Introduction on Situational Calculus

Transcription

1 Introduction on Situational Calculus Fangkai Yang Department of Computer Sciences The University of Texas at Austin September 24, 2010 Fangkai Yang September 24, / 27

2 Outline 1 Reasoning with Actions 2 Situational Calculus 3 Frame Problem Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

3 Outline 1 Reasoning with Actions 2 Situational Calculus 3 Frame Problem Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

4 Goal of Artificial Intelligence 1 Goal of artificial intelligence: building programs capable of acting intelligently in the world. 2 Specifically, a program that has a general representation of the world cf. data structures are also the abstract representation of the world for all real problem. can decide what to do to achieve its goal by inferring in a formal language the goal that AI aims for! 3 Therefore, in order to achieve its goal, an intelligent program needs to manipulate what it knows: facts about the world what it can do: conditions and effects of the actions that it can execute what it wants to achieve: goal and some reasoning problems Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

5 Monkey and Banana Problem A monkey is in a room. Suspended from the ceiling is a bunch of bananas, beyond the monkey s reach. In the corner of the room is a box. How can the monkey get the bananas? The solution is that the monkey must push the box under the bananas, then stand on the box, and then grab the bananas. how to build an intelligent monkey program that can decide what to do to achieve its goal, once the goal is given? Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

6 A Monkey Program 1 Something must be told to the program: what it knows: the location of banana and the box, the initial location of itself what it can do: it can walk to a location, move a box, climb up to the box, and grasp the banana what it wants to do: grasp the banana 2 Something must NOT be told to the program: how to grasp the banana what actions should be executed one by one to achieve its goal. 3 We need the program to find such solution by itself: a planning problem, intelligence! Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

7 Epistemic Method John McCarthy s 1969 paper: SOME PHILOSOPHICAL PROBLEMS FROM THE STANDPOINT OF ARTIFICIAL INTELLIGENCE proposed that First Order Logic should be used as a formal language for world model and inferring, which later becomes the central problem of Knowledge Representation and Reasoning (KR&R). Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

8 Why logic? 1 Mathematically sound its formulas and sentences are inductively defined and unambiguous 2 Well-defined formal semantics the semantics of logics are mostly based on an algebra structure /set theory 3 Philosophical considerations metaphysically adequate and epistemically adequate (further refer to the literature) Specifically, he proposed an formal language called Situational Calculus to represent dynamic world, reason with action and generate plans to achieve goals. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

9 Outline 1 Reasoning with Actions 2 Situational Calculus 3 Frame Problem Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

10 Situational Calculus The basic ideas of situational calculus are situations,actions,fluents. - Actions are what makes the dynamic world change from one situations to another when performed by agents - Fluents are situation-dependent functions used to describe the effect of actions. - A situation is a complete state of the universe at an instance of a time A situation calculus is a first-order language with the above sorts on its objects. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

11 Blocks World The following is a classical AI problem called blocks world Some blocks of equal size (namely, blocks A, B, C,...) can be arranged into a set of towers on a table. A one-handed robot can be directed to grasp any block that is on the top of a tower, and either add it to the top of another tower or put it down on the table to make a new tower. Given a pattern of a set of towers, the agent should decide a sequence of actions, executing which can build up the set of towers. To represent blocks world by situational calculus, one needs to decide The set of actions available for the agents to execute The set of fluents needed to describe the changes the actions will have on the world. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

12 Actions By analyzing the above domain, it is easy to see that the agent can perform the following actions: stack(x, y): put block x on block y, provided the robot is holding x, and y is clear, i.e. there is no other block on it; unstack(x, y): pick up block x from block y, provided the robot s hand is empty, x is on y, and x is clear; putdown(x): put block x down on the table, provided the robot is holding x; pickup(x):pick up block x from the table, provided the robot s hand is empty, x is on the table and clear. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

13 Fluents and Situations The fluents are used to describe the changes that the actions have on the world. We introduce the following fluents: handempty: true in a situation if the robot s hand is empty holding(x): true in a situation if the robot s hand is holding block x; on(x, y): true in a situation if block x is on block y; ontable(x): true in a situation if block x is on the table; clear(x): true in a situation if block x is the top block of a tower, i.e. the robot is not holding it, and there is no other block on it. The situations are a set of labels, i.e., s 1, s 2,...,. Note: fluents, actions and situations are all objects in the first order language. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

14 Axiomatizations In order to express certain actions are executed/executable at certain situations or certain fluent holds at certain situation, we introduce the following domain-independent predicates and functions: Holds(p, s): fluent p is true in situation s; do(a, s): the situation that results from performing action a in situation s; Poss(a, s): action a is executable in situation s. Now we can express handempty is true at situation s 1 as Holds(handempty, s 1 ) and stack(a,b) is executed at situation s 2, which makes B unclear at the resulted situation as Holds(clear(B), do(stack(a, B), s 2 )) Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

15 Initial State Assume the blocks world starts from the following initial state: ontable(c, S 0 ) on(b, C, S 0 ) on(a, B, S 0 ) ontable(d, S 0 ) clear(a, S 0 ) clear(d, S 0 ) holding(e, S 0 ) (1) (2) (3) (4) (5) (6) (7) For the time being, we only specify positive facts, because we assume we seldom use negative facts such as clear(b, S 0 ). Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

16 A Complete Axiomatization In the following, all free variables are universally quantified. First, the sufficient and necessary conditions for each action to be executed: Poss(stack(x, y), s) holding(x, s) clear(y, s) (8) Poss(unstack(x, y), s) on(x, y, s) clear(x, s) handempty(s) (9) Poss(pickup(x), s) ontable(x, s) clear(x, s) handempty(s) (10) Poss(putdown(x), s) holding(x, s) (11) Fangkai Yang September 24, / 27

