A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to a set is developed. To do this, we extend the sufficient condition of [13] that imposes the additional requirement of rigidity of the level mapping on the call set, to the case of general quasi orderings. First we adapt the notion of rigidity to general orderings. Definition 7. see also [8]) The term or atom A 2 U P [B P is called rigid with respect to a quasi ordering if for any substitution , A A . In this case is said to be rigid on A. The notion of the rigidity on a term (an atom) is naturally extended to the notion of rigidity on a set of atoms (terms) In particular, we ....

....for proving termination are unable to generate them. In such cases, generating an appropriate ordering, replacing the level mapping, may often be much easier, especially since we can reuse the impressive machinery on orderings developed for term rewrite systems. In some other cases, such as turn [8], simple level mappings do exist (in the case of turn: a norm counting the number of 0s before the first occurrence of 1 in the list is sufficient) but most systems based on level mappings will not even find this level mapping, because they only consider mappings based on term size or list length ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, February 1994.

....FC(P ) associated to a program P are the sets of symbols PC(P ) fp i j p=n 2 P redP i 2 f1; ngg, and FC(P ) ff i j f=m 2 FunP i 2 f0; 1; mgg. Here, all norms and level mappings will be of a speci ed form (for the norms this is a slight variant of the semi linear norms [3]) De nition 12 (symbolic norm and level mapping, symbol mapping) The symbolic norm k : k S and symbolic level mapping j:j S are de ned as: k X k S = k c k S = 0 for X 2 V arP ; c 2 ConstP ; k f(t 1 ; t m ) k S = f 0 P m i=1 f i k t i k S jp(t 1 ; t n )j S ....

....In particular, in [8] it is required that the level mapping is rigid on the call set of the program w.r.t. the set of queries for which one wants to prove termination. A level mapping is rigid on a set of atoms, if the value of an atom of the set is invariant under substitutions. As was shown in [3] (see also [15] a level mapping j:j s is rigid on a set S if it does not take into account too many argument positions of predicates and functors in S (more precisely, for every atom A in S and every variable X in A, no occurrence of X in A is taken into account, i.e. s is 0 there) As follows ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297-328, 1994.

....w.r.t. a set is developed. To do this, we extend the sufficient condition of [8] that impose the additional requirement of rigidity of the level mapping on the call set, to the case of general term orders. First we adapt the notion of rigidity to general term orders. Definition 8. see also [6]) The term or atom A 2 U E P [ B E P is called rigid w.r.t. a quasi order if for any substitution , A = A . In this case is said to be rigid on A. 7 The notion of the rigidity on a term (an atom) is naturally extended to the notion of rigidity on a set of atoms (terms) In particular, we ....

....termination are unable to generate them. In such cases, generating an appropriate term ordering, replacing the level mapping, may often be much easier, especially since we can reuse the impressive machinery on term orders developed for term rewrite systems. In some other cases, such as turn [6], simple level mappings do exist (in the case of turn: a norm counting the number of 0s before the first occurrence of 1 in the list is sufficient) but most systems based on level mappings will not even find this level mapping, because they only consider mappings based on term size or list length ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, February 1994.

....language underlying P only contains a finite number of predicate symbols) So, the question is: which norms are finitely partitioning Let us first assume that the language underlying P contains only a finite number of constants and functions symbols. In that case, consider any semilinear norm [3], which are norms that can be defined as: n(f(t 1 ; t k ) c f X i2If n(t i ) where c f 2 IN , I f [1; k] and depend on f=k only. Given the restriction on the language underlying P , any semi linear norm which has c f 0 and I f = 1; k] for all f=k in FunP , in other words ....

A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and their use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124(2):297-- 328, February 1994.

....the simple semantics of logic programs made the search for sufficient conditions for termination a challenge for the research community. Research on the first two topics of the list above led to completely automated tools for verifying termination [12, 22] based on the use of linear norms (cf. [6, 14, 27, 34]) These systems are powerful enough to deal with a large fraction of the programs that have appeared in the literature [3, 10, 14, 21] Moreover, most of the examples can be proved using term size or, less often, list size. When other linear norms are necessary, the user is expected to provide ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, February 1994.

