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1 Automatic Verication Based on Abstract Interpretation Mizuhito Ogawa 1 NTT Communication Science Laboratories 3-1 Morinosato-Wakamiya Atsugi Kanagawa, Japan Abstract. This paper reconstructs and extends the automatic verication technique of Le Metayer, Proving properties of programs dened over recursive data structures (ACM PEPM '95), based on a backward abstract interpretation. To show the eectiveness of extensions, we show two examples of the declarative specications of sorting and formatting programs, which are directly and concisely expressed in our specication language. 1 Introduction Program errors cause failures during execution that can be classied into three categories. 1. Execution eventually stops as a result of illegal operations. 2. Execution does not terminate. 3. Execution results are not what was intended. Errors of the rst kind are detected by type inference, with such languages as ML. In addition, although termination is in general undecidable, errors of the second kind can be automatically prevented by several techniques, such as simple termination [12, 13], termination analysis [16], and dependency pairs [2]. The third kind of error cannot be prevented without a specication language, and there is always a trade-o between expressiveness and feasibility. If the aim is to express everything, it is easy to fall into the trap of undecidability. Moreover, too much expressiveness may make users hard to learn. For compiletime error detection, an automatic verier that functions without any human guidance is desirable even if it veries only partial specications. Then the user can concentrate on what kind of properties, under the limitation of a simple and restricted specication language, properly approximate the program behavior. By restricting both properties and languages, Le Metayer developed an automatic verication technique [19]. Its target language is a strongly-typed rstorder functional language with product types and recursive types. The important restriction is that the conditional part of an if-expression contains only basic predicates (such as null, leq, geq, and equal) without any functional symbols.
2 He denes a language which prescribes a class of uniform predicates over recursive types. These predicates are constructed by predicate constructors from basic predicates on base types. As an example, his system expresses that a sort program returns a list of decreasing values (if the sort program terminates) and automatically veries it. This property is called orderedness of the sort program, which is expressed by true!rgeq(sort X) in our specication language. Note that the termination of the sort program is not veried; this verication is left to a termination analysis. Similar ideas to those of uniform predicates are also found in Refs. [15, 17, 3, 21]; however, the signicant dierences are that { binary predicates are allowed in constructing predicates, and { free variables in binary predicates are allowed. The former extends the expressiveness of target properties from other ow analyses. The latter maintains the power of inter-functional inferences. However, the expressive power of the specication language is still fairly restricted as a verication; for instance, the input-output relation cannot be described. This paper reconstructs and extends the automatic verication technique of Le Metayer [19] based on a backward abstract interpretation [11, 1, 7]. The termination and soundness proofs of the verication are naturally derived from the formalization as a backward abstract interpretation. Extensions are achieved by (1) using the input variable in function properties, (2) introducing new predicate constructors, and (3) using uninterpreted function/predicate symbols. They are demonstrated by verifying the sorting and formatting programs. The rst and the second extensions expand the ability of the specication language so that it covers another major specication of the sorting program; namely, weak preservation, i.e., the input and the output are the same set. This is expressed by true!8 l 9 r equal ^8 r 9 l equal(sort X): Note that since our specication language cannot express the number of elements in a list, our algorithm cannot detect the full specication of sort, called preservation, i.e., the input and the output are the same multiset. The third extension expands the range of both target programs and the specication language. The expansion of target programs is achieved by loosening the restrictions on the conditional part of an if-expression. The running example is format, which formats a given sentence (expressed as a list of strings) to a specied width. The technique behind this extension is the use of uninterpreted functions. We also show how partial evaluation will cooperate with the verication. Other major specications of format become expressible by the use of uninterpreted predicates. This technique drastically expands the expressive ability, such as the specication that the order of words is preserved by format. This paper is organized as follows: Section 2 denes programming and specication languages. Section 3provides the verication algorithm based on a backward abstract interpretation. The termination and soundness proofs are also given. Section 4 demonstrates the verication of orderedness of the (simple but inecient) sort program to explain the algorithm. Section 5 presents extensions
3 and demonstrates the verication of major specications of the sorting and formatting programs. Section 6 discusses related work and Section 7 concludes the paper and discusses future work. 2 Preliminaries 2.1 Programming Language The target language is a strongly-typed rst-order functional language with MLlike syntax, in which product types and recursive types, such as lists list(a) = :nil + A 2 list(), are allowed. We use :: to mean inx to mean inx append, and [ ] to mean a list, namely, [a 1 ;a 2 ;a 3 ]=a 1 :: (a 2 :: (a 3 :: nil)). The semantics of the language is given by an ordinary least x-point computation. We assume that the language is strict, but the same technique can be applied to a lazy language as well. The precise syntax and semantics of the language are shown in Fig. 1 and Fig. 2. Parentheses in the syntax are used for either making pairs or clarifying the order of applications of inx operators. Basic concrete domains D Bool and D Int are at cpo's (as usual), and the other concrete domains D of type are constructed by the list and pair constructors. The interpretation of expressions has the hidden argument, i.e., for simplicity the environment fve of function variables are omitted in Fig. 2. The language of expressions E = x j C j (E 1 ;E 2 ) j E 1 :: E 2 j fej op E j (E) j if Cond then E 1 else E 2 j let val x = E 1 in E 2 end j let val (x; y) =E 1 in E 2 end j let val x :: xs = E 1 in E 2 end Cond = p u x j p b (x; y) 8 < E 2 Exp expressions op 2 Prim primitive functions where C 2 Const constants p u ;p b 2 Pred basic predicates : x 2 Bv bound variables f 2 Fv functional variables The syntax of programs Prog = f fun f i x i = E i ; g The language of types T = T G j T F T F = T G! T G T G = T U j T P j T R T P = T G 2 T G T R = :nil + T G :: T U = (basic types) Fig. 1. Syntax of Programming Language.
