Conservative Extension Research Articles

Refutational theorem proving for hierarchic first-order theories

We extend previous results on theorem proving for first-order clauses with equality to hierarchic first-order theories. Semantically such theories are confined to conservative extensions of the base models. It is shown that superposition together with variable abstraction and constraint refutation is refutationally complete for theories that are sufficiently complete with respect to simple instances. For the proof we introduce a concept of approximation between theorem proving systems, which makes it possible to reduce the problem to the known case of (flat) first-order theories. These results allow the modular combination of a superposition-based theorem prover with an arbitrary refutational prover for the primitive base theory, whose axiomatic representation in some logic may remain hidden. Furthermore they can be used to eliminate existentially quantified predicate symbols from certain second-order formulae.

Definable types in -minimal theories

Let L be a first order language. If M is an L-structure, let LM be the expansion of L obtained by adding constants for the elements of M.Definition. A type is definable if and only if for any L-formula , there is an LM-formula so that for all iff M ⊨ dθ(¯). The formula dθ is called the definition of θ.Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for M ≺ N, the property that every M-type realized in N is definable is equivalent to N being a conservative extension of M, whereDefinition. If M ≺ N, we say that N is a conservative extension of M if for any n and any LN -definable S ⊂ Nn, S ∩ Mn is LM-definable in M.Van den Dries [Dl] studied definable types over real closed fields and proved the following result.0.1 (van den Dries), (i) Every type over (R, +, -,0,1) is definable.(ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent:(a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F.(b) K is a conservative extension of F.

MLOG: a strongly typed confluent functional language with logical variables

A new programming language called MLOG is introduced. MLOG is a conservative extensions of ML with logical variables. To validate our concepts, a complier named CAML Light FLUO was implemented. Numerous examples are presented to illustrate the possibilities of MLOG. The pattern matching of ML is kept for λ-calculus bindings and an unification primitive is introduced for the logical variables bindings. A suspension mechanism allows cohabitation of pattern-matching and logical variables. Although the evaluation strategy for the application is fixed, the order for the evaluation of the parts of pairs and application remains free. MLOG programs can be evaluated in parallel with the same result obtained irrespective of the particular order of evaluation. This is guaranteed by the Church-Rosser property observed by the evaluation rules. As a corollary, a strict λ-calculus with explicit substitution on named variables is shown to be confluent. A completely formal operational semantics of MLOG is given in this paper.

Kelley-Morse+Types of well order is not a conservative extension of Kelley Morse

Assuming the consistency ofZF + “There is an inaccessible number of inaccessibles”, we prove that Kelley Morse theory plus types is not a conservative extension of Kelley-Morse theory.

Simplified semantics for relevant logics (and some of their rivals)

This paper continues the work of Priest and Sylvan in Simplified Semantics for Basic Relevant Logics, a paper on the simplified semantics of relevant logics, such as B+ and B. We show that the simplified semantics can also be used for a large number of extensions of the positive base logic B+, and then add the dualising'*' operator to model negation. This semantics is then used to give conservative extension results for Boolean negation.

Conservativity of Equational Theories in Typed Lambda Calculi1

In programming languages that feature unrestricted recursion, the equational theory corresponding to evaluation of data type expressions must be distinguished from the classical theory of the data as given by, say, algebraic specifications. Aiming to preserve classical reasoning about the underlying data types, that is, for the equational theory of the programming language to be a conservative extension of the theory given by the data type specification, we investigate, alternative computational settings given by typed lambda calculi, specifically here by the Girard-Reynolds polymorphic lambda calculus ( λ ∀ ). We prove that the addition of just the λ ∀ -constructions to arbitrary specifications, as given by algebraic theories, and even simply typed lambda theories, is conservative. This suggests that polymorphic constructs and reasoning can be superimposed on familiar data-type definition features of programming languages without changing the behavior of these features. Using purely syntactic methods, we give transformational proofs of these results for certain systems of equational reasoning. A new technique for analyzing polymorphic equational proofs is developed to this purpose. Finally, we prove, with a semantics argument, that it is possible to combine arbitrary algebraic data type specifications and the λ ∀ -constructions into functional programming languages that both conserve algebraic reasoning about data. and ensure, over arbitrary algebraically specified observables, a computing power equivalent to that of the pure λ ∀ . The corresponding problem for simply typed specifications remains open.

