Annals Of Pure Research Articles

Lower bounds on β ( α ) and other properties of α-register machines

This paper extends our paper (In Revolutions and Revelations in Computability. CiE 2022 (2022)) for the conference “Computability in Europe” 2022. After Infinite Time Turing Machines (ITTM) were introduced in Hamkins and Lewis (Journal of Symbolic Logic 65 (2000)), a number of machine models of computability have been generalized to the transfinite. While for some of these models the computational strength has been successfully determined, there are still several white spots on the map of transfinite computability. In this paper, we contribute to the understanding of the computational strength of transfinite machine models by (i) proving lower bounds on the computational strength of α-Infinite Time Register Machines (α-ITRMs) for certain values of α, refuting a conjecture about their strength made in (Annals of Pure and Applied Logic 173 (2022)), (ii) showing that the computational strength of cardinal-recognizing ITRMs is equal to that of ITRMs and (iii) showing that non-solvability of the bounded halting problem, existence of a universal machine and an increase of computational power by allowing machines to recognize cardinals are equivalent for α-ITRMs for all relevant values of α. Finally, we give some results indicating how the picture changes when the use of parameters is dropped or restricted.

A variety of algebras closely related to subordination algebras

We introduce a variety of algebras in the language of Boolean algebras with an extra implication, namely the variety of pseudo-subordination algebras, which is closely related to subordination algebras. We believe it provides a minimal general algebraic framework where to place and systematise the research on classes of algebras related to several kinds of subordination algebras. We also consider the subvariety of pseudo-contact algebras, related to contact algebras, and the subvariety of the strict implication algebras introduced in Bezhanishvili et al. [(2019). A strict implication calculus for compact Hausdorff spaces. Annals of Pure and Applied Logic, 170, 102714]. The variety of pseudo-subordination algebras is term equivalent to the variety of Boolean algebras with a binary modal operator. We exploit this fact in our study. In particular, to obtain a topological duality from which we derive the known topological dualities for subordination algebras and contact algebras.

Extending Hrushovski's groupoid-cover correspondence using simplicial groupoids

Hrushovski's suggestion, given in [3], to capture the structure of the 1-analysable covers of a theory T using simplicial groupoids definable in T is realized here. The ideas of Haykazyan and Moosa, found in [“Functoriality and uniformity in Hrushovski's groupoid-cover correspondence,” Annals of Pure and Applied Logic, 2018] are used, and extended, to define an equivalence of categories. Finally, a couple of examples are studied with these new tools.

Access-based intuitionistic knowledge

Abstract We introduce the concept of $\textit{access-based}$ intuitionistic knowledge which relies on the intuition that agent $i$ knows $\varphi$ if $i$ has found $\textit{access to a proof}$ of $\varphi$. Basic principles are distribution and factivity of knowledge as well as $\square\varphi\rightarrow K_i\varphi$ and $K_i(\varphi\vee\psi) \rightarrow (K_i\varphi\vee K_i\psi)$, where $\square\varphi$ reads $`\varphi$ is proved'. The formalization extends a family of classical modal logics (Lewitzka, 2017, Journal of Logic and Computation, 27, 201--212) designed as combinations of $\mathit{IPC}$ and $\mathit{CPC}$ and as systems for the reasoning about proof, i.e. intuitionistic truth. We adopt a formalization of common knowledge from (Lewitzka, 2011, Studia Logica, 97, 233--264) and interpret it here as access-based common knowledge. We compare our proposal with recent approaches to intuitionistic knowledge (Artemov and Protopopescu, 2016, The Review of Symbolic Logic, 9, 266--298; Lewitzka, 2019, Annals of Pure and Applied Logic, 170, 218--250) and bring together these different concepts in a unifying semantic framework based on Heyting algebra expansions.

Introducing article numbering to Annals of Pure and Applied Logic

Introducing article numbering to Annals of Pure and Applied Logic

Bridging Ranking Theory and the Stability Theory of Belief

In this paper we compare Leitgeb’s stability theory of belief (Annals of Pure and Applied Logic, 164:1338-1389, 2013; The Philosophical Review, 123:131-171, [2014]) and Spohn’s ranking-theoretic account of belief (Spohn, 1988, 2012). We discuss the two theories as solutions to the lottery paradox. To compare the two theories, we introduce a novel translation between ranking (mass) functions and probability (mass) functions. We draw some crucial consequences from this translation, in particular a new probabilistic belief notion. Based on this, we explore the logical relation between the two belief theories, showing that models of Leitgeb’s theory correspond to certain models of Spohn’s theory. The reverse is not true (or holds only under special constraints on the parameter of the translation). Finally, we discuss how these results raise new questions in belief theory. In particular, we raise the question whether stability (a key ingredient of Leitgeb’s theory) is rightly thought of as a property pertaining to belief (rather than to knowledge).

