Syntactic Equivalence Research Articles

Extending monads with pattern matching

Sequencing of effectful computations can be neatly captured using monads and elegantly written using do notation. In practice such monads often allow additional ways of composing computations, which have to be written explicitly using combinators. We identify joinads, an abstract notion of computation that is stronger than monads and captures many such ad-hoc extensions. In particular, joinads are monads with three additional operations: one of type m a -> m b -> m ( a , b ) captures various forms of parallel composition, one of type m a -> m a -> m a that is inspired by choice and one of type m a -> m ( m a) that captures aliasing of computations. Algebraically, the first two operations form a near-semiring with commutative multiplication. We introduce docase notation that can be viewed as a monadic version of case . Joinad laws imply various syntactic equivalences of programs written using docase that are analogous to equivalences about case. Examples of joinads that benefit from the notation include speculative parallelism, waiting for a combination of user interface events, but also encoding of validation rules using the intersection of parsers.

Maintaining consistent results of continuous queries under diverse window specifications

Continuous queries applied over nonterminating data streams usually specify windows in order to obtain an evolving–yet restricted–set of tuples and thus provide timely and incremental results. Although sliding windows get frequently employed in many user requests, additional types like partitioned or landmark windows are also available in stream processing engines. In this paper, we set out to study the existence of monotonic-related semantics for a rich set of windowing constructs in order to facilitate a more efficient maintenance of their changing contents. After laying out a formal foundation for expressing windowed queries, we investigate update patterns observed in most common window variants as well as their impact on adaptations of typical operators (like windowed join, union or aggregation), thus offering more insight towards design and implementation of stream processing mechanisms. Furthermore, we identify syntactic equivalences in algebraic expressions involving windows, to the potential benefit of query optimizations. Finally, this framework is validated for several windowed operations against streaming datasets with simulations at diverse arrival rates and window specifications, providing concrete evidence of its significance.

A Proposal for the Cooperation of Solvers in Constraint Functional Logic Programming

This paper presents a proposal for the cooperation of solvers in constraint functional logic programming, a quite expressive programming paradigm which combines functional, logic and constraint programming using constraint lazy narrowing as goal solving mechanism. Cooperation of solvers for different constraint domains can improve the efficiency of implementations since solvers can take advantage of other solvers' deductions. We restrict our attention to the cooperation of three solvers, dealing with syntactic equality and disequality constraints, real arithmetic constraints, and finite domain (FD) constraints, respectively. As cooperation mechanism, we consider to propagate to the real solver the constraints which have been submitted to the FD solver (and viceversa), imposing special communication constraints to ensure that both solvers will allow the same integer values for all the variables involved in the cooperation.

A computably stable structure with no Scott family of finitary formulas

One of the goals of computability theory is to find syntactic equivalences for computational properties. The Limit Lemma is a classic example of this type of equivalence: X ⊆ ω is computable from 0′ if and only if it is arithmetically definable by a ∆2 formula. A more relevant example for this paper was proved independently by Ash, Knight, Manasse and Slaman [1] and by Chishom [2]: a computable structure is relatively computably categorical if and only if it has a computably enumerable Scott family of finitary existential formulas. (These terms are defined below.) Our main theorem is a negative result which states that there is a computably stable rigid graph which does not have a Scott family of finitary formulas. Therefore, any attempt to modify the syntactic characterization of relative computably categoricity to capture computable stability using notions such as a Scott family must involve an infinitary language. In this section, we give the background definitions and motivation for this theorem. In the second section, we construct a countable family of sets A with a specific list of enumeration properties. In the last section, we code A into a rigid graph G, prove a partial quantifier elimination theorem for G, and use the enumeration properties of A and the quantifier elimination result for G to show that G does not have a Scott family of finitary formulas. Let M be a countable structure in a computable language whose domain |M| is a subset of ω. The degree of M (denoted deg(M)) is the Turing degree of the atomic diagram of

The "Perpetual Exercise of an Interminable Quest": "The Biographia Literaria" and the Kantian Revolution

