Conservative Extension Research Articles

An extension of HM(X) with bounded existential and universal data-types

We propose a conservative extension of HM( X ), a generic constraint-based type inference framework, with bounded existential (a.k.a. abstract) and universal (a.k.a. polymorphic) data-types. In the first part of the article, which remains abstract of the type and constraint language (i.e. the logic X ), we introduce the type system, prove its safety and define a type inference algorithm which computes principal typing judgments. In the second part, we propose a realistic constraint solving algorithm for the case of structural sub-typing, which handles the non-standard construct of the constraint language generated by type inference: a form of bounded universal quantification.

New sequent calculi for Visser's Formal Propositional Logic

AbstractTwo cut‐free sequent calculi which are conservative extensions of Visser's Formal Propositional Logic (FPL) are introduced. These satisfy a kind of subformula property and by this property the interpolation theorem for FPL are proved. These are analogies to Aghaei‐Ardeshir's calculi for Visser's Basic Propositional Logic.

A Logic for Schema-Based Program Development

Abstract. We show how a theory of specification refinement and program development can be constructed as a conservative extension of our existing logic for Z. The resulting system can be set up as a development method for a Z-like specification language, or as a generalisation of a refinement calculus (with a novel semantics). In addition to the technical development we illustrate how the theory can be used in practice.

A binary modal logic for the intersection types of lambda-calculus

Intersection types discipline allows to define a wide variety of models for the type free lambda-calculus, but the Curry–Howard isomorphism breaks down for this kind of type systems. In this paper we show that the correspondence between types and suitable logical formulas can still be recovered appealing to the fact that there is a strict connection between the semantics for lambda-calculus induced by the intersection types and a Kripke-style semantics for modal and relevant logics. Indeed, we present a modal logic hinted by the analysis of the sub-typing relation for intersection types, and we show that the deduction relation for such a modal system is a conservative extension of the relation of sub-typing. Then, we define a Kripke-style semantics for the formulas of such a system, present suitable sequential calculi, prove a completeness theorem and give a syntactical proof of the cut elimination property. Finally, we define a decision procedure for theorem-hood and we show that it yields the finite model property and cut-redundancy.

Abstract Data Types Can HaveInequations 1

Abstract. This work proposes a typed functional wide-spectrum language, Funiq, for both programming and specification as an extension of Cardelli-Wegner’s Fun enriched with the fixed-point construction on expressions and inequational refinements (as assertions) for types. The type system of our wide-spectrum Funiq language is shown to be a conservative extension of that of the original Fun and sound with respect to the complete PER semantics for types.

On the equational definition of the least prefixed point

We propose a method to axiomatize by equations the least prefixed point of an order preserving function. We discuss its domain of application and show that the Boolean modal μ-calculus has a complete equational axiomatization. The method relies on the existence of a “closed structure” and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed structure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for defining by equations the least prefixed point. We stress the interplay between closed structures and fixed point operators by showing that the theory of Boolean modal μ-algebras is not a conservative extension of the theory of modal μ-algebras. The latter is shown to lack the finite model property.

A flux-split algorithm applied to conservative models for multicomponent compressible flows

In this paper we consider a conservative extension of the Euler equations for gas dynamics to describe a two-component compressible flow in Cartesian coordinates. It is well known that classical shock-capturing schemes applied to conservative models are oscillatory near the interface between the two gases. Several authors have addressed this problem proposing either a primitive consistent algorithm [J. Comput. Phys. 112 (1994) 31] or Lagrangian ingredients (Ghost Fluid Method by Fedkiw et al. [J. Comput. Phys. 152 (1999) 452] and [J. Comput. Phys. 169 (2001) 594]). We solve directly this conservative model by a flux-split algorithm, due to the first author (see [J. Comput. Phys. 125 (1996) 42]), together with a high-order (WENO5) flux reconstruction [J. Comput. Phys. 115 (1994) 200; 83 (1989) 32]. This algorithm seems to reduce the oscillations near the interfaces in a way that does not affect the physics of the experiments. We validate our algorithm with the numerical simulation of the interaction of a Mach 1.22 shock wave impinging a helium bubble in air, under the same conditions studied by Haas and Sturtevant [J. Fluid Mech. 181 (1987) 41] and successfully simulated by Quirk and Karni [J. Fluid Mech. 318 (1996) 129].

