Results 1  10
of
13
Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
Abstract

Cited by 18 (6 self)
 Add to MetaCart
Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
On the complexity of proof deskolemization
 J. Symbolic Logic
"... Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of for ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. We consider the following problem: Given a proof of the Skolemization of a formula F, what is the length of the shortest proof of F? For the restriction of this question to cutfree proofs we prove corresponding exponential upper and lower bounds. §1. Introduction. The Skolemization of formulas is a standard technique in logic. It consists of replacing existential quantifiers by new function symbols whose arguments reflect the dependencies of the quantifier. The Skolemization of a formula is satisfiabilityequivalent to the original formula. This transformation has a number of applications, it is for example crucial for automated theorem
Positive HigherOrder Queries
"... We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition, secondorder query functionals, which allow the transformation of CQs and UCQs in a generic (i.e., syntaxindependent) way. We investigate the equivalence and containment problems for this calculus, which subsumes traditional CQ/UCQ containment. Query functionals are said to be equivalent if the output queries are equivalent, for each possible input query, and similarly for containment. These notions of containment and equivalence depend on the class of (ordinary relational algebra) queries considered. We show that containment and equivalence are decidable when query variables are restricted to positive relational algebra and we identify the precise complexity of the problem. We also identify classes of functionals where containment is tractable. Finally, we provide upper bounds to the complexity of the containment problem when functionals act over other classes.
Deciding FORewritability in EL
"... Abstract. We consider the problem of deciding, given an instance query A(x), an ELTBox T, and possibly an ABox signature Σ, whether A(x) is FOrewritable relative to T and ΣABoxes. Our main results are PSPACEcompleteness for the case where Σ comprises all symbols and EXPTIMEcompleteness for the ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
Abstract. We consider the problem of deciding, given an instance query A(x), an ELTBox T, and possibly an ABox signature Σ, whether A(x) is FOrewritable relative to T and ΣABoxes. Our main results are PSPACEcompleteness for the case where Σ comprises all symbols and EXPTIMEcompleteness for the general case. We also show that the problem is in PTIME for classical TBoxes and that every instance query is FOrewritable into a polynomialsize FO query relative to every (semi)acyclic TBox (under some mild assumptions on the data). 1
A constructive proof of Skolem theorem for constructive logic
, 2004
"... Abstract. If the sequent Γ ⊢ ∀x∃y A is provable in first order constructive natural deduction, then the theory Γ, ∀x (f(x)/y)A, where f is a new function symbol, is a conservative extension of Γ. ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. If the sequent Γ ⊢ ∀x∃y A is provable in first order constructive natural deduction, then the theory Γ, ∀x (f(x)/y)A, where f is a new function symbol, is a conservative extension of Γ.
Beth Definability in Expressive Description Logics
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2011
"... The Beth definability property, a wellknown property from classical logic, is investigated in the context of description logics (DLs): if a general LTBox implicitly defines an Lconcept in terms of a given signature, where L is a DL, then does there always exist over this signature an explicit def ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The Beth definability property, a wellknown property from classical logic, is investigated in the context of description logics (DLs): if a general LTBox implicitly defines an Lconcept in terms of a given signature, where L is a DL, then does there always exist over this signature an explicit definition in L for the concept? This property has been studied before and used to optimize reasoning in DLs. In this paper a complete classification of Beth definability is provided for extensions of the basic DL ALC with transitive roles, inverse roles, role hierarchies, and/or functionality restrictions, both on arbitrary and on finite structures. Moreover, we present a tableaubased algorithm which computes explicit definitions of at most double exponential size. This algorithm is optimal because it is also shown that the smallest explicit definition of an implicitly defined concept may be double exponentially long in the size of the input TBox. Finally, if explicit definitions are allowed to be expressed in firstorder logic then we show how to compute them in EXPTIME.
Query Rewriting over Shallow Ontologies
"... Abstract. We investigate the size of conjunctive query rewritings over OWL 2 QL ontologies of depth 1 and 2 by means of a new formalism, called hypergraph programs, for computing Boolean functions. Both positive and negative results are obtained. All conjunctive queries over ontologies of depth 1 ha ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Abstract. We investigate the size of conjunctive query rewritings over OWL 2 QL ontologies of depth 1 and 2 by means of a new formalism, called hypergraph programs, for computing Boolean functions. Both positive and negative results are obtained. All conjunctive queries over ontologies of depth 1 have polynomialsize nonrecursive datalog rewritings; treeshaped queries have polynomialsize positive existential rewritings; however, for some queries and ontologies of depth 1, positive existential rewritings can only be of superpolynomial size. Both positive existential and nonrecursive datalog rewritings of conjunctive queries and ontologies of depth 2 suffer an exponential blowup in the worst case, while firstorder rewritings can grow superpolynomially unless NP ⊆ P/poly. 1