Situation Calculus and Golog

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1 Situation Calculus and Golog Institute for Software Technology 1

2 Recap Situation Calculus Situation Calculus allows for reasoning about change and actions a dialect of the second-order logic uses the concept of situations allows for proving properties solves the frame problem Basic Action Theory implements the situation calculus Foundation Axioms & Action Precondition & Successor Sate Axioms & Unique Name Assumption 2

3 Golog 3

4 Golog Situation Calculus is yet only a theoretical construct Golog (algol for Logic) is based on the Situation Calculus it is a program language for dynamic systems it allows a balance between reasoning/planning and imperative programming (i.e., planning is expensive) it allows complex actions there exist Prolog interpreters for Golog 4

5 Golog a Golog program is based SC it uses the macro Do(,s,s ) Do(,s,s ) will be macro-expand to a SC formula the formula Do states that s is reachable from s by executing the program the syntax supports primitive and complex actions 5

6 Primitive Action: a Golog Syntax (1) has a precondition Poss and the effects are modeled in the successor state axioms walk(r,l) Test action: φ? tests if φ holds in a situation, does not change the situation ( ( x,y).nextto(x,y))? Sequence: 1 ; 2 executes 1 and 2 one after each other walk(r,l);pickup(r,k) 6

7 Golog Syntax (2) Non-deterministic Choice of Actions: 1 2 (randomly) action 1 or 2 will be executed walk(r,a) walk(r,b) Non-deterministic Choice of Arguments: (π x) (x) (randomly) choose an argument x for the action (π x)pickup(r,x) Non-deterministic Iteration: * executes for a not defined number of times (n 0) (pickup(r,l);drop(r,l))* 7

8 Golog Syntax (3) Conditionals: if then 1 else 2 endif based on the truth value of either 1 or 2 is executed if low_battery(r) then recharge(r) else walk(r,party) endif Loops: while do endwhile as long as is true repeat action while low_battery(r) do pickup(r,l);drop(r,l) endwhile Procedures: proc P(x) endproc defines the procedure P with the parameters x proc d(n) (n=0)? d(n-1) endproc 8

9 On the Semantics of Golog Golog programs are macro-expanded to Situation Calculus formulas using the macro Do(,s,s ) what is the meaning of Golog and a program D ( s).do(,s 0,s) the Basic Action Theory entails if a given program lead to the situation s starting from S 0 drawback: the macros are less expressive a program trace can be obtained by a constructive proof of the above sentence some properties are provable: e.g., termination 9

10 Semantics of Golog Parts (1) Primitive Action: a Do(a,s,s )=Poss(a[s],s) s =do(a[s],s) a[s] denotes restoring of all situation arguments in the functional fluents mentioned in a a=goto(location(sam)), a[s]=goto(location(sam,s)) Test action: φ? Do(?,s,s )= [s] s=s [s] denotes restoring of all situation arguments in the fluents mentioned in =( x).ontable(x), [s]=( x).ontable(x,s) Sequence: 1 ; 2 Do( 1 ; 2,s,s )=( s ). Do( 1,s,s ) Do( 2,s,s ) 10

11 Semantics of Golog Parts (2) Non-deterministic Choice of Actions: 1 2 Do(( 1 2 ),s,s )=Do( 1,s,s ) Do( 2,s,s ) Non-deterministic Choice of Arguments: (π x) (x) Do((π x) (x),s,s )=( x)do( (x),s,s )) Non-deterministic Iteration: * Do( *,s,s )=( P).{ s 1.P(s 1,s 1 ) s 1,s 2,s 3.[P(s 1,s 2 ) Do(,s 2,s 3 ) P(s 1,s 3 )]} P(s,s ) 11

12 Semantics of Golog Parts (3) Conditionals: if then 1 else 2 endif expressed by the previous constructs Do(if then 1 else 2 endif)=do([?; 1?; 2 ],s,s ) Loops: while do endwhile expressed by the previous constructs Do(while do endwhile,s,s )=Do(((?; )*;?),s,s ) 12

13 Semantics of Golog Parts (4) A Program: proc P 1 (x 1 ) 1 endproc; ;proc P n (x n ) n endproc; 0 a sequence of procedure declarations plus a main program 0 we define Do(P(t 1,,t n ),s,s )=P(t 1 [s],,t n [s],s,s ) defines a procedure call t i [s] denotes the evaluation of t i in situation s before passing to P represents a call by value Do({ proc ( P,, P 1 P ( v ) endproc; ; proc 1 n 1 ). i n 1 1 ( s 1, s Do(, s, s ) 0 2, v ). Do(, s i i P ( v 1 n, s 2 n ) endproc; }, s, s ) ) n P( v, s i i 1, s 2 0 ) 13