On naturally continuous non-dcpo domains

After works of Normann and the author on sequentiality (Normann 2006Mathematical Structures in Computer Science16 (2) 279–289; Normann and Sazonov 2012Annals of Pure and Applied Logic163 (5) 575–603; Sazonov 2007Logical Methods in Computer Science3 (3:7) 1–50), the necessity and possibility of a non-dcpo domain theory became evident. In this paper, the category of continuous dcpo domains is generalized to a category of ‘naturally’ continuous non-dcpo domains with ‘naturally’ continuous maps as arrows. A full subcategory of the latter, assuming a kind of bounded-completeness requirement of domains and presence of ⊥ in each, proves to be Cartesian closed and equivalent to a subclass of Ershov's general A-spaces (Ershov 1974Algebra and Logics12 (4) 369–416). This extends a non-dcpo generalization of Scott (algebraic) domains introduced and proved to be equivalent to Ershov's general f-spaces (Ershov 1972Algebra and Logic11 (4) 367–437) in Sazonov (2007 op. cit.; 2009 Annals of Pure and Applied Logic159 (3) 341–355).The current approach to natural domains (v-domains) is different from f-spaces and A-spaces in that it has arisen in Sazonov (2007 op. cit.) in a different way from defining fully abstract models for some versions of the language PCF over Integers, whereas the Ershov's approach was not initially related with full abstraction, and non-dcpo version of f-spaces and A-spaces were originally considered in an abstract (mainly topological) style. In this paper devoted to naturally continuous natural domains (v-continuous v-domains), we also work in an abstract (mainly order-theoretic) style but with the hope to relate it in the future with the ideas of PCF over Reals by exploring and adapting the ideas in Escardó (1996Theoretical Computer Science162 (1) 79–115), Escardó et al. (2004Mathematical Structures in Computer Science, 14 (6), Cambridge University Press 803–814), Marcial-Romero and Escardó (2007Theoretical Computer Science379 (1-2) 120–141), Sazonov (2007 op. cit.).

Bouligand–Severi tangents in MV-algebras

In their important recent paper published in the Annals of Pure and Applied Logic, Dubuc and Poveda call an MV-algebra A strongly semisimple if all principal quotients of A are semisimple. All boolean algebras are strongly semisimple, and so are all finitely presented MV-algebras. We show that for any 1-generator MV-algebra, semisimplicity is equivalent to strong semisimplicity. Further, a semisimple 2-generator MV-algebra A is strongly semisimple if and only if its maximal spectral space \mu(A)\subseteq [0,1]^2 does not have any rational Bouligand–Severi tangents at its rational points. In general, when A is finitely generated and \mu(A)\subseteq [0,1]^n has a Bouligand–Severi tangent then A is not strongly semisimple. An MV-algebra A is strongly semisimple if and only if so is every 2-generator subalgebra of A .

L. A. Levin. Some theorems on the algorithmic approach to probability theory and information theory (1971 Dissertation directed by A. N. Kolmogorov). Annals of Pure and Applied Logic, vol. 162 (2010), pp. 224–235

L. A. Levin. Some theorems on the algorithmic approach to probability theory and information theory (1971 Dissertation directed by A. N. Kolmogorov). Annals of Pure and Applied Logic, vol. 162 (2010), pp. 224–235 - Volume 19 Issue 3

André Nies. Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305. - Bjørn Kjos-Hanssen, André Nies, and Frank Stephan. Lowness for the class of Schnorr random reals. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647–657. - Noam Greenberg and Joseph S. Miller. Lowness for Kurtz randomness. The Journal of Symbolic Logic, vol.

