61,041 research outputs found
A Decision Procedure for Herbrand Formulas without Skolemization
This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with
A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs.
The described algorithm illustrates how, in the case of Herbrand formulae,
decision problems can be solved through a systematic search for proofs that
reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I.
Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction
The differential point of view of the infinitesimal calculus in Spinoza, Leibniz and Deleuze
In Hegel ou Spinoza,1 Pierre Macherey challenges the influence of Hegelâs reading of Spinoza by stressing the degree to which Spinoza eludes the grasp of the Hegelian dialectical progression of the history of philosophy. He argues that Hegel provides a defensive misreading of Spinoza, and that he had to âmisread himâ in order to maintain his subjective idealism. The suggestion being that Spinozaâs philosophy represents, not a moment that can simply be sublated and subsumed within the dialectical progression of the history of philosophy, but rather an alternative point of view for the development of a philosophy that overcomes Hegelian idealism. Gilles Deleuze also considers Spinozaâs philosophy to resist the totalising effects of the dialectic. Indeed, Deleuze demonstrates, by means of Spinoza, that a more complex philosophy antedates Hegelâs, which cannot be supplanted by it. Spinoza therefore becomes a significant figure in Deleuzeâs project of tracing an alternative lineage in the history of philosophy, which, by distancing itself from Hegelian idealism, culminates in the construction of a philosophy of difference.
It is Spinozaâs role in this project that will be demonstrated in this paper by differentiating Deleuzeâs interpretation of the geometrical example of Spinozaâs Letter XII (on the problem of the infinite) in Expressionism in Philosophy, Spinoza,2 from that which Hegel presents in the Science of Logic.
Lambda theories of effective lambda models
A longstanding open problem is whether there exists a non-syntactical model
of untyped lambda-calculus whose theory is exactly the least equational
lambda-theory (=Lb). In this paper we make use of the Visser topology for
investigating the more general question of whether the equational (resp. order)
theory of a non syntactical model M, say Eq(M) (resp. Ord(M)) can be
recursively enumerable (= r.e. below). We conjecture that no such model exists
and prove the conjecture for several large classes of models. In particular we
introduce a notion of effective lambda-model and show that for all effective
models M, Eq(M) is different from Lb, and Ord(M) is not r.e. If moreover M
belongs to the stable or strongly stable semantics, then Eq(M) is not r.e.
Concerning Scott's continuous semantics we explore the class of (all) graph
models, show that it satisfies Lowenheim Skolem theorem, that there exists a
minimum order/equational graph theory, and that both are the order/equ theories
of an effective graph model. We deduce that no graph model can have an r.e.
order theory, and also show that for some large subclasses, the same is true
for Eq(M).Comment: 15 pages, accepted CSL'0
Existential witness extraction in classical realizability and via a negative translation
We show how to extract existential witnesses from classical proofs using
Krivine's classical realizability---where classical proofs are interpreted as
lambda-terms with the call/cc control operator. We first recall the basic
framework of classical realizability (in classical second-order arithmetic) and
show how to extend it with primitive numerals for faster computations. Then we
show how to perform witness extraction in this framework, by discussing several
techniques depending on the shape of the existential formula. In particular, we
show that in the Sigma01-case, Krivine's witness extraction method reduces to
Friedman's through a well-suited negative translation to intuitionistic
second-order arithmetic. Finally we discuss the advantages of using call/cc
rather than a negative translation, especially from the point of view of an
implementation.Comment: 52 pages. Accepted in Logical Methods for Computer Science (LMCS),
201
Two loop detection mechanisms: a comparison
In order to compare two loop detection mechanisms we describe two calculi for theorem proving in intuitionistic propositional logic. We call them both MJ Hist, and distinguish between them by description as `Swiss' or `Scottish'. These calculi combine in different ways the ideas on focused proof search of Herbelin and Dyckhoff & Pinto with the work of Heuerding emphet al on loop detection. The Scottish calculus detects loops earlier than the Swiss calculus but at the expense of modest extra storage in the history. A comparison of the two approaches is then given, both on a theoretic and on an implementational level
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