4 '[[ f fun f i x i = E i ; g ]] [[C]]bve [[x]bve [[op E]]bve [[p u x]]bve [[p b (x; y)]]bve [[f E]]bve [[(E 1 ;E 2 )]]bve [[E 1 :: E 2 ]]bve = fve whererec fve =[(y 1 111y n :if (bottom? y 1 111y n ) then? else [[E i ]][y j =x j ])=f i ] = c [[C]] = bve[[x] = f [[op]( [[E]]bve) = p [[p u ]](bve[[x]) = p [[p b ]](bve[[x];bve[[y]]) = fve[[f]]( [[E]]bve) =( [[E 1 ]]bve; [[E 2 ]]bve) =( [[E 1 ]]bve) ::( [[E 2 ]]bve) [[if Cond then E 1 else E 2 ]]bve = if (bottom? ( [[Cond]]bve)) then? elsif [[Cond]]bve then [[E 1 ]]bve else [[E 2 ]]bve [[let val x = E 1 in E 2 ]]bve = [[E 2 ]](bve[ [[E 1 ]]bve=x]) [[let val (x; y) =E 1 in E 2 ]]bve = [[E 2 ]](bve[ [[E 1 ]]bve=(x; y)]) [[let val x :: xs = E 1 in E 2 ]]bve = [[E 2 ]](bve[ [[E 1 ]]bve=x :: xs]) bottom? y 1 111y n =(y 1 =?) _111_(y n =?) 8 :(Fve!)Exp! Bve! D >< p : Prog! Fve where fve 2 Fve = Fv! D! D >: bve 2 Bve = Bv! D f : Prim! D! D p : Pred! D! Bool c : Const! D Fig. 2. Semantics of Programming Language. An important restriction is that the conditional part of an if-expression must consist only of basic predicates without any functional symbols. Section 5 discusses how this requirement can be loosened. Until then, we use only null as the unary basic predicate on lists, leq, geq, and equal as the binary basic predicates on integers, :: as a binary primitive function, and nil as a constant. The type description in a program is often omitted if it can be easily deduced by type inference. For technical simplicity, we also set the following restrictions. { Basic types are Int and Bool. { Product types and recursive types are pairs and lists, respectively. { Each function is unary, i.e., pairs must be used to compose variables. The third restriction means that binary functions and predicates are respectively regarded as unary functions and predicates which accept the argument of pair
5 type. This assumption can easily be extended to a more general setting, for instance, with tuple types. Values are denoted by a; b; c; 111, lists by as; bs; cs; 111, and lists of lists by ass; bss; css; We also assume that variable names of input variables and locally dened variables are dierent. The following functions present a sorting program with types sort : int list! int list and max : int list! int2int list. fun sort as = if null as then nil else let val (b,bs) = max as in b::sort bs end; fun max cs = let val d::ds = cs in if null ds then (d,nil) else let val (e,es) = max ds in if leq(e,d) then (d,e::es) else (e,d::es) end end; 2.2 Specication Language The language for specifying properties is constructed by using predicate constructors 8; 8 l ; 8 r, and r on basic predicates, constants, free variables, and variables appearing in a program. Predicate constructors will be extended in Section 5. A basic unary predicate is denoted by p U, a basic binary predicate by p B, a unary predicate by P U, and a binary predicate by P B. Indexes U and B are often omitted when they are clear from the context. As convention, bound variables are denoted by a; b; c; 111;x;y;z;111; as; bs; cs; 111; xs; ys; zs; 111,freevariables by X; Y; Z; 111, constants by C; M; 111, and expressions by E;E 1 ;E 2 ; 111. A binary predicate P is transformed into a unary predicate P E by substituting an expression E for the second argument. That is, P E (E 0 )=P (E 0 ;E). P is dened by P (E1 ;E 2 )=P (E 2 ;E 1 ). The grammar of the construction of predicates is shown in Fig. 3. Specication of a function f is expressed with a free variable by Q(X)! P (f X); which means if input X satises Q then output fxsatises P, when P; Q 2 P 0 G. Each input X is a distinct free variable for each function and property to avoid name crash. Note that negation is not allowed. The meanings and examples of the predicate constructors 8; 8 l ; 8 r ; and r are given as follows. { 8P U (xs) i either xs is nil or the unary predicate P U (x) holds for each element x in xs. { rp B (xs) i either xs is nil or 8 P y B (ys) ^rp B(ys) for xs = y :: ys. 8 l P B (xs; y) and8 r P B (x; ys) are dened by 8P y (xs) and8 B P x B (ys), respectively. The examples are shown in the table below. 1 as is a reserved word of ML, but we ignore it.
6 P F = P 0 X! P 0 G G (fx) properties of functions P 0 = predicates in P G G without bound variables P G = P S j P P j true j false ground properties P S = P U j P R unary predicates P R = 8P U jrp G j PP E j P R ^ P R j P R _ P R properties of lists P P = P B j PP j P S 2 P S j8 r P P j8 l P P j properties of pairs P P ^ P P j P P _ P P P B = p b j PB j P B ^ P B j P B _ P B basic binary predicates P U = p u j PB X j P U ^ P U j P U _ P U basic unary predicates E expressions V = X j x j c basic expressions X (nitely many) free variables x bound variables C constants Fig. 3. Language for specication of properties. predicate true false geq 3 (a) 4 2 8geq 3 (as) [3,6,4], nil [3,6,1] 8 l leq(as; a) ([3,6,4], 8), (nil, 8) ([3,6,4], 5) 8 r leq(a; as) (3, [3,6,4]), (3, nil) (5, [3,6,4]) rgeq(as) [6,4,3], nil [4,6,3] For instance, the sorting program is fully specied by the following conditions. 1. Output must be ordered (called orderedness). 2. Input and output are the same multiset (called preservation). Orderedness is expressed as true!rgeq(sort X): That is, the output of the sorting program is decreasing if the input satises true (i.e., empty assumptions). For preservation, the weaker condition called weak preservation, i.e., the input and the output are the same set, is expressed by true! (8 l 9 r equal ^8 r 9 l equal) X (sort X): with the introduction of additional predicate constructors 9; 9 l, and 9 r (which intuitively mean :8:; :8 l :, and :8 r :, respectively), which will be discussed in Section 5. Note that only the input variable of a function remains in the scope when a function call occurs, thus our denition of properties of functions (P F ) is possible (which was neglected in [19]).
