1. DERI Research Seminar „Some philosophical problems from the standpoint of artificial intelligence“ – John McCarthy & Patrick J. Hayes, 1969 Uwe Keller

2. DERI Research Seminar 03/2004 Overview Motivation Philosophical questions Formalisms: Situation Calculus Remarks & open problems Discussion of literature

3. DERI Research Seminar 03/2004 Motivation A dream of CS researchers: Intelligent Agents  Requires … General representation of the world in terms of which the inputs are interpreted Commitment on: what is knowledge and how is it obtained? ► Some of the major traditional problems of philosophy arise in AI too! More specifically: We want to build a program that …  decides what to do by inferring in a formal language that a certain strategy will achieve its assigned goal. ► Hence, we have to formalize the notions  causality (“causes”), ability (“can”) and knowledge (“knows”).

4. … Philosophical questions Adequate representation of the world „can“, „causes“, „knows“

5. DERI Research Seminar 03/2004 Representation of the world We have to decide …  What structure is the world to be regarded as having  How to represent information about … the world & its laws of change in the machine Adequacy of such a representation ? Def.: A representation is metaphysically adequate if the world could have that form without contradicting the facts of the aspect of reality that interests us.  Examples (for different aspects of reality) … Representation of the world as a collection of particles interacting through forces between each pair of particles Representation of the world as a giant quantum-mechanical wave function Representation as a system of interacting discrete automata

6. DERI Research Seminar 03/2004 Representation of the world (II) Def.: A representation is epistemologically adequate for a person or machine if it can used be practically to express the facts that one actually has about the world. ► None of the given examples are adequate for representing e.g. „John is at home“ or „dogs chase cats“!  Ordinary language … is obviously adequate to express facts that people communicate to each other in ordinary language, but … is not adequate e.g. for expressing what people know about how to recognize a particular face.  Later: Give an epistemologically adequate formal representation of common-sense facts for causality, ability and knowledge

7. DERI Research Seminar 03/2004 Automaton representation Let S be a system of interacting discrete finite automata: S = (Subautomata, Signals, Statefunctions, Outputfunctions) Statefunction: a i (t+1) = A i (a i (t), s 1 (t), …, s k (t)), t=0,1, … (time) Outputfunction: s i (t+1) = S i (a i (t)) Input/Output of S are defined as expected  Simplest examples of systems that interact over time; determ.  Automaton representation: Consider the world as such an S Difficulties with this model in some situations:  Huge number of states, e.g. representing all the knowledge of a person  Geometric info about objects are hard to represent  System of fixed interconnections is inadequate  Most serious objection: Not epistemologically adequate: We don‘t ever know a person well enough to list all his internal states. The kind of information we do have about him has to be represented in some other way („know“) But: We may use it for concepts of „can“, „causes“ and „believes“

8. DERI Research Seminar 03/2004 Notion of „can“ Let S be a system of automata (SOA) without external inputs, p be a subautomata with m output signals. Let S* be the SOA which is obtained by disconnecting these m signals from p and replacing them by external input lines Then  The new system S* has always the same set of states as S  Let π be a condition on the state - „the box is under the bananas“  We‘ll write can(p, π,s) („subautomaton p can bring about the condition π in state s“) … if there is a sequence of outputs of S* which that will eventually put S into a state a‘ that satisfies π (a‘) In other words: In determining what p can achieve, we consider the effects of sequences of ist actions, quite apart form the conditions what it actually will do! This seems to be in line with our common-sense notion of „can“ But needs further elaboration for actual coverage of common-sense notion and for practical purposes!

9. DERI Research Seminar 03/2004 Notion of „can“ (II)  Now suppose that S admits external inputs.  Then, there are two possible definitions of „can“: 1. Assert that can(p, π,s) if p can achieve π regardless of what signals appear on external inputs („cana“)  Note, that we‘re not requiring here that p have any way of knowing what the external inputs were 2. p can achieve a goal if the goal would be achieved for arbitrary inputs by some automaton put in place of p („canb“) Counter-intuitive, since „can“ only depends on the place instead of the function of automaton p

10. DERI Research Seminar 03/2004 Notion of „causes“  Besides the explication of „can“, the automata representation of the world is very suited for defining notions of causalty: causes(p, π,s) = subautomaton p caused the condition π in s causes(p, π,s) if changing the output of p would prevent π.

11. … Formalisms for proving that a strategy will achieve a goal Situation, fluent, future operator action, strategy, result of a strategy & knowledge

12. DERI Research Seminar 03/2004 Situation Definition  A situation s is the complete state of the universe at an instant of time.  Sit := {s | s is situation} Since the universe is too large, we‘ll never describe the universe completely, but give only facts  These facts will be used to deduce further facts about that situation, future situations, situations that persons can bring about  Requires also to consider hypothetical situations  These situations can‘t be completely described, but with sufficient details for some purposes. (Jones-sell-car)

13. DERI Research Seminar 03/2004 Fluents Idea: Partial description of a situation s Definition  A fluent f is a function with domain Sit  Range of f {true,false}: propositional fluent Sit: situational fluent  Fluents are often the values of functions raining(x) is a fluent f s.t. raining(x)(s) is true iff. it is raining at the place x in situation s Also written as: raining(x,s), …

