Rule formats for bounded nondeterminism in structural operational semantics

Transcription

1 Rule formts for bounded nondeterminism in structurl opertionl semntics Luc Aceto, Álvro Grcí-Pérez, nd Ann Ingólfsdóttir ICE-TCS, School of Computer Science, Reykjvík University, Menntvegur 1, IS-101, Reykjvík, Icelnd. Abstrct We present rule formts for structurl opertionl semntics tht gurntee tht the ssocited lbelled trnsition system hs ech of the three following finiteness properties: finite brnching, initils finiteness nd imge finiteness. Keywords: structurl opertionl semntics, lbelled trnsition systems, rule formts, bounded nondeterminism 1 Introduction Structurl opertionl semntics (SOS) [25, 27] is widely used formlism for defining the forml semntics of computer progrms nd for proving properties of the corresponding progrmming lnguges. In the SOS formlism trnsition system specifiction (TSS) [13], which consists of signture together with set of inference rules, specifies lbelled trnsition system (LTS) [16] whose sttes (i.e., processes) re closed terms over the signture nd whose trnsitions re those tht cn be proved using the inference rules. Rule formts [2, 21] re syntcticlly checkble restrictions on the inference rules of TSS tht gurntee some useful property of the ssocited LTS. The properties ensured by such rule formts vry from compositionlity of behviourl equivlences [7,13,14,30] to finiteness of the number of outgoing trnsitions from given stte [6,9,32]. This pper focuses on the finiteness property, which is referred to s bounded nondeterminism in [12]. Brodly, bounded nondeterminism is tken s synonym of finite brnching [9]. Finite brnching breks down into the more elementry properties of initils finiteness nd imge finiteness [1] (see Section 2 for forml definitions). Vndrger [32] introduced the notion of bounded TSS nd proved tht bounded TSS in de Simone formt [30] induces n LTS tht is finite brnching. Bloom [6] used notion of bounded TSS reminiscent of tht of Vndrger nd showed tht bounded TSS in his higher-order-gsos formt [6] induces n LTS tht is finite brnching. Finlly, Fokkink nd Vu [9] used yet nother notion of bounded TSS nd introduced less restrictive rule formt tht they clled This reserch hs been supported by the project Nominl Structurl Opertionl Semntics (nr ) of the Icelndic Reserch Fund.

2 bounded nondeterminism formt. They dpted the notion of strict strtifiction from [14] nd showed tht bounded TSS in bounded nondeterminism formt tht hs strict strtifiction induces n LTS tht is finite brnching. In this pper we tke Fokkink nd Vu s progrmme further nd present rule formts for initils finiteness nd for imge finiteness. For initils finiteness we relx the requirement tht the η-types of [9] be finitely inhbited nd we introduce the initils finite formt, which replces the bounded nondeterminism formt of [9]. For imge finiteness, we introduce the notion of θ-type. Unlike the η-types of [9], which crry informtion bout the sources of positive premisses in rules, the θ-types lso keep trck of the ctions tht lbel positive premisses. Moreover, we introduce uniformity requirement on the trgets of positive premisses, which strengthens the requirement in [9] tht the vribles in rule hve to be used uniformly. We introduce the ccompnying notions of initils-bounded TSS nd imge-bounded TSS nd show the following results. An initils-bounded TSS in initils finite formt tht hs strict strtifiction induces n LTS tht is initils finite (Theorem 2). An imge-bounded TSS in bounded nondeterminism formt tht hs strict strtifiction induces n LTS tht is imge finite (Theorem 3). The results nd the techniques we employ in this pper touch upon some of the min topics in the reserch of Flemming Nielson nd Hnne Riis Nielson over the yers, nmely opertionl semntics [23], sttic nlysis [22] nd type systems [3]. This study contributes to the development of generl theory of opertionl semntics bsed on rule formts, which my be seen s providing some stticlly checkble, lrgely syntctic, conditions gurnteeing tht the specified lnguges fford some semntic properties of interest. The vrious notions of types tht we use in the definition of the rule formts discussed in this pper llow us to clssify the inference rules in lnguge specifiction. Informlly, types contribute to gurnteeing tht composite processes hve the finiteness property of interest, if their components do so. The rest of the pper is orgnised s follows. Section 2 revisits preliminries nd bsic notions from [9] nd dpts some of its definitions. Definition 10 formlises the notion of uniform TSS nd Proposition 2 shows tht closed term p unifies only with finitely mny rules in uniform TSS. Section 3 provides n lterntive proof of Theorem 1 in [9] tht removes the reductio d bsurdum rgument tht is used there. Theorem 1 shows tht bounded TSS in bounded nondeterminism formt tht hs strict strtifiction induces n LTS tht is finite brnching. The proof of Theorem 1 here is direct nd fully constructive. Section 4 discusses the vrible flow in trnsition rule nd Definition 18 introduces the initils finite formt, which requires tht ech vrible in the source of positive premiss occur lso in the source of the rule. Definition 20 introduces the notion of initils-bounded TSS, which relxes the η-types of [9] by requiring tht the ctions of n η-type re finite, insted of requiring the η-type to be finitely inhbited. Theorem 2 shows tht n initils-bounded TSS in initils finite formt tht hs strict strtifiction induces n LTS tht is initils finite. Section 5 discusses the logicl content of the η-types under the prism of intuitionistic 2

