bluuebunny/crossref_metadata_2025_split huggingface dataset:
Minimal models of Heyting arithmetic
Authors: Moerdijk Ieke , Palmgren Erik 
Published: May 2006

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be calledHAIandHAIωrespectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].
Canonical formulas for K4. Part II: Cofinal subframe logics
Authors: Zakharyaschev Michael 
Published: May 2006

This paper is a continuation of Zakharyaschev [25], where the following basic results on modal logics with transitive frames were obtained:• With every finite rooted transitive frame and every set of antichains (which were called closed domains) in two formulas α (, , ⊥) and α(, ) were associated. We called them the canonical and negation free canonical formulas, respectively, and proved the Refutability Criterion characterizing the constitution of their refutation general frames in terms of subreduction (alias partial p-morphism), the cofinality condition and the closed domain condition.• We proved also the Completeness Theorem for the canonical formulas providing us with an algorithm which, given a modal formula φ, returns canonical formulas α(i, i), ⊥), for i = 1,…, n, such thatif φ is negation free then the algorithm instead of α(i, i, ⊥) can use the negation free canonical formulas α(i, i). Thus, every normal modal logic containing K4 can be axiomatized by a set of canonical formulas.In this Part we apply the apparatus of the canonical formulas for establishing a number of results on the decidability, finite model property, elementarity and some other properties of modal logics within the field of K4.Our attention will be focused on the class of logics which can be axiomatized by canonical formulas without closed domains, i.e., on the logics of the formAdapting the terminology of Fine [11], we call them the cofinal subframe logics and denote this class by . As was shown in Part I, almost all standard modal logics are in .
Boolean algebras, Stone spaces, and the iterated Turing jump
Authors: Jockusch Carl G. , Soare Robert I. 
Published: May 2006

AbstractWe show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, is a countable structure with finite signature, and d is a degree, we say that has αth-jump degreed if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of with universe ω in which the functions and relations have degree at most c. We show that every degree d ≥ 0(ω) is the ωth jump degree of a Boolean algebra, but that for n < ω no Boolean algebra has nth-jump degree d < 0(n). The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties.
The critical number of a variable in a function
Authors: Takeuti Gaisi 
Published: May 2006

LetL0be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), andμx≤ ∣s∣t(x). Hereμx≤ ∣s∣t(x) is the smallest numberx≤ ∣s∣ satisfyingt(x) > 0 and 0 if there exist no suchxand we stipulate that ifsandt(a) are terms inLo, thenμx≤ ∣s∣t(x) is also a term inLo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:LetLa language on ℕ with only predicate constant = andL0⊆L. Letf(b,a1,…,am) be a function for ℕm+1into ℕ. We say “fis weakly expressed by termst1(b,a1,…,am),…,tr(b,a1,…,am) inL” if for everyb,a1,…,am∈ ℕ,f(b,a1,…,am) is equal to one ofti(b,a1, …,am). The critical number ofbinfwith respect toLis the minimum numbernsuch that wheneverf(b,a1,…,an) is weakly expressed by termst1(b,a1, …,an),…, the number of occurrences ofbin someti(b,a1,…,an) is at leastn.As is defined in [1], a functionfis defined by a limited iteration fromgandhwith respect toLiff the following holds: Letτbe defined aswith the condition; andis defined bywhereandare terms inL. We say “fis defined by a short limited iteration fromgandh” ifis defined bywhereτ,s, tare the same as above satisfying the condition.
Undecidability in diagonalizable algebras
Authors: Shavrukov V. Yu. 
Published: May 2006

AbstractIf a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.
A special class of almost disjoint families
Authors: Leathrum Thomas E. 
Published: May 2006