An alternative formulation of operational conservativity with binding terms

In a previous paper, the approach to structural operational semantics using transition system specifications (TSSs) was extended to deal with variable binding operators. It was shown that in the new setting a generalization of the transition rule format known as the panth format guarantees that bisimulation equivalence is a congruence for meaningful TSSs. In the current paper, it is shown that certain syntactic criteria to determine whether a TSS is an operational conservative extension of another TSS, originating from Fokkink and Verhoef, are applicable to the new setting as well. This result can for example be used to simplify proofs of axiomatic conservativity and completeness in the case where an existing process calculus is extended with new features.

A Linear Logical Framework

We present the linear type theory λ Π⊸&⊤ as the formal basis for LLF , a conservative extension of the logical framework LF . LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of Mini-ML with mutable references including its type system and its operational semantics and describe how to take practical advantage of the representation of its computations.

Timed Term Rewrite Systems

We propose a new class of rewrite systems that are conservative extensions of first-order conditional term rewrite systems together with time features such as clocks, signals, timed terms, timed atoms and timed rules. We define first timed term rewrite systems by their syntax and illustrate them through some examples. We provide then a semantics based on labelled transition systems. Finally, we show how our framework compares to related work.

Entailment and Bivalence

My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics.

An axiomatic presentation of the nonstandard methods in mathematics

AbstractA nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC.

On the Beth Property in Extension of Lukasiewicz Logics

It is proved that a necessary condition for conservative axiomatic extensions of the infinite-valued (or n-valued) Lukasiewicz logic to possess the Beth definability property consists in the presence in the language of these extensions of a countable set (resp. a set of power n) of constant terms nonequivalent with respect to the given extensions.

La logique linéaire non commutative et le calcul de Lambek

The paper analyzes the correspondence existing between the Syntactic Calculus (Lambek 1958) and (multiplicative) non-commutative linear logic (Abrusci 1991), both in its intuitionistic and classical formulation. Particular attention is paid to the fragment of classical non-commutative linear logic (or classical bilinear logic) that is a conservative extension of the Syntactic Calculus. We present some linguistic applications of this system and propose a way to build up planar graphs for its type logical formulas.

Coercion completion and conservativity in coercive subtyping

Coercive subtyping offers a general approach to subtyping and inheritance by introducing a simple abbreviational mechanism to constructive type theories. In this paper, we study coercion completion in coercive subtyping and prove that the formal extension with coercive subtyping of a type theory such as Martin–Löf's type theory and UTT is a conservative extension. The importance of coherence conditions for the conservativity result is also discussed.

The logic of tasks

The paper introduces a semantics for the language of classical first order logic supplemented with the additional operators ⊓ and ⨅. This semantics understands formulas as tasks. An agent (say, a machine or a robot), working as a slave for its master (say, the user or the environment), can carry out the task α⊓β if it can carry out any one of the two tasks α, β , depending on which of them was requested by the master; similarly, it can carry out ⨅xα(x) if it can carry out α( x) for any particular value for x selected by the master; an agent can carry out α→ β if it can carry out β as long as it has, as a slave (resource), an agent who carries out α; finally, carrying out P, where P is an atomic formula, simply means making P true; in particular, ⊥ is a task that no agent can carry out. When restricted to the language of classical logic, the meaning of formulas is isomorphic to their classical meaning, which makes our semantics a conservative extension of classical semantics. This semantics can claim to be a formalization of the resource philosophy associated with linear logic, if resources are understood as agents carrying out tasks. The classical operators of our language correspond to the multiplicative operators of linear logic, while ⊓ and ⨅ correspond to the additive conjunction and universal quantifier, respectively. Our formalism may also have a potential to be used in AI as an alternative logic of planning and action. Its main appeal is that it is immune to the frame problem and the knowledge preconditions problem. The paper axiomatically defines a logic L in the above language and proves its soundness and completeness with respect to the task semantics in the following intuitive sense: L ⊢ α iff α can be carried out by an agent who has nothing but its intelligence (i.e. no physical resources or external sources of information) for carrying out tasks. This logic is shown to be semidecidable in the full language and decidable when the classical quantifier (but not ⨅) is forbidden in it.