On the asymptotic range of cocyles for shifts of finite type

AbstractWe study the problem of lifting ergodicity to certain skew-product extensions over shifts of finite type. This can be done by computing the asymptotic range of the defining cocycle. For a class of ergodic shift-invariant measures and a class of functions (defining conservative extensions) we give a characterization of ergodicity in terms of weights of periodic orbits in the shift space.

Solving divergence in Knuth-Bendix completion by enriching signatures

The Knuth-Bendix completion algorithm is a procedure which generates confluent and terminating sets of rewrite rules. The algorithm has many applications: the resulting rules can be used as a decision procedure for equality or, in the case of program synthesis, as a program. We present an effective algorithm to solve some cases of divergence in the Knuth-Bendix completion algorithm, starting from a grammar characterising the infinite rule set. We replace an infinite set of rewrite rules by a finite complete set by enriching the original (order-sorted) signature with new sorts and new operator arities, while remaining within a conservative extension of the original system and within the original term set. The complexity of the new rewriting system is no worse than that of the original system. We characterise the class of examples for which this approach is applicable and give some sufficient conditions for the algorithm to succeed.

Fixed points in Peano arithmetic with ordinals

Jäger, G., Fixed points in Peano arithmetic with ordinals, Annals of Pure and Applied Logic 60 (1993) 119-132. This paper deals with some proof-theoretic aspects of fixed point theories over Peano arithmetic with ordinals. It studies three such theories which differ in the principles which are available for induction on the natural numbers and ordinals. The main result states that there is a natural theory in this framework which is a conservative extension of Peano arithmetic.

Classically complete modal relevant logics

AbstractA variety of modal logics based on the relevant logic R are presented. Models are given for each of these logics and completeness is shown. It is also shown that each of these logics admits Ackermann's rule γ and as a corollary of this it is proved that each logic is a conservative extension of its counterpart based on classical logic, hence we call them “classically complete”. MSC: 03B45, 03B46.

Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations

This paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.

M-LISP: a representation-independent dialect of LISP with reduction semantics

In this paper we introduce M-LISP, a dialect of LISP designed with an eye toward reconciling LISP's metalinguistic power with the structural style of operational semantics advocated by Plotkin [28]. We begin by reviewing the original definition of LISP [20] in an attempt to clarify the source of its metalinguistic power. We find that it arises from a problematic clause in this definition. We then define the abstract syntax and operational semantics of M-LISP, essentially a hybrid of M-expression LISP and Scheme. Next, we tie the operational semantics to the corresponding equational logic. As usual, provable equality in the logic implies operational equality. Having established this framework we then extend M-LISP with the metalinguistic eval and reify operators (the latter is a nonstrict operator that converts its argument to its metalanguage representation). These operators encapsulate the matalinguistic representation conversions that occur globally in S-expression LISP. We show that the naive versions of these operators render LISP's equational logic inconsistent. On the positive side, we show that a naturally restricted form of the eval operator is confluent and therefore a conservative extension of M-LISP. Unfortunately, we must weaken the logic considerably to obtain a consistent theory of reification.

The revised report on the syntactic theories of sequential control and state

The syntactic theories of control and state are conservative extensions of the λ υ-calculus for equational reasoning about imperative programming facilities in higher-order languages. Unlike the simple λ υ-calculus, the extended theories are mixtures of equivalence relations and compatible congruence relations on the term language, which significantly complicates the reasoning process. In this paper we develop fully compatible equational theories of the same imperative higher-order programming languages. The new theories subsume the original calculi of control and state and satisfy the usual Church–Rosser and Standardization Theorems. With the new calculi, equational reasoning about imperative programs becomes as simple as reasoning about functional programs.