AbstractThe collection of branches (maximal linearly ordered sets of nodes) of the tree <ωω (ordered by inclusion) forms an almost disjoint family (of sets of nodes). This family is not maximal — for example, any level of the tree is almost disjoint from all of the branches. How many sets must be added to the family of branches to make it maximal? This question leads to a series of definitions and results: a set of nodes is off-branch if it is almost disjoint from every branch in the tree; an off-branch family is an almost disjoint family of off-branch sets; and is the minimum cardinality of a maximal off-branch family.Results concerning include: (in ZFC) , and (consistent with ZFC) is not equal to any of the standard small cardinal invariants or = 2ω. Most of these consistency results use standard forcing notions—for example, in the Cohen model.Many interesting open questions remain, though—for example, whether .
A sheaf representation and duality for finitely presented Heyting algebras
Authors: Ghilardi Silvio , Zawadowski Marek 
Published: May 2006

AbstractA. M.Pitts in [Pi] proved that is a bi-Heyting category satisfying the Lawvere condition. We show that the embedding Φ: → Sh(P0, J0) into the topos of sheaves, (P0 is the category of finite rooted posets and open maps, J0 the canonical topology on P0) given by H ↦ HA(H, (−)) : P0 → Set preserves the structure mentioned above, finite coproducts, and subobject classifier; it is also conservative. This whole structure on can be derived from that of Sh(P0, J0) via the embedding Φ. We also show that the equivalence relations in are not effective in general. On the way to these results we establish a new kind of duality between and a category of sheaves equipped with certain structure defined in terms of Ehrenfeucht games. Our methods are model-theoretic and combinatorial as opposed to proof-theoretic as in [Pi].
A model in which every Boolean algebra has many subalgebras
Authors: Cummings James , Shelah Saharon 
Published: May 2006

AbstractWe show that it is consistent with ZFC (relative to large cardinals) that every infinite Boolean algebraBhas an irredundant subsetAsuch that 2∣A∣= 2∣B∣. This implies in particular thatBhas 2∣B∣subalgebras. We also discuss some more general problems about subalgebras and free subsets of an algebra.The result on the number of subalgebras in a Boolean algebra solves a question of Monk from [6]. The paper is intended to be accessible as far as possible to a general audience, in particular we have confined the more technical material to a “black box” at the end. The proof involves a variation on Foreman and Woodin's model in which GCH fails everywhere.
The hierarchy theorem for generalized quantifiers
Authors: Hella Lauri , Luosto Kerkko , Väänänen Jouko 
Published: May 2006

AbstractThe concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.
Fibred semantics and the weaving of logics. Part 1: Modal and intuitionistic logics
Authors: Gabbay D. M. 
Published: May 2006

AbstractThis is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systemsLi,i∈I, to form a new systemLI. The methodology ‘fibres’ the semanticsiofLiinto a semantics forLI, and ‘weaves’ the proof theory (axiomatics) ofLiinto a proof system ofLI. There are various ways of doing this, we distinguish by different names such as ‘fibring’, ‘dovetailing’ etc, yielding different systems, denoted byetc. Once the logics are ‘weaved’, further ‘interaction’ axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics.The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property.Some results on combining logics (normal modal extensions ofK) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20–30 years. We hope our methodology will help organise the field systematically.
Uniformization, choice functions and well orders in the class of trees
Authors: Lifsches Shmuel , Shelah Saharon 
Published: May 2006

AbstractThe monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don't have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection to the uniformization problem.
A star-free semantics for R
Authors: Mares Edwin D. 
Published: May 2006

AbstractThe purpose of this paper is to show that semantics for relevance logic, based on the Routley-Meyer semantics, can be given without using the Routley star operator to treat negation. In the resulting semantics, negation is treated implicationally. It is shown that, by the use of restrictions on the ternary accessibility relation, simplified by the use of some definitions, a semantics can be stipulated over which R is complete.
The undecidability of the Π4-theory for the r.e. wtt and Turing degrees
Authors: Lempp Steffen , Nies André 
Published: May 2006

