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....as the meaning of an arithmetic operation requires a messy denotational description; one needs to check for failure at each argument, and propagate accordingly. The story is similar for other cases, including I O and assignments, two of the most popular e#ects found in many programming languages [76, 77]. It was Moggi s influential work on monads that revolutionized the semantic treatment of e#ects, which he referred to as notions of computation. Moggi showed how monads can be used to model programming language features in a uniform way, providing an abstract view of programming languages [62, ....

....to see more clearly why this default definition is not appropriate for implementing value recursion. 1. 5 The basic framework and notation For most of this thesis we investigate value recursion in the usual domain theoretic semantics of programming languages, where types are modeled by domains [77, 82]. We write for the least element of the domain representing the type # , dropping the subscript whenever unambiguous. Functions are modeled by continuous (and hence monotonic) maps between domains, not necessarily strict. Recursion is modeled via least fixed points. We use monads to model ....

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Schmidt, D. A. Denotational Semantics. Allyn and Bacon, Boston, 1986. (1, 10, 32)

....last ten years. The goal is clear: From a language de nition expressed in a not too cumbersome formalism automatically to derive an ecient implementation that is faithful to the semantic de nition. An important formalism for assigning meanings to programs is denotational semantics [Stoy 1977, Schmidt 1986] founded by Scott and Strachey. A denotational de nition assigns to each program a lambda expression denoting the input output function computed by the program. Denotational semantics was intended to be the mathematical theory of programming languages, saying what the meaning of a program is, ....

....self application, and on how it is done, via the two level lambda calculus and binding time analysis. Part II of this article is more mathematical, showing why mix functions correctly. 1. 2 Prerequisites, Overview and Outline The reader should be familiar with denotational semantics (e.g. [Schmidt 1986]) and should have some knowledge of partial evaluation. After a summary of this article s contents, the rest of section 1 is devoted to a review and overview of the basic theory of partial evaluation and compiler generation (more details may be found in [Jones et al. 1985, Jones et al. 1989] A ....

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D. A. Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....and 5 for more detailed comments comparing these functions. Gibbons does not consider expressing graph folds in an eager language, nor does he develop any theory for his two folding functions. The theory of fold builds on classical results of fixed point theory from denotational semantics (e.g. [19]) The fixed point iteration performed by cycfold is similar to that performed in traditional compiler data flow analysis; see [17] for a nice summary of of this area. 2. THE THEORY OF UNFOLD AND FOLD ON REGULAR TREES 2.1 Mathematical Denitions 2.1.1 Numbers, Sets, Binary Relations, Posets, ....

D. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....i.e. there are several functions satisfying the recursion equation. In every introduction to the semantics of programming languages one nds that the intended semantics is given by the least xed point of the recursion equation (with respect to the de nedness order) cf. e.g. Manna [21] Schmidt [26] or Jones [16] Hence we need a powerful principle to prove statements about recursive programs. Probably the most famous such principle is xed point induction introduced by Scott [27] which is based on a CPO interpretation of terms. For a good overview of Scott s induction principle and its ....

David Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....T P semantics will not distinguish the two systems. Our position is that di erent strategies determine di erent logics and that a good semantics must make such distinctions. Second, a denotational semantics of the execution of Prolog programs can be given in the usual domain theoretic style [32]. Our position is that a good semantics of logic programs must account for both the logical and operational aspects of the meaning of a program. 6 Conclusion We have introduced the problem of providing a semantics for reductive proof theory, i.e. for proof search. We have sketched a solution ....

Schmidt, D. \Denotational Semantics", Allyn and Bacon, 1986.

....of Prolog derivation stops at the level of resolution [Llo87] For the purposes of verification within our transformational approach to annotating full Prolog, we needed a more detailed notion of a derivation. So we came up with a novel continuationbased semantics of Prolog execution (cf. JM84] [Sch86], NF89] And92] BR93] Pet95] Some highlights of our approach: a novel linear representation of the Prolog tree traversal, enabling small and readable specifications closer to the spirit of logic programming: by treating if then else and disjunction simply and compatibly with cut, ....