3 logic [19], nd shows tht the η-types relise the intuitionistic interprettion of the property of initils finiteness. Definition 21 introduces the θ-types, which re nlogous to the η-types in tht they relise the intuitionistic interprettion of the property of imge finiteness. Definition 22 introduces uniformity in the trgets of positive premisses, which prevents the θ-types to be infinitely mny s result of using infinitely mny different nmes for vrible occurring in the trget of some positive premiss, nd Definition 23 introduces the notion of imge-bounded TSS. Theorem 3 shows tht n imge-bounded TSS in bounded nondeterminism formt tht hs strict strtifiction induces n LTS tht is imge finite. Section 6 discusses venues for future work nd concludes. 2 Preliminries We give n overview of the structurl opertionl semntics formlism (SOS for short). We follow the nottion nd the presenttion in [9]. For set S, we write P(S) for the collection of ll the subsets of S, nd P ω (S) for the collection of ll the finite subsets of S. Definition 1 (Signture nd Term). We ssume countbly infinite set of vribles V, rnged over by x, y, z. A signture Σ is set of function symbols, disjoint from V, together with n rity mp tht ssigns nturl number to ech function symbol. We use f to rnge over Σ. Function symbols of rity zero, which my be rnged over by c, d, re clled constnts. Function symbols of rity one nd two re clled unry nd binry functions respectively. The set T(Σ) of (open) terms over signture Σ, rnged over by t, u, v, is the lest set such tht: 1. ech vrible is term, nd 2. if f is function symbol of rity n nd t 1,..., t n re terms, then f(t 1,..., t n ) is term. The function vr : T(Σ) P ω (V ) delivers, for term t, the set of vribles tht occur in t. A term t is closed iff vr(t) =. The set of closed terms over Σ, rnged over by p, q, is denoted by T (Σ). Definition 2 (Formul). We consider set of ctions A, rnged over by, b (nd c when no confusion rises with the constnts). The set of positive formule over signture Σ nd ctions A is the set of triples (t,, t ) T(Σ) A T(Σ). We use the more suggestive nottion t t in lieu of (t,, t ). The set of negtive formule over signture Σ nd ctions A is the set of pirs (t, ) T(Σ) A. We use the more suggestive nottion t in lieu of (t, b). Definition 3 (Substitution). A substitution is prtil mp σ : V T(Σ). The substitutions re rnged over by σ, τ. A substitution is closed if it mps vribles to closed terms. A substitution extends to mp from terms to terms in the usul wy, i.e., the term σ(t) is obtined by replcing the occurrences in b 3

4 t of ech vrible x in the domin of σ by σ(x). When pplying substitutions σ nd τ successively, we my bbrevite τ(σ(t)) to τσ(t). We sy term u is substitution instnce of t iff there exists substitution σ such tht σ(t) = u. In wht follows, we shll sometimes use the nottion {x i t i i I}, where I is n n index set nd the x i s re pirwise distinct vribles, to denote the substitution tht mps ech x i to the term t i (i I). A substitution σ extends to formule t t nd u in the usul wy, by pplying the substitution to the term components of the formule, i.e., σ(t) σ(t b ) nd σ(u) respectively. The notion of substitution instnce extends similrly. Definition 4 (Lbelled trnsition system). Let Σ be signture nd A set of ctions. A lbelled trnsition system (LTS for short) is pir (T (Σ), ) where T (Σ) is the set of processes, i.e., closed terms, nd T (Σ) A T (Σ) is the set of trnsitions, i.e., closed positive formule. We sy tht p p is trnsition of the LTS iff (p,, p ). Lbelled trnsition systems [16] re fundmentl model of computtion nd re often used to describe the opertionl semntics of progrmming nd specifiction lnguges see, for instnce, [20, 26, 27, 29]. Trnsition system specifictions, which we now proceed to define, describe the LTS giving the semntics of lnguge by mens of signture (nmely, the collection of term constructors offered by the lnguge) nd set of inference rules tht cn be used to prove the vlid trnsitions between terms in the lnguge. Definition 5 (Trnsition system specifiction). Let Σ be signture nd A set of ctions. A trnsition rule ( rule, for short) ρ is of the form H t t (bbrevited s H/t t ) where H is set of positive premisses of the form u b u c nd negtive premisses of the form v, nd t t is the conclusion of the rule (with t, t, u, u, v T(Σ) nd, b, c A). We sy t is the source, is the ction, nd t is the trget of ρ. We sy ρ is n xiom iff ρ hs n empty set of premisses, i.e., H =. A trnsition system specifiction (TSS for short) is set of trnsition rules. A substitution mp extends to rule ρ by pplying the substitution to the formule in ρ. The notion of substitution instnce extends similrly to rules. Definition 6 (Unify with rule). Let R be TSS. We sy tht trnsition p p unifies with rule ρ R iff ρ hs conclusion t t nd p p is substitution instnce of t t. b 4

5 Definition 7 (Proof tree). Let R be TSS without negtive premisses. A proof tree in R of trnsition p p is n upwrdly brnching tree without pths of infinite length whose nodes re lbelled by trnsitions such tht 1. the root is lbelled by p p, nd 2. if K is the set of lbels of the nodes directly bove node with lbel q b then K/q q is substitution instnce of some rule H/t We sy tht p b t R. p is provble in R iff p p hs proof tree in R. b q, The set of provble trnsitions in R is the lest set of trnsitions tht stisfies the rules in R. Notice tht if p p unifies with n xiom (i.e., rule of the form /t t ) then, trivilly, p p hs proof tree in R which consists of root node lbelled by p p. A TSS without negtive premisses induces n LTS in strightforwrd wy. Definition 8 (TSS induces LTS). Let R be TSS without negtive premisses nd T n LTS. R induces T (or T is ssocited with R) iff the set of trnsitions of T is the set of provble trnsitions in R. The phrses 1. p p is provble in R, 2. p p is trnsition of T, nd 3. p cn perform n -trnsition to p in T re synonyms. For brevity, we my omit the R nd/or the T when they re cler from the context. In [28], Przymusinsky introduced three-vlued stble models, which cn be used to ssocite n LTS to TSS with negtive premisses. Ech TSS hs lest three-vlued stble model, which coincides with the well-founded semntics from [11]. We consider the set of sentences tht re certinly true in the lest three-vlued stble model, which, for TSS without negtive premisses, coincides with the set of provble trnsitions in Definition 8. As Fokkink nd Vu noticed in [9], if R is TSS nd R is obtined by removing ll the negtive premisses from the rules in R, then the LTS ssocited with R is included in the LTS ssocited with R. In prticulr, if the LTS ssocited with R hs ny of the finiteness properties considered in this pper, then the LTS ssocited with R hs the property too. We follow [9] nd ignore the negtive premisses in the TSSs. None of the rule formts tht we introduce here impose ny restrictions on negtive premisses. The notion of uniform TSS stems from [9]. We introduce the notion of structure of term, nd provide forml definition of uniform TSS, to which we refer s uniform TSS in the sources becuse the focus is on the sources of the rules. Definition 9 (Structure of term). Let R be TSS. The terms t nd u hve the sme structure iff 5