1. Mr. Flosky, Mr. Coleridge, Professor Kant, and Jacques Lacan THOMAS LOVE PEACOCK'S SEND UP OF SAMUEL TAYLOR COLERIDGE IN Nightmare Abbey says more than its apparently reductive satire at first suggests. Through Nightmare Abbey's character Mr. Flosky, Peacock's novel links Coleridge's political apostasy, philosophically abstruse prevarications, and argumentative lacunae with (1) The singularity of Peacock's satire is that it is not content to simply present Mr. Flosky as a long-winded fool who has merely substituted his own self-justifying mystifications for a deep scholarly understanding of Kantian thought. Rather, Mr. Flosky's interest in Kant, regardless of its accuracy, serves as a crucial point of reference under whose aegis personal beliefs and critical pronouncements on the subjects of politics, religion, and philosophy take on a certain ideological consistency. Further, Mr. Flosky claims that the obscurity and inaccessibility of his Kantian method is the secret to mental health; it is a process of losing your way, and keeping your mind in perfect health, by perpetual exercise of an quest (67). His mental health and intellectual pursuits, Mr. Flosky claims, are sustained by the pure enjoyment of an impossible philosophical task. In Nightmare Abbey, ideological consistency and mental health emerge as intimately related to one another, the philosophical subject and the subject of philosophy bound up in the same interminable quest. Beyond identifying Coleridge's treatment of Kantian philosophy as a predominant humour upon which to hang its characterization of Mr. Flosky, a technique whereby the subject of discourse becomes the subject as discourse, Nightmare Abbey also provides a critique of one of the perennial sites of critical interest in Coleridge scholarship, one which has developed a particularly sophisticated richness over the last thirty years: the gap in the Biographia Literaria's deduction of the imagination. This trouble spot in Coleridge's monumental work has been understood variously as an instance of philosophical obscurity, brilliant irony, intellectual dishonesty, and as a sign of its author's mental health. (2) In his desynonymizing lecture to the hapless Marionetta O'Carrol, Mr. Flosky demonstrates that the gap in the Biographia's philosophical argument is, indeed, the object of Peacock's satire. Here, Mr. Flosky insists: Think is not synonymous with believe--for believe, in many most important particulars, results from the total absence, the absolute negation of thought, and is thereby the sane and condition of the mind; and and are both essentially different from fancy, and fancy, again, is distinct from imagination. This distinction between fancy and imagination is one of the most abstruse and important points of metaphysics. I have written seven hundred pages of promise to elucidate it.... (83) In Peacock's satire, the allusion to Coleridge's promised deduction of the imagination in Chapter 13 becomes a specific manifestation of Mr. Flosky's enjoyment-in-deferral of an interminable quest. Further, the gap occasioned by the non-sequitur association of the Biographia's philosophical endeavor with a preference for belief over thought is critically astute insofar as it performs its critique of the gap in the Biographia's deduction of the imagination as a gap in the Biographia's logic. What seems merely deferred in the Biographia, Nightmare' Abbey identifies simultaneously as an instance of circumlocutious prevarication and as a site of ideological consistency; belief makes orthodoxy and sanity interchangeable in the assertion of their syntactic equality. In the fractured Floskean logic of the passage, belief is the condition of the orthodox and the sane, not because it is philosophically sound or even logically coherent, but precisely because the ideological consistency it produces is the antithesis of rational thought. …

On the complexity of equational problems in CNF

Equational problems (i.e. first-order formulae with quantifier prefix ∃ ∗ ∀ ∗ , whose only predicate symbol is syntactic equality) are an important tool in many areas of automated deduction, e.g. restricting the set of ground instances of a clause via equational constraints allows the definition of stronger redundancy criteria and hence, in general, of more efficient theorem provers. Moreover, the inference rules themselves can be restricted via constraints. In automated model building, equational problems play an important role both in the definition of an appropriate model representation and in the evaluation of clauses in such models. Also, many problems in the area of logic programming can be reduced to equational problem solving. Finally, equational problems over a finite domain correspond to the evaluation of certain queries over relational databases. The goal of this work is a complexity analysis of the satisfiability problem of equational problems. The main results will be a proof of the NP-completeness (and, in particular, the NP-membership) of equational problems in CNF over an infinite domain and of the Σ 2 p -completeness in the case of CNF over a finite domain.