7 3 Automatic Verication as Abstract Interpretation 3.1 Verication Algorithm as Abstract Interpretation An abstract interpretation consists of an abstract domain, its order, and an interpretation (on an abstract domain) of primitive functions [11, 1, 7]. Our choice is a backward abstract interpretation with abstract domain a set of predicates (in Fig. 3) satisfying the type, order the entailment relation dened in Fig. 4, and interpretation dened in Fig. 5. Let f fun f i x i = E i ; g be a program. The verication algorithm is the least xed point computation to solve whererec equations in Fig. 5. Section 4 explains how this algorithm performs on the sorting program. The entailment relation v (in Fig. 4) is intuitively the opposite of the logical implication. That is, P v Q means that Q implies P. By denition, true is the least element and false is the greatest element in the abstract domain. We denote by P Q if P v Q and P w Q. Theentailment relation may be used to trim at each step of interpretation 9. Formally, the entailment relation consists of axioms on basic predicates/predicate constructors, and ordinary logical rules, and their extensions by predicate constructors, as dened in Fig. 4. In Fig. 5, let Formula be a set of all formulae constructed from predicates in Fig. 3and bound variables (in the scope of an expression) with logical connectives ^ and _. The disjunction _ is regarded as composing branches of the verication, i.e., each branch (conjunctive formula) is analyzed independently. The projection # extracts a predicate P, in which a bound variable x (in the scope of an expression E) is substituted in a formula. When regarding a conjunctive formula as an assignment from bound variables to predicates, # x coincides with the restriction to x. For instance, (leq(x; y) ^r(xs)) # x = leq y (x). Note that the # operator is used when the local denition of x in a let-expression is analyzed, thus P = # x must exclude x. In our specication language, if such case occurs then P (x) can be reduced true or false using the entailment relation. The :-elimination operator : is dened by : [(Cond ^ P) _ (:Cond ^ P 0 )]=_ i Q i where each Q i satises Cond ^ P v Q i and :Cond ^ P 0 v Q i.for instance, : [(leq(e; d) ^8leq d (es)) _ (:leq(e; d) ^8leq e )]=8leq d (es) _8leq e (es). The interpretation of a function call 9[[f E]]P = 9[[E]]((fvp[[f]]P ) ) requires another operation, called -expansion (similar to the substitution calculus in higher-order rewrite systems [24]). When a function f is called in an expression E, bound variables (except for an input variable to f) become out of the scope. Thus, if a predicate P contains a bound variable, then it must be replaced with a free variable and the substitution to the free variable must be kept. They are P and.for instance, 9[[f E]]8leq b creates P = 8leq Z and =[Z b].
8 Axioms on basic predicates equal equal leq geq null(nil) true null(x :: xs) false equal(x; x) true geq(x; x) true leq(x; x) true equal v equal x 2 equal x geq v geq x 2 geq x leq v leq x 2 leq x P v P x 2 P x 8P y v8p z ^ P (z;y) P v P x 2 P x Transitivity 8 l 8 r P v (8 l 8 r P ) xs xs 2 (8 l 8 r P) Ordinary logical rules on logical connectives P ^ P P P 1 v P 1 ^ P 2 true ^ P P false ^ P false P _ P P P 1 _ P 2 v P 1 true _ P true false _ P P P 1 v P 2 P 0 1 v P 0 2 P 1 ^ P 0 1 v P 2 ^ P 0 2 P 1 v P 2 P 0 1 v P 0 2 P 1 _ P 0 1 v P 2 _ P 0 2 Entailment relation of predicate constructors P 1 v P 2 P 1 v P 2 P 1 v P 2 yp 1 vyp 2 list y (P 1 ^ P 2 ) yp 1 ^yp 2 with y2f8; 8 l ; 8 r ; rg P 1 v P 0 1 P 2 v P 0 2 P 1 2 P 2 v P P 0 2 pair (P 1 2 P 2 ) ^ (P P 0 2 ) (P 1 ^ P 0 1 ) 2 (P 2 ^ P 0 2 ) P P P 1 ^ P 2 P1 ^ P2 P 1 _ P 2 P1 _ P2 P 1 2 P 2 P 2 2 P 1 8 l P 8 r P 8r P 8 l P (8l P) E 8(P E ) 8 l 8 r P 8 r 8 l P 8P (a :: as) P (a) ^8P (as) rp (a :: as) 8 P a (as) ^rp(as) 8P (nil) true rp (nil) true Fig. 4. Entailment relation Theorem 1. The verication algorithm always terminates. (Sketch of proof) Basic predicates, variables, and constants appearing in a program are nite. A free variable is introduced as a substitute for a bound variable only when function-calls; thus, only nitely many free variables are used during verications. Since each predicate constructor enriches types, once types of functions are xed only nitely many applications of predicate constructors are possible. The niteness of an input-dependent abstract domain is then guaranteed. The algorithm is therefore formulated as the least x-point computation on a nite abstract domain, so that it terminates.