6 1. t = x nd u = y, where x nd y re vribles, or 2. t = f(t 1,..., t n ) nd u = f(u 1,..., u n ), where f is function symbol of rity n 0, nd the terms t i nd u i hve the sme structure for ech 1 i n. Intuitively, two terms t nd u hve the sme structure iff their syntx trees differ only in the nme of the vribles. For exmple f(x, y) nd f(x, x) hve the sme structure. Two closed terms hve the sme structure iff they re the sme term. Definition 10 (Uniform in the sources). A TSS R is uniform in the sources iff t = u holds whenever t nd u hve the sme structure nd re sources of ny two rules in R. In Section 5 we will introduce the nlogous notion of uniform TSS in the trgets of positive premisses, in which the focus is on the trgets of positive premisses. When no confusion rises, we my bbrevite nd sy uniform TSS for uniform TSS in the sources. The rtionle behind uniformity in [9] is to enforce tht in uniform TSS, ech closed term is substitution instnce of the sources of t most finitely mny trnsition rules. In order to show this property, we introduce the notion of prtil term nd the less-defined-thn reltion. Definition 11 (Prtil term). The set T (Σ) of prtil terms over signture Σ, rnged over by r, s, is the set of terms tht results by extending Σ with the constnt symbol. The symbol, which stnds for undefined, is different from the other symbols in Σ. Notice tht T (Σ) T(Σ) T (Σ). The notion of structure of term from Definition 9 is extended to prtil terms in strightforwrd wy by considering the symbol s vrible. For instnce, f(x, y) hs the sme structure s f(, z). Definition 12 (Less-defined-thn reltion). The reltion (which we refer to s the less-or-eqully-defined-thn reltion) is the lest binry reltion over prtil terms such tht 1. r for ech prtil term r, 2. x x for ech vrible x, nd 3. f(s 1,..., s n ) f(r 1,..., r n ) where f is function symbol of rity n 0 iff s i r i for ech 1 i n. We sy s is n pproximnt of r iff s r. The less-defined-thn reltion is the binry reltion over prtil terms defined thus: s r iff s r nd s r. It is esy to see tht induces prtil order nd induces strict prtil order over prtil terms. Proposition 1. The less-defined-thn reltion,, is well-founded reltion. 6

7 Proof. We prove tht there exists no infinite decresing chin r 1 r To this end, we first define the size of prtil term r s follows: 1. the size of is zero, 2. the size of vrible is one, nd 3. the size of f(r 1,..., r n ) with f function symbol of rity n 0 is one plus the sum of the sizes of the r i s with 1 i n. Let r nd s be prtil terms. If s r then s is obtined by replcing by one or more mximlly disjoint subterms of r tht re different from, nd hence the size of s is strictly smller thn tht of r. Since the size of every prtil term is finite, ech decresing chin is lso finite nd we re done. Proposition 2. Let R be uniform TSS. For ech closed term p the set of pirs (t, σ) with σ : vr(t) T (Σ) such tht σ(t) = p nd t is the source of some rule in R is finite. Proof. We prove the generlised proposition: Let R be uniform TSS. For ech pir (p, r) where p is closed term nd r is n pproximnt of p (i.e., r p), the set of pirs (t, σ) with σ : vr(t) T (Σ) such tht there exists r n pproximnt of r (i.e., r r) with the sme structure s t, nd σ(t) = p nd t is the source of some rule in R, is finite. The originl proposition follows from the generlised proposition by fixing r = p. We construct the set S of pirs (t, σ) tht meet the conditions of the generlised proposition nd show tht S is finite. We proceed by well-founded induction on the set of pproximnts of r ordered by the less-defined-thn reltion ( ). We first check tht the generlised proposition holds for the -miniml prtil terms in the set of pproximnts of r. The only such prtil term is r 0 =. There exists only one r 0 n pproximnt of r 0 (i.e., r 0 = = r 0 ) nd the only terms tht hve the sme structure s r 0 re the vribles. Since R is uniform, ll the rules whose source is vrible (if there re ny) hve the sme vrible x s source. If there exist no rules whose source is vrible, then the set we re looking for is the empty set. Otherwise, the set we re looking for is S = {(x, {x p})}. Both sets re finite. Now we check tht the generlised proposition holds for n rbitrry prtil term r in the set of pproximnts of r. Notice tht the prtil term r is such tht r r p. By the induction hypothesis, for every r x such tht r x r, the set S x of pirs (t, σ) such tht there exists r x n pproximnt of r x (i.e., r x r x ) with the sme structure s t, nd σ(t) = p nd t is the source of some rule in R, is finite. Since R is uniform, ll the rules whose source hs the sme structure s r x (if there re ny) hve the sme source u. Since r x r, then r x is obtined by replcing by one or more mximlly disjoint subterms of r tht re different from. We let x j (with j rnging over some index set J) be the vribles tht occur in u in the positions corresponding to the occurrences of in r x. (Notice tht no other vribles could occur in u, since u hs the sme 7

8 structure s r x, nd r x p.) We let t j be the terms such tht p results from replcing respectively the x j by the t j in u. If there exist no rules with source u, then the set we re looking for is S = S x. Otherwise, the set we re looking for is S = S x {(u, {x j t j j J})}. Both sets re finite. The next exmple shows tht Proposition 2 does not hold for TSSs tht re not uniform. Exmple 1. Let Σ consist of constnt c nd ssume A = {}. Let the x i with i N be infinitely mny distinct vribles. Consider the TSS with rules x i c, i N. All the x i in the instntitions of the rule templte bove hve the sme structure, but x j x k for j, k N nd j k. Therefore, the TSS is not uniform. Notice tht for c there exist infinitely mny pirs (x i, σ i ) (with i N nd σ i = {x i c}) such tht σ i (x i ) = c. We focus on the properties of finite brnching, initils finiteness, nd imge finiteness [1], which we define next. Definition 13 (Bounded nondeterminism). Let T be n LTS nd p closed term in T. We sy 1. p is finite brnching iff the set {(, p ) p p } is finite, 2. p is initils finite iff the set { p s.t. p p } is finite, nd 3. p is imge finite iff for every ction, the set {p p p } is finite. An LTS T is finite brnching (resp. initils finite nd imge finite) iff every closed term in T is finite brnching (resp. initils finite nd imge finite). We cll { p s.t. p p } the set of initils of p. We cll {p p p } the set of imges of p for ction. 3 Finite brnching The rule formt in [9], which restricts TSS to be bounded, to be in bounded nondeterminism formt, nd to hve strict strtifiction, ensures tht the ssocited LTS is finite brnching. Intuitively, the restrictions such formt plces on the llowed rules ensure tht, for ech closed term p, 1. the rules in the TSS do not llow one to simulte ungurded recursion for p, 2. only finitely mny rules cn be employed to derive trnsitions from p, nd 3. ech rule cn only be used to infer finitely mny trnsitions from p. 8