André Nies. Lowness properties and randomness. Advances in Mathematics, vol. 197 (2005), no. 1, pp. 274–305. - Bjørn Kjos-Hanssen, André Nies, and Frank Stephan. Lowness for the class of Schnorr random reals. SIAM Journal on Computing, vol. 35 (2005), no. 3, pp. 647–657. - Noam Greenberg and Joseph S. Miller. Lowness for Kurtz randomness. The Journal of Symbolic Logic, vol. 74 (2009), no. 2, pp. 665–678. - Laurent Bienvenu and Joseph S. Miller. Randomness and lowness notions via open covers. Annals of Pure and Applied Logic, vol. 163 (2012), no. 5, pp. 506–518. - Johanna N. Y. Franklin, Frank Stephan, and Liang. Yu Relativizations of randomness and genericity notions. The Bulletin of the London Mathematical Society, vol. 43 (2011), no. 4, pp. 721–733. - George Barmpalias, Joseph S. Miller, and André Nies. Randomness notions and partial relativization. Israel Journal of Mathematics, vol. 191 (2012), no. 2, pp. 791–816. - Volume 19 Issue 1

George Barmpalias, Andrew E. M. Lewis and Keng Meng NG. The importance of Π01 classes in effective randomness. The Journal of Symbolic Logic, vol. 75 (2010), pp. 387–400. - George Barmpalias, Andrew E. M. Lewis and Frank Stephan. Π01 classes, LR degrees and Turing degrees. Annals of Pure and Applied Logic, vol. 156 (2008), pp. 21–38. - Antonin Kučera. Measure,

George Barmpalias, Andrew E. M. Lewis and Keng Meng NG. The importance of Π01 classes in effective randomness. The Journal of Symbolic Logic, vol. 75 (2010), pp. 387–400. - George Barmpalias, Andrew E. M. Lewis and Frank Stephan. Π01 classes, LR degrees and Turing degrees. Annals of Pure and Applied Logic, vol. 156 (2008), pp. 21–38. - Antonin Kučera. Measure, Π01 classes and complete extensions of PA. Recursion Theory Week (Oberwofach, 1984). Lecture Notes in Mathematics, vol. 1141. Springer, Berlin, 1985, pp. 245–259. - Frank Stephan. Martin-Löf randomness and PA complete sets. Logic Colloquium '02. Lecture Notes in Logic, vol. 27, Association for Symbolic Logic, La Jolla, CA, 2006, pp. 342–348. - Volume 18 Issue 3

S. S. Goncharov. Autostability and computable families of constructivizations. Algebra and Logic, vol. 14 (1975), no. 6, pp. 392–409. - S. S. Goncharov. The quantity of nonautoequivalent constructivizations. Algebra and Logic, vol. 16 (1977), no. 3, pp. 169–185. - S. S. Goncharov and V. D. Dzgoev. Autostability of models. Algebra and Logic, vol. 19 (1980), no. 1, pp. 28–37. -

S. S. Goncharov. Autostability and computable families of constructivizations. Algebra and Logic, vol. 14 (1975), no. 6, pp. 392–409. - S. S. Goncharov. The quantity of nonautoequivalent constructivizations. Algebra and Logic, vol. 16 (1977), no. 3, pp. 169–185. - S. S. Goncharov and V. D. Dzgoev. Autostability of models. Algebra and Logic, vol. 19 (1980), no. 1, pp. 28–37. -

The strength of extensionality II — Weak weak set theories without infinity

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 (2009) 234–268] which measures them in the presence. These investigations provide a uniform framework in which three different kinds of reverse mathematics–Friedman–Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory–can be reformulated within one language so that we can compare them more directly.

Eskolemization in Intuitionistic Logic

In Baaz and Iemhoff (2006, Annals of Pure and Applied Logic, 142, 269–295), an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore, eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in Baaz and Iemhoff (2006, Annals of Pure and Applied Logic, 142, 269–295) were semantical. In this article, a proof-theoretic proof of the completeness of eskolemization with respect to existential quantifiers is presented.

A Note on Expressive Coalgebraic Logics for Finitary Set Functors

This article has two purposes. The first is to present a final coalgebra construction for finitary endofunctors on Set that uses a certain subset L* of the limit L of the first ω terms in the final sequence. L* is the set of points in L which arise from all coalgebras using their canonical morphisms into L, and it was used earlier for different purposes in Kurz and Pattinson (2005, Mathematical Structures in Computer Science, 15, 543–473). Viglizzo (2005, PhD Dessertation, Indiana University) showed that the same set L* carried a final coalgebra structure for functors in a certain inductively defined family. Our first goal is to generalize this to all finitary endofunctors; the result is implicit in Worrell (2005, Theoritical Computer Science, 338, 184–199). The second goal is to use the final coalgebra construction to propose coalgebraic logics similar to those in Lawrence S. Moss (1999, Annals of Pure and Applied Logic, 96, 277–317) but for all finitary endofunctors F on Set. This time one can dispense with all conditions on F, construct a logical language LF directly from it, and prove that two points in a coalgebra satisfy the same sentences of LF iff they are identified by the final coalgebra morphism. The language LF is very spare, having no boolean connectives. This work on LF is thus a re-working of coalgebraic logic for finitary functors on sets.