9 8[[ f fun f i x i = E i ; g ]] = fvp whererec fvp =[(P 1 111P n :(9[[E i ]]P i )# xi )=f i ] true if P (C) 9[[C]]P = false otherwise 9[[x]P 9[[op E]]P = P (x) = 9[[E]](4[[op]]P) 9[[f E]]P = 9[[E]]((fvp[[f]]P ) ) 9[[E1 ]]P 9[[(E 1 ;E 2 )]]P = 1 ^ 9[[E 2 ]]P 2 if P = P 1 2 P 2 9[[E 1 ]]P E 2 _ 9[[E 2 ]] P E 1 otherwise 9[[E1 ]]Q ^ 9[[E 9[[E 1 :: E 2 ]]P = 2 ]]8Q if P = 8Q 9[[(E 1 ;E 2 )]]8 r Q ^ 9[[E 2 ]]rq if P = rq 9[[if Cond then E 1 else E 2 ]]P = : [(Cond ^ 9[[E 1 ]]P) _ (:Cond ^ 9[[E 2 ]]P)] 9[[let val x = E 1 in E 2 ]]P = 9[[E 1 ]](9[[E 2 ]]P)# x 9[[let val (x; y) =E 1 in E 2 ]]P = 9[[E 1 ]](9[[E 2 ]]P)# (x;y) 9[[E1 ]]rq if (9[[E 9[[let val x :: xs = E 1 in E 2 ]]P = 2 ]]P)# (x;xs) = 8 r Q 9[[E 1 ]]8Q if (9[[E 2 ]]P)# x = Q or (9[[E 2 ]]P)# xs = 8Q 8 9 :(Fvp!)Exp! Pred! Formula 4 : Prim! Pred! Pred >< where 8 : Prog! Fvp #: Formula! Bv! Pred >: fvp 2 Fvp = Fv! Pred! Pred : : F ormula! Formula Fig. 5. Abstract semantics of verication 3.2 Soundness by Domain-Induced Abstract Interpretation In this section, we will show how the abstract interpretation 8 is obtained as a domain-induced abstract interpretation, i.e., an abstract interpretation induced from domain abstractions. As a consequence, the soundness proof is given. Note that an automatic verication cannot be complete by nature. Let the domain and codomain of a function f of type! be D and D, respectively. Let the power domain PD[D ]ofd be fcl # D (X ) jx D g with the order v 01 =, where cl # D is the downward closure operator in D. 8 (in Fig. 5) is expressed as the two-step domain-induced abstract interpretation (as indicated in Fig. 6). The rst step is backward and consists of { the abstract domain PD[D ] { the concretization map conc 1 = id D { the abstraction map abs 1 = cl # D This step precisely detects how much of the input is enough to produce the output satisfying the specication. The next step approximates according to the specication language in order to make the analysis decidable. Let Pred be a
10 D f! D abs 1 f01 conc 1 PD[D ] PD[D ] abs 1 (cl # D ) conc 1 (id D ) abs 2 conc 2 PD[D ] PD[D ] Pred Pred abs 1 f 01 conc 1 abs 2 abs1 f 01 conc 1 conc2 Fig. 6. Two steps of domain-induced abstract interpretation set of predicates on D generated as P 0 G and consists of in Fig. 3. The second step is forward { the abstract domain Pred. { the concretization map conc 2 (P )=cl # D (fx 2 D j P (x)g) for P 2 Pred. { the abstraction map abs 2 (X )=u(fp 2 Pred j conc 2 (P) Xg) for X 2 PD[D ]. Note that the abstract domain Pred is a lattice wrt the entailment relation. For instance, P t Q and P u Q always exists as P ^ Q and P _ Q, respectively. Thus an abstract interpretation 4 on a primitive function op of type! is dened by 4(op) =abs 1 op 01 1 conc, where abs = abs 2 1 abs 1 and conc = conc 1 1 conc2. Similar to 9 on expressions. The abstract interpretation 8 on recursively dened functions f i 's is obtained by the least x-point computation. Denition 2. For an abstract interpretation 8, a function f is safe if f satises 8(f) v abs 1 f 01 1 conc. An abstract interpretation 9 is safe if each primitive function is safe. Theorem 3. The verication algorithm is sound (i.e., the detected property of a program always holds if a program terminates). (Sketch of proof) Since the concretization map conc and the abstraction map abs satisfy abs 1 conc = id D and conc 1 abs cl # D, a recursively dened function is safe. Thus the detected property is sound. 4 Example: Verifying Orderedness of Sorting The verication algorithm is explained here by an example of orderedness true! rgeq(sort X). When unknown properties of user-dened functions are required, new conjectures are produced. For instance, when verifying true!rgeq(sort X), it automatically produces and proves the lemmata; 8leq Z (X)!8leq Z (sort X), :null^8leq Z (Y )! leq Z 28leq Z (max Y ), and :null(y )!8 r geq(max Y ). The generation of lemmata is shown at the top of Fig. 7. The vertical wavy arrow indicates an iterative procedure, the double arrow indicates the creation of a conjecture, and the arrow returns the resulting lemma.
11 true!rgeq(sort X) true!8leq Z (sort X) 8leq Z (X)!8leq Z (sort X) true!8 rgeq(max Y ) :null(y )!8 rgeq(max Y ) rst iteration second iteration true!8leq Z (sort X) true! leq Z 2 true(max Y ) :null(y ) ^8leq Z! leq Z 2 true(max Y ) 8leq Z (X)!8leq Z (sort X) true! leq Z 28leq Z (max Y ) :null(y ) ^8leq Z (Y )! leq Z 28leq Z (max Y ) success Fig. 7. Generation of lemmata for true!rgeq(sort X) 8leq Z (X)!8leq Z (sort X) (converged) For instance, 8leq Z (X)!8leq Z (sort X) means that if an input of sort is less-than-or-equal-to any given Z, anoutputisalsoless-than-or-equal-to Z. This lemma is generated as a conjecture true!8leq Z (sort X) attheelse-branch of the if-expression in sort (9[[b::sort bs]]rgeq) as follows. rgeq(b :: sort bs) 8 r geq(b; sort bs) ^rgeq(sort bs) 8leq b (sort bs) ^rgeq(sort bs) Since there are no conjectures related to 8leq b (sort X) in the recursion hypothesis, a new conjecture is created. But properties of functions (P F in Fig. 3) exclude bound variables. Thus, by the-expansion, 8leq b (sort X) is transformed to 8leq Z (sort X) with the substitution [Z b], and true!8leq Z (sort X) is created. This means that no local information on b is used during the verication of true!8leq Z (sort X). This conjecture does not hold; instead, we obtain 8leq Z (X)!8leq Z (sort X) as a lemma. A typical example of the use of the entailment relation appears in the verication of :null(y )!8 r geq(max Y ). At the second if-expression in max, 9[[if leq(e,d) then (d,e::es) else (e,d::es)]]8 r geq is created. Thus, (leq(e; d)^8leq d (es)) _ (:leq(e; d)^8leq e (es)) is obtained. From the transitivity of leq, leq(e; d) ^8leq d (es)) v leq(e; d) ^8leq e (es) (see the underlined parts), therefore we obtain 8leq e (es) bythe:-elimination. Note that the :-elimination also creates 8leq d (es), but only 8leq e (es) branch is successful, i.e., from the recursion hypothesis 9[[max ds]]8 r geq is reduced to :null(ds), as desired. Thus 8leq d (es) is omitted. 5 Extensions 5.1 New Predicate Constructors In this section we introduce new predicate constructors and extend the entailment relation to make it possible to verify weak preservation of sort programs.