9 The third property is checkble for ech rule in isoltion nd is embodied in the requirement tht the TSS be in bounded nondeterminism formt (see Definition 16 to follow). On the other hnd, the first nd the second properties re globl nd need to be checked for sets of rules. The existence of strict strtifiction (see Definition 17) enforces the first property, while the second is gurnteed by the requirement tht the TSS be bounded (see Definition 15 below). In order to define the notion of bounded TSS, Fokkink nd Vu clssify the trnsition rules in TSS ccording to their so-clled η-types. Intuitively, rules hving the sme η-type re those tht could potentilly be used to derive trnsitions from closed term p tht unifies with the source of the rules. The requirement tht the TSS be uniform nd tht the η-types be finitely inhbited ensures therefore tht only finitely mny rules cn be employed to derive trnsitions from p. We now dpt the definition of η-types in [9], on which the notion of bounded TSS is bsed, nd recll the bounded nondeterminism formt nd the notion of strict strtifiction in [9]. We let η : T(Σ) P(T(Σ)) be the mps tht prmetrise the η-types of Definition 14 to follow. The mps η deliver, for given term t, predefined set of sources of positive premisses in rules tht hve source t. We sy tht η(t) is the support of the sources for source t. 1 Definition 14 (η-type). Let R be TSS, ρ R rule with source t nd positive premisses {t i i t i i I}, nd η mp with type T(Σ) P(T(Σ)). We define ψ : η(t) P(A) s the mp tht delivers, for ech term u in the support of the sources for t, i.e., u η(t), the ctions of the positive premisses of ρ with source u. More formlly, ψ(u) = { i i I t i = u}. The tuple t, ψ is sid to be the η-type of rule ρ. Differently from [9], our definition of η-type does not require tht ech set in the codomin of ψ be finite. This requirement is not necessry for the rule formt to ensure finite brnching, s we explin in Remrk 1 to Theorem 1. The η-types distinguish rules bsed on their source nd on the set of ctions of their positive premisses whose source belongs to the predefined set specified by the mp η. For instnce, ll the rules without positive premisses tht hve the sme source belong to the sme η-type, regrdless of their ction nd trget. Intuitively, s mentioned bove, rules tht hve the sme η-type might ll be used to derive trnsitions from closed instntition of their source. As the following exmple indictes, the presence of infinitely mny rules with the sme η-type might yield infinite brnching. Exmple 2. Let A be n infinite set of ctions nd Σ = {c}. Consider the TSS c c, A. 1 We beg the reder to ber with us in the repetition of sources nd source in sentences like the bove. The sources refers to the positive premisses nd the source to the conclusion of the rule. 9

10 All the infinitely mny instntitions of the rule templte bove hve η-type c, ψ, where ψ mps ech term in η(c) (if ny) to the empty set. Note tht c is not finite brnching. Definition 15 (Bounded). A TSS R is bounded iff R is uniform nd there exists η with codomin P ω (T(Σ)) (i.e., the set η(t) is finite for ech t) such tht for every rule ρ R with η-type t, ψ, the η-type t, ψ is finitely inhbited. The requirement tht the function η hve codomin P ω (T(Σ)) in Definition 15 mens tht in bounded TSS only finite support of the sources for source cn be distinguished. Consider Exmple 5 on pge 508 of [9], which we reproduce next. Exmple 3. Let A be n infinite set of ctions nd Σ consist of constnts A {c} where c A. Consider the TSS, A y c y, A. If we llowed η to hve codomin P(T(Σ)), e.g., η() = (with A) nd η(c) = A, then it would be possible to distinguish the infinite support of the sources in the rule templte on the right, nd ech η-type c, ψ (with A), where ψ () = {} nd ψ (b) = for b, would correspond to exctly one rule. If insted we require η to hve codomin P ω (T(Σ)), e.g., η() = with A nd η(c) = B for some B P ω (A), then n infinite number of sources of premisses A \ B will be excluded from the support for source c, i.e., η(c) (A \ B) =. The sources A \ B cnnot be distinguished, nd thus the infinitely mny instntitions of the rule templte on the right with sources of premisses A \ B will hve the sme η-type c, ψ where ψ(t) = with t B. Therefore, the TSS is not bounded. Notice tht c is not finite brnching. Definition 16 (Bounded nondeterminism formt). A rule b {u i i u i i I} t t is in bounded nondeterminism formt iff 1. vr(u i ) vr(t) for ech i I, tht is, ll the vribles occurring in the source of its positive premisses lso occur in its source, nd 2. vr(t ) vr(t) {vr(u i ) i I}, tht is, ll the vribles occurring in its trget lso occur in its source, or in the trget of some of its positive premisses. A TSS R is in bounded nondeterminism formt iff every rule in R is in bounded nondeterminism formt. The bounded nondeterminism formt enforces tht the trget of trnsition ultimtely comes from the source, i.e., the rules cnnot introduce vribles spuriously. The following exmple illustrtes this fct. 10

11 Exmple 4. Let Σ consist of constnt c nd binry function symbol f, nd let A = {}. Consider the TSS c c x z f(x, y) f(z, y). The TSS is in bounded nondeterminism formt. Note tht vrible z in the premiss of the rule on the right comes neither from the source of the premiss nor from the source of the rule. However, in every ppliction of tht rule in proof trees llowing one to derive trnsitions from closed terms of the form f(p, q), the vrible z will lwys be instntited to some closed term p such tht p p. Therefore, the rule does not introduce vribles spuriously. On the other hnd, consider the rule f(x, y) z. Such rule is not in bounded nondeterminism formt becuse the vrible z in the trget of the rule does not pper in its source. The bove rule cn be used to prove trnsitions of the form f(p, q) r for ll closed terms p, q nd r, so the trget r of trnsition does not necessrily stnd for process tht cn be reched from either p or q. Therefore, the rule introduces the vrible z spuriously. Definition 17 (Strict strtifiction). Let R be TSS. A strict strtifiction of R consists of mp S from closed terms T (Σ) to ordinl numbers such tht for every trnsition rule H/t t R nd for every closed substitution σ, S(σ(u)) < S(σ(t)) for every u b u H. The conditions of Theorem 1 on pge 509 of [9] define the rule formt for finite brnching. We prphrse Theorem 1 of [9] nd its proof, nd remove the reductio d bsurdum rgument tht is used there, providing direct nd fully constructive proof. Theorem 1 (Theorem 1 of [9]). Let R be bounded TSS in bounded nondeterminism formt tht hs strict strtifiction S. The LTS ssocited with R is finite brnching. Proof. We prove tht ech closed term p in the LTS ssocited with R is finite brnching. Since R is uniform, for given p there re only finitely mny distinct terms t i nd substitutions σ i : vr(t i ) T (Σ) (i.e., the i rnges over finite index set I) such tht σ i (t i ) = p nd the rules tht unify with trnsitions from p hve some t i s source. We proceed by induction on S(p). The initil cse is when S(p) = 0. Since S(σ i (t i )) = 0, the rules with source t i re xioms of the form t i j, t j i I, j J i 11