Strong Normalization with Singleton Types

This paper presents a new lambda-calculus with singleton types, called λ≤{}βδ. The main novelty of λ≤{}βδ is the introduction of a new reduction, the δ-reduction, replacing any variable declared of singleton type by its value, and the definition of equality as the syntactic equality of βδ-normal forms. The δ-reduction has a very odd behavior on untyped terms, which renders its metatheoretical study difficult since the usual proof method for subject-reduction and Church-Rosser property are inapplicable. Nevertheless, these properties can be proved simultaneously with strong normalization on typed terms using a proof method à la Coquand-Gallier, borrowing ideas to Goguen. In spite of its complex metatheory, our calculus enjoys a simple, sound and complete type-inference algorithm.

Explicit versus implicit representations of subsets of the Herbrand universe

In Lassez and Marriott (J. Automat. Reson. 3 (3) (1987) 301–317), explicit and implicit generalizations were studied as representations of subsets of some fixed Herbrand universe H. An explicit generalization E=r 1∨⋯∨r l represents all ground terms that are instances of at least one of the terms t i , whereas an implicit generalization I=t/t 1∨⋯∨t m represents all H-ground instances of t that are not instances of any term t i . More generally, a disjunction I=I 1∨⋯∨I n of implicit generalizations contains all ground terms that are contained in at least one of the implicit generalizations I j . Implicit generalizations have applications to many areas of Computer Science like machine learning, unification, specification of abstract data types, logic programming, functional programming, etc. In these areas, the so-called finite explicit representability problem plays an important role, i.e. given a disjunction of implicit generalizations I=I 1∨⋯∨I n , does there exist an explicit generalization E, s.t. I and E are equivalent? We shall prove the coNP-completeness of this decision problem. Implicit generalizations can be represented as equational formulae, i.e., first-order formulae whose only predicate symbol is syntactic equality. Closely related to the finite explicit representability problem is the so-called negation elimination problem of equational formulae, i.e. given an arbitrary equational formula P , is P semantically equivalent to an equational formula without universal quantifiers and negation. In this work we study the negation elimination problem of equational formulae with purely existential quantifier prefix. We prove the coNP-completeness for such formulae in DNF and the Π 2 p -hardness in case of CNF.

Sur un lexique-grammaire comparé d’adverbes figés

Summary This paper presents some of the findings which emerged during the construction of a French-Italian lexicon-grammar of frozen adverbs. A French lexicon-grammar compiled by Maurice Gross (1990) was compared with an Italian lexicon-grammar (De Gioia 1994b) with a view to finding semantically equivalent entries for translation purposes. The results showed that 65% of the French entries can be translated with equivalent adverbial and idiomatic expressions in Italian, giving the lie to the traditionally held view that idiomatic expressions are generally untranslatable. A further analysis of cases of semantic and syntactic equivalence showed that the two languages share many families of frozen adverbs, which can be represented by automata and transducers.

Solvable Set/Hyperset Contexts: II. A Goal-Driven Unification Algorithm for the Blended Case

A universe composed by rational ground terms is characterized, both constructively and axiomatically, where the interpreted construct with which designates the operation of adjoining one element to a set, coexists with free Herbrand functors. Ordinary syntactic equivalence must be superseded by a bisimilarity relation ≈, between trees labeled over a signature, that suitably reflects the semantics of with. Membership (definable as “d∈t =Def (t with d≈t”) meets the non-well-foundedness property characteristic of hyperset theory A goal-driven algorithm for solving the corresponding unification problem is provided, it is proved to be totally correct, and exploited to show that the problem itself is NP-complete. The results are then extended to the treatment of the operator less, designating the one-element removal operation. Applications to the automaton matching and type-finding problems are illustrated.

