of RRS • the knowledge can be usefully described in terms of individuals and relations among individuals • the knowledge base contains only definite and positive clauses • definit clause: atom (fact) or of the form „b1  …  bm a” (rule) • environment is static (ignore change) • finite number of individuals of interest in the domain, each individual can be given a unique name DATALOG (special case of rule sets) Reasoning and search

9. Rule-based reasoning Reasoning and search

10. Knowledge-base The knowledge base (KB) of rule-based system consists of: • facts (predicates) • declarative knowledge about the given problem • statements with true or false values • values can change in time and during reasoning • rules (conditional statements) • represent heuristics or „rules of thumb” • modelling human’s thinking • describing experts’ knowledge (heuristics) • specify actions taking in a given situation • generally valid part of the practical knowledge • rules are operated by the inference engine • meta-rules (rules about how to use rules) Reasoning and search

11. Rules and facts Rules: IF <condition> THEN <conclusion> THEN <conclusion> IF <condition> condition/premise conclusion/consequence/antecedent/action • condition, conclusion: • statements • and/or connections of statements • procedural elements (executable actions) ex. IF level is above max_level THEN close valve_in Facts:statements without conditions ex. valve_in is closed valve_out connected_to the output_port of water-heater Reasoning and search

12. Reasoning 1 inference engine uses rules, derives new knowledge the reasoning algorithm: • pattern matching • finding applicable rules • fireable rules  conflict set • conflict resolution • selecting the most appropriate rule from conflict set • conflict resolution strategies • firing • executing the selected rule  new knowledge • watching termination conditions • restart of the cycle Reasoning and search

13. Reasoning 2 • new facts can be deduced during reasoning • aim: • proving a goal statement • achieving a goal state • task: • finding a solution (reasoning path, chain of rules) between the initial and the goal states • reasoning tool: • applying rules/ matching • modus ponens (MP) A A  B B premise conclusion Reasoning and search

14. Reasoning 3 • modus ponens can be used in two ways  two different forms of reasoning: • data-driven (forward) reasoning • aim: reach or construct a goal state starting from the initial state • new conclusions are generated by MP • until termination conditions satisfied or no more applicable rules (no more conclusion) • goal-driven (backward) reasoning • aim: prove a goal statement using initially valid facts • new subgoals are generated by MP • until all subgoals proved or no more applicable rules (no more provable subgoals) Reasoning and search

15. Forward reasoning 1 • application of modus ponens: A, A  B  B • new fact into KB (conclusions of data) • the reasoning algorithm: • selecting the applicable rules: pattern matching • a rule is applicable when its condition part is true • matching: conditional part of rules/ facts • selecting the most appropriate rule: conflict resolution • conflict resolution strategies • executing the selected rule: firing • conclusion part of rule is executed (set to true) • watching termination conditions (a goal state is reached) • restart of the cycle Reasoning and search

16. Reasoning and search

17. Forward reasoning 3 the inference chain produced by the example Reasoning and search

18. Forward reasoning 4 • the problems of forward reasoning: • forward reasoning with defined goal • given: the initial state of fact-base, the rule-base, the goal state(s) of fact-base • question: is goal state a consequence of initial state? can goal state be derived from initial state by the rules? decision problem: the whole search tree must be traversed in the worst case NP-complete (the size of the tree increases the number of computational steps exponentially) • forward reasoning (without defined goal) • given: the initial state of fact-base, the rule-base • compute: all the possible consequences of initial state search problem: the whole search tree must be traversed NP-complete (follows from the problem specification) Reasoning and search

19. Backward reasoning 1 • application of modus ponens: prove B with A  B, need A • new subgoals (conditions of rule) • the reasoning algorithm: • selecting the applicable rules: pattern matching • a rule is applicable when its conclusion part matches (sub)goal • matching: facts orconclusion part of rules/ (sub)goals • selecting the most appropriate rule: conflict resolution • conflict resolution strategy usually the first applicable rule • executing the selected rule: firing • condition part of rule is executed (set to new subgoal) • watching termination conditions (all subgoals proved) • restart of the cycle Reasoning and search

20. Backward reasoning 2 • backtracking: trying next alternative • cancel the last part of the path • go back to the previous state with more applicable rules • continue with another rule (next alternative) • backtrack is needed when: • there is no applicable rule for the actual state (came to dead-end) • not worth examining the actual state (on the basis on heuristics) • all of the rules applicable to the actual state is examined (mouth of dead-end) • reach a state is kept on the actual path (loop) • the state is too far from the initial state (depth limit) Reasoning and search