12 where the J i re tken to be disjoint to void prolifertion of indices. Since R is bounded, there exists η such tht for ech i nd for ech j J i the instntition of the rule templte bove hs η-type t i, ψ j, nd t i, ψ j is finitely inhbited. By Definition 14, ll the ψ j mp ech term in η(t i ) (if ny) to the empty set. Since R is in bounded nondeterminism formt, vr(t j ) vr(t i), nd thus the σ i (t j ) re closed. Since the rules bove re xioms, the trnsitions σ i (t i ) j σ i (t j ) re provble in R. Since ll the ψ j in the η-types t i, ψ j with j J i re equl, nd since the η-types re finitely inhbited, then the J i re finite. Therefore, for ech i I the set {( j, σ i (t j)) σ i (t i ) j σ i (t j)} is finite. By the finiteness of I it follows tht the set {(, p ) p p } is finite nd we re done. The generl cse is when S(p) > 0. The rules with source t i such tht σ i (t i ) = p re of the form b {u k k u k k K j } j, i I, j J i t j t i where the J i nd the K j re tken to be disjoint to void prolifertion of indices. Since R is bounded, there exists η such tht for ech i nd for ech j J i, the instntition of the rule templte bove hs η-type t i, ψ j, the set η(t i ) is finite, nd t i, ψ j is finitely inhbited. For ech i, we show tht there re only finitely mny distinct ψ j with j J i such tht rules with η-type t i, ψ j give rise to trnsitions from σ i (t i ). By Definition 14, ech rule of η-type t i, ψ j contins premiss of the form v c v for ech v η(t i ) nd ech c ψ j (v). Since R is in bounded nondeterminism formt, vr(v) vr(t i ), nd thus the σ i (v) re closed. By Definitions 7 nd 8, for ech trnsition in the node of proof tree, if the trnsition unifies with rule of η-type t i, ψ j then for ech v η(t i ) the process σ i (v) cn perform, t lest, c-trnsition for ech c ψ j (v). The ψ j giving rise to trnsitions from σ i (t i ) re c dependent functions of type Π v η(ti){c σ i (v) τσ i (v )} with substitutions τ : (vr(v ) \ vr(v)) T (Σ). For ech i the refined type of the ψ j with j J i is finitely inhbited, since the codomin of dependent function depends on the inputs of the function. Ech imge of ψ j cnnot be n rbitrry subset of A, but only the one tht is determined by the input v nd by the ssocited LTS. Tht c is, the only elements in the codomin of ψ j re the sets {c σ i (v) τσ i (v )} where v η(t i ). Since the η(t i ) re finite sets, both the domin nd the codomin of ψ j re finite. Therefore, for ech i there re only finitely mny distinct ψ j with j J i such tht rules with η-type t i, ψ j give rise to trnsitions from σ i (t i ). Since R is in bounded nondeterminism formt, vr(u k ) vr(t i ) nd therefore the σ i (u k ) re closed terms. As S is strict strtifiction, S(σ i (u k )) < S(p). By the induction hypothesis the σ i (u k ) re finite brnching, nd therefore for ech i I the set {(b k, τ l σ i (u k)) σ i (u k ) b k τ l σ i (u k)} 12

13 is finite, with τ l : (( k K j vr(u k )) \ vr(t i)) T (Σ) closed substitutions where l rnges over some index sets L j nd where j J i. Since R is in bounded nondeterminism formt, vr(t j ) (vr(t i) ( k K j vr(u k ))) nd therefore the τ l σ i (t j ) re closed terms. Since for ech i there re only finitely mny distinct ψ j with j J i such tht rules with η-type t i, ψ j give rise to trnsitions from σ i (t i ), nd since the η-types t i, ψ j re finitely inhbited, then the L j re finite. Therefore, for ech i I the set {( j, τ l σ i (t j)) σ i (t i ) j τ l σ i (t j)} (j J i, l L j ) is finite. By the finiteness of I it follows tht the set {(, p ) p p } is finite nd we re done. Remrk 1. The requirement in [9] tht ech set in the codomin of ψ in n η-type t, ψ must be finite (i.e., the codomin of ψ must be P ω (A)) is superfluous. In TSS tht induces finite-brnching LTS such tht η witnesses tht the TSS is bounded, there could be rules with η-type t, ψ where ech set in the codomin of ψ is infinite, but since the LTS is finite brnching, the trnsitions in the nodes of proof tree will never unify with these rules. Remrk 2. The proof bove follows tht of Theorem 1 in [9], with the most notble difference being tht [9] uses reductio d bsurdum rgument to show tht the distinct ψ j for given i re finitely mny. The proof in [9] ssumes tht there exists m I such tht there re infinitely mny ψ n with n J m such tht rules with η-type t m, ψ n give rise to trnsitions from σ m (t m ), nd then shows tht this ssumption contrdicts the induction hypothesis. We believe direct proof is preferble over proof by contrdiction. Our proof not only estblishes the desired conclusion bove, but lso the intermedite conclusion tht the ψ j such tht rules with η-type t i, ψ j give rise to trnsitions c from σ i (t i ) re dependent functions of type Π v η(ti){c σ i (v) τσ i (v )} with substitutions τ : (vr(v ) \ vr(v)) T (Σ). This is n interesting observtion in its own right tht could be used to drw further conclusions. Besides, our proof is fully constructive, nd thus it is better suited for the purpose of mechnising it. Exmple 5. Let A consist of n ction. Consider TSS whose signture contins the constnts c i, with i 1, nd whose rules re x y y z x, i 1. z c i c i+1 This TSS is neither in bounded nondeterminism formt nor strictly strtified, nd therefore does not stisfy the conditions of Theorem 1. It is esy to see tht every constnt c i (i 1) hs infinitely mny outgoing trnsitions. Indeed, c j is provble for ll j > i 1. c i Severl exmples of pplictions of the rule formt defined by the conditions of Theorem 1 cn be found in [9]. In the next section we dpt the conditions of Theorem 1 to ccount for initils finiteness. 13