AbstractWe show that the Π4-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.
More about relatively lawless sequences
Authors: Moschovakis Joan Rand 
Published: May 2006

AbstractIn the author's Relative lawlessness in intuitionistic analysis [this Journal, vol. 52 (1987), pp. 68–88] and An intuitionistic theory of lawlike, choice and lawless sequences [Logic Colloquium ’90, Springer-Verlag, Berlin, 1993, pp. 191–209] a notion of lawlessness relative to a countable information base was developed for classical and intuitionistic analysis. Here we simplify the predictability property characterizing relatively lawless sequences and derive it from the new axiom of closed data (classically equivalent to open data) together with a natural principle of invariance under finite translation. We characterize relative lawlessness in terms of a notion of forcing. Finally, we study relative lawlessness on an arbitrary fan and show that the collection of lawless binary sequences (which is comeager in the sense of Baire) has probability measure zero. The reasoning is predominantly constructive.
Chains and antichains in interval algebras
Authors: Bekkali M. 
Published: May 2006

AbstractLet κ be a regular cardinal, and let B be a subalgebra of an interval algebra of size κ. The existence of a chain or an antichain of size κ in ℬ is due to M. Rubin (see [7]). We show that if the density of B is countable, then the same conclusion holds without this assumption on κ. Next we also show that this is the best possible result by showing that it is consistent with 2ℵ0 = ℵω1 that there is a boolean algebra B of size ℵω1 such that length(B) = ℵω1 is not attained and the incomparability of B is less than ℵω1. Notice that B is a subalgebra of an interval algebra. For more on chains and antichains in boolean algebras see. e.g., [1] and [2].
Almost weakly 2-generic sets
Authors: Fenner Stephen A. 
Published: May 2006

AbstractThere is a family of questions in relativized complexity theory—weak analogs of the Friedberg Jump-Inversion Theorem—that are resolved by 1-generic sets but which cannot be resolved by essentially any weaker notion of genericity. This paper defines aw2-generic sets, i.e., sets which meet every dense set of strings that is re. in some incomplete r.e. set. Aw2-generic sets are very close to 1-generic sets in strength, but are too weak to resolve these questions. In particular, it is shown that for any set X there is an aw2-generic set G such that NPG ∩ co-NPG ⊈ PG⊕x. (On the other hand, if G is 1-generic, then NPG ∩ co-NPG ⊆ PG⊕SAT, where SAT is the NP-complete satisfiability problem [6].) This result runs counter to the fact that most finite extension constructions in complexity theory can be made effective. These results imply that any finite extension construction that ensures any of the Friedberg analogs must be noneffective, even relative to an arbitrary incomplete r.e. set. It is then shown that the recursion theoretic properties of aw2-generic sets differ radically from those of 1-generic sets: every degree above 0′ contains an aw2-generic set: no aw2-generic set exists below any incomplete r.e. set: there is an aw2-generic set which is the join of two Turing equivalent aw2-generic sets. Finally, a result of Shore is presented [30] which states that every degree above 0′ is the jump of an aw2-generic degree.
Stabilité polynômiale des corps différentiels
Authors: Portier Natacha 
Published: May 2006

AbstractA notion of complexity for an arbitrary structure was defined in the book of Poizat Les petits cailloux (1995): we can define P and NP problems over a differential field K. Using the Witness Theorem of Blum et al., we prove the P-stability of the theory of differential fields: a P problem over a differential field K is still P when restricts to a sub-differential field k of K. As a consequence, if P = NP over some differentially closed field K, then P = NP over any differentially closed field and over any algebraically closed field.
The complexity of ODDnA
Authors: Beigel Richard , Gasarch William , Kummer Martin , Martin Georgia , Mcnicholl Timothy , Stephan Frank 
Published: May 2006

