Csirmaz, László

English

Csirmaz László

University of Szeged

Structure of program runs of non-standard time B y L . CSIRMAZ J 1. Introduction In this section the set of natural numbers is denoted by J f , and the set of Peano axioms (with+and-only) by PA. In our point of view, a program is a finite se-quence of labelled statements. The labels are (distinct) natural numbers. Each state-ment is either an assignment of the form "v*-x" where v is a variable symbol and T is a term of Peano arithmetic containing operator symbols+and-only, or is an if-statement of the form " IF x THEN / " where x is a quantifier free formula of PA and IdJV is a label. Denote by Vp the (finite) set of variable symbols occuring in the program p and let Lp be the set of the labels of the statements and h£Jf\Lp (the "halt" label). A run of the program p is a sequence (/,-, where (i) /;€Lp U {A} and / ¡ : Vp-~Jf is a valuation of the variables for every i ^ J f ; (ii) if l=h then / ,+!=/;, fi+i=fi-, (iii) if the statement labelled by /,• is " V ^ T " then |Z ; +1 if Z ,+ l€£ p , I h otherwise, ifi(w) if w£Vp, w v, ,+ l , other ise, (iv) if the statement labelled by /, is " IF x THEN / " then I if l£Lp and xlfii is t rue, /,-+1 = • h + 1 if h + ^ L p and xUi] is false, h otherwise, fl +1 ~ fi-i +1 The run of the program halts, if lt = h for some i [cf. 4]. 326 L. Csirmaz It is well-known that every program can be written in this form because we have made no restriction on the content of variables [5]. Moreover, there is a straighforward way to prove partial correctness of programs of this type: assign formulas to every element of Lp U {h} and prove (say, from the Peano axioms) that if the formula assigned to /,• is satisfied then after executing the statement belonging to the formula assigned to / i + 1 will be satisfied, too. Then, if the run halts, the formula assigned to H is satisfied, i.e. the program is partially correct. This method is the so called Floyd—Hoare derivation [6]. It is easy to give a rigorous proof that if a program has a Floyd—Hoare deriva-tion, i.e. if we may assign formulas and prove what we have to prove then the program is partially correct. But, alas, there are programs which always halt, al-ways give the same result but have no Floyd—Hoare derivation. For example let «jo be a formula of Peano arithmetic such that neither q> nor its negation are prov-able from PA. Our program checks whether its only input is the Godel number of a proof of <p from PA. If it is, it prints 1, if not, i.e. if the input is either not a Godel number of a proof or does not prove cp, it prints 0. Our program always halts because it is a decidable property to be a proof of a concrete formula, and always prints 0 because there is no proof of cp. Moreover, we can not prove this (from PA) because then we would be able to prove in PA that there is a non-provable formula, i.e. that PA is consistent, which is impossible. This difficulty vanishes if we allow the program to operate not only on J f but on any model of PA and to run not only through finite time but through non-standard time. This idea is behind the concept of continuous trace, it simulates the non-standard runs of programs, see [1], section 3 of [7], and [8]. 2. Notation, definitions Denote by L the set of classical first order formulas of type t, where t is the similarity type of arithmetic, i.e. it consists of " + , 0,1" with arities "2, 2, 0, 0", respectively. PA denotes the following (infinite) set of axioms: PI x + l j i 0 P2 Jt+1 = y + 1 — x = y P3 J t+0 = x P4 * + ( y + l ) = (x+y) + l P5 x - 0 = 0 P6 * ' ( y + l ) = ( * ^ ) + JC -P7 for all formulas <p with x as free variable |>(0)A V*($>(*) - <p(x+\))] - V*<?