14 4 Initils finiteness As shown by Exmple 4 in Section 3, the bounded nondeterminism formt enforces tht no vribles re introduced spuriously, thus preventing infinite brnching coming from replcing the vribles in the trget of rule by infinitely mny distinct terms. In trnsition rule, there re three kinds of vrible flow tht it is worth considering: 1. vribles from the source of the rule tht flow to the sources of the positive premisses, 2. vribles from the source of the rule tht flow to the trget of the rule, nd 3. vribles from the trgets of positive premisses tht flow to the trget of the rule. By the bounded nondeterminism formt, ll the vribles in rule (except for the vribles in the source of the rule nd in the trgets of positive premisses) come from some of the vrible flows described bove. By induction on the proof tree, it is esy to show tht the circultion of the vribles is closed in the leves of the proof tree (i.e., by the second kind of vrible flow bove) nd thus no vribles cn be introduced spuriously. This requirement is too strong for initils finiteness, which is only concerned with the ctions of trnsitions tht re provble from given process. For initils finiteness it is immteril whether the rules introduce vribles in the trget spuriously, nd the bounded nondeterminism formt cn be relxed. However, s the following exmple shows, dropping ll the requirements on the vrible flow does not ensure initils finiteness. Exmple 6. Let A be n infinite set of ctions nd let Σ = A {c, f} with c constnt, f unry function symbol nd f, c A. Consider the TSS f() f(), A f(x) y, A. c y The TSS is uniform nd hs strict strtifiction given by S(c) = 1 S(f(p)) = 0. We let η(f()) = (with A) nd η(c) = {f(x)}. For ech A, the instntition of the rule templte on the left hs η-type f(),, nd the instntition of the rule on the right hs η-type c, ψ where ψ (f(x)) = {}. However, the ssocited LTS is not initils finite becuse the set of initils of c is A. Vrible x in the rule templte on the right does not come from the source of the rule. Thus, there exist infinitely mny substitutions τ : {x} A such tht the trnsitions from τ(f(x)) unify with some instntition of the rule templte on the left. For initils finiteness, it is enough to prevent spurious vribles in the sources of positive premisses. We now introduce the initils finite formt, which tkes cre of the first kind of vrible flow described bove. 14

15 Definition 18 (Initils finite formt). A rule b {u i i u i i I} t t is in initils finite formt iff ll the vribles occurring in the sources of its positive premisses lso occur in its source, tht is, vr(u i ) vr(t) for ech i I. A TSS R is in initils finite formt iff every rule in R is in initils finite formt. The following exmple shows tht the requirements on the vrible flow, except for the one enforced by the initils finite formt, cn be dropped. Exmple 7. Let A = {} nd Σ consists of infinitely mny constnts {c, d,...}. Consider the TSS with rule c x. The system is uniform nd hs trivil strict strtifiction. The rule bove hs η-type c, ψ where ψ mps ech term in η(c) (if ny) to the empty set, nd thus the TSS is bounded. Vrible x comes neither from the trget of ny positive premiss, since there re none, nor from the source of the rule, nd hence the TSS is not in bounded nondeterminism formt. However, the TSS is in initils finite formt. Notice tht the ssocited LTS is initils finite. However, replcing the bounded nondeterminism formt by the initils finite formt is not enough to cover ll the TSSs in which we re interested. Some initils-finite LTSs re induced by TSSs which re not bounded, despite being in initils finite formt. This is shown in the following exmple. Exmple 8. Let A = {} nd let Σ consist of infinitely mny constnts {c, d,...}. Let P = {p i i I} (with the p i distinct nd I n infinite index set) be proper subset of T (Σ), i.e., P T (Σ). Consider the TSS with rules c, i I. p i All the rules bove hve η-type c, ψ where ψ mps ech term in η(c) (if ny) to the empty set, nd hence the η-type c, ψ is infinitely inhbited nd the TSS is not bounded. However, the ssocited LTS is initils finite. Exmple 8 implements bounded quntifiers by mens of rule templte nd n d hoc infinite index set I. The use of bounded quntifiers 2 is different from 2 Notice tht bounded in bounded quntifiers does not hve the connottion of finite tht is present in bounded nondeterminism. The bounded quntifiers restrict the rnge of the quntified vrible, but this rnge could still be infinite. Exmples 7 nd 8 illustrte the difference between universl quntifiers nd bounded quntifiers with n infinite rnge. 15

16 the implicit universl quntifiers for vribles in the rules of TSS, s illustrted in Exmple 7. The TSS of Exmple 7 consists of single rule whose trget x rnges over the set of closed terms T (Σ). On the contrry, the TSS of Exmple 8 consists of rule templte such tht the trgets p i with i I rnge over n infinite proper subset of the set of closed terms, i.e., {p i i I} T (Σ). Techniclly, the sentences x. c x nd x {p i i I}. c x re respectively Π 1 -sentence nd Π 0 -sentence in the Lévy hierrchy [18]. Bounded quntifiers re conventionl nd useful, nd we wish our rule formt to llow for TSSs like the one of Exmple 8. To this end, we need more refined notion of bounded TSS, which disregrds the crdinlity of the set of inhbitnts of n η-type nd tkes into ccount the ctions of rules. We now define the ctions of n η-type nd introduce the notion of initilsbounded TSS. Definition 19 (Actions of n η-type). Let R be TSS. We define χ : η-type P(A) s the mp tht delivers, for ech η-type t, ψ, the set of ctions of the rules tht hve η-type t, ψ. More formlly, χ(t, ψ) = { ρ hs η-type t, ψ nd is the ction of ρ}. The set χ(t, ψ) is sid to be the ctions of η-type t, ψ. Definition 20 (Initils bounded). A TSS R is initils bounded iff R is uniform nd there exists η with codomin P ω (T(Σ)) (i.e., the set η(t) is finite for ech t) such tht for every rule ρ R with η-type t, ψ, the η-type t, ψ hs finitely mny ctions, i.e., χ(t, ψ) P ω (A). In the TSS of Exmple 8, the η-type c, ψ is infinitely inhbited but it hs finitely mny ctions s χ(c, ψ) = {}. Therefore, the TSS of Exmple 8 is initils bounded. Intuitively, since rules hving the sme η-type re those tht could potentilly be used to derive trnsitions from closed term p tht unifies with the source of the rules, requiring tht η-types hve finitely mny ctions cn help one to ensure tht p be initils finite. However, s the following exmple shows, hving strict strtifiction is lso needed to ensure initils finiteness s it intuitively disllows ungurded recursion. Exmple 9. Let Σ consist of constnt c nd unry function symbol f, nd let A = { 1, 2,...} be n infinite set of ctions. Consider the TSS with rules f(x) 1 c f(x) i y f(x) i+1 y, i N. The TSS is uniform nd in initils finite formt. We let η(f(x)) = {f(x)}. The rule on the left hs η-type f(x), ψ where ψ(f(x)) =, nd for ech i N, the instntition of the rule templte on the right hs η-type f(x), ψ i where ψ i (f(x)) = { i }. The set of ctions of f(x), ψ is { 1 }, nd for ech i N 16

