First-Order Logic and Plans Reading: C. 11 (Plans)

2 Grades Handed back at end of class. Midterms will only be given to owners All of your grades are online in the Courseworks gradebook. You can check. Class participation grade Represents your effort to date For remainder of semester, I will use class list to call on people, particularly those who are less vocal. This will give everyone a chance. Of course, you must be present to take advantage of your turn.

3 Tournament Start date: Tuesday, Nov.16 th You must sign up by Nov. 11 th, midnight to participate. It’s optional. 2 person games only 1 minute time limitation per move Passing to second round: Each program will play two times against every other program, as first and as second player Top 50% will move on to second round Number of rounds will be determined in part by number of entrants If you decide to enter, you must include a script to run your program with proper parameters You determine search depth that you will play at

4 Extra Credit 5 points for entering and finishing without faults No crashes < 50% timeouts 5 points for moving to round 2 10 points for moving to round 3 (or top 50% of round 2) 10 points for top 4 10 points for #1 finish

5 Example Family trees What does the model look like? Father-of Mother-of Sibling What can we infer? Cousin Ancestors

6 To Make Inferences in FOL Method 1 Unification of variables with literals (in the KB) Generalized Modus Ponens Forward-chaining or Backward-chaining Method 2 Resolution

7 Unification We want to find a substitution  such that x and y match literals Unify ( ,  ) =  if  =  Some examples

8 Unification for example

9 P`1= father-of(Kathy)=Michael P 1 = father-of(x)=y  ={x/Kathy,y/Michael} q=ancestor(x,y) q`=ancestor(Kathy,Michael)

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11 Example inference using forward chaining (production systems)

12 Properties of forward-chaining Sound and complete for first-order definite clauses Datalog is first-order definite clauses and no functions May not terminate in general if is not entailed This is unavoidable: entailment with definite clauses is semi-decidable Forward chaining is widely used in deductive databases

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14 Example inference using backward chaining

15 Properties of backward-chaining Depth-first recursive proof search: space is linear in size of proof Incomplete due to infinite loops Fix by checking current goal against every goal on stack Inefficient due to repeated subgoals (both success and failure) Fix using cache of previous results (extra space!) Widely used (without improvements!) for logic programming (e.g., Prolog)

16 Refutation and generalized resolution Inference with generalized modus ponens is not complete for FOL Proof by refutation and generalized resolution is complete But slow and only semi-decidable (cannot know that a sentence is not entailed)

17 Refutation/Resolution Introduce negated sentence Convert to a CNF (disjunction of terms, or “literals”) Apply resolution search to determine satisfiability (SAT) or unsatsifiability (UNSAT) SAT, then not entailed Semi-decidability implies there may be a SAT solution we can never find UNSAT, then entailed

18 Conversion to CNF 1. Eliminate  using ⌐ and V 2. Push ⌐ inward using de Morgan’s laws 3. Standardize variables apart 4. Eliminate  using Skolem functions 5. Move  to front, and drop 6. Distribute Λ over V to get CNF 7. Split into separate clauses (remove Λ) 8. Standardize variables apart (across clauses)

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20 Example using Resolution

21 Planning An approach to problem solving that combines logic with search Cognitively motivated? Suited for more complex problems

22 Problem solving so far Typical Problem -> language -> representation + algorithm -> solution For example, recall the 8-puzzle Choose a representation Choose a heuristic Choose a search algorithm

23 General Problem solving Ideally, we would like to describe a problem at a high level, let AI take care of the rest Problem -> representation -> solutions generalgeneral language algorithm

24 Required Planning Components Expressive Formal Language: Initial state of world Description of the agent’s goal Description of the possible actions that can be performed Solver generates a sequence of actions which achieve the goal Key idea will be to use structure to extract admissable heuristics automatically

25 STRIPS language (Fikes&Nilsson 71) State is a conjunction of positive ground literals No variables and no functions AT(,Kathy,Gate70), Weight-limit-lifted(CO36) Goal is a conjunction of positive ground literals At(Kathy,Denver)Λ⌐onboard(Kathy,C036) Action schemas Conjunction of positive literals as preconditions Conjunction of positive and negative literals as effects

26 Action Schema Action: Go(p,x,y) Precond: At(P,x) Effect: ⌐At(p,x)At(p,y) delete list add list

27 Comments: STRIPS Closed World assumption Unmentioned literals are false (no need to explicitly list out) w/an open world assumption unmentioned literals are unknown STRIPS assumption Every literal not mentioned in the “effect” of an action schema remains unchanged

28 Travel Example

29 STRIPS Expressiveness Literals are function free: Any action scheme can be propositionalized E.g., 10 planes, 5 airports, then Fly(p,from,to) can be expressed as 250 purely propositional actions Cannot address the ramifications of an action The people, packages, wine, etc. on an airplane also change location when the plane flies Cannot express disjunctive goals At(Kathy,Denver)ΛDay(Friday) V At(Kathy,Denver)ΛDay(Saturday) No conditional effects No way to say “At(Kathy,Denver)” an effect of Fly(Kathy,Co36,Denver) only if ⌐crash(CO36)

30 Planning Algorithms Forward vs. Backward search Partial-order vs. Total-order Partial: search in plan space Can work on several subgoals independently, solve and then combine subplans Total: search in state space Search linear sequence of actions directly connected to the start or the goal

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32 Forward State Space Search State: positive ground literals Initial state: set of initial world literals Search operator: Choose an action A that 1.preconditions are satisfied (perhaps finding a unifier) Construct the new search state Add all positive effects; remove all negative effects Solution: when current state is goal state