....of [JM84] and [NF89] The language constructs covered include the cut operator, but more advanced control constructs like catch and throw or database predicates are not treated. Among the first to specify backtracking and cut as an elegant use of continuations was also D. Schmidt in his book [Sch86]. Our approach is also based on continuations, but their contents and handling is (to our knowledge) novel, which appears to dramatically simplify the semantics. Whereas continuations play a central role in denotational semantics, it can also be beneficial to have first order continuations at ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....employed in actual Prolog systems is modelled by the denotational semantics approaches of Jones and Mycroft [12] Nicholson and Foo [18] Debray and Mishra [8] de Vink [7] or Baudinet [4] Among the first to specify backtracking and cut as an elegant use of continuations was also D. Schmidt [21]. The language constructs covered in these approaches include the cut operator, but further advanced control constructs like catch throw or database predicates are not treated. Continuations play a central role in denotational semantics, but it can also be beneficial to have them at the source ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....that more and more working professionals are being confronted with. This paper is proposing a domain theoretic foundation for frameworks within which systems that are possibly incomplete can be speci ed and analyzed. Domain theory [2] as developed within the project of denotational semantics [42, 41, 36], had the original intent of providing a mathematical theory and framework for a formal programming language semantics. This project relied on domain theory s conceptual contribution of viewing complete information (often some in nite structure) as something that can be approximated, in a ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

.... Lego David Espinosa Columbia University Department of Computer Science New York, NY 10027 espinosa cs.columbia.edu Draft March 20, 1995 Abstract Denotational semantics [Sch86] is a powerful framework for describing programming languages; however, its descriptions lack modularity: conceptually independent language features influence each others semantics. We address this problem by presenting a theory of modular denotational semantics. Following Mosses [Mos92] we ....

....a computation either terminates, producing a value, or pauses, producing a computation with which to continue. The typical use of resumptions is to interleave several computations by executing one until it pauses, then executing the next, etcetera. The standard parallel semantics, described in [Sch86] has T (A) Sto List(A Theta Sto) so that computations accept and return stores and can fork into multiple computations. Thus the complete type of denotations is Den = fix (X) Sto List( Val X) Theta Sto) Figure 1.27 shows the SL specification for this language, along with several ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, New York, NY, 1986.

....parts of Verdi would be implemented. The present work complements the model theory by presenting an alternative definition of the semantics of the executable portion of Verdi, with a proof of correspondence between the new semantics and the old. The new semantic definition is denotational in style [4, 13, 14]. I tried as much as possible, within the other constraints of the project, to adopt the so called standard style of definition. However, I avoided many of the usual typographical and notational conventions, as these were inconsistent with the existing model theory. Since the bulk of this report ....

....complications in the congruence proofs. The use of conditions allows for some easier proofs. I believe that converting the present denotational definition to the continuation EVES Project TR 90 5446 02 3 style, with a proof of congruence, would be a fairly simple step (see, for example, 8] and [13]) the difficulties mentioned above were a result of considering this transformation together with all the others. A third departure from standard semantics is in the use of powerdomains to admit nondeterminism in the denotational definition. This is particularly useful in the areas of storage ....

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David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....will see examples of function definitions in functional programming languages like Standard ML [MTH90, Pa91] and Miranda [BW88] which are translated easily to domain theory, and vice versa. 1. 1 Domain Theory Denotational semantics was pioneered by Christopher Strachey in the early 60 s (see e.g. [Mo90, Sc86]) In his work, Strachey used the untyped calculus as a way of writing denotations, though, at the time, the untyped calculus did not have a formal model in which terms represented mathematical functions. It was a fundamental breakthrough when Dana Scott discovered a model of the untyped ....

....of domain theory can be formalized by giving their semantic definitions in HOL. In this section we concentrate on how this is done; in particular, the section does not provide a thorough introduction on domain theory such can be found in the textbooks by Winskel [Wi93] Gunter [Gu92] and Schmidt [Sc86]. A partial order (po) is a pair consisting of a set and a binary relation such that the relation is reflexive, transitive and antisymmetric on the set. We could formalize such pairs in various ways in HOL. The set component can be denoted by a HOL type ff such that a partial order is represented ....

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D.A. Schmidt, Denotational Semantics. Allyn and Bacon, 1986.

....there are several functions satisfying the recursion equation. In every introduction to the semantics of programming languages one finds that the intended semantics is given by the least fixed point of the recursion equation (with respect to the definedness order) cf. e.g. Manna [21] Schmidt [26] or Jones [16] # This author was working on the final version of this paper during a visit at Oberwolfach supported by the Volkswagen Stiftung (RiP program at Oberwolfach) ## Research supported by the Swiss National Science Foundation. 1 Hence we need a powerful principle to prove statements ....

David Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....sake of clarity, we will present the theory on which our approach is based in a less abstract way than what is usually done. The drawback of this description is its lack of generality. However the reader interested in a general description of the static analysis of recursive procedures can consult [Sch86] and [Apt81] To oversimplify, we consider that any recursive method m can be written in the following way: method m (parameters) in class fif base case then i else recursive callg Where base case denotes the test to stop recursion and recursive call denotes a call to m with an object ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, Inc., 1986.

....e 0 1 ; xn = e 0 n in e 0 , and x also occurs in p or fx 1 ; xn g, respectively; otherwise the occurrence of x is free. FV (e) denotes the set of variables with at least one free occurrence in e. The semantics of functional languages are typically specified denotationally [13], whereby each expression is denoted by some member of a semantic domain this is the value of the expression. The semantic domain contains for each type t an element t which denotes any expression of type t whose evaluation fails to terminate and outputs no information (below, we omit the ....

.... 0 , provided that: 1) g e 0 = h e, 2) g ffi f 0 = h ffi f , 3) g ( h s 1 ) op 0 (h s 2 ) h ( g s 1 ) op (g s 2 ) We show this equivalence by structural induction over s (this is a valid proof technique even for infinite s because all functions in our language are continuous see [13]) If s = Empty then LHS Empty = g (h Empty) g e 0 RHS Empty = h (g Empty) h e and these are equal by (1) If s = then LHS = RHS = by coll=1. If s = single a for some expression a, then LHS s = g (h (single a) g (f 0 a) RHS s = h (g (single a) h (f a) and these are equal ....

D.A.Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....of interpreting the dynamic aspects of the construction of proofs so as, for example, to be able to distinguish between breadth and depth rst strategies. A rst example is, perhaps, provided by Schmidt s denotational treatment of backtracking using Success continuations and Failure continuations [158]. Given such a semantic account, we may, for example, aim to focus on di erent speci c models for di erent purposes. For example, we may wish to work with non provability via the generation of counter models. Semantic Foundations for Search Spaces. We have seen, in x 2.3, that a derivation of a ....

D.A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....actual mapping to concrete machines. In this section we describe the necessary abstract algebras; then define the meanings of EM code programs using semantic functions for instructions (E) programs (D) and term rewriting applications (H) We adopt roughly the same notational conventions used in [Sch86] Double square brackets [ are used to delimit syntactic objects. The symbol is 0 branch[R0, g : 101 f : 110 DEFAULT 136] 136 return[R0] 101 R1 : down[R0, 1] 102 R2 : call[0, R0: R1] 103 branch[R2, a : 104 DEFAULT 109] 109 return[R0] 104 R4 : build[1, 105 R3 : build[d, R4] ....

.... replace[R10] 119 R12 : call[0, R0: R10] 120 return[R12] Figure 4: EM code program for automaton example. used here to indicate an absence of information; is lifted so that Theta = x: and = Sum, product, and function domains are constructed from other domains as in [Sch86] We define most functions in curried form, except when there is a strong sense that certains pairs of arguments should not be broken up. We use the following abbreviation to construct functions from other functions: u v]f abbreviates x: ae v if x = u f x otherwise A number of the domains ....

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David A. Schmidt. Denotational Semantics. Allyn and Bacon, Inc., 1986.

....i.e. all monofunctors defined using the above basic functors or any of the map functors introduced in x5. Then there exists a type L and two strict functions in F 2 LF L and out F 2 L LF (omitting subscripts whenever possible) which are each others inverse and even id = in F out) [6, 23, 16, 24, 30, 12]. We let F denote the pair (L; in) and say that it is the least fixed point of F . Since in and out are each others inverses we have that LF is isomorphic to L, and indeed L is upto isomorphism a fixed point of F. For example taking XL = 1 j AkX, we have that (A; in) L defines the ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

.... note that the additional condition in the above proposition: there does not exist any comprehension derived function h = fejp 1 e 00 1 ; p n e 00 n ; Qg in above two sets such that some e 00 k is empty guarantees that the proposition can be expressed as an inclusive predicate [Sch86] and that the fixed point induction principle [Sch86] can be used to prove it. Detailed discussion of this and formal proofs of the propositions are beyond the scope of this paper and will appear in a forthcoming technical report. To illustrate, for the derived functions in databases ICTRAIN and ....