Cost recurrences for DML programs

A cost recurrence describes an upper bound for the running time of a program in terms of the size of its input. Finding cost recurrences is a frequent intermediate step in complexity analysis, and this step requires an abstraction from data to data size. In this article, we use information contained in dependent types to achieve such an abstraction: Dependent ML (DML), a conservative extension of ML, provides dependent types that can be used to associate data with size information, thus describing a possible abstraction. We automatically extract cost recurrences from first-order DML programs, guiding the abstraction from data to data size with information contained in DML type derivations.

The Pleasures of Anticipation: Enriching Intuitionistic Logic

We explore a relation we call ‘anticipation’ between formulas, where AanticipatesB (according to some logic) just in case B is a consequence (according to that logic, presumed to support some distinguished implicational connective →) of the formula A→B. We are especially interested in the case in which the logic is intuitionistic (propositional) logic and are much concerned with an extension of that logic with a new connective, written as “a”, governed by rules which guarantee that for any formula B, aB is the (logically) strongest formula anticipating B. The investigation of this new logic, which we call ILa, will confront us on several occasions with some of the finer points in the theory of rules and with issues in the philosophy of logic arising from the proposed explication of the existence of a connective (with prescribed logical behaviour) in terms of the conservative extension of a favoured logic by the addition of such a connective. Other points of interest include the provision of a Kripke semantics with respect to which ILa is demonstrably sound, deployed to establish certain unprovability results as well as to forge connections with C. Rauszer's logic of dual intuitionistic negation and dual intuitionistic implication, and the isolation of two relations (between formulas), head-implication and head-linkage, which, though trivial in the setting of classical logic, are of considerable significance in the intuitionistic context.

Recursion over objects of functional type

This paper presents an extension of the simply typed λ-calculus that allows iteration and case reasoning over terms of functional types that arise when using higher order abstract syntax. This calculus aims at being the kernel for a type theory in which the user will be able to formalize logics or formal systems using the LF methodology, while taking advantage of new induction and recursion principles, extending the principles available in a calculus such as the Calculus of Inductive Constructions. The key idea of our system is the use of modal logic S4. We present here the system, its typing rules and reduction rules. The system enjoys the decidability of typability, soundness of typed reduction with respect to the typing rules, the Church–Rosser and strong normalization properties and it is a conservative extension over the simply typed λ-calculus. These properties entail the preservation of the adequacy of encodings.

Dependent Types with Subtyping and Late-Bound Overloading

We present a calculus with dependent types, subtyping, and late-bound overloading. Besides its theoretical interest this work is motivated by several practical needs that range form the definition of logic encodings to proof specialization and reuse and to object-oriented extension of the SML module system. The theoretical study of this calculus is not straightforward. While confluence is relatively easy to prove, subject reduction is much harder. We were not able to add overloading to any existing system with dependent types and subtyping, and prove subject reduction. This is why we also define here as by-product a new subtyping system for dependent types that improves previous systems and enjoys several properties (notably the transitivity elimination property). The calculus with overloading is then obtained as a conservative extension of this new system. Another difficult point is strong normalization, which is a necessary condition to the decidability of subtyping and typing relations. The calculus with overloading is not strongly normalizing. However, we show that a reasonably useful fragment of the calculus enjoys this property and that its strong normalization implies the decidability of its subtyping and typing relations. The article is divided into two parts: the first three scetions provide a general overview of the systems and its motivations and can be read separately; the remaining sections develop the formal study.