....j f=m 2 FunP i 2 f0; 1; mgg , PC(P ) fp i j p=n 2 P redP i 2 f1; ngg . Similar as in the constraint based approach of [14] towards LD termination, we will consider norms and level mappings of a speci ed form (for the norms this is a slight variant of the semi linear norms [8]) We rst introduce these symbolic forms for the norm and level mapping, with symbols in FC(P ) PC(P ) A concrete norm or level mapping is then obtained by giving values to the symbols of FC(P ) PC(P ) in IN. 14 So e Verbaeten, Danny De Schreye De nition 17 (symbolic norm and level mapping) ....

....= 0; n, for all f=n 2 FunP , n 0. In the rest of this subsection, we introduce the notions of rigid and nitely partitioning level mapping. As we will see, these notions turn out to be very useful in the context of a termination analysis. In particular, the notion of rigid level mapping [8] deals with the problem of backpropagation of bindings in the calls. Instead of reasoning on calls in the termination condition, a level mapping which is rigid on the call set al..lows us to reason fully at the clause level, which is important in the context of a constraint based, automatic ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297-328, 1994.

....defined below measures the length of both open and closed lists of integers. jvj List(Int) v jN ilj List(Int) 0 jCons(t 1 ; t 2 )j List(Int) 1 jt 2 j List(Int) 2 It is appropriate at this point to review the important concept of rigidity. This idea was originally introduced in [5] in order to prove termination for a class of goals with possibly nonground terms. A rigid term is one whose size, as determined by a norm, is not affected by substitutions applied to the term. Definition 3.2 (rigid term) Let j:j be a typed norm for and t be a term of type . Then t is rigid ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.

....method can be viewed as a relativization of the above termination proof methods with respect to the intended queries; this facilitates in many examples the required reasoning, in that uninteresting input queries are not to be taken into account. The method of Bossi and Cocco has been extended in [15] to reason on termination. They define level mappings on non ground atoms as well. However, this leads to complications, since correctness can be proved only using a restricted class of level mappings, called rigid , which limits the expressiveness of the method. In conclusion, the proof method ....

A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and their use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124:297--328, 1994.

....work has been recently devoted to universal termination. For instance [1, 2, 3, 4] suggest new theoretical characterizations. In addition there are many proposals of effective methods, such as those in [12, 14, 15, 16] which infer interargument relations using AND OR Dataflow Graphs, and those in [6, 7], where the termination is proved for programs enriched with assertions. Up to now, very little attention has been devoted to existential termination. 5] proposes a semantic approach using inductive proof techniques, 11] studies an effective method for function free programs, while [10, 19] just ....

A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and Their Use in Proving Universal Termination of a Logic Programs. TCS, 124(2):297--328, 1994.

....is a generalization of the method for proving termination in [7, 2] The proposed method combines the two approaches [10] and [7] into a unified one, which thus provides a uniform framework for addressing the various verification issues. The mentioned method in [10] has been also extended in [11] to deal with termination, by adopting level mappings on non ground atoms. This leads to complications, since restrictions are to be imposed on the level mappings, which are prevented in our approach. There are several symmetries in the Definitions 3. In particular, we want to point out that the ....

A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and their use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124:297--328, 1994. 38

....P ost 0 a model of P which is not present in our approach. In addition, confusion can arise due to the fact that acceptability analysis acts at a ground level, whilst well m assertedness acts at a non ground level. Also, we mention that well m assertedness has been extended by Bossi et al. [17] to reason on termination. They de ne level mappings j j on non ground atoms as well, and require that for every A B 1 ; Bn instance of a clause of P , for every i 2 [1; n] pre j= A post j= B 1 ; B i 1 implies jAj jB i j: 6) However, this leads to complications, since ....

A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and their use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124:297-328, 1994.

....the simple semantics of logic programs made the search for sufficient conditions for termination a challenge for the research community. Research on the first two topics of the list above led to completely automated tools for verifying termination [21, 11] based on the use of linear norms (cf. [25, 31, 13, 6]) These systems are powerful enough to deal with a large fraction of the programs that have appeared in the literature [3, 20, 13, 9] Moreover, most of the examples can be proved using term size or, less often, list size. When other linear norms are necessary, the user is expected to provide ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, February 1994.

