1. F oundations for the Situation Calculus ● Paper by: Hector Levesque, Fiora Pirri, and Ray Reiter ● Year of Publication: 1998 Presented on 5 th Nov. 2009 By Akshar Prabhu Desai Salil Joshi Subhajit Datta

2. Abstract ● The language of the situation calculus. ● Foundational axioms ● Axioms for an underlying domain theory. ● Axioms for knowledge and sensing actions.

3. Motivation ● Limitations of First Order Logic (FOL) – Expressive Power ● Representing Dynamic Domains – Where state of system is changing – e.g. A robot world wher robot can move and pick up objects.

4. Overview ● Situation Calculus as SOL ● Situations as action sequences ● Foundational Axioms for Situations ● Domain Axioms ● Frame Problem ● Current Scenario ● Example

5. Situation Calculus ● Actions that can be performed in the world – Actions can be quantified ● Fluents that describe the state of the world ● Situations represent a history of action occurrences – A dynamic world is modeled as progressing through a series of situations as a result of various actions being performed within the world – A finite sequence of actions – A situation is not a state, but a history [Reiter]

6. Language of SitCal ● S 0 : The Initial Situation ● do : action x situation -> situation represented as do(a,S) ● : situation x situation S 1 S 2 means that S 1 is a proper ● subhistory of S 2.

7. Language of SitCal contd.. Operations

8. Foundational Axioms

9. Domain Axioms ● Action Precondition Axiom Example ● Successor State Axiom

10. Domain Axioms cntd.. Example

11. Frame Problem The frame problem How can we derive the non-effects of axioms? E.g. How can we derive that after picking up an object, the robot’s location remains unchanged? This requires a formulae like Poss(pickup(o),s)  location(s) = (x,y) → location(do(pickup(o),s)) = (x,y) Problem: too many of such axioms, difficult to specify all

12. Solution The solution: Successor state axioms Specify all the ways the value of a particular fluent can be changed Poss(a,s)  γ + F (x,a,s) → F(x,do(a,s)) Poss(a,s)  γ - F (x,a,s) → ¬F(x,do(a,s)) γ + F describes the conditions under which action a in situation s makes the fluent F become true in the successor situation do(a,s). γ - F describes the conditions under which performing action a in situation s makes fluent F false in the successor situation. Poss(a,s) → [F(x,do(a,s)) ↔ γ + F (x,a,s)  (F(x,s)  ¬γ - F (x,a,s))]

13. Current Advancements ● SitCal has found applications in fields like – Robotics – Databases – Business Information Systems ● GOLOG (alGOl in LOGic)

14. Example The Problem Ram is standing outside KResit wearing a white T- shirt. Ram picks up a stone, waits for a few minutes, and pelts it at a squirrel.

15. Fluents ● isStanding(place) – whether Ram is standing at the given place or not. ● isWhite – whether Ram is wearing a white t-shirt or not. ● isHarmed – whether the squirrel is harmed. ● hasPicked – whether Ram has picked up a stone

16. Actions ● Pick – Ram picks up a stone or not. ● Wait – Ram waits for a few seconds. ● Throw – Ram throws the stone.

17. Initial State ● holds(isStanding(KResit), S0). ● holds(isWhite, S0). ● -holds(isHarmed, S0). ● -holds(hasPicked, S0).

18. Relating actions with situations ● If Ram pelts the stone and if he had picked it up then the squirrel is harmed. – For all s holds(isHarmed, do(Throw,s)) → holds(hasPicked,do(Wait,s)) ● If Ram pelts the stone it is not in his hand anymore. – For all s -holds(hasPicked, do(Throw, s)). ● If Ram waits after picking up the stone, the stone remains in his hand – For all s holds(hasPicked, s) → holds(hasPicked, do(wait, s)).

19. Frame Problem ● Picking up a stone doesn’t change the color of Ram’s T-shirt. ● Picking up a stone doesn’t harm the squirell. ● Occurance of an action may not affect all the fluents. Therefore we need to specify all the ways the value of a particular fluent can be changed. ● Such list has to be provided manually by the designer.

20. Frame Problem cntd.. ● Tackling the problem: Poss(a,s) → [F(x,do(a,s)) ↔ γ + F (x,a,s)  (F(x,s)  ¬γ - F (x,a,s))] – forall a,s,f holds(f,do(a,s)) ↔ causes(a,s,f) \/ [ holds(f,s) & ~cancels(a,s,f) ] – define causes and cancels predicate. – forall a,s,f causes(a,s,f) ↔ [ a=pick & f=hasPicked] \/ [ a=throw & f = isHarmed & holds(hasPicked,s)] – forall a,s,f cancels(a,s,f) ↔ [ a=Throw & f=hasPicked & holds(hasPicked,s)]

21. Conclusion ● SitCal is expressive enough to represent dynamical domains however it comes with a price. i.e. Undecidability of SOL ● The foundational axioms of SitCal provide sufficient framework to describe dynamic domains and with successor state axiom we can even get rid of Frame problem.

22. References ● McCarthy [SOME PHILOSOPHICAL PROBLEMS FROM THE STANDPOINT OF ARTIFICIAL INTELLIGENCE] 1969 ● Anjum Gupta [Discussion on SitCal] 2004 ● Vassilis Papataxiarhis [Foundations of Databases] 2006

23. Thank You! Questions?