.... there does not exist any comprehension derived function h = fejp 1 e 00 1 ; p n e 00 n ; Qg in above two sets such that some e 00 k is empty guarantees that the proposition can be expressed as an inclusive predicate [Sch86] and that the fixed point induction principle [Sch86] can be used to prove it. Detailed discussion of this and formal proofs of the propositions are beyond the scope of this paper and will appear in a forthcoming technical report. To illustrate, for the derived functions in databases ICTRAIN and LOCTRAIN , by syntactically comparing their ....

D.A. Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....Computation in the calculus proceeds by syntactically transforming terms using fi reduction. This rewrites a function application of the form (x:e)e 0 to the expression e[e 0 =x] obtained by replacing all free occurrences of x in e by e 0 . The denotational semantics of the calculus (see [Sch86]) assigns to each expression a value in a semantic domain this is the meaning of the expression. fi reduction is semantically sound in that it does not alter the meaning of an expression. The language that we will be optimising is the calculus extended with constructors, let expressions and ....

....be useful at its outset (in order to generate an overall expression to optimise) Functions are defined by equations of the form f = e. If f 2 FV (e) i.e. if f is recursively defined, the meaning of f is given by the least fixed point of the higher order (and non recursive) function f:e (see [Sch86]) This meaning may just be nontermination for some arguments e.g. for the function f = x:not(f x) Thus, the semantic domain contains for each type t an element t which denotes no information and represents a non terminating computation (sometimes we omit the subscript t when it can be ....

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Schmidt, D.A. Denotational Semantics, Allyn and Bacon, 1986.

....Lego David A. Espinosa Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences. COLUMBIA UNIVERSITY c fl 1995 David A. Espinosa ALL RIGHTS RESERVED ABSTRACT Semantic Lego David A. Espinosa Denotational semantics [Sch86] is a powerful framework for describing programming languages; however, its descriptions lack modularity: conceptually independent language features influence each others semantics. We address this problem by presenting a theory of modular denotational semantics. Following Mosses [Mos92] we ....

....that the usual style of writing interpreters isn t modular, shows two styles of writing modular interpreters, and exercises SL with a series of examples. 1.1 Denotational Semantics In this section, we discuss denotational semantics, the theory on which SL is based. Excellent references are [Gun92, Sch86]. In general, semantics is the study of meaning. Our goal is to determine, for example, that x = x 1 means add one to x . Programming language semantics is actually a branch of mathematical logic; the main difference is that logics are for reasoning, while programming languages are for ....

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David A. Schmidt. Denotational Semantics. Allyn and Bacon, New York, 1986.

....are not publicised. Considerable progress has been made towards the goal of automatically generating programming language compilers. Using specialised notations it is possibleto specify compilation sub tasks such as lexical analysis [22, 17, 9, 16, 1] parsing [18, 23, 2, 8] and semantic analysis [24, 3, 21, 27]. Other compiler tasks such as code generation for complex architectures have partially yielded to attack and are the subject of continuing research [4, 6] Libraries of existing infrastructure can be used to combine solutions for sub tasks into a complete compiler implementation [11] Proponents ....

SCHMIDT, D. A. Denotational Semantics. Allyn and Bacon, Newton, MA, 1986.

....in this paper is an example of a semantics directed compiler generator. It accepts language definitions written in action notation, and it outputs compilers that emit code in an abstract RISC machine language. The traditional approach to compiler generation is based on denotational semantics [53]. Examples of existing compiler generators based on this idea include Mosses Semantics Implementation System (SIS) 29] Paulson s Semantics Processor (PSP) 45, 46] and Wand s Semantic Prototyping System (SPS) 62] In SIS, the lambda expressions are executed by a direct implementation of ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....and its input as well. The ability to interpret programs with different numbers of inputs requires that the interpreter itself can accept input sequences of varying lengths 3 . We use a universal value domain V defined as follows. Appropriate mathematical interpretations are described in [17] and other works on denotational semantics. As is customary we omit summand projections and injections. 1 Defined as the least set such that D = Atom [ D Theta D, where an atom is any sequence of one or more letters, digits or characters excluding parentheses, comma and blank. 2 In fact any ....

....to another, usually involving a change of signature (a signature morphism is an example) Derivors must be compositional: the translation of a composite term is a combination of the translations of its components. The requirement of compositionality is very natural for denotational semantics [17]. Derivors over term algebras can also be thought of as finite state transducers that have been extended to operate on trees (strings are a special case) Derivors can also be composed symbolically in a way very analogous to finite transducers. For a small example, consider the three term algebras ....