AbstractFor a fixed set A. the number of queries to A needed in order to decide a set S is a measure of S's complexity. We consider the complexity of certain sets defined in terms of A:and, for m > 2,where #nA. (x1….. xn) = A(x1) + A(xn)(We identify with , where χA is the characteristic function of A.)If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODDnA and MODmnA:• ODDnA can be decided with n parallel queries to A, but not with n − 1.• ODDnA can be decided with ⌈log(n + 1)⌉ sequential queries to A but not with ⌈log(n + 1)⌉ − 1.• MODmnA can be decided with ⌈n/m⌉ + ⌊n/m⌋ parallel queries to A but not with ⌈n/m⌉ + ⌊n/m⌋ − 1.• MODmnA can be decided with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈n/m⌉ + ⌊n/m⌋ + 1)⌉ − 1.The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.)In particular, every nonzero truth-table degree contains a set A such that ODDnA cannot be decided with n − 1 parallel queries to A. Since every truth-table degree also contains a set B such that ODDnB can be decided with one query to B, a set's query complexity depends more on its structure than on its degree.For a fixed set A,Q(n, A) = {S: S can be decided with n sequential queries to A}.Q∥ (n, A) = {S : S can be decided with n parallel queries to A}.We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies:• Q(0,A) ⊂ Q (1, A) ⊂ Q(2, A) ⊂ …Q∥ (0, A) ⊂ Q∥ (1, A) ⊂ Q∥ (2, A) ⊂ …The same is true if A is a jump.
Infinite time Turing machines
Authors: Hamkins Joel David , Lewis Andy 
Published: May 2006

AbstractWe extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Everyset. for example, is decidable by such machines, and the semi-decidable sets form a portion of thesets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators.
A note on the model theory of generalized polygons
Authors: Tent Katrin 
Published: May 2006

AbstractUsing projectivity groups, we classify some polygons with strongly minimal point rows and show in particular that no infinite quadrangle can have sharply 2-transitive projectivity groups in which the point stabilizers are abelian. In fact, we characterize the finite orthogonal quadranglesQ(4,2).Q−(5.2) andQ(4,3) by this property. Finally we show that the sets of points, lines and flags of any ℵ1-categorical polygon have Morley degree 1.
On fixed-point logic with counting
Authors: Flum Jörg , Grohe Martin 
Published: May 2006

One of the fundamental results of descriptive complexity theory, due to Immerman [13] and Vardi [18], says that a class of ordered finite structures is definable in fixed-point logic if, and only if, it is computable in polynomial time. Much effort has been spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered finite structures by a logic in a similar way.The most obvious shortcoming of fixed-point logic itself on unordered structures is that it cannot count. Immerman [14] responded to this by adding counting constructs to fixed-point logic. Although it has been proved by Cai, Fürer, and Immerman [1] that the resulting fixed-point logic with counting, denoted by IFP+C, still does not capture all of polynomial time, it does capture polynomial time on several important classes of structures (on trees, planar graphs, structures of bounded tree-width [15, 9, 10]).The main motivation for such capturing results is that they may give a better understanding of polynomial time. But of course this requires that the logical side is well understood. We hope that our analysis of IFP+C-formulas will help to clarify the expressive power of IFP+C; in particular, we derive a normal form. Moreover, we obtain a problem complete for IFP+C under first-order reductions.
Jónsson cardinals, Erdős cardinals, and the core model
Authors: Mitchell W. J. 
Published: May 2006

AbstractWe show that if there is no inner model with a Woodin cardinal and the Steel core model K exists, then every Jónsson cardinal is Ramsey in K, and every δ-Jónsson cardinal is δ5-Erdős in K.In the absence of the Steel core model K we prove the same conclusion for any model L[] such that either V = L[] is the minimal model for a Woodin cardinal, or there is no inner model with a Woodin cardinal and V is a generic extension of L[].The proof includes one lemma of independent interest: If V = L[A], where A ⊂ κ and κ is regular, then Lκ[A] is a Jónsson algebra. The proof of this result. Lemma 2.5, is very short and entirely elementary.