12 The new predicate constructors are 9; 9 l ; 9 r,and4. The predicates are extended by updating part of the grammar in Fig. 3with P R = 8P U j9p U jrp G j4p G j P E j P P R ^ P R j P R _ P R properties of lists P P = P B j PP j P S 2 P S j8 r P B j8 l P B j9 l P B j9 r P B j properties of pairs P P ^ P P j P P _ P P where the underlined parts are newly added. Their meanings are shown by examples in the table below. The entailment relation is enriched as in Fig. 8. predicate true false 9geq 5 (as) [3,6,4] ([3,2,4]), nil 9 l leq(as; a) ([3,6,4], 5) ([3,6,4], 2), (nil, 5) 9 r leq(a; as) (5, [3,6,4]) (7, [3,6,4]), (5, nil) 4leq(as) [3,2,4] [3,6,4], [3], nil Then weak preservation of sort is expressed by true! 8 l 9 r equal ^8 r 9 l equal X (sort X): During the verication of true! 8 l 9 r equal X (sort X), the key step is at 8 l 9 r equal(as; b :: sort(bs)) in sort. By transitivity 8 l 9 r equal(as; b :: bs) ^ 8 l 9 r equal(b :: bs; b :: sort(bs)) is inferred. To solve the second component, the entailment relation 8 l 9 r P (a :: as; b :: bs) v P (a; b) ^8 l 9 r P(as; bs) is used. This is obtained as a transitive closure by 8 l 9 r P (a :: as; b :: bs) 9 r P (a; b :: bs) ^8 l 9 r P (as; b :: bs) v (P (a; b) _9 r P (a; bs)) ^8 l 9 r P (as; bs) v P (a; b) ^8 l 9 r P(as; bs): Thus 8 l 9 r equal(b :: bs; b :: sort(bs)) is reduced to 8 l 9 r equal(bs; sort(bs)) which is a recursion hypothesis. The rest 8 l 9 r equal(as; b :: bs) creates the conjecture true! 8 l (equal 2 true _ true 29 r equal) Y (max Y ) at (b,bs) = max as, and similar approximations occur in max at expressions (d,e::es) and (e,d::es). true! 9 l 8 r equal X (sort X) is similarly veried. 5.2 Uninterpreted Functions and Predicates This section extends the range of conditional expressions that can be included in the programs to be veried. Function symbols (either primitive or user-dened) in the conditional part of an if-expression are allowed. They are left uninterpreted during the verication, and the result will be rened by partial evaluation of these function symbols. The example is a formatting program format that formats a sentence (expressed by a list of strings) as a list of sentences each of which has a width less-than-or-equal-to a specied number M. Its specications are as follows.
13 P 1 v P 2 4P 1 v4p 2 P 1 v P 2 yp 1 vyp 2 list y (P 1 _ P 2 ) yp 1 _yp 2 with y2f9; 9 l ; 9 r g 9 l P 9 r P 9r P 9 l P (9l P ) E 9(P E ) 9 l 9 r P 9 r 9 l P P v P x 2 P x P v P x 2 P x 8 l 9 r P v (8 l 9 r P ) xs 2 (8 l 9 r P ) xs 9 l 8 r P v (9 l 8 r P ) xs xs 2 (9 l 8 r P ) Transitivity 9P (a :: as) P (a) _9P (as) 9P (nil) false 8 l 9 r P (as; b :: bs) v8 l 9 r P (as; bs) 8 r 9 l P(a :: as; bs) v8 r 9 l P (as; bs) 4P (a :: b :: bs) 9 P a (b :: bs) ^ (null(bs) _4P (b :: bs)) 4P (a :: nil) false Fig. 8. New entailment relation 4P (nil) false { Each sentence of the output must have a width less-than-equal to M. { The order of each word in the input must be kept in the output. { Each word of the input must appear in the output, and vice versa. fun format as = f (as,nil); fun f (bs,cs) = if null bs then [cs] else let val d::ds = bs in if leq (width then f else cs::f (ds,[d]) end; fun width es = if null es then 0 else let val f::fs=es in if null fs then size f else 1+size f+width fs end; In this example, string is added to base types. Basic functions + and constants 0; 1 also are used in the program, but they are not directly related to the verication. Thus, their interpretation and entailment relations are omitted. The rst specication of format states that an output must satisfy 8(leq M 1 width). Note that this predicate allows a function symbol width in it. Verication starts with true!8(leq M 1 width)(format X); which is immediately reduced to true!8(leq M 1 width)(f Y): The result of the verication is (8leq M 1 width 1 [])2 (leq M 1 width)(y )!8(leq M 1 width)(f Y); and this deduces (8leq M 1 width 1 [])(X) ^ (leq M 1 width)(nil)!8(leq M 1 width)(format X):
14 Note that the result is not aected by whatever width is, since width is left uninterpreted. The key steps are at if leq (width then... Since the throughout of then-branches leq(width holds, the second argument of f (ds; always satises leq M 1width. These steps depend only on _-elimination so that the function symbol width remains uninterpreted. With the aid of partial evaluation which leads to width nil =0,width [x] = size x, we obtain 8(leq M 1size)(X)^leq(0;M)!8(leq M 1width)(formatX): For the partial evaluation, only the relation between size and width is important. The information on the function size is required only when the nal result above is interpreted by a human being. Note that in general a partial evaluation may not terminate. However, this nal step is devoted to transforming the detected property into a more intuitive form for a human being, and even if it fails the detected property is correct. The second and the third specication of format are similar to orderedness and weak preservation of sort, respectively. They require further extensions. The second specication of format is expressed by a fresh binary predicate Rel on pairs of strings as rrel (X)!8rRel ^r2rel (format X) where 2 is an abbreviation of 8 l 8 r. Note that throughout the verication the predicate Rel is left uninterpreted. This implies that the specication above holds for any binary relation Rel. Finally, the meaning of Rel is assumed by a human being, and in this case it is suitable to be interpreted as the appearance order of strings. The third specication is expressed by true! 8 l 9 r 9 r equal X (format X) and true! 8 r 8 r 9 l equal X (format X): Our algorithm detects the latter, but for the former we also need a new transitivity-like entailment relation of type list() 2 list(list()), i.e., P v P x 2 P x 6 Related Work 8 l 9 r 9 r P v (8 l 9 r P ) xs 2 (8 l 9 r 9 r P ) xs : Many studies have been undertaken on verication. Most are based on theorem provers, for example, Coq, LCF, Boyer-Moore prover, Larch, and EQP. They require either complex heuristics or strong human guidance (or both), either of which is not easy to learn. However, for huge, complex, and critical systems, this price is worth paying. The complementary approach uses intelligent compile-time error detection for easy debugging. For imperative programs, Bourdoncle proposed an assertionbased debugging called Abstract debugging [5, 4]. For logic programs, Comini, et. al. [8] and Bueno, et. al. [6] proposed extensions of declarative diagnosis based on abstract interpretation. Cortesi, et. al. [10, 18] proposed the automatic verication based on abstract interpretation. Levi and Volpe proposed the framework based on abstract interpretation to classify various verication methods [20]. Among them, target specications primarily focus on behavior properties, such as termination, mutual exclusion of clauses, and size/cardinality relation between inputs and outputs.