D. Schmidt: Denotational Semantics: a Methodology for Language Development. Allyn and Bacon, 1986

....with its specifications. A model theory (semantics) of the language is defined and used to show the adequacy of the proof obligations. This paper will be most easily appreciated by the reader with some prior knowledge of Mathematical Logic [8, 19] Set Theory [11] and Denotational Semantics [9, 18, 20]. Verdi differs from its predecessor m Verdi [4] in several significant ways: ffl The logical basis is untyped first order logic, with Zermelo Fraenkel set theory built in. Typing is not, however, totally absent from the language: executable constructs are still subject to type checks. The type ....

....component of biffle B. 6 EVES Project TR 90 5429 10b 2.2 Chain Complete Partial Orders As usual in denotational definitions or imperative languages, fixed points are used in expressing the meanings of looping or recursive constructs. The basics of fixed point theory can be found in [6] or [18]. Only the following elements of the theory of fixed points are required in this paper. Definition 2.1 A partially ordered set (X; is called chain complete iff every totally ordered subset of X has a least upper bound in X. We call a totally ordered subset of X a chain, and denote the least ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....language. Such a compiler can then be composed with a language definition to yield a correct compiler for the language, see figure 1. This approach is usually called semantics directed compiler generation. The traditional approach to compiler generation is based on denotational semantics [37]. Examples of existing compiler generators based on this idea include Mosses Semantics Implementation System (SIS) 17] Paulson s Semantics Processor (PSP) 31, 32] and Wand s Semantic Prototyping System (SPS) 44] Denotational semantics has achieved much popularity as a vehicle for ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....sake of clarity, we will present the theory on which our approach is based in a less abstract way than what is usually done. The drawback of this description is its lack of generality. However the reader interested in a general description of the static analysis of recursive procedures can consult [Sch86] and [Apt81] To oversimplify, we consider that any recursive method m can be written in the following way: method m (parameters) in class fif base case then i else recursive callg Where base case denotes the test to stop recursion and recursive call denotes a call to m with an object ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, Inc., 1986.

.... note that the additional condition in the above proposition: there does not exist any comprehension derived function h = fejp 1 e 00 1 ; p n e 00 n ; Qg in above two sets such that some e 00 k is empty guarantees that the proposition can be expressed as an inclusive predicate [Sch86] and that the fixed point induction principle [Sch86] can be used to prove it. Detailed discussion of this and formal proofs of the propositions are beyond the scope of this paper and will appear in a forthcoming paper. 3.3 CDB Integration Thus far, we have obtained Schema, EDB and IDB for the ....

.... there does not exist any comprehension derived function h = fejp 1 e 00 1 ; p n e 00 n ; Qg in above two sets such that some e 00 k is empty guarantees that the proposition can be expressed as an inclusive predicate [Sch86] and that the fixed point induction principle [Sch86] can be used to prove it. Detailed discussion of this and formal proofs of the propositions are beyond the scope of this paper and will appear in a forthcoming paper. 3.3 CDB Integration Thus far, we have obtained Schema, EDB and IDB for the integrated database D. Our next integration step is to ....

D.A. Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....must switch to the more difficult continuation semantics. We do no longer have compositionality of semantics which results in increased complexity in any program analysis that models this semantics (such as data flow analysis) More information on direct and continuation semantics can be found in [Sch86] Third, the specific method of control flow normalization used in our architecture makes it unnecessary to perform interval analysis as part of the data flow analysis. Interval analysis is a technique to divide a (control flow) graph into sets of nodes (intervals) so that analysis of the ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, Boston, 1986.

....sequence and the computation sequence in a uniform way. It provides a formal background for process monitoring. This appendix is an introduction to execution calculus. A. 2 Formal Approaches to Program Behavior A common formal approach to describe program behavior is denotational semantics [141]. Denotational semantics takes a high level view of programs; a program is a function. The object of study is program text from which program behavior is derived. Denotational semantics is useful for formal program development. It is of limited value in monitoring, where dynamic process behavior ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, Boston, 1986. 186

....Theory has been around for a quarter century, no booklength treatment of it has yet been published. Quite a number of books on semantics of programming languages, incorporating substantial introductions to domain theory as a necessary tool for denotational semantics, have appeared [Stoy, 1977, Schmidt, 1986, Gunter, 1992b, Winskel, 1993] but there has been no text devoted to the underlying mathematical theory of domains. To make an analogy, it is as if many Calculus textbooks were available, offering presentations of some basic analysis interleaved with its applications in modelling physical and ....

