1. The Situation Calculus KR and Planning Some final topics in KR
CS 2710, ISSP 2160
The Situation Calculus
KR and Planning
Some final topics in KR

2. Situation Calculus
Planning in propositional logic: Section 7.7 through 7.7.2
Section
Handouts
Note Fall 2015: you are only responsible for some of these notes, specifically what we covered 11/3/2015

3. Other topics in KR Semantic Networks: 12.5.1
[Description Logic: : we didn’t cover this]
[Satisfiability and WalkSat: Intro to Section 7.6; 7.6.2: we didn’t cover this]

4. Actions, Situations, and Events The Situation Calculus
The robot is in the kitchen.
in(robot,kitchen)
He walks into the living room.
in(robot,livingRoom)
in(robot,kitchen,2:02pm)
in(robot,livingRoom,2:17pm)
But what if you are not sure when it was?
We can do something simpler than rely on time stamps…
The Situation Calculus is a logic formalism for representing and reasoning about dynamic domains.

5. Situation Calculus Ontology
Actions: terms, such as “forward” and “turn(right))”
Situations: terms; initial situation, say s0, and all situations that are generated by applying an action to a situation. result(a,s) names the situation resulting when action a is done in situation s.

6. Situation Calculus Ontology continued
Fluents: functions and predicates that vary from one situation to the next. By convention, the situation is the last argument of the fluent. ~holding(robot,gold,s0)
Atemporal or eternal predicates and functions do not change from situation to situation. gold(g1). lastName(wumpus,smith). adjacent(livingRoom,kitchen).

7. Sequences of Actions Also useful to reason about action sequences
All S resultSeq([],S) = S
All A,Se,S resultSeq([A|Se],S) = resultSeq(Se,result(A,S))
resultSeq([a,b,c],so) is
result(c,result(b,result(a,s0)

8. Modified Wumpus World Fluent predicates: at(O,X,S) and holding(O,S)
In our simple world, only the agent can hold a piece of gold, so for simplicity, only the gold and situation are arguments
Initial situation: at(agent,[1,1],s0) ^ at(g1,[1,2],s0)
But we want to exclude possibilities from the initial situation too…

9. Initial KB
All O,X (at(O,X,s0)  [(O=agent ^ X = [1,1]) v (O=g1 ^ X = [1,2])])
All O ~holding(O,s0)
Eternals:
gold(g1) ^ adjacent([1,1],[1,2]) ^ adjacent([1,2],[1,1]) etc.

10. Goal: g1 is in [1,1] Planning by answering the query:
Exists S at(g1,[1,1],resultSeq(S,s0))
Solution:
At(g1,[1,1],resultSeq([go([1,1],[1,2]),grab(g1),go([1,2],[1,1])],s0))
The situation designated by the second term:
result(go([1,1],[1,2]),result(grab(g1),result(go([1,1],[1,2]),s0)))
Let’s look at what has to go in the KB for such queries to be answered...

11. Possibility and Effect Axioms
Possibility axioms:
Preconditions  poss(A,S)
Effect axioms:
poss(A,S)  changes that result from that action

12. Axioms for our Wumpus World
For brevity: we will omit universal quantifies that range over entire sentence. S ranges over situations, A ranges over actions, O over objects (including agents), G over gold, and X,Y,Z over locations.

13. Possibility Axioms The possibility axioms that an agent can
go between adjacent locations,
grab a piece of gold in the current location, and
release gold it is holding

14. Effect Axioms
If an action is possible, then certain fluents will hold in the situation that results from executing the action
Going from X to Y results in being at Y
Grabbing the gold results in holding the gold
Releasing the gold results in not holding it

15. Frame Problem We run into the frame problem
Effect axioms say what changes, but don’t say what stays the same
A real problem, because (in a non-toy domain), each action affects only a tiny fraction of all fluents

16. Frame Problem (continued)
One solution approach is writing explicit frame axioms, such as:
(at(O,X,S) ^ ~(O=agent) ^ ~holding(O,S))  at(O,X,result(Go(Y,Z),S))
If something is at X in S, and it is not the agent, and also it is not something the agent holds, then O is still at X if the agent moves somewhere.
F fluents and A actions: O(FA) axioms needed
We can do something more efficient than this