Woordverwerving in Een Vreemde Taal

In communicative foreign Language teaching there is more emphasis on vocabulary than in traditional structural approaches where it takes second place to grammar. Two important questions concerning the vocabulary are: what words to present and how to present them? The first question can be answered on the basis of a (socio)linguistic analysis of the communicative needs of the target group. The answer to the second question must be derived from psycholinguistic and semantic theories about how words are learned and stored in the mental lexicon. This paper deals with one aspect of the first question (viz. how many words?) and discusses some possible applications of psychological theories about word learning and word storage. As to the first question it is argued for the intermediate levels that to (partially) avoid word selection problems (what and how many?) the syllabus should include twice as many words (say 6.000) as are traditionally presented. If students master 50% of this list, they should be able to handle semi-authentic reading and listening material by contextually guessing the unknown words since most of the texts (about 95%) will be covered by the words they know. As to the second question (how to present words) it is argued that for intentional word learning, a contextualised presentation is preferable since it provides the student with more possiblities to embed the word in the interrelated networks of various kinds that constitute our memory. A distinction should be made however between "easy" and "difficult" words, easy implying a direct syntactic and semantic equivalence between the LI and L2 (i.e. same concept, different label) and difficult referring to cases where there is no such similarity (i.e. concept and label different). Some emperical evidence is discussed that leads to the conclusion that it is more efficient (taking time and output into consideration) to present difficult words only in context and easy words without context. Finally an experimental technique is discussed (called graded contextual desambiguation) that tries to grade the mental operations necessary for working out the meaning of an unknown word.

The categorical abstract machine

The Cartesian closed categories have been shown by several authors to provide the right framework of the model theory of λ-calculus. The second author developed this as a syntactic equivalence between two calculi, giving rise to a new kind of combinatory logic: the categorical combinatory logic, where computations can be done through simple rewrite rules, and, as usual with combinators, avoiding problems with variable name clashes. This paper goes further (though not requiring a previous knowledge of categorical combinatory logic) and describes a very simple machine where categorical terms can be considered as code acting on a graph of values (the essential actions are LISP's “cons” and “cdr”, as well as “rplacd” to implement recursion). The only saving mechanism is a stack containing pointers on code or on the graph. Abstractions are handled in the very same way as in P. Landin's SECD machine, using closures. The machine is called categorical abstract machine or CAM. The CAM is easier to grasp and prove than the SECD machine. The paper discusses the implementation of a real functional programming language, ML, through the CAM. A basic acquintance with λ-calculus is required.

Categorical combinators

Our main aim is to present the connection between λ -calculus and Cartesian closed categories both in an untyped and purely syntactic setting. More specifically we establish a syntactic equivalence theorem between what we call categorical combinatory logic and λ -calculus with explicit products and projections, with β and η -rules as well as with surjective pairing. “Combinatory logic” is of course inspired by Curry's combinatory logic, based on the well-known S , K , I . Our combinatory logic is “categorical” because its combinators and rules are obtained by extracting untyped information from Cartesian closed categories (looking at arrows only, thus forgetting about objects). Compiling λ -calculus into these combinators happens to be natural and provokes only n log n code expansion. Moreover categorical combinatory logic is entirely faithful to β -reduction where combinatory logic needs additional rather complex and unnatural axioms to be. The connection easily extends to the corresponding typed calculi, where typed categorical combinatory logic is a free Cartesian closed category where the notion of terminal object is replaced by the explicit manipulation of applying (a function to its argument) and coupling (arguments to build datas in products). Our syntactic equivalences induce equivalences at the model level. The paper is intended as a mathematical foundation for developing implementations of functional programming languages based on a “categorical abstract machine,” as developed in a companion paper ( Cousineau, Curien, and Mauny, in “Proceedings, ACM Conf. on Functional Programming Languages and Computer Architecture,” Nancy, 1985 ).