14 A Simple Example =(π x){a(x);?;[b(x) C(x);?]} Do(,S 0,s)? ( x).{( s 1 ).Poss(A(x),S 0 ) s1=do(a(x),s 0 ) ( s 2 ). [s 1 ] s 1 =s 2 [Poss(B(x),s 2 ) s=do(b(x),s 2 ) ( s 3 ).Poss(C(x),s 2 ) s 3 =do(c(x),s 2 ) [s 3 ] s 3 =s]} ( x).{poss(a(x),s 0 ) [do(a(x),s 0 )] [Poss(B(x), do(a(x),s 0 )) s=do(b(x),do(a(x),s 0 )) Poss(C(x), do(a(x),s 0 )) [do(c(x),do(a(x),s 0 ))] s = do(c(x),do(a(x),s 0 ))]} two possible traces: do(b(x),do(a(x),s 0 )), do(c(x),do(a(x),s 0 )) assuming and holds in the related situations 14

15 Situation Calculus vs. Golog Programs Situation Calculus and Golog Programs are related macro expansion uses the basic action theory (e.g., preconditions and successor state axioms) Golog is not a classical program language there are no side effects or states program traces come from theorem proving Axioms ( s).do(,s 0,s) allows for proving of properties of the program 15

16 Executing a Golog Program Golog Program Macro Expansion Logic Sentence in s and S 0 Theorem Prover Basic Action Theory Situation s Constructive Proof 16

17 Planning versus Programming proc toh() while solved do ( o) ( d) move(o,d) endwhile endproc do(toh(),s0) proc toh(n,o,d,h) toh(n-1,o,h,d) move(o,d) toh(n-1,h,d,o) endproc do(toh(3,a,b,c),s0) a b c

18 The famous Elevator Example Elevator several floors each floor has a door each floor has a call button Possible Actions up(n) down(n) trunoff(n) open close Task serve all requests 18

19 Fluents Elevator Part 1 currentfloor(n,s), elevator is on floor n on(n,s), call button is on at floor n Primitive Action Preconditions Poss(up(n),s) m.currentfloor(m,s) m<n Poss(down(n),s) m.currentfloor(m,s) m>n Poss(open,s) true Poss(close,s) true Poss(turnoff(n),s) on(n,s) 19

20 Successor State Axioms Elevator Part 2 currentfloor(m,do(a,s)) a=up(m) a=down(m) currentfloor(m,s) ( n).[(a=up(n) a=down(n)) m n] on(m,do(a,s)) on(m,s) a turnoff(m) Initial Situation currentfloor(4,s 0 ), on(3,s 0 ), on(5,s 0 ) 20

21 Golog Procedures Elevator Part 3 proc serve(n) gofloor(n); turnoff(n);open;close endproc proc gofloor(n) (currentfloor(n))? up(n) down(n) endproc proc serveafloor (π n)[serve(n)] endproc proc control [while ( n)on(n) do serveafloor endwhile]; park endproc proc park if currentfloor(0) then open else down(0);open endproc 21

22 Running the Program Elevator Part 4 prove the following entailment : Axioms ( s).do(,s 0,s) s=do(open,do(down(0),do(close,do(open,do(turnoff(5), do(up(5),do(close,do(open,do(turnoff(3),do(down(3,s 0 )))))))))) action sequence: down(3),turnoff(3),open,close,up(5), turnoff(5),open,close,down(0),open there are multiple solutions 22

23 Golog - Summary Golog is logic program language Golog is based on Situation Calculus axioms its interpreter is a general purpose theorem prover Golog programs are executed for their side effects Golog programs are executed off-line 23

24 Implementation of Golog 24

25 Prolog as Implementation of Logic Prolog is a logic program language Prolog allows for declarative programming (i.e., concentrates on the problem formulation) Prolog can be used to implement a logical theory if some properties hold there is mapping between Prolog and the logical theory behind use a definitional theory use a proper Prolog interpreter Theorem of Clark 25

26 Prolog Interpreter Prolog represents a clause as A:-L 1,,L m. Prolog represents a goal as L 1,,L k. a proper Prolog interpreter has the following properties it interprets the negation not A as negation as failure it does so only if A is a ground literal if A is not ground the interpreter suspend the evaluation until A become ground or it may abort the computation the Clark s Theorem guarantees the right return value of a Prolog program 26

27 A Golog Interpreter we will show a simple reference implementation (Raymond Reiter) it is based on Prolog for theorem proving the Situation Calculus is based on second-order logic Prolog does not support the whole power of secondorder logic the interpreter use some assumptions there are no functional fluents the initial database is closed special assumptions for of action preconditions and relational fluents 27

28 an initial database D S 0 Closed Initial Database is closed iff for every relation fluent F there is only on sentence in the form F(x 1,,x n,s 0 ) F (x 1,,x n,s 0 ) for every non-fluent predicate P there is only on sentence in the form P(x 1,,x n ) P (x 1,,x n ) where P is situation independent for two distinct function symbols: f(x) g(y) for a function symbol: f(x 1,,x n )= f(y 1,,y n ) x 1 = y 1 x n = y n for every term t[x] that mentions the variable x t[x] x (objects) for every term [a] that mentions the variable a [a] a (actions) a restriction on the expressiveness, not allowed: F(A,S 0 ) F(B,S 0 ) ( x) F(x,S 0 ) murderer(caesar)=brutus 28