17. Frame Problem What stays the same?
A actions, F fluents, and E effects/action (worst case). Typically, E << F
That is, the effects of an action are typically only a small set of all the things that could change
Want O(AE) versus O(AF) solution

18. “Solving” the Frame Problem
For each fluent, have successor-state axioms:
Action is possible 
(fluent is true in result state 
action’s effect made it true v
it was true before and action left it alone)
Each of the E effects of each of the A actions is mentioned exactly once, so O(AE) axioms needed
Note: we will return to this point later, after going through the wumpus world example

19. Trace through reasoning so far on board;
Initial KB (reminder)
All O,X at(O,S,s0)  [O=agent ^ X = [1,1]) v (O=g1 ^ X = [1,2])]
All O ~holding(O,s0)
Eternals:
gold(g1) ^ adjacent([1,1],[1,2]) ^ adjacent([1,2],[1,1]).
Trace through reasoning so far on board;
state space handed out
At this point, we are switching to variables being small case, constants upper case, following the text

20. 4-5 are Successor-State Axioms
At(Agent, x, s)  Adjacent(x,y)  Poss(Go(x,y),s)
Gold(g)  At(Agent,x,s)  At(g, x, s)  Poss(Grab(g),s)
Holding(g,s)  Poss(Release(g),s)
Poss(a,s)  Holding(g,Result(a,s))  a = Grab(g) v (Holding(g,s)  a  Release(g)))
Poss(a,s)  (At(o,y,Result(a,s))  (a = Go(x,y)  (o = Agent v Holding(o,s))) v (At(o,y,s)  ¬(z y  z  a = Go(y,z)  (o = Agent v Holding(o,s))))

21. More explicit version; Replaced existential with universal in 5
All x,y,s ((At(Agent, x, s)  Adjacent(x,y))  Poss(Go(x,y),s)
All g,x,s ((Gold(g)  At(Agent,x,s)  At(g, x, s))  Poss (Grab(g),s))
All g,s (Holding(g,s)  Poss(Release(g),s))
All a,s,g (Poss(a,s)  (Holding(g,Result(a,s))  (a = Grab(g) v (Holding(g,s)  a  Release(g))))
All a,s,o,y,z (Poss(a,s)  (At(o,y,Result(a,s))  ((a = Go(x,y)  (o = Agent v Holding(o,s))) v (At(o,y,s)  ¬(a = Go(y,z)  y  z  (o = Agent v Holding(o,s)))))))
Justification: previous 5 has ¬(z … the change is justified because this is equivalent to all z ¬…

22. Same as previous, but without comments
All x,y,s ((At(Agent, x, s)  Adjacent(x,y))  Poss(Go(x,y),s)
All g,x,s ((Gold(g)  At(Agent,x,s)  At(g, x, s))  Poss (Grab(g),s))
All g,s (Holding(g,s)  Poss(Release(g),s))
All a,s,g (Poss(a,s)  (Holding(g,Result(a,s))  (a = Grab(g) v (Holding(g,s)  a  Release(g))))
All a,s,o,y,z (Poss(a,s)  (At(o,y,Result(a,s))  ((a = Go(x,y)  (o = Agent v Holding(o,s))) v (At(o,y,s)  ¬(a = Go(y,z)  y  z  (o = Agent v Holding(o,s)))))))

23. A return to complexity
Each of the E effects of each of the A actions is mentioned exactly once, so O(AE) axioms needed
Notes:
an effect may be to make a fluent true (add it) or to make it false (delete it)
Counting axioms is a bit arbitrary, since a single axiom may mention a disjunction of add effects and/or a disjunction of delete effects (see the holding axiom for the blocks world)
It is true that each of the add or delete effects of each action is mentioned once

24. A return to complexity
Each of the E effects of each of the A actions is mentioned exactly once, so O(AE) axioms needed total action mentions
Another schema for successor state axioms:
Action is possible 
(fluent is true in result state 
(action is one of the actions that makes it true (add effect) v
it was true before and action is not one of the actions that makes it false (delete effect))
Thus, for each fluent (possible effect), we mention exactly once each action that has it as an effect (either making it true or making it false)
Or, for each action, it is mentioned exactly once in a successor state axiom for each of its effects
Or, for each action, it is mentioned exactly once in the successor state axioms for each of its effects

25. Fall 2014
In class exercise – the blocks world [handout]

26. Qualification Problem
Ensuring that all necessary conditions for an action’s success have been specified. No complete solution in logic. KR/planning designers have to decide how much detail to go into.