On the form and meaning of Hindi passive sentences

Passive sentences in Hindi share distinctive verbal morphology, and case marking of the agent phrase, if it is expressed. But in meaning and use, the sentences fall into three not totally distinct sub-groups, each with somewhat different meaning, and each loosely associated with a typical form. Since there is no absolute association of particular sentence form with one of the three meanings (ability, prohibition, neutral), it is not possible to relate form and meaning solely with transformations, optional or obligatory, abstract elements, or interpretive rules. If, as is proposed, the passive form is ‘marked’ by comparison with the active or usual form, the speaker's choice of the passive form is the basis for conversational inference, to derive from the literal contents a closely related conveyed meaning. Formal devices like transformations state the syntactic equivalence of active and passive sentences, but particular shades of meaning are represented as possible inferences conversationally and not linguistically related to particular sentence structures.

Twenty-sixth annual meeting of the Association for Symbolic Logic

S OF PAPERS 38 of the m-adic notations for x and y is the m-adic notation for z. Replacing addition and multiplication by the single relation of m-adic concatenation in the above description of CA, we have (Smullyan, ibid.) a description of the m-rudimentary relations. It is shown that these are merely alternative descriptions of the same class of relations; i.e., for each m, the class of m-rudimentary relations is precisely CA. The ternary relation of exponentiation is in CA. In fact, the quaternary relation SP(n, x, y) = z is in CA where SP is the doubly but not primitive recursive function defined by SP(O,x,y) = x + y; SP(w + 1, x, O) = x; and SP(w + 1, x,y + 1) = SP(n,x, SP. (n + 1, x, y)). However, CA is a small subclass of the primitive recursive relations in the sense that there is a fixed positive integer K such that the characteristic function of every relation in CA is definable from the successor function by at most K uses of primitive recursion. Progress in relating CA to other small classes of relations defined in the literature is also described. (Received November 28, 1961.) R. B. ANGELL. A logical notation with two primitive signs. This paper presents two systems of logical notation, each using just two typographic signs or shapes, namely, the leftand right-hand parentheses. The first system is easily shown adequate for Quine's Mathematical Logic (1940, 1958). It contains: I. Primitive signs:) (. II. Symbols (or well-formed signs): 1. '()' is a symbol. 2. If S and S' are symbols, rSS'1 is a symbol. 3. If S is a symbol, r(S)p is a symbol. A. Atomic symbols: 1. '(()' is an atomic symbol. 2. If r(S)1 is an atomic symbol, F(SO()1 is an atomic symbol. (Variables are atomic symbols containing 2n (n >1) symbols '('.) B. Well-formed formulae: 1. If a and fi are variables, r((())(zfi))1 is a wff (in this case alone, an atomic formula). 2. If 0 is a wff, and a is a variable, r((a)I))l is a wff. 3. If 1D and T are wffs, then F((D) (T))1 is a wff. It is shown that Quine's variables, atomic formulae, quantified expressions, and stroke applications can be correlated unambiguously with expressions resulting from formation rules 3A, 3BI, 3B2 and 3B3 respectively. The class of wffs defined above are syntactically equivalent to Quine's class of logical formulae (cf. Quine, p. 124) and are thus adequate for all his definitions and statements. The second system is syntactically simpler, eliminating a special notation for quantifiers in favor of a single class constant. Formation rules are the same except that X, 1-3 are replaced by B'1. If a is a variable and f is either a variable or '((0()', then r((())-, is a wff. B'2. If T is a wff, F(o)l is a wff. B'3. If 4 and T are wffs, then r(QD*)1 is a wff. Rules B'2 and B'3 are associated with denial and conjunction respectively. By the following definitions membership is introduced, as well as a class constant 'E' which eliminates the need of quantifier notation: Dl. ot e fi1 for F((0)(,B)) . D2. FE0l for rx e En or r((()(,(( ())1l. An interpretation is provided such that every standard quantified statement is equivalent on the usual interpretation to a statement in this notation, and every statement in this notation is either equivalent to a standard quantified statement in its usual interpretation or else an unobjectionable addition. (The interpretation introduced requires a new interpretation of expressions usually called propositional functions). The paper does not attempt to provide a set of axioms for this system, as it establishes only syntactical equivalence of symbolisms. (Received April 23, x962.) G. KREISEL and W. W. TAIT. Induction and recursion. Let R be primitive recursive arithmetic (PRA) with the rule A (0), t (a') - A (a)