17 A Complete Axiomatization (ctd) Second, how executing action stack affects each fluent holding(u, do(stack(x, y), s)) holding(u, s) u x (12) handempty(do(stack(x, y), s)) (13) on(u, v, do(stack(x, y), s)) (u = x v = y) on(u, v, s) (14) clear(u, do(stack(x, y), s)) u = x (clear(u, s) u y) (15) ontable(u, do(stack(x, y), s)) ontable(u, s) (16) Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

18 Reasoning with Actions Now we have sentences specifying the initial state (1) (7), denoted as, and the axioms for stack (13) (16). We can perform the following reasoning with action stack. Denote these sentences as Σ stacking E onto D will make E above D in the resulted situation. Show the following:, Σ = on(e, D, do(stack(e, D), S 0 )) stacking E onto D will not change the fact that A is on B. Show the following:, Σ = on(a, B, do(stack(e, D), S 0 )) Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

19 Problem 1 Try to write first order sentences to describe how executing other actions (i.e. unstack, pickup, and putdown) affects each fluent. There should be 15 sentences in all. 2 Based on your axiomatization and the initial state (1) (6), assuming initially the hand of the robot is empty, try to show that: (i)moving A onto table will make B clear; (ii) moving A onto table will make D still on the table. 3 Introduce the following action: move(x, y): move block x to position y, provided that block x is clear to move, where a position is either a block or the table and try to axiomatize it by situational calculus, and reason with the new action over the problem 2. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

20 Using Situational Calculus for Planning Problem 1 By situation calculus, one can write a complete logical specification for changes caused by actions in a world, by a set of first order sentences, thus it forms a complete representation on the world the agent locates 2 The agent can decide whether a given goal can be achieved in the world by performing a sequence of actions, which is formulated as a theorem proving task in situation calculus: T = s.g(s) 3 For the time being, we will focus on the representational issues of Situational Calculus, rather than computational issues. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

21 Outline 1 Reasoning with Actions 2 Situational Calculus 3 Frame Problem Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

22 An Observation 1 We have formulated a complete axiomatization for the block world, in which we specify the sufficient and necessary of each action to execute in any situations, and fully specify the effects of each action 2 However, such formulation is not natural. In real life, we don t really remember to say something which. for example, implies by (16). 3 (16) implies this: if we stack x onto y, and initially y is on the table, then y is still on the table. (16) also says this: if we stack x onto y, for a block u other than x, if u is initially on the ground, u is still on the ground afterwards. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

23 Frame Problem 1 In real life, when talking about the effects of an action, we usually only talk about the change it will bring, but seldom say (or remember!) what are not changed by the action. 2 Specifically, when talking about the effect of action stack, the following effect axioms are much more natural: on(x, y, do(stack(x, y), s)) Poss(stack(x, y), s) (17) clear(x, do(stack(x, y), s)) Poss(stack(x, y), s) (18) clear(y, do(stack(x, y), s)) Poss(stack(x, y), s) (19) handempty(do(stack(x, y), s)) Poss(stack(x, y), s) (20) holding(x, do(stack(x, y), s)) Poss(stack(x, y), s) (21) Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

24 Is it correct? Now we have facts for initial state (1) (7), denoted as, the precondition of stack (8), and a revised version of effect axioms (17) (21), denoted as Σ. Can we entail the following? (Show it!), Σ = on(e, D, do(stack(e, D), S 0 )) Yes! Can we entail the following? (Show it!), Σ = on(a, B, do(stack(e, D), S 0 )) No... So Σ is no longer a complete logical axiomatization of action stack. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

25 Commonsense Law of Inertia We try to formalize the effect axioms of stack like this: apart from (17) (21), we add one more statement: nothing else changes. This is called the commonsense law of inertia. for example, we can add one more sentence saying that stack(x, y) doesn t change ontable(y) if y is initially on table, and also doesn t change block u if u is not x: ontable(y, s) ontable(y, do(stack(x, y), s)) ontable(u, s) ontable(u, do(stack(x, y), s)) x u However, this solution is cumbersome. Once we add one more fluent into domain, e.g. color(x), we need to specify that color doesn t change for all actions because usually actions only change a small number of fluents. If we later add one more action, we also need to check all fluents to see if any frame axioms are needed. So it is not practical to enumerate explicitly the nothing else. Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

26 McCarthy s attempt Towards the statement nothing else changes, McCarthy proposed the following statement: Normally, given any action and any fluent, the action doesn t change the fluent. So we can specify stack is abnormal relative to the fluent it changes: s.ab(on(x, y), stack(x, y), s) (22) s.ab(clear(x), stack(x, y), s) (23) s.ab(handempty, stack(x, y), s) (24) s.ab(holdings(x), stack(x, y), s) (25) and for any fluent p, situation s, if action a is not abnormal, then p continues to hold after the action is executed, i.e., Holds(p, s) ab(p, a, s) Holds(p, do(a, s)) (26) Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27

27 Problem: Does it work? Now we have initial state, the precondition of stack (8) and a revised version of effect axioms, denoted as Σ, and declarations about abnormality and generic frame axiom (22) (25), (26), denoted as Γ. Now can we entail the following?, Σ, Γ = ontable(d, do(stack(e, A), S 0 )) If we can, show it. If we can t, what do we miss here? Fangkai Yang (fkyang@cs.utexas.edu) September 24, / 27