15 In contrast, Metayer's [19] and our specication language (for functional programs) directly express the programmer's intention in a concise and declarative description. This point is more desirable for some situation, such as, when a novice programmer writes a relatively small program. As an abstract interpretation, our framework is similar to inverse image analysis [14]. The signicant dierence is that inverse image analysis determines how much of the input is needed to produce a certain amount of output and computes Scott's open sets. Our framework, in contrast, determines how much of the input is enough to produce a certain amount of output and computes Scott's closed sets. In terms of [22], the former is expressed by a HOMT (id; 0; v 00 ; min), and the latter is expressed by (id; 0; v 01 ; max). Similar techniques that treat abstract domain construction as a set of predicates are found in several places. However, predicates are either limited to unary [15, 17, 3, 21] (such as null and :null), or are limited to propositions corresponding to variables appearing in a (logic) program [9]. 7 Conclusion This paper reconstructs and extends the automatic verication technique of Le Metayer [19] based on a backward abstract interpretation. To show the eectiveness, two examples of the declarative specications of the sorting and formatting programs are demonstrated. Although we adopted the simple and inecient sorting program here, we also tried ecient sort programs, such as orderedness of quick-sort and merge-sort (both topdown and bottomup), and weak preservation of the topdown merge-sort. These verications are quite messy by hand [23]. Future work will include the implementation and the exploration of its use on more complex examples. An ecient implementation may require an ecient reachability test algorithm (as well as a congruence-closure algorithm) and a strategy to prune highly-nondeterministic :-eliminations and transitivity. Acknowledgments The author would like to thank CACA members for their useful comments and discussions. References 1. S. Abramsky and C. Hankin, editors. Abstract interpretation of declarative languages. Ellis Horwood Limited, T. Arts and J. Gisel. Termination of term rewriting using dependency pairs. Theoretical Computer Science, to appear. 3. P.N. Benton. Strictness properties of lazy algebraic datatypes. In Proc. 3rd WSA, pages 206{217, Springer LNCS F. Bourdoncle. Abstract debugging of higher-order imperative programs. In ACM SIGPLAN PLDI 1993, pages 46{55, F. Bourdoncle. Assertion-based debugging of imperative programs by abstract interpretation. In 4th ESEC, pages 501{516, Springer LNCS 717.
16 6. F. Bueno et al. On the role of semantic approximations in validation and diagnosis of constraint logic programs. In Proc. AADEBUG '97, pages 155{169, G.L. Burn. Lazy Functional Languages: Abstract Interpretation and Compilation. The MIT Press, M. Comini, G. Levi, M. C. Meo, and G. Vitiello. Proving properties of logic programs by abstract diagnosis. In M. Dams, editor, Analysis and Verication of Multiple-Agent Languages, 5th LOMAPS Workshop, number 1192 in LNCS, pages 22{50. Springer-Verlag, A. Cortesi, G. File, and W. Winsborough. Prop revisited: Propositional formula as abstract domain for groundness analysis. In Proc. 6th LICS, pages 322{327, A. Cortesi, B. Le Charlier, and S. Rossi. Specication-based automatic verication of Prolog programs. In Proc. LOPSTR '96, pages 38{57, Springer LNCS P. Cousot and R. Cousot. Abstract interpretation: A unied lattice model for static analysis of programs by construction of approximation of xpoints. In Proc. 4th ACM POPL, pages 238{252, N. Dershowitz and J.-P. Jouannaud. Rewrite systems. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B, chapter 6, pages 243{320. Elsevier Science Publishers, D. Detlefs and R. Forgaard. A procedure for automatically proving the termination of a set of rewrite rules. In Proc. 1st RTA, pages 255{270, Springer LNCS P. Dybjer. Inverse image analysis generalizes strictness analysis. Information and Computation, 90(2):194{216, C. Ernoult and A. Mycroft. Uniform ideals and strictness analysis. In Proc. 18th ICALP, pages 47{59, Springer LNCS J. Gisel. Termination of nested and mutually recursive algorithms. Journal of Automated Reasoning, 19:1{29, T.P. Jensen. Abstract interpretation over algebraic data types. In Proc. Int. Conf. on Computer Languages, pages 265{276. IEEE, B. Le Charlier, C. Leclere, S. Rossi, and A. Cortesi. Automatic verication of behavioral properties of Prolog programs. In Proc. ASIAN '97, pages 225{237, Springer LNCS D. Le Metayer. Proving properties of programs dened over recursive data structures. In Proc. ACM PEPM '95, pages 88{99, G. Levi and P. Volpe. Derivation of proof methods by abstract interpretation. In ALP'98, pages 102{117, Springer LNCS M. Ogawa and S. Ono. Deriving inductive properties of recursive programs based on least-xpoint computation. Journal of Information Processing, 32(7):914{923, in Japanese. 22. M. Ogawa and S. Ono. Transformation of strictness-related analyses based on abstract interpretation. IEICE Trans., E 74(2):406{416, L.C. Paulson. ML for the working programmer. Cambridge University Press, 2nd edition, V. Van Oostrom. Conuence for Abstract and Higher-Order Rewriting. PhD thesis, Vrije universiteit, Amsterdam., This article was processed using the LaT E X macro package with LLNCS style
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DSSSB AE JE Manager Answer Sheet 2015, DSSSB AE JE Exam Answer Key 2015, DSSSB AE JE Manager Score Card 2015, DSSSB Exam Answer key 2015, Download DSSSB Answer Sheet of AE/ JE/ Manager Exam 2015, DSSSB Answer Sheet 2015, DSSSB Engineering Exam Answer Key, DSSSB AE Answer key, DSSSB JE Answer Key
DSSSB which refer to Delhi Subordinate Service Selection Board has uploaded the admit card for Assistant Engineer (AE)/ Junior Engineer (JE) and Various other manager posts written exam which will be conducted on 31st May 2015 on sunday. Admit card can be downloaded from DSSSB official website (www.dsssbonline.nic.in) so candidates who have applied for post code 2/12 to 15/12 / 46/13 to 70/13 & 197/2014 to 217/2014 can download their admit card from official portal. After the exam all students or we can say candidates will eagerly wait for DSSSB AE JE Exam Answer Key 2015. In this post we are providing info releted to DSSSB Engineering Exam Answer Key.