....use the information from B select to update B 9. The following abstract sequence is obtained: B10 = betaref: frm = 5 [6 7] mo = 6 ground,7 ground ty = 2 list, 5 list,6 any,7 list betaout: frm = 5 [7 8] mo = 1 ground, 3 ground, 6 ground,7 ground, 8 ground ty = 1 any,2 list,3 list, 5 list,6 list,7 any, 10 8 list ps = inref = 1 1,2 2,3 3,4 4,5 5,6 6,7 7 inout = 1 8,2 9,3 10,4 11,5 12,6 13,7 14,8 15 Erefout = sz(13) sz(7) Esol = sol=sz(7) 1 It ....

.... betaref: sv = X 1,L 2,LS 3; frm = 2 [4 5] 5 [6 7] mo = 1 var,2 ground,3 var,4 ground,5 ground, 6 ground,7 ground ty = 1 anylist,2 list,3 anylist,4 any,5 list,6 any, 7 list ps = 1,1) 3,3) betaout: sv= X 1,L 2,LS 3; frm = 2 [4 5] 3 [4 6] 5 [7 8] mo = 1 ground,2 ground,3 ground,4 ground,5 ground, 6 ground,7 ground,8 ground ty = 1 any,2 list,3 list,4 any,5 list,6 list,7 any, 8 list ps = inref = 1 1,2 2,3 3,4 4,5 5,6 6,7 7 inout = 1 8,2 9,3 10,4 11,5 12,6 13,7 14,8 15 Erefout = sz(13) sz(7) Esol = ....

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A. Bossi, N. Cocco, and M. Fabris. Norms on Terms and their Use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124(2):297--328, 1994. 37

....) associated to a program P are the sets of symbols ffl FC(P ) ff i j f=m 2 FunP i 2 f0; 1; mgg , ffl PC(P ) fp i j p=n 2 P redP i 2 f1; ngg . Here, all norms and level mappings will be of a specified form (for the norms this is a slight variant of the semi linear norms [8]) We first introduce these symbolic forms for the norm and level mapping, with symbols in FC(P ) PC(P ) A concrete norm or level mapping is then obtained by giving values to the symbols in FC(P ) PC(P ) in IN. Let V arP denote the set of variables in a program and ConstP denote the set of ....

....0; n, for all f=n 2 FunP , n 0. In the rest of this subsection, we introduce the notions of rigid and finitely partitioning level mapping. As we will see, these notions turn out to be very useful in the context of a termination analysis. In particular, the notion of rigid level mapping [8] deals with the problem of backpropagation of bindings in the calls. Instead of reasoning on calls in the termination condition, a level mapping which is rigid on the call set al..lows us to reason fully at the clause level, which is important in the context of an automatic termination analysis ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297-- 328, februari 1994.

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A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.

....1 Introduction When developing a logic program we, more or less explicitly, have to analyze it with respect to correctness, termination properties and the existence of successful computations. On correctness and termination verification there have been rather many proposals such as [21, 8, 3, 12, 7, 24]. While for distinguishing failing and successful computations or for ensuring the presence of successes in a computation, very little is available, both in terms of methodology and tools. We can illustrate the kind of information on derivations we would like to infer through a few simple ....

....query. An LD derivation ending with 2 is a successful LD derivation, one ending with fail is a failing one (FD) We consider the property of universal termination for a query Q in a program P , which means that the LD tree of Q in P is finite. This termination property has been widely studied [7, 12, 24] and it ensures termination with a Prolog interpreter. We denote sequences by bold characters and we call p(X) a general atom when its terms are all distinct variables. We use identifiers to label clauses and derivations. Then l : Q j oe Gamma P R stands for an LD derivation, l, of the query ....

Bossi, A., Cocco, N., Fabris, M.: Norms on Terms and their Use in Proving Universal Termination of a Logic Program. Theoretical Computer Science, 124 (1994) 297--328

....11] LD resolution [10, 23] LDNF resolution [9, 12] SLD resolution with dynamic scheduling [14, 26] or with tabling [30, 29] The second one is intended to automatize the veri cation by de ning sucient conditions for termination wrt. a speci c interpreter, e.g. the standard Prolog interpreter [31, 20, 13, 22, 19, 21]. In this paper we follow the rst approach: we de ne and characterize the class of well typed typed terminating programs, namely well typed general programs terminating wrt. LDNF resolution for any well typed general query. These programs and queries may contain negated literals; they are moded ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297-328, 1994.