14. DERI Research Seminar 03/2004 Causality Make assumptions about causality by means of a fluent F( π ) (where π is a fluent itself! )  F( π,s) asserts that the situation s will be followed (after some time) by a situation that satisfies the fluent π  Example: All x, All p [ raining(x) and at(p,x) and outside(x) -> F(wet(p))] Expression of physical laws, … Other useful operators  F( π,s): For some situation s‘ in the future of s holds π(s‘)  G( π,s): For all situation s‘ in the future of s holds π(s‘)  P( π,s): For some situation s‘ in the past of s holds π(s‘)  H( π,s): For all situation s‘ in the past of s holds π(s‘)

15. DERI Research Seminar 03/2004 Causality (II) It seems also useful to to define the situational fluent next( π )  The next situation s‘ in the future of s for which π(s‘) holds  If there is no such fluent (not F(π,s)) then next(π) is undefined

16. DERI Research Seminar 03/2004 Actions Fundamental role in our study of actions:  Situational fluent: result(p,σ, s) where … p:Person, σ: Action/Strategy, s:Situation  The value of result(p,σ, s) is the situation that results when p carries out σ, starting in situation s  When σ is nonterminating in that context, this value is undefined With this fluent, we can express certain laws of ability:  e.g.: has(p,k,s) and fits(k,sf) and at(p,sf,s) -> open(sf, result(p,open(sf,k)),s))

17. DERI Research Seminar 03/2004 Strategies Actions can be combined into strategies Simplest case: Finite sequence of actions More general: ALGOL-like compound statement  containing actions written in the form of procedure calling assignement statements and conditional GOTO statements Example strategy: begin face(South) n:=0; b:if n = 17 then goto a; walk-a-block;n:=n+1; goto b; a:turn-right; c:walk-a-block; if name-on-street-sign <> ‚Chestnut Street‘ then goto c; end;

18. DERI Research Seminar 03/2004 Strategies applied (I) In order to apply mathematical theory of computation (Manna)  Write program differently: replace each occurrence of an action A by an assignement statement „s:=result(p,A,s)“ For our example strategy this means … Try to show: John can go home, provided that he initially is in his office begin s:= result(p, face(South), s); n:=0; b:if n = 17 then goto a; s:=result(p,walk-a-block,s);n:=n+1; goto b; a:s:=result(p,turn-right,s); c:s:=result(p,walk-a-block,s); if name-on-street-sign <> ‚Chestnut Street‘ then goto c; end;

19. DERI Research Seminar 03/2004 Strategies applied (II) Given  initial condition: at(John, office(John), s0)  final condition: at(John, home(John), s)  and our strategy (program) S The Theory of Manna allows us to generate a sentence W of FOL Proving W will show that S terminates and that when it terminates s will satisfy at(John, home(John), s) begin s:= result(p, face(South), s); n:=0; b:if n = 17 then goto a; s:=result(p,walk-a-block,s);n:=n+1; goto b; a:s:=result(p,turn-right,s); c:s:=result(p,walk-a-block,s); if name-on-street-sign <> ‚Chestnut Street‘ then goto c; end; Ex s0 ( at(John, office(John), s0) and q1(0, result(John, face(South),s0)) ) -> Ex n,s ( q1(n,s) and if n = 17 then q2(result(John, walk-a-block, result (John,turn-right,s))) else not q1(n+1, result(John, walk-a-block, s))) or Ex s ( q2(s) and if name-on-street-sign <> ‚Chestnut Street‘ then not q2(result(John,walk-a-block, s)) else at(John, home(John), s))

20. DERI Research Seminar 03/2004 Strategies applied (III) Needed facts for proving W  Facts of geography  Fluent „name-on-street-sign“ will have the value ‚ChS‘ at that point  Facts giving the effects of action A expressed as predicates about result(p,A,s) deducible from sentences about s  Axiomschema for induction Ex s0 ( at(John, office(John), s0) and q1(0, result(John, face(South),s0)) ) -> Ex n,s ( q1(n,s) and if n = 17 then q2(result(John, walk-a-block, result (John,turn-right,s))) else not q1(n+1, result(John, walk-a-block, s))) or Ex s ( q2(s) and if name-on-street-sign <> ‚Chestnut Street‘ then not q2(result(John,walk-a-block, s)) else at(John, home(John), s))

21. … Remarks & open problems in extenting the discussed formalism

22. DERI Research Seminar 03/2004 result(p,o,s) … Approximate character of result(p,o,s)  Using the situational fluent result(p,o,s) has 2 main advantages over the can(p, π,s) from section 2: permits more compact and transparent sentences lends itself to the application of the mathematical theory of computation  However it‘s only an approximation to say that an action (other than that which will actually occur) leads to a definite situation (Spontaneous, emotional decision on actions in the real-world vs. artifical operations under partial knowledge)

23. DERI Research Seminar 03/2004 Possible meanings for „can“ A computer program can readily be given much more powerful introspection than a person has (self-simulation possible!) Various possible notions of „can(Program, Cond)“ for a computer program:  The is a subprogram S and room for it in the memory which would achieve Cond if it where in the memory and control were transfered to S. No assertion is made that Program knows S or even knows that S exists  S exists as above and that S will achieve Cond follows from information in memory according to a proof that Program is capable of checking  Program‘s standard problem-solving procedure will find S if achieving Cond is ever accepted as a subgoal

24. DERI Research Seminar 03/2004 Frame Problem As often for formalisms that „hypothetically“ execute State-Transitions:  What changes and what stays unchanged?  Explicit unchanged statements might be needed to proof something. This can be a tedious task

25. Questions ?