DSSSB AE-JE-Manager Answer Sheet 2015
Delhi Subordinate Service Selection Board will going to conduct written examination for the posts which are as :- Asst Engineer (Civil) / Jr Engineer (Civil) / Jr Engineer (Electrical) / Administrative Officer / Asst Assessor and Collector / Administrative Officer / Zonal Revenue Officer / Manager (Civil) / Manager (Electrical) / Dy Vigilance Officer & Dy Security Officer under various Posts so all candidates can get their admit card from the official portal.
DSSSB AE-JE-Manager Examination Details :-
Name of the Organisation:- Delhi Subordinate Service Selection Board
Examination Name :- DSSSB AE-JE-Manager examination
Examination Date :- 31/5/2015
About :- Answer Key / Answer Sheet
Tomorrow DSSSB AE – JE exam will be conducted on monday in various examination Centers so all candidate will eagerly wait for the DSSSB Exam Answer key 2015 and within one week DSSSB AE JE Manager Answer Sheet 2015 will be announced by the officially by the Organisation. Then all candidates can take an idea of their result or DSSSB AE JE Manager Score Card 2015.
Procedural steps to Download DSSSB Answer Sheet of AE/ JE/ Manager Exam 2015 :-
- All candidates must have to log on to the official portal of the Organisation.
- Then go to the Home Page Of the official site of the Organisation.
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Good luck and best wishes to all candidates for their exam. To get more update or latest information keep in touch with us we will keep you update. If you have any querry releted to this post then please put your comment under comment Box. Let us know we will solve your problme as soon as we can.
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The founder, William Russell Kelly, was among the first businessmen to provide temporary staffing services. It was back in 1946 that he opened a small office in Detroit, Michigan. Together with two employees, Kelly provided business services such as typing and bookkeeping to local companies. Soon more and more clients wanted the employees to work directly for them. Kelly thus specialized in the leasing of personnel and laid the foundation for modern staffing agencies.
"Every customer should be treated as if he were the only one" – this was W. R. Kelly's vision. We have been maintaining and fostering this loyal attitude for more than six decades. It is the foundation of a trusting relationship with our clients. We take all our promises seriously and work with passion and commitment. Working in partnership with our clients is the cornerstone of mutual success.
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Report Ad
Class A CDL Experienced Truck Drivers Are Valued Employees! (No Experience Needed) - CRST EXPEDITED
Posted: Tuesday, July 4, 2017 6:04 PM
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There's always a job out there for a truck driver and it pays a lot more than you may think. With your trucking license, you can find work all over the country, from shipyards to oilfields to local delivery.
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text:
Kelsey Keel
My Louisiana Sky
Newcomer Kelsey Keel stars in this touching coming-of-age drama directed by Adam Arkin. Tiger Ann (Keel) is a 12-year-old girl in 1950s Louisiana who's coming to terms with living with mentally handicapped parents. Old enough now to realize the gravity of the situation, she struggles to cope with these unsettling revelations. Her lively aunt (Juliette Lewis) soon comes to the rescue, helping to ease both Tiger Ann's mind and her parents' burden.


text:
|0 sq. ft.
|Property type
|Minimum Stay
|5 nights
The villa is located within a resort offering a hotel and spa. The resort is located in the lovely town of Parga in a secluded bay of the Ionian Sea in the Epirus region of Greece.
Upon entering this stylishly decorated villa you will find an open plan living/dining and kitchen area. The living area features two 2 seater settees, coffee table and a fireplace. For entertainment you will find a flat screen TV with satellite TV and sports channels, DVD player and HI-FI radio/CD player. The villa also offers the luxury of high speed wireless internet connection.
The dining area is beautifully furnished with a small table and 4 high back chairs with direct access to the patio area and the swimming pool.
The fully equipped kitchen features 4 ring hob, oven, fridge/freezer, dishwasher, kettle, coffee machine and espresso coffee machine.
The master bedroom of the villa is a beautiful double bedroom with direct access to the patio area and the swimming pool. The master bedroom offers the luxury of a transparent glass shower room.
The second bedroom of the villa is a twin bedroom offering the luxury of an en suite bathroom decorated with mosaics featuring a shower and a bath tub with air massage.
This villa is air conditioned throughout. Bathrobes and slippers are provided. Furthermore available is a safety deposit box, hair dryer and a mini bar. A baby cot and high chair is available on request.