17 the set of ctions of f(x), ψ i is { i+1 }. Therefore, the TSS is initils bounded. However, the TSS does not hve strict strtifiction. Notice tht the ssocited LTS is not initils finite, since the set of initils of f(c) is A. The conditions of the following theorem define the rule formt for initils finiteness. Theorem 2. Let R be n initils-bounded TSS in initils finite formt tht hs strict strtifiction S. The LTS ssocited with R is initils finite. Proof. We prove tht ech closed term p in the LTS ssocited with R is initils finite. Since R is uniform, for given p there re only finitely mny distinct terms t i nd substitutions σ i : vr(t i ) T (Σ) (i.e., the i rnges over finite index set I) such tht σ i (t i ) = p nd the rules tht unify with trnsitions from p hve some t i s source. We proceed by induction on S(p). The initil cse is when S(p) = 0. Since S(σ i (t i )) = 0, the rules with source t i re xioms of the form t i j, t k i I, j J i, k K j where the J i nd the K j re tken to be disjoint to void prolifertion of indices. Since R is initils bounded, there exists η such tht for ech i, for ech j J i, nd for ech k K j, the instntition of the rule templte bove hs η-type t i, ψ k, nd t i, ψ k hs finitely mny ctions, i.e., χ(t i, ψ k ) P ω (A). By Definition 14, ll the ψ k mp ech term in η(t i ) (if ny) to the empty set, so ll the rules bove with source t i hve the sme η-type. Since the rules bove re xioms, j the trnsitions σ i (t i ) τσi (t k ) re provble in R for ech substitution τ : (vr(t j ) \ vr(t i)) T (Σ). Since ll the ψ k in the η-types re equl, nd since the η-types hve finitely mny ctions, the sets J i re finite. Therefore, for ech i I the set { j p s.t. σ i (t i ) j p } = χ(t i, ψ k ) is finite. By the finiteness of I it follows tht the set { p s.t. p p } is finite nd we re done. The generl cse is when S(p) > 0. The rules with source t i such tht σ i (t i ) = p re of the form b {u l l u l l L k } j, i I, j J i, k K j t k t i where the J i, the K j, nd the L k re tken to be disjoint to void prolifertion of indices. Since R is initils bounded, there exists η such tht for ech i, for ech j J i, nd for ech k K j, the instntition of the rule templte bove hs η-type t i, ψ k, the set η(t i ) is finite, nd t i, ψ k hs finitely mny ctions. For ech i, we show tht there re only finitely mny distinct ψ k with k K j nd j J i such tht rules with η-type t i, ψ k give rise to trnsitions from σ i (t i ). By Definition 14, ech rule of η-type t i, ψ k contins premiss of the 17

18 form v c v for ech v η(t i ) nd ech c ψ k (v). Since R is in initils finite formt, vr(v) vr(t i ), nd thus the σ i (v) re closed. By Definitions 7 nd 8, for ech trnsition in the node of proof tree, if the trnsition unifies with rule of η-type t i, ψ k, then for ech v η(t i ) the processes σ i (v) cn perform, t lest, c-trnsition for ech c ψ k (v). The ψ k giving rise to trnsitions c from σ i (t i ) re dependent functions of type Π v η(ti){c σ i (v) τσ i (v )} with substitutions τ : (vr(v ) \ vr(v)) T (Σ). For ech i the refined type of the ψ k with k K j nd i J i is finitely inhbited, since the codomin of dependent function depends on the inputs of the function. Ech imge of ψ k cnnot be n rbitrry subset of A, but only the one determined by the input v nd by the ssocited LTS. Tht is, the only elements in the codomin of ψ k re the sets c {c σ i (v) τσ i (v )} where v η(t i ). Since the η(t i ) re finite sets, both the domin nd the codomin of ψ k re finite. Therefore for ech i there re only finitely mny distinct ψ k with k K j nd j J i such tht rules with η-type t i, ψ k give rise to trnsitions from σ i (t i ). Since for ech i there re only finitely mny distinct ψ k with k K j nd j J i such tht rules with η-type t i, ψ k give rise to trnsitions from σ i (t i ), nd since the η-types t i, ψ k hve finitely mny ctions, then for ech i I the set { j p s.t. σ i (t i ) j p } is finite. By the finiteness of I it follows tht the set { p s.t. p p } is finite nd we re done. Remrk 3. In Theorem 2 the TSS R is not required to be in bounded nondeterminism formt. The terms t k my hve vribles which re neither in t i nor in l L k vr(u l ). Consider τ m : (vr(t k ) \ vr(t i)) T (Σ) closed substitutions with m rnging over index sets M k such tht σ i (t i ) j τ m σ i (t k ). For ech σ i (t i ) there my be infinitely mny trnsitions σ i (t i ) j τ m σ i (t k ) becuse the M k my be infinite. This is illustrted by Exmple 7. Remrk 4. In Theorem 2 the η-types re not required to be finitely inhbited. For ech η-type t i, ψ k there could be infinitely mny rules with conclusions j t i t k, nd the K j need not be finite. The TSS R could be in bounded nondeterminism formt, nd then there would be τ m : (( l L k vr(u l )) \ vr(t i)) T (Σ) closed substitutions with m M k nd M k re finite index sets such tht σ i (t i ) j τ m σ i (t k ). But, lthough the M k my be finite, for ech σ i (t i ) there my be infinitely mny trnsitions σ i (t i ) j τ m σ i (t k ), becuse the K j my be infinite. This is illustrted by Exmple 8. We now present n exmple of ppliction of the rule formt defined by the conditions of Theorem 2. Exmple 10. Let Σ contin constnts c nd 0 nd the unry ction prefixing opertion. from Milner s CCS [20]. Consider the TSS with rules.x }{{ x } b c. }. {{.... } 0, i 0. i times 18