....1993, Ong, 1993, Hennessy, 1993, Jensen, 1992, Jensen, 1991, Smyth, 1983b] Applications of Domain Theory There is by now an enormous literature on the semantics of programming languages, much of it using substantial amounts of Domain Theory. We will simply list a number of useful textbooks: [Schmidt, 1986, Tennent, 1991, Gunter, 1992b, Winskel, 1993] In addition, a number of other applications of Domain Theory have arisen: in Abstract Interpretation and static program analysis [Abramsky, 1990a, Burn et al. 1986, Abramsky and Jensen, 1991] see also the article on Abstract Interpretation in this ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

.... TR[ICTRAIN:idirectreach] TR[LOCTRAIN:indirectreach] TR[ICTRAIN:reachable] TR[LOCTRAIN:reachable] The resulting IDB is given in Appendix C. 1 This condition guarantees that the proposition can be expressed as an inclusive predicate which can be proved by means of fixed point induction [14]. 3.4 CDB Integration The next integration step is to integrate CDB 1 ; CDBn to give CDB. We recall that no integrity constraint calls any other integrity constraint, which means that each integrity constraint is defined in terms of data functions and derived functions only. Before ....

D.A. Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....Theory has been around for a quarter century, no book length treatment of it has yet been published. Quite a number of books on semantics of programming languages, incorporating substantial introductions to domain theory as a necessary tool for denotational semantics, have appeared [Stoy, 1977, Schmidt, 1986, Gunter, 1992b, Winskel, 1993] but there has been no text devoted to the underlying mathematical theory of domains. To make an analogy, it is as if many Calculus textbooks were available, offering presentations of some basic analysis interleaved with its applications in modelling physical and ....

....1993, Hennessy, 1993, Boudol, 1991, Jensen, 1992, Jensen, 1991, Smyth, 1983b] Applications of Domain Theory There is by now an enormous literature on the semantics of programming languages, much of it using substantial amounts of Domain Theory. We will simply list a number of useful textbooks: [Schmidt, 1986, Tennent, 1991, Gunter, 1992b, Winskel, 1993] In addition, a number of other applications of Domain Theory have arisen: in Abstract Interpretation and static program analysis [Abramsky, 1990a, Burn et al. 1986, Abramsky and Jensen, 1991] see also the article on Abstract Interpretation in this ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....interpreting the dynamic aspects of the construction of proofs so as, for example, to be able to distinguish between breadth and depth first strategies. A first example is, perhaps, provided by Schmidt s denotational treatment of backtracking using Success continuations and Failure continuations [158]. Given such a semantic account, we may, for example, aim to focus on different specific models for different purposes. For example, we may wish to work with non provability via the generation of counter models. Semantic Foundations for Search Spaces. We have seen, in x 2.3, that a derivation of ....

D.A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....a priori bounded numbers of the computational reals by the finite and a priori bounded numbers of the machine. The development of a sound understanding of the number systems starts with Wilkinson[36, 37] The concept of the Wilkinson set fits very nicely with the ideas of denotational semantics[4, 6, 25, 29, 31]. This development should be primarily algebraic in nature, adding a level to the traditional algebraic hierarchy. The constructive program might also shift in emphasis in development of numerical mathematics. For example, we can achieve some results by replacing limits with extrapolations. In ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....for programming languages (the other two being operational and axiomatic semantics) While it is a beautiful theory, which established a solid mathematical foundation for the enterprise of programming language semantics, it has not yet had much practical impact. There have been some applications [45], and concern for an elegant semantics has guided the design of some languages (Standard ML, Scheme) but it is usually the case that the language comes first, and then, perhaps, small pieces of it will be given a denotational semantics. Part of the reason for this is the fact that denotational ....

.... still has some advantages: it is defined compositionally and permits algebraic reasoning (to show that two programs have the same meaning, we need only show that have the same denotation) and it enables the use of well known techniques for reasoning about programs, such as fixed point induction [45]. 2.3 An example: Primitive recursion and the lazy natural numbers As a simple example, we exhibit an intensional semantics for the primitive recursive (PR) algorithms [33] A PR algorithm can only recur on one input. The presentation of the algorithms is as a rewrite system following Colson ....