29 Implementation Theorem Let D be a basic action theory and P a Prolog program obtained by Lloyd-Topor rules for each non-fluent predicate of D S of form 0 P(x 1,,x n ) P (x 1,,x n ) : P (x 1,,x n ) P(x 1,,x n ) for each relational fluent predicate of D S of form 0 F(x 1,,x n,s 0 ) F (x 1,,x n,s 0 ) : F (x 1,,x n,s 0 ) F(x 1,,x n,s 0 ) for each action precondition axiom of D ap of form Poss(A(x 1,,x n ),s) A (x 1,,x n,s) : A (x 1,,x n,s) Poss(A(x 1,,x n ),s) for each successor state axiom of D ssa of form F(x 1,,x n,do(a,s)) F (x 1,,x n,a,s) : F (x 1,,x n,a,s) F(x 1,,x n,do(a,s)) then P provides a correct Prolog implementation of D 29

30 Directives and Definitions :- dynamic proc/2, restoresitarg/3. /* Compiler directives. Be sure some SWI Prolog related directives :- op(800, xfy, [&]). % Conjunction :- op(850, xfy, [v]). % Disjunction :- op(870, xfy, [=>]). % Implication :- op(880, xfy, [<=>]). % Equivalence :- op(950, xfy, [:]). % Action sequence :- op(960, xfy, [#]). % Nondeterministic action choice definition of operators 30

31 Definition of the do Function do(e1 : E2,S,S1) :- do(e1,s,s2), do(e2,s2,s1). do(?(p),s,s) :- holds(p,s). do(e1 # E2,S,S1) :- do(e1,s,s1) ; do(e2,s,s1). do(if(p,e1,e2),s,s1) :- do((?(p) : E1) # (?(-P) : E2),S,S1). do(star(e),s,s1) :- S1 = S ; do(e : star(e),s,s1). do(while(p,e),s,s1):- do(star(?(p) : E) :?(-P),S,S1). do(pi(v,e),s,s1) :- sub(v,_,e,e1), do(e1,s,s1). do(e,s,s1) :- proc(e,e1), do(e1,s,s1). do(e,s,do(e,s)) :- primitive_action(e), poss(e,s). a program e is a list of expressions a query of the form do(e,s 0,S) will be answered by binding the final situation S 31

32 Definition of Substitution sub(x1,x2,t1,t2) :- var(t1), T2 = T1. sub(x1,x2,t1,t2) :- \+ var(t1), T1 = X1, T2 = X2. sub(x1,x2,t1,t2) :- \+ T1 = X1, T1 =..[F L1], sub_list(x1,x2,l1,l2), T2 =..[F L2]. sub_list(x1,x2,[],[]). sub_list(x1,x2,[t1 L1],[T2 L2]) :- sub(x1,x2,t1,t2), sub_list(x1,x2,l1,l2). sub(name,new,term1,term2) Term2 is Term1 with name replaced by new 32

33 Test Conditions (Lloyd-Topor Transformation) holds(p & Q,S) :- holds(p,s), holds(q,s). holds(p v Q,S) :- holds(p,s); holds(q,s). holds(p => Q,S) :- holds(-p v Q,S). holds(p <=> Q,S) :- holds((p => Q) & (Q => P),S). holds(-(-p),s) :- holds(p,s). holds(-(p & Q),S) :- holds(-p v -Q,S). holds(-(p v Q),S) :- holds(-p & -Q,S). holds(-(p => Q),S) :- holds(-(-p v Q),S). holds(-(p <=> Q),S) :- holds(-((p => Q) & (Q => P)),S). holds(-all(v,p),s) :- holds(some(v,-p),s). holds(-some(v,p),s) :- \+ holds(some(v,p),s). holds(-p,s) :- isatom(p), \+ holds(p,s). holds(all(v,p),s) :- holds(-some(v,-p),s). holds(some(v,p),s) :- sub(v,_,p,p1), holds(p1,s). 33

34 Treatment of the hold Predicate holds(a,s) :- restoresitarg(a,s,f), F ; \+ restoresitarg(a,s,f), isatom(a), A. isatom(a) :- \+ (A = -W ; A = (W1 & W2) ; A = (W1 => W2) ; A = (W1 <=> W2) ; A = (W1 v W2) ; A = some(x,w) ; A = all(x,w)). restoresitarg(poss(a),s,poss(a,s)). restoresitarg have to be defined for all atoms taking a situation argument (i.e., action preconditions and fluents) 34

35 The famous Elevator Example Again A Demo in SWI Prolog 35

36 Golog Summary Golog is a program language for dynamic systems Golog used the principles of the Situation Calculus execution of a Golog program is related to theorem proving there exists a reference Golog interpreter for Prolog if some assumptions hold the interpreter correctly executes the program 36

37 Questions? 37

Gerald Steinbauer Institute for Software Technology

Gerald Steinbauer Institute for Software Technology Situation ti Calculus l and Golog Institute for Software Technology 1 Recap Situation Calculus Situation Calculus allows for reasoning about change and actions a dialect of the second-orderorder logic