21. Reasoning and search

22. Backward reasoning 4 the inference chain produced by the example Reasoning and search

23. Backward reasoning 5 • the problems of backward reasoning: • backward reasoning with defined state • given: the goal state of fact-base, the rule-base, one or more given states („s”) of fact-base • question: can „s” be a reason of the goal state? (can goal state be derived from „s” by the rules?) decision problem: the whole search tree must be traversed in the worst case NP-complete (the size of the tree increases the number of computational steps exponentially) • backward reasoning (without defined state) • given: the goal state of fact-base, the rule-base • compute: all the possible reasons of the goal state search problem: the whole search tree must be traversed NP-complete (follows from the problem specification) Reasoning and search

24. Selection of reasoningtechnique Forward or backward technique can be applied? • the determinant elements are: • the number of possible initial and final states prediction in a real-time expert system, diagnosis in diagnostic system • which direction has greater branching factor proving a theorem in mathematics • explanation is needed or not medical diagnostic system • time-honoured observation: • in case of giving answer to a question: backward reasoning • in case of reaching new facts or states: forward reasoning Reasoning and search

25. Bidirectional reasoning • problems where neither forward nor backward chaining is efficient, but they operate efficiently at an early stage • combination of backward and forward techniques  bidirectional reasoning • the goal state can be reached from two directions at same time • terminates when the reasoning bridge is built up Reasoning and search

26. Conflict resolution 1 • problems: • no exact solution strategy optimal to every possible reasoning task • test every possible way of solution  combinatorial explosion • but: • no need to produce all possible solutions for most of the real practical problems  aim: „good enough” solution in a „short enough” time • conflict resolution: • choosing a rule to apply next from the applicable ones (conflict set) • almost always contains heuristic knowledge (extra knowledge about the structure of the rule-base) Reasoning and search

27. Conflict resolution 2 • heuristic knowledge • no exact definition for heuristics • properties of a heuristic procedures: • „good enough” solution is found in most cases • optimal solution or any solution is not guaranteed • considerably improve the efficiency of problem solving (reduce the number of attempts to reach the solution) • properties of a good heuristics: • it is used and computed efficiently • it is a good estimate Reasoning and search

28. Conflict resolution 3 • conflict resolution strategies: • using first applicable rule • rules are placed in order of importance • refractoriness • control reusing of rules (to release cycles in executing) • ex. an instance of a rule can only fire once or cannot executed in the following step • recency • attach time-stamps to facts • prefer rule instances referring to facts with fresh time-stamps • specificity • prefer rules with more conditions (specific rules, describe exceptions) • assigning priority Reasoning and search

29. Advantages and disadvantages • advantages of rule-based systems • modularity • universal representation • naturalness • easy to complete with uncertainty handling methods • disadvantages of rule-based systems • endless chaining/loop • new rule or modification of rule  may be contradictory • rules and meta-rules are in the same syntax (not differ from each other), the two types of information are mixed • language of rules is not standardized  difficult to transfer it to another system Reasoning and search

30. Solving problems by search Reasoning and search

31. Reasoning and search 1 • reasoning problems are solved by search in the state space (r1): p1=t  p2=t (r2): p2=t  p3=t (r3): p3=u  p1=u initial state: {p1=t, p2=f, p3=u}, (a0) states: {p1=u, p2=f, p3=u}, (a1) {p1=t, p2=t, p3=u}, (a2) {p1=t, p2=t, p3=t}, (a3) {p1=u, p2=t, p3=u}, (a4) Reasoning and search

32. Reasoning and search 2 • the state space of reasoning problem can be represented by a reasoning graph • reasoning problems are solved by search on the reasoning graph • search: general problem solving method or mechanism • search is needed: • difficult (non-trivial, complex, large or complicated) problem • an algorithm for problem solving is not given • non-trivial, human-like problem solving • efficiency of a search strategy can be measured by • whether (optimal) solution is founded • the cost of the solution • the cost of search (time and memory) Reasoning and search

33. Search 1 • basic elements of problem definition (states and actions) • initial state(s) • set of possible actions (operators) • state space (set off all states reachable from the initial state by any set of actions – implicit) • goal test (or explicit set of possible goals) • cost of the actions • aim: to find a path from an initial state to a goal state (satisfies the goal test) • search space • generated on the fly • represented by directed acyclic graph (searching tree) Reasoning and search

34. Search 2 • some notions: • node/arc • root • parent/children • branching factor • level • leaf • goal node • path • cost Reasoning and search

35. Search strategies 1 • conflicts: search strategies needed for decision making • search strategies can be grouped according to modification • irrevocable strategies/non-modifiable strategies • no opportunity to withdraw the application of an action • no opportunity to try another applicable action • supposing that all of the chosen action have been selected properly • tentative strategies/modifiable strategies • able to recognize the erroneous or improper application of an action • algorithm can go back to an earlier state to try a new direction when • reaches a stage which does not lead to a goal state • it does not seem promising to resume the search in that direction Reasoning and search