(*)• We will use other relation and function symbols, as e.g. x<y or rem (x, y) which are definable in PA. We introduce the bounded quantifiers (Vx~=:y)<p(y)'~ -^Vx(x<y—<p(x)), etc., too. The following reformulation of the axiom of in-duction Structure of program runs of non-standard time 327 P8 [(Vy < x)(p(y) - <p(x)] - \/x<p(x) may be obtained from P7 substituting cp(x) by (\/y<x)(p(y) [cf. 2]. The inclusion AcB allows the sets A and B be equal. To save space we use vector notation in place of sequence of symbols of same type. E.g. we write (p(x) instead of cp(x1, x2, ..., xn), etc. The dimension of vectors is always clear from the context. Definition. Let A be any model of PA with universe A. Let <p (x1; ..., x„, yu ..., y„) be a formula of PA so that P A h V * 1 . . - . V * , B ! f t . . . 3 l w ( x 1 ^ (2.1) Let qa = (ql, ..., q") be a sequence of length n of elements of A for every a£A. We say that the sequence (qa)aiA is a continuous trace (ct in short) of <p if A N ?fl+1) for every a£A; (2.2 for every formula \j/ of PA and every sequence p of elements of A A N [ > ( W ) A A (ip(.qa,P) - A t(§a,P)- (2-3) aZA aiA In the remaining part of this paper we fix the model A of PA, the formula cp, the sequence q0 and its length n. Whenever we speak about continuous traces we mean ct of cp with first element q0 in A. We shall need the notion of coding function of sequences. Let rem (x, y) be the remainder when x is divided by y and define the ordered pair (x,y) as (x+y) • • Let moreover the triplet (x,y, z)=(x, (y, z)> and define the formulas PAIR (z), SEQ (u) and the functions LENGTH (it), ELEM (u, i) as follows. PAIR(z) = V u ( i c u S zA(u + l ) - ( u + l) > z — z ^ u -u + u); SEQ («) = PAIR (u)A Vx Vy(u — (x, y) — PAIR (y)); {n if SEQ (u) and u = (x, y, n), 0 otherwise, frem (m, 1 + ( / + 1 ) • b) if SEQ (u) and u = (m, b, n) and i < n, ELEM (m, 0 = | 0 o t h e r w . s e A straighforward proof shows that PA | - PAIR (z) - 3 ! x 3 ! y ( z = <x, j;» PA t- Vu 3 !n (LENGTH (u) = n) PA | - V« V i 3 !x(ELEM («, i) = x). We say that m6 A is a sequence if A N SEQ (w), its length is n if A1= LENGTH (M) = n and its i'-th element is a if AN ELEM (w, i)—a. Note that 0 is a sequence of length 0. The following theorem says that every sequence can be lengthened by 1 [3]. 328 L. Csirmaz Theorem 1. P A | - V " Vz 3 D ( S E Q ( M ) -{SEQ (v) A LENGTH (v)=LENGTH (M) +1A (V/<LENGTH(w))(ELEM(w, i)=ELEM(t; , /))A ELEM (d, LENGTH («))=2}). • 3. The result First we prove some lemmas. We remind that A, q>, q0 and n are fixed. Lemma 1. There are a formula $ of PA and a unique sequence (qa)aiA of se-quences of length n of elements of A with the given q0 such that P A I - V M V X J . . . VX„ 3 . . . 3 I y„$(m, xx, ...,x„, ..., y„) PA f- Mm Mx V j Vz($(m, x,y)h$(m + \, x,z) - <p(y, z)) A |= <P(a, q0, qa) for every a£A A 1= <p(9fl,ia+i) for every a£A. Proof. Let <P1(m) be Vx1...Vx„ Bu%(x1, ..., x„,u, m). where 1 is "u is a sequence of length m + 1 such that every element of u is a sequence of length n, ihe elements of the 0-th element of u are x1, ..., x„ in this order and for every i<m the z'-th element y and the (7+ l)-st element z of u satisfy cp(y, z)". It is clear that P A | - # i ( 0 ) , one only have to use Theorem 1 n times. In view of (2.1) and Theorem 1, P A | - ^ ( w ) — + 1 ) holds. Therefore, by the induc-tion axiom, PA l- V m ^ i W . A very similar argument shows that the following formula, denoted by <P2, is also PA provable (by induction on i): ... V*„ V« V^ V' ("if u and v are sequences as above and z'^min (LENGTH (m), LENGTH (y)) then the elements of the z-th element of u and v coincide"). Now let <P be 3u(%(xj, . . . , xn, u, m)A "the elements of the m-th element of u are yt, ...,yn in this order"). The existence in (3.1) is ensured by 4>1, the uniqueness is by <P2- (3-2) is trivial. Consider now the valuation of in A where the values of Xx, ..., x„ are ql, . . . , respectively while m has the value a£A. Denote the values of ylt ...,yn for which <P(m,x,y) holds in A by ql, ..., q", respectively. The qa's are determined uniquely by (3.1) and (3.3) is satisfied by definition. (3.4) follows immediately from (3.2). • Definition. The sequence (qa)aiA defined previously is called the standard con-tinuous trace (set