....interpreter, such as termination wrt. SLD resolution [3, 11] LD resolution [10, 22] LDNFresolution [9, 12] SLD resolution with dynamic scheduling [14, 25] The second one is intended to automatize the veri cation by de ning sucient conditions for termination wrt. the standard Prolog interpreter [27, 20, 13, 21, 19]. In this paper we follow the rst approach: we de ne and characterize the class of well typed typed terminating programs, namely well typed general programs terminating wrt. LDNF resolution for any well typed general query. These programs and queries may contain negated literals; they are moded ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297-328, 1994.

....considering semilinear norms and rigid terms [BCF91] instead of linear norms and ground terms in input, and his condition to be safe corresponds to our condition to be well acceptable. An automatic termination analysis derived by these sufficient conditions has been adopted by Mercury [SSS97] In [BCF91, BCF94] Pre Post conditions on the atoms are used for proving termination. Such Pre Post conditions deal with the rigidity of terms. Rigidity of terms is a generalization of groundness and a consequence of the mode and type properties of the atom. In [DDSB92, DSF93, SV95, Dec97] modes are essential for ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.

....well typing and in the termination proofs we can make use of typing information. The typical situation is as follows. We have a program P and a query Q which are well typed wrt a given typing. The typing specifies the structure of input terms. By associating some measure to input terms (a norm in [15], a level mapping in [5] we may prove that Q universally terminates in P . But also a termination by failure is possible, in fact the previous examples show that the well typing of a query and a program does not imply the absence of failures. Also the correct typing of the query does not imply ....

....Ys) reverse(Xs, Rs) app(Rs, X] Ys) With typing reverse( List; Gamma : List) and app( List; List; Gamma : List) any well typed query reverse(x; y) namely with x 2 List, universally terminates. This may be proved by means of any technique for proving universal termination such as [5, 15, 31, 39, 38]. By applying our safe transformation sequence, extended by the safe replacement, we obtain a new well typed program which is equivalent to the original one wrt that query, but more efficient. This is a well known example of optimization by introduction of an accumulator [8, 16, 27, 34, 29, 40] ....

A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.

....well moded and simply moded. Thus B j 0 is ground in its input positions and its output positions are filled in by distinct variables and then B j 0 is a renaming of B j 0 . 12] Since B j is an instance of B j 0 , we observe that jB j j is defined and jB j j = jB j 0 j. [13] Then we have that 12 jH j definition of j j = nodes P (H ) by Lemma 5.1 (iii) nodes P (B 1 ; Bn ) by [10] and Lemma 5.1 (i) nodes P (B 1 0 ; Bn 0 ) by [11] and Lemma 5.2 nodes P (B 1 0 ; B j Gamma1 0 ) nodes P (B j 0 ; Bn 0 ....

....P (B 1 ; Bn ) by [10] and Lemma 5.1 (i) nodes P (B 1 0 ; Bn 0 ) by [11] and Lemma 5.2 nodes P (B 1 0 ; B j Gamma1 0 ) nodes P (B j 0 ; Bn 0 ) nodes P (B j 0 ; Bn 0 ) by Lemma 5. 1 (ii) nodes P (B j 0 ) by [12] and [13] = jB j j which proves [9] and the thesis. 2 6. CONCLUSIONS AND RELATED WORK In this paper we have studied how mode information can be used for characterizing termination properties. We have defined the class of well terminating programs, namely programs for which all well moded queries have ....

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A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297-- 328, 1994.

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A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124:297--328, 1994.

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A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, 1994.

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A. Bossi, N. Cocco, and M. Fabris. Norms on terms and their use in proving universal termination of a logic program. Theoretical Computer Science, 124(2):297--328, February 1994.

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BOSSI, A., COCCO,N.,AND FABRIS, M. 1994. Norms on terms and their use in proving universal termination of a logic program. Theor. Comput. Sci. 124, 2 (Feb.), 297--328.

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