Additional services offered from the hotel at the resort include Turn-Down service (subject to a small extra charge) and wake up calls.
To the front of the villa you will find the patio area surrounding the 8m x 4m private swimming pool featuring plenty of garden chairs and sun beds allowing you to enjoy the amazing sea and mountain views.
Parga is definitely one of the most beautiful towns in Greece. It is located 68 km north west from Preveza and 50 km from Igoumenitsa. Parga is built amphitheatrically on the slopes of a mountain. It is an idyllic resort that uniquely combines the beautiful mountain side with the sea. It has traditional architecture with picturesque paved streets, beautiful churches, monuments, monasteries and all kind of shops.
One of the most remarkable sites of Parga is the Medieval Castle built in 14th century with the help of the Normans when the locals asked for protection against the Ottoman Turks. The castle is situated at the north side of the port and it offers a wonderful view.
A small green island with a white chapel stands at the entrance of the port and a religious festival takes place every August to honor this chapel. The beaches around Parga are a pole of attraction for the thousands of visitors coming every year.
The beaches of Parga are not only numerous but also very beautiful, so most of the times the visitor has the dilemma of which one to visit.
The transparent water is their common element and the usual beauty always attracts the guests.
One can start from the beach in the center of Parga, Krioneri Beach, and then visit the further 3 known beaches, namely Valtos, Lichnos and Sarachinico.
Valtos beach is the biggest beach of Parga located approximate 2 kilometres from the town centre. The view that can be enjoyed at Valtos is unique; high on the left is the imposing castle which seems to rule the bay. On the right, looking at the sea you can admire the nearby islands of Paxos and Antipaxos which are reachable by small boats. During the summer this service is given every day.
Lichnos Beach, located approximately 4 kilometres from the town centre, is the second longest beach of Parga. Along the descent, which leeds to the beach, one can enjoy the amazing view of teh crystal clear water. The beach is partly sandy and partly pebbles. During the day one can take water-skiing lessons, ride a jet-ski or enjoy oneself with the famous bananas and rings. One should also visit the wonderful cave of Lychnos (Aphrodite's cave).
The road leading to the Sarachiniko Beach is not the best drive; full of slopes and bends. One is however rewarded for the difficulty to reach the beach by the enchanting beauty of the place. The beach is partly sandy and part pebbles and the water is transparent. A beautifully organized beach which kindly offers you a welcome feeling of relaxation.
There are also ideal remote beaches along the coastline of Parga, which one can only visit by boat.
*Plus compulsory or optional charges based on actual occupancy or consumption. The payment has to be made in the currency offered by the property manager, before use of conversion tool. Payments in other currencies have to be negotiated beforehand with the property manager. The currency conversion tool is provided for your convenience only and is non-binding.


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Lullaby Medley
Language Focus
The Lullaby Medley is a quiet instrumental medley of songs from Super Simple Songs Three. It's great for nap time or creating a calming atmosphere for story time or other relaxed activities. You'll notice the kids singing along at times. It's a great way to keep the songs (and language) fresh in the children's heads.


text:
Popular Now
Popular Now
Popular Now
Popular Now
01803 865913
+44 1803 865913
By: Anna Claybourne(Author)
64 pages, colour illustrations
'Close to my house, there's a beautiful, huge, spreading tree. I see it every day. We walk past it on the way to school, and hear the wind rustling through it at night. How long has that tree been standing there? If the tree wasn't there any more, I would miss it all the time. But it's not just me – so many animals depend on it too...' I Love This Tree is a book which looks at the life of an old living deciduous tree and through it explores the amazing life of a tree and the creatures that inhabit it. Using a mixture of photos and artwork I Love This Tree shows how the tree has changed and how it acts as a habitat for other life.
This beautiful book explores how important trees are to the planet. Packed with information and pictures it provides a wealth of information about trees and their importance and inspires a love of these amazing plants.
There are currently no reviews for this product. Be the first to review this product!
Anna Claybourne has written over 150 children's books on all kinds of subjects, from sharks, black holes and splitting the atom to how to draw, Shakespeare's life, and Greek myths and legends. She likes space, sewing and music, plays the trombone and loves going camping.
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Originally Posted by judesmama
I'd like to know, too! Dh has TONS of shirts and most of them he never wears. We'd have so much more closet space if he would downsize.
Originally Posted by shirabira
Are you married to my husband by any chance? DH's thing is old clothes, especially old Tshirs. His side is much bigger than mine. He will never wear these clothes again but he refuses to throw them away. Everytime I put his clean laundry in the closet I'm afraid the pile of shirts will fall on my head . And yes, if I throw away something he notices
|How do you pry those "precious treasures" (i.e. JUNK that he's forgotten about) from your DH's grasp? I really need him to downsize the clothing and knicknacks.
Originally Posted by Greenie
Shira, would he be okay with maybe putting some in storage? Maybe just the ones he hasn't worn in 12 months? Then, they might mysteriously vanish from there....
Rhu - mother,grandmother,daughter,sister,friend-foster,adoptive,and biological;not necessarily in that order. Some of it's magic, some of it's tragic, but I had a good life all the way (Jimmy Buffet)
Originally Posted by mamarhu
Once upon a time, I made a patchwork quilt from the printed area of a million XDH's tee shirts. Just an idea - it went back to concerts from the 70's and was way cool.
Originally Posted by kewb
Also, a lot of his clothes that I hate never seem to return from the laundry room. I have never been able to figure out where they go. Hmmmmm.
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Question: Why is the MPR for credit card receivable backed securities important
Why is the MPR for credit card receivable-backed securities important?
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Answer to relevant QuestionsWhat are the advantages of pooled investment vehicles relative to the direct purchase of fixed-income assets? How does the management fee structure of ahedge fund differ from that of an asset manager of a mutual fund? Why do mutual funds have different classes of shares? Comment on the following statement: "If an interest-rate model allows the possibility of negative interest rates, then it is not useful in practice." Answer the below questions. (a) What are the general characteristics of the Ho-Lee arbitrage-free interest-rate model? (b) How does the Ho-Lee arbitrage-free interest-rate model differ from the Hull-White arbitrage-free ...
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