in short). (We will see later that this sequence forms a continuous trace indeed.) Lemma 2. Let \f/ be any formula of PA, p a fixed sequence of elements of A and A \j/(a, qa,p) (3.1) (3.2) (3.3) (3.4) Structure of program runs of non-standard time 329 for some adA. Then there is a least suffix with this property, i.e. ad A such that Proof. Suppose the contrary, i.e. whenever A \=ij/(b,cjb ,p) for every b<a then At=i\>(a,qa,~p). Denote by W(m,x,z) the formula x, y)A A\j/(m,y,z)). Then, by the reformulation of the induction axiom, PA h [(Vn < m ) f (n, 3c, z) - f ( m , 3c, I ) ] - V m W(m, 3c, z). Now valuate this in A putting q0 instead of 3c an p instead of z. Notice, that the implication in the square bracket holds, therefore the second half of the implication holds also, i.e. At=i¡i(a, qa,p) for every ad A, which is a contradiction. • COROLLARY. In virtue of this lemma, we may use induction type proofs for the sequence (qa)a€A. • Lemma 3. Let u, v£A, 0<v. If qu = qu+E then for every x, yd. A. Proof. Let y= 1 and use induction by the Corollary on x. After this fix x and use induction on y. • Lemma 4. There exists an EczA such that Proof. If all of the elements of the sequence <qa)a(A are different, the set E—A satisfy (3.5)—(3.7). If not, there is an ad A so that qa occurs at least twice in the sequence. This property is expressible by an L-formula, so, by Lemma 2, we can assume that this a is the minimal one, i.e. qb is unique if b<a. There are other occurrences of qa , hence, also by Lemma 2, there is a second one, i.e. there is an such that qa = qe+i but ¡Ja^i]b if a < b ^ e . We claim that the set E={adA: a^e} satisfies (3.5) and (3.6). It is sufficient to see (3.5) in case only. Suppose qbl=^b2- Lemma 3 with the cast u=b1,v—bi—bi, x = rem (e+1 — b1} b2-b1) gives q e + 1 = qbl+x9£qa which is a contradiction. (3.6) is an easy consequence of Lemma 3. • Now we have all of the tools for the proof of the main result of this paper. First we need some more preliminaries. Definition. The subset SczA is a slice if ad S implies a+ Id S and bd S for all ¿><0. The subset TczA is a thread if adT implies, a+ldT, a-IdT (for a^O) and a,bdT, a<b imply that for some natural number n A \j/(a, qa,p) but A N \j/(b, qb,p) if b < a. if b2dE, b^bz then qbl ^ qb%; for every adA there exists bdE such that qa=qb, either E=A or for some edE, E— {adA: a^e}. (3.5) (3.6) (3.7) b = a + l + ... + l (n times). 330 L. Csirmaz The function f : A-+A is a projector if / ( 0 ) = 0 and f(a+ l)=f(a)+1 for every a£A. The sequence (pa)aiA is a projection of the sequence (qa)aeA if there exists a projector f and a slice S such that Theorem 2. The standard continuous trace (<ja)a£A forms a continuous trace. Proof. (2.2) of the definition is satisfied by (3.4) of Lemma 1. (2.3) is immediate from the Corollary of Lemma 2. • Theorem 3. The projections of the standard continuous trace are continuous traces. Moreover every continuous trace is a projection of the standard one. REMARKS. An easy consequence of Theorem 3 is that if A is a non-standard model of PA then for all <p and q0, the cardinality of ct's of (p with first element q0 is 2M. Now consider a ct (pa)azA the defining formula of which is and let Po=0 for the initial position p0. Then pl+1=pl +1 for every a£A but one can not hope for p\=a in general. Actually, by Theorem 3, the standard ct is the only ct which has this property. We may interpret this'phenomenon as follows. We add a "clock" to a continuous trace and suppose that the clock works well (i.e. it jumps by 1 at every step). If we require the clock to show the correct time then the ct is unique. Compare with Theorem 3.3 of [7]. Proof of the theorem. First let / : A— A be the projector and S the slice of a projection. We prove that {pa)aiA is a ct. (2.2) of the definition follows from f(a+ \)=f («) +1 . To prove (2.3) let i//£L be arbitrary and assume By the hypotheses, pa—qa, for all ¿>£5 there is an a£A such that qb=p„ and if qb=Pa then qb + 1—p a + 1 . So we know that A N iK<?0>P) and A t= ^ (§„,/>) - il>(qa+i,p) for all a£S (3.11) and it is enough to show that this implies Suppose the contrary. Then, by Lemma 2, there is a least counter-example, say b. But this b belongs to S because S is a slice, which contradicts (3.11). Now we turn to the second and longer part of the proof. Let (pa)aiA be any ct, /50=<70. We claim that for any a£A there exists a least b£A such that ¡7),.