36. Search strategies 2 • search strategies can be grouped according to used knowledge • random search • goal achievement is not insured in finite time • blind search/uninformed search • all of the paths are traversed in a systematic way • no information about „goodness” of the path or node • algorithm distinguishes a goal state from a nongoal state • heuristic search/informed search • specific knowledge about the given problem (heuristic) • estimate the distance from a node to a solution Reasoning and search

37. Search strategies 3 • irrevocable strategies/non-modifiable strategies • no possibility for withdraw/modification of a selected action • no possibility going back on the path from start node (to goal node) • algorithm stores information only the actual node on the path (without any earlier branch) • the applicable actions = the actions applicable on the actual node  selection (with local knowledge, the most promising child) • finding a(n optimal) solution is not guaranteed Reasoning and search

38. Hill-climbing search 1 • the most known non-modifiable search strategy • choosing the next node  heuristic function (minimal in the initial node, maximal in the goal node) • special maximum search: • selects the child of the current node with the highest heuristic value • stops when no child with a higher value than the current node • known as gradient method beyond AI • applicable as minimum search, too Reasoning and search

39. Hill-climbing search 2 Reasoning and search

40. Hill-climbing search 3 • difficulties during hill-climbing search • foothills: local maximum/ global maximum • plateaus: the evaluation function around the current node is essentially flat • ridges: the values of the children nodes are lower, but node with higher value can be reached by the combination of steps • advantage of hill-climbing search: • small memory requirement • getting rid of difficulties • starting from some position (random restart hill-climbing) • selecting the child node at random (simulated annealing) Reasoning and search

41. Backtracking strategy 1 • one of the most significant modifiable search strategies in AI • algorithm stores information about one – the actual path (nodes with other possible branches) • the applicable actions = the actions applicable on the last node of the path  selection (first alternative) • in case of dead-end (the path cannot be pursued)  backtrack • cancel the last part of the path • go back to the previous node with branching • continue with another direction (next alternative) • pursue until a goal node is reached or all paths are examined Reasoning and search

42. Backtracking strategy 2 Reasoning and search

43. Backtracking strategy 3 • advantages of backtracking strategy: • simple, easily implemented • small memory requirement • is great of importance in AI systems: • strategy of inference engines in rule-based expert systems • Prolog systems • disadvantages of backtracking strategy: • (optimal) solution is not guaranteed (endless depth or cycle) • wrong direction is detected only in the dead-end • a part of the path to the same dead-end can be examined several times (it has not memory) Reasoning and search

44. Graph search strategies 1 • the other of the most significant modifiable search strategies in AI • algorithm stores information about all of the examined path (in some depth) from the initial node • move along the path which promises to the best from the aspects of reaching a goal node • all the successors of the node in the end of the selected path are produced  expanding the node • a subgraph is constructed • nodes in the end all of the examined path  frontier/open nodes (nodes that are waiting to be expanded) • does not forget the examined part of the searching graph Reasoning and search

45. Graph search strategies 2 • the main steps of the general algorithm: • Add the initial node to L. (L: list of open nodes) • If L is empty, return failure; otherwise choose a node n from L. • If n is a goal node, stop and return the path from the initial node to n; otherwise, remove n from L, expand n, add the successors of n to L, return to step 2. • different graph search strategies according to: • how to select a node from open list • how to add a node to the open list Reasoning and search

46. Graph search strategies 3 • some important graph search strategies: • uninformed strategies: • depth-first search • breadth-first search • uniform-cost search/lowest-cost-first search • informed strategies: • best-first search • A* search Reasoning and search

47. Depth-first search 1 • uninformed graph search strategy • modification of the general algorithm: • n is the first node from L in step 2. • the successors of n is added to the front of L in step 3. • the open list is used as a stack • one of the nodes at the deepest level is expanded • only in case of dead-end (a nongoal node with no expansion) goes back and expands nodes at shallower levels Reasoning and search

48. Depth-first search 2 Reasoning and search

49. Depth-first search 3 • advantages of the method: • easy implementation • modest memory requirement (b*d, b: branching factor, d: depth) • disadvantages of the method: • can get stuck going down the wrong path (very deep or infinite search tree)  incomplete • can find solution that is more expensive than the optimal ones  not optimal Reasoning and search

50. Breadth-first search 1 • uninformed graph search strategy • modification of the general algorithm: • n is the first node from L in step 2. • the successors of n is added to the end of L in step 3. • the open list is used as a queue • one of the nodes at the shallowest level is expanded • all the nodes at depth d in the search tree are expanded before the nodes at depth d+1 • considers all the path of length 1 first, then all those of length 2, and so on Reasoning and search