D.A. Schmidt, Denotational Semantics (Allyn and Bacon, 1986).

....output language contains single entry, single exit while loops but neither goto s nor labels, and represents the normalized form of the program. 2. 1 Denotational Semantics The semantics of a programming language is a precise mathematical specification of the meaning of programs in the language [42, 45, 40]. The idea of this approach is to define functions which map syntactic constructs into algebraic ones. This method is based on Scott and Strachey s work and has been used to define languages like ALGOL 60 [31] PASCAL [45] and CLU [39] Our input language contains branching instructions. Simple ....

D.A. Schmidt. Denotational Semantics. Allyn and Bacon, Newton, MA, 1986.

....as saying that an abstract domain is fully determined by a certain kind of properties definable over the domain together with the logical entailment relation associated with these properties. In this paper we shall take an abstract interpretation to be a denotational semantics (see e.g. [13]) for which the denotation of any program is effectively computable. One way of achieving effectiveness is by requiring the domains used in the interpretation to be finite. This on the other hand introduces a loss of information that in general makes it impossible to calculate the condition in a ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

....type of the result. As a consequence of the previous choice, function types are elements of the domain defined by the following recursive domain equation: G = Relational (G Omega G) Phi Relational (G) where the symbol Relational denotes the constructor of the relational powerdomain [Sch86] [GuSc90] The initial approximation is the flat domain G 0 containing all the elements fnum; bool; num Phi boolg and the bottom element . The powerdomain Relational (G Omega G) denotes the relations over elements of G. Since functions are denoted by their functional graph, functions are ....

D. A. Schmidt, Denotational Semantics, Allyn and Bacon, 1986.

....HASKELL provide an excellent framework for exploring relationships between evaluators, using abstract interpretation as a motivating example. 1. Introduction Evaluators, also called interpreters , play a variety of roles in the study of programming languages. They define the meaning of programs [7], support the development of new language paradigms [3] relate abstract semantics of programs to the corresponding concrete semantics [2] provide debugging information [5] and yet are simple enough to teach students about the implementation of programming languages. Given this, it s surprising ....

Schmidt, D. A., Denotational Semantics, Allyn and Bacon, Inc., Massachusetts, 1986.

....and copy propagation does the rest. Chapter 9 A Cost Semantics A way of proving a transformation s correctness is by presenting expressions before and after the transformation is applied and then showing that the two forms are semantically equivalent, e.g. using denotational semantics [Sch86] But for the class of transformations we are interested in (code improving program transformations) we would like to prove not only that the transformations are correct, but also that they are indeed improving the code. All the transformations we have presented were suggested by our intuitions ....

D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986. Bibliography 208

....procedure determines it has failed it is not necessary for anyone else to test for a failure condition. A table lookup program written in this style is shown below. Because the caller wants to know the value, not just whether one was found, the value is an argument to the success continuation [13], k. Since in the failure case there is no information that needs to be communicated, the failure continuation, q, takes no arguments. define lookup (lambda (key table k q) cond [ null table) q) eq key (key of (first table) k (value of (first table) else (lookup key (rest table) k ....

Schmidt, D.A., Denotational Semantics. Allyn and Bacon, Boston, 1986. pp. 202-204.

....must switch to the more difficult continuation semantics. We do no longer have compositionality of semantics which results in increased complexity in any program analysis that models this semantics (such as data flow analysis) More information on direct and continuation semantics can be found in [Sch86] Third, the specific method of control flow normalization used here makes it unnecessary to perform interval analysis as part of the data flow analysis. Interval analysis is a technique to divide a (control flow) graph into sets of nodes (intervals) so that analysis of the graph can be composed ....

David A. Schmidt. Denotational Semantics. Allyn and Bacon, Boston, 1986.

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D. A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

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D. A. Schmidt, Denotational Semantics, Allyn and Bacon. New York, 1986.

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David A. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

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D. Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

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D. A. Schmidt. Denotational Semantics. Allyn and Bacon, Boston, 1986.

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David Schmidt. Denotational Semantics. Allyn and Bacon, 1986.

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Schmidt86. Schmidt, D., Denotational Semantics, Allyn and Bacon, Inc., Boston, MA (1986).

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Schmidt, D.A. Denotational Semantics, Allyn and Bacon, 1986.

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