19 Intuitively, the constnt c is kin to rndom ssignment [4]. The TSS is uniform nd hs trivil strict strtifiction. We let η(.x) = nd η(c) =. The rule on the left hs η-type.x, nd ech instntition of the rule templte on the right hs η-type c,. The rule templte on the right implements bounded quntifiers s illustrted in Exmple 8. Although the η-type c, is infinitely inhbited, the set of its ctions is {}. The ssocited LTS is initils finite. In the next section we develop rule formt for imge finiteness. 5 Imge finiteness Consider the properties of n LTS in Definition 13, which we prphrse here in mthemticl nottion: Finite brnching: p. {(, p ) p p } P ω (A T (Σ)). Initils finiteness: p. { p. p p } P ω (A). Imge finiteness: p.. {p p p } P ω (T (Σ)). We consider the Brouwer-Heyting-Kolmogorov interprettion of intuitionistic logic (BHK interprettion for short) [15]. According to the BHK interprettion, the proof of ny of the properties bove consists of function tht tkes one rgument for ech of the universlly quntified symbols nd returns proof of the triling proposition fter the quntifiers, 3 which sserts tht some set is finite. As proof of ech ssertion, it is enough to exhibit the set in point. For exmple, given TSS R the proof tht the LTS ssocited with R is initils finite consists of function tht tkes n element p T (Σ) nd delivers the finite set of ctions such tht p p (with p T (Σ)) is provble in R. The BHK interprettion provides profitble insight on the notion of η-types. In essence, the η-types re sort of syntctic fingerprint of the BHK interprettion of initils finiteness. Recll from Definition 14 tht in n η-type t, ψ the mp ψ tkes term nd delivers set of ctions. This mp represents the function corresponding to the BHK interprettion. The disciplined focus on the positives premisses (e.g., through the finite support of the sources defined by η nd with the vrible flow enforced by the initils finite formt) is only n instrument to construct the intuitionistic proof from ψ, by induction on the strict strtifiction of the TSS. This is exemplified by our proof of Theorem 2. It my seem odd tht the η-types, which correspond to the BHK interprettion of initils finiteness, re lso used in the rule formt tht ensures finite brnching. The mp ψ delivers set of ctions b, insted of the set of pirs of ctions nd terms (b, u ) tht would be expected from the BHK interprettion of finite brnching. The reson lies in the fct tht the dditionl requirements of the rule formt mke keeping trck of the trgets u redundnt. To see this, let the positive premisses be of the shpe u b u. Since the TSS is required to be in 3 Recll tht in intuitionistic logic universl quntifier x. is kin to big lmbd Λx., i.e., binding opertor t the level of types. 19

20 bounded nondeterminism formt, then vr(u) vr(t) nd the σ(u) re closed. Thus, for ech u η(t) there re t most finitely mny pirs (b, u ) such tht b b ψ(u) nd σ(u) τσ(u ) with substitutions τ : (vr(u ) \ vr(u)) T (Σ). Therefore, keeping trck of the trgets u of positive premisses is redundnt becuse the requirements of the rule formt ensure tht the ssocited LTS is finite brnching. The different requirements for bounded TSS nd for initils-bounded TSS complete the picture, respectively for the BHK interprettion of finite brnching nd of initils finiteness. In bounded TSS, it is required tht there exists n η such tht ech η-type t, ψ is finitely inhbited. This enforces tht if for ech term t nd for ech substitution σ such tht σ(t) is closed there re only finitely mny ψ such tht the rules tht give rise to trnsitions from σ(t) hve η-type t, ψ, then the set of pirs (, t ) such tht σ(t) τσ(t ) with substitutions τ : (vr(t ) \ vr(t)) T (Σ) is finite. The bounded nondeterminism formt ensures tht there re only finitely mny substitutions τ, nd then the set of pirs (, τσ(t )) is lso finite nd the ssocited LTS is finite brnching. In n initils-bounded TSS, it is only required tht there exists n η such tht ech η-type t, ψ hs finite ctions. This enforces tht if for ech term t nd for ech substitution σ such tht σ(t) is closed there re only finitely mny ψ such tht the rules tht give rise to trnsitions from σ(t) hve η-type t, ψ, then the set of ctions such tht σ(t) τσ(t ) with substitutions τ : (vr(t ) \ vr(t)) T (Σ) is finite. The bounded nondeterminism formt cn be replced by the initils finite formt becuse the number of substitutions τ is immteril in order to keep the number of ctions finite, nd thus for the ssocited LTS to be initils finite. We now introduce the θ-types, which re glened from the BHK interprettion of imge finiteness. Unlike the η-types of [9], which crry informtion bout the sources of positive premisses in rules, the θ-types lso keep trck of the ctions tht lbel positive premisses. We let θ : (T(Σ) A) P(T(Σ) A) be the mps tht prmetrise the θ-types of Definition 21 to follow. The mps θ deliver, for given term t nd ction, predefined set of sources nd ctions of positive premisses in rules tht hve source t nd ction. We sy tht θ(t, ) is the support of the sources nd of the ctions for source nd ction (t, ). 4 Definition 21 (θ-type). Let R be TSS, ρ R rule with source t, ction, nd positive premisses {t i i t i i I}, nd θ mp with type (T(Σ) A) P(T(Σ) A). We define φ : θ(t, ) P(T(Σ)) s the mp tht delivers, for ech term u nd ction b in the support of the sources nd of the ctions for (t, ), i.e., (u, b) θ(t, ), the trgets of the positive premisses of ρ with source u nd ction b. More formlly, φ(u, b) = {t i i I t i = u i = b}. 4 We beg the reder to ber with us in the repetition of sources, ctions, source, nd ction in sentences like the bove. The sources nd ctions refer to the positive premisses, nd the source nd ction to the conclusion of the rule. 20