-In-deed, let \]/(y, x) be "3m&(m, x, y)". It is clear that A\=ip(p0, q0) with m=0 and Pa = bw for every Rng ( / ) c S for every b£ S there is an a£A such that qb = q/iay. (3.8) (3.9) (3.10) (p(xu ...,x„, >>!, ...,>'„) = = Xi+IA... A N \ll{p0,p)h A ( ' K p a , p ) - Pa + l'P))• A N ip(qa,p) for all a£S. Structure of program runs of non-standard time 331 if At= 4>(Pa> 9o) with some m, then A.\=ijj(pa+1,q0) with (m + 1) because the successors are unique. Then by (2.3), A i=^ (p a , go ) for all a£A which states the existence of b. Finally, Lemma 2 ensures a least one. Denote this b by / ( a ) and denote S the range of the function / . Let moreover E and e be as defined in Lemma 4. It is clear that S c £ , / ( 0 ) = 0 and if f(a)?e then f(a+])=f(a)+\. Now let b£A, b$S and let \p{x,y,m) be the formula " ( E 3 m ' < m ) x , y)". Since 0 < 6 and if f(a)~zb then / ( a + l ) = s / ( a ) + l<&, we know ANIMPO,<io,b) and A |= q0, b) - ^(pa+1, q0, b). By (2.3) of the definition of ct, A\=^(pa,q0,b), i.e. /(a)<Z> for all a£A. This means that if b£ S and c<b t h e n c f S . We distinguish tvfo cases. 1. E—A or E^A but S. In this case / is a projector and S is a slice, there-fore {pa)aZA is a projection. _ _ 2. eZS, i.e. S=E. Let b£E so that qb—qe+1- By Lemma 3 qu—Qe-y\.=^b if u=a+y • (e+1 —a), so if we choose y large enough then the thread Tu of u does not contain b. For each thread T define the function gT: T-»A as follows. If T is the thread of 0 then let gT(a)=a. Otherwise if f(v)—b for some v£T then let gT(a)=b+a—v. Otherwise if f(v)—e for some y ^ r t h e n let gT(a)=u—l+a—v, otherwise let gr(fl)=f (a). Finally, let g(a)=gT(a) if a is in the thread T. It is clear from the definition of gT that g is a projector and ;?0=<7 /(fl)=#9(a). By Lemma 4, every qb equals to some pa, i.e. in this case the projector g and the whole A as a slice shows that (pa)aiA is a projection. • Abstract Continuous traces are introduced to simulate program runs when time is measured by the ele-ments of a non-standard model of Peano axioms. This concept is a very useful one in considerations of program verification. We give here a full description of continuous traces in every model of PA. It turns out that there is exactly one continuous trace definable by a formula of PA and every other one can be got from this by a simple transformation. MATHEMATICAL INSTITUTE OF THE H U N G A R I A N ACADEMY OF SCIENCES REÁLTANODA U. 13—15. BUDAPEST, H U N G A R Y H—1053 References [1] ANDRÉKA H . , I . NÉMETI, Completeness of Floyd logic, Bull, of Section of Logic v. 7, 1978, pp. 115—121. [2] CHANG, C . C . , H. J. KEISLER, Model theory, North Holland, 1973 . [3] CSIRMAZ, L., On definability in Peano arithmetic, Bull, of Section of Logic, v. 8 , 1 9 7 9 , pp. 1 4 8 — 1 5 3 . [4] MANNA, Z., Mathematical theory of computation, McGraw-Hill, 1974. [5] PÉTER, R., Rekursive Funktionen in der Komputer-Theorie, Akadémiai Kiadó, 1976. [6] PRATT, V . R . , Semantic considerations on Floyd-Hoare logic, 17-th IEEE Symp. Found. Comp. Sci., 1 9 7 6 . [7] SAIN, I., There are general rules for specifying semantics, Observations on Abstract Model Theory, CL & CL, 1979, No. 2, to appear. [8] SÁNTÁNÉ—TÓTH, E., M. SZŐTS, Colloquium on Logic in Programming, Salgótarján, Rundbrief der Fachgruppe der Gesellschaft für Informatik, 1979, No. 17. (Received April 5, 1979) 

Structure of program runs of non-standard time

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Structure of program runs of non-standard time

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