11,091 research outputs found
A deductive system for existential least fixpoint logic
Existential least fixpoint logic (ELFP) is a logic with a least fixpoint operator but only existential quantification. It arises in many areas of computer science including logic programming, database theory, program verification, complexity theory, and recursion theory on abstract structures. A sequent calculus (Gentzen-style deductive system) for this logic is presented and proved to be complete. Basic model theoretic facts about ELFP are derived from the completeness theorem and the construction used in its proof. The relationship of these model theoretic facts to logic programming and database queries is explored
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
On axiom schemes for T-provably Δ1 formulas
This paper investigates the status of the fragments of Peano Arithmetic obtained by restricting induction, collection and least number axiom schemes to formulas which are Δ1 provably in an arithmetic theory T. In particular, we determine the provably total computable functions of this kind of theories. As an application, we obtain a reduction of the problem whether IΔ0+¬exp implies BΣ1 to a purely recursion-theoretic question.Ministerio de Ciencia e Innovación MTM2008–0643
Relational semantics of linear logic and higher-order model-checking
In this article, we develop a new and somewhat unexpected connection between
higher-order model-checking and linear logic. Our starting point is the
observation that once embedded in the relational semantics of linear logic, the
Church encoding of any higher-order recursion scheme (HORS) comes together with
a dual Church encoding of an alternating tree automata (ATA) of the same
signature. Moreover, the interaction between the relational interpretations of
the HORS and of the ATA identifies the set of accepting states of the tree
automaton against the infinite tree generated by the recursion scheme. We show
how to extend this result to alternating parity automata (APT) by introducing a
parametric version of the exponential modality of linear logic, capturing the
formal properties of colors (or priorities) in higher-order model-checking. We
show in particular how to reunderstand in this way the type-theoretic approach
to higher-order model-checking developed by Kobayashi and Ong. We briefly
explain in the end of the paper how his analysis driven by linear logic results
in a new and purely semantic proof of decidability of the formulas of the
monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte
Provably Total Primitive Recursive Functions: Theories with Induction
A natural example of a function algebra is R (T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kind of function algebras is obtained. This provides a useful tool for studying the class of primitive recursive functions in R (T). We prove that this is the class of p.t.c.f. of the theory axiomatized by the induction scheme restricted to (parameter free) Δ1(T)–formulas (i.e. Σ1–formulas which are equivalent in T to Π1–formulas).
Moreover, if T is a sound theory and proves that exponentiation is a total function, we characterize the class of primitive recursive functions in R (T) as a function algebra described in terms of bounded recursion (and composition). Extensions of this result are related to open problems on complexity classes. We also discuss an application to the problem on the equivalence between (parameter free) Σ1–collection and (uniform) Δ1–induction schemes in Arithmetic.
The proofs lean upon axiomatization and conservativeness properties of the scheme of Δ1(T)–induction and its parameter free version
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
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