On conservative extensions of syntax in system development

In a method called denorational engineering [6] the development of a denotational model of a software system consists of two major steps: first we develop an algebra of denotations Den, then we derive an appropriate algebra of syntax Syn. The latter is constructed in such a way that there exists a unique homomorphism (a denotational semantics) from Syn to Den. As usual, we assume that the syntax is described by a context-free grammar. However, unlike in traditional approaches to denotational semantics, we allow the grammar to be ambiguous. This frequently contributes to the clarity of the underlying definition of semantics and can also make parsing more efficient [2]. The YACC system is a well-known example of a standard computersupport tool where some form of such a possibility has been incorporated: YACC allows for grammars with typical ambiguities justified, for example, by the associativity of an operation or by the use of priority rules. In real-life applications the construction of syntax is not necessarily a one-way process. It may involve backtracking to denotations since after having derived a syntax we may wish to modify denotations and thus have to modify the syntax as well. This may happen both at the stage of the original system design and later, when new versions of an already existing and used system are developed. Every modification of a software system may be described as a combination of two steps: a restriction, where we remove some existing mechanisms, and an extension, where we add some new ones. In the former case the modification of syntax is rather easy: we just have to remove some productions from the corresponding grammar. The latter case, however, may be more complicated. Any extension

A second order version of S2i and U21

In [1] S. Buss introduced systems of bounded arithmetic , , , (i = 1, 2, 3, …). and are first order systems and and are second order systems. and are closely related to and respectively in the polynomial hierarchy, and and are closely related to PSPACE and EXPTIME respectively. One of the most important problems in bounded arithmetic is whether the hierarchy of bounded arithmetic collapses, i.e. whether = or = for some i, or whether = , or whether is a conservative extension of S2 = ⋃i. These problems are relevant to the problems whether the polynomial hierarchy PH collapses or whether PSPACE = PH or whether PSPACE = EXPTIME. It was shown in [4] that = implies and consequently the collapse of the polynomial hierarchy. We believe that the separation problems of bounded arithmetic and the separation problems of computational complexities are essentially the same problem, and the solution of one of them will lead to the solution of the other.

Semantic file systems

A semantic file system is an information storage system that provides flexible associative access to the system's contents by automatically extracting attributes from files with file type specific transducers . Associative access is provided by a conservative extension to existing tree-structured file system protocols, and by protocols that are designed specifically for content based access. Compatiblity with existing file system protocols is provided by introducing the concept of a virtual directory . Virtual directory names are interpreted as queries, and thus provide flexible associative access to files and directories in a manner compatible with existing software. Rapid attribute-based access to file system contents is implemented by automatic extraction and indexing of key properties of file system objects. The automatic indexing of files and directories is called "semantic" because user programmable transducers use information about the semantics of updated file system objects to extract the properties for indexing. Experimental results from a semantic file system implementation support the thesis that semantic file systems present a more effective storage abstraction than do traditional tree structured file systems for information sharing and command level programming.

Annotations on the consistency of the closed world assumption

The treatment of negation and negative information in a logic programming environment has turned out to be a major problem. We introduce a relativized version of Reiter's closed world assumption and study it from a logical point of view. In particular, we look at the questions of consistency and conservative extension.

Completeness proofs for propositional logic with polynomial-time connectives

We introduce a conservative extension of propositional calculus which allows conjunctions and disjunctions of variable length. Consistency, completeness and decidability for both classical and intuitionistic systems are given provided some simple conditions on the formation of propositional letters are satisfied. We describe an application to PROLOG programming in the context of databases.

A1 is not a conservative extension of S4 but of S5

In [1], D. W. Hart and C. Mcginn considered two logics A1 and A2. These logics embody part of a tradition about a priori knowledge and necessity. They proved that A2 is a conservative extension of a well-known modal logic S5 but left the problem whether A1 is a conservative extension of S4 open. In this note, we shall show that A1 is not a conservative extension of S4 but of S5, and also correct an inadequate proof.

Abstraction and Empty Process in Process Algebra1

In this paper, we combine the hidden step η of the authors’ paper [2] with the empty process ε of VRANCKEN [12] and the authors’ [3]. We formulate a system ACPc, which is a conservative extension of the systems ACPη, ACP√, but also of ACPτ. This is a general system, in which most relevant issues can be discussed. Abstraction from internal steps can be achieved in two ways, in two stages: we can abstract to the hidden step η, and then from η to Milner’s silent step τ.