27. What did we see? A sophisticated KR scheme
Important Problems in planning:
Addressed by successor-state axioms (Reiter 1991)
Frame Problem (what stays the same?)
Ramification Problem (implicit effects, such as that gold moves too if the agent moves and it is holding the gold)
Not addressed completely in logic
Qualification Problem
Concepts for planning, such as fluents and situations
Planning as search

28. Stopped here, Fall 2014

29. Semantic Networks Graphical aids for visualizing the knowledge base
Efficient algorithms for inferring properties based on category membership
Often, correspond to a subset of first-order logic
Many variants
All distinguish among individual objects, categories of objects and relations among objects

30. Example See figure 12.5 (next slide) Specify what edges and nodes mean
In Figure 12.5, indivs and categories look the same
memberOf(indiv,category)
sisterOf(indiv,indiv)
subsetOf(category,category)
hasMother(indiv,indiv)

32. (Note: this does not say that each person has a mother)
Semantic Networks
Is hasMother(persons,femalePersons) consistent with the representation?
Nope: hasMother is a relation between individuals
cat1-- label  cat2 means:
all X (X in cat1  (all Y label(X,Y)  Y in cat2))
(Note: this does not say that each person has a mother)

33. Semantic Networks cat – label  value
All X (X in cat  label(X,value))

34. Inheritance Inheritance is efficient and convenient
Trace paths from individuals to categories, inheriting properties as you go
In Figure 12.5, how many legs does John have? Most specific (nearest) information wins

35. Semantic Networks
In this type of semantic network, only binary relations are possible
A richer representation is possible by reifying propositions and events (example: SNePS)
This forces creation of a rich ontology of reified concepts; many current ideas originated in semantic network systems

36. Description Logics http://en.wikipedia.org/wiki/Description_logic
This won’t be tested on the exam, but I want you to know what description logics are
Subset of full first order logic; a family of logics of increasing expressiveness; well studied; most are decidable; good link between theory and practice.

37. We stopped here Fall 2012.

38. Proof methods Proof methods divide into (roughly) two kinds:
Application of inference rules:
Legitimate (sound) generation of new sentences from old.
Resolution
Forward & Backward chaining
Model checking
Searching through truth assignments.
Improved backtracking: Davis--Putnam-Logemann-Loveland (DPLL)
Heuristic search in model space: Walksat.

39. Model Checking Two families of efficient algorithms:
Complete backtracking search algorithms: DPLL algorithm. You read this on your own for the midterm.
Incomplete local search algorithms
WalkSAT algorithm

40. The DPLL algorithm
Determine if an input propositional logic sentence (in CNF) is
satisfiable. This is just backtracking search for a CSP.
Improvements:
Early termination
A clause is true if any literal is true.
A sentence is false if any clause is false.
Pure symbol heuristic
Pure symbol: always appears with the same "sign" in all clauses.
e.g., In the three clauses (A  B), (B  C), (C  A), A and B are pure, C is impure.
Make a pure symbol literal true
Unit clause heuristic
Unit clause: only one literal in the clause
The only literal in a unit clause must be true.
Note: literals can become a pure symbol or a
unit clause when other literals obtain truth values. e.g.

41. The WalkSAT algorithm Incomplete, local search algorithm
Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses
Balance between greediness and randomness
See figure 7.18 (on your own)

42. WrapUp: Situation Calculus
Planning in propositional logic: Section 7.7 through 7.7.2
Section
Handouts

43. WrapUp: Other topics in KR
Semantic Networks: [not covered Fall 2014]