On lengths of proofs in non-classical logics

We give proofs of the effective monotone interpolation property for the system of modal logic K , and others, and the system I L of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations.

More on SOP 1 and SOP 2

This paper continues the work in [S. Shelah, Towards classifying unstable theories, Annals of Pure and Applied Logic 80 (1996) 229–255] and [M. Džamonja, S. Shelah, On ◃ ∗ -maximality, Annals of Pure and Applied Logic 125 (2004) 119–158]. We present a rank function for NSOP 1 theories and give an example of a theory which is NSOP 1 but not simple. We also investigate the connection between maximality in the ordering ◃ ∗ among complete first order theories and the (N)SOP 2 property. We prove that ◃ ∗ -maximality implies SOP 2 and obtain certain results in the other direction. The paper provides a step toward the classification of unstable theories without the strict order property.

Partially Ordered Connectives and Monadic Monotone Strict NP

Motivated by constraint satisfaction problems, Feder and Vardi (SIAM Journal of Computing, 28, 57---104, 1998) set out to search for fragments $${\mathcal{L}}$$ of $$\Sigma_1^1$$ satisfying the dichotomy property: every problem definable in $${\mathcal{L}}$$ is either in P or else NP-complete. Feder and Vardi considered in this connection two logics, strict NP (or SNP) and monadic, monotone, strict NP without inequalities (or MMSNP). The former consists of formulas of the form $$\exists \vec{X}\forall \vec{x} \phi$$ , where $$\phi$$ is a quantifier-free formula in a relational vocabulary; and the latter is the fragment of SNP whose formulas involve only negative occurrences of relation symbols, only monadic second-order quantifiers, and no occurrences of the equality symbol. It remains an open problem whether MMSNP enjoys the dichotomy property. In the present paper, SNP and MMSNP are characterized in terms of partially ordered connectives. More specifically, SNP is characterized using the logic D of partially ordered connectives introduced in Blass and Gurevich (Annals of Pure and Applied Logic, 32, 1---16, 1986), Sandu and Vaananen (Zeitschrift fur Mathematische Logik und Grundlagen der Mathematik, 38, 361---372 1992), and MMSNP employing a generalization C of D introduced in the present paper.

Programs from proofs using classical dependent choice

This article generalises the refined A -translation method for extracting programs from classical proofs [U. Berger,W. Buchholz, H. Schwichtenberg, Refined program extraction from classical proofs, Annals of Pure and Applied Logic 114 (2002) 3–25] to the scenario where additional assumptions such as choice principles are involved. In the case of choice principles, this is done by adding computational content to the ‘translated’ assumptions, an idea which goes back to [S. Berardi, M. Bezem, T. Coquand, On the computational content of the axiom of choice, JSL 63 (2) (1998) 600–622] and has been further elaborated in [U. Berger, P. Oliva, Modified bar recursion and classical dependent choice, in: M. Baaz, S.D. Friedman, J. Kraijcek (Eds.), Logic Colloquium ’01, in: Lecture Notes in Logic, vol. 20, Springer, Berlin, 2005, pp. 89–107]. We further investigate the applicability of this extension by means of a small but non-trivial example, and discuss the formalisation as well as some optimisations necessary in order to extract terminating programs. Both formalisation and term extraction have been implemented in the interactive proof assistant Minlog.

Isolation, Infima and Diamond Embeddings

In (1993, Annals of Pure and Applied Logic, 62, 207–263), Kaddah pointed out that there are two d.c.e. degrees a,b forming a minimal pair in the d.c.e. degrees, but not in the Δ20 degrees. Kaddah's; result shows that Lachlan's; theorem, stating that the infima of two c.e. degrees in the c.e. degrees and in the Δ20 degrees coincide, cannot be generalized to the d.c.e. degrees. In this article, we apply Kaddah's; idea to show that there are two d.c.e. degrees c,d such that c cups d to 0', and caps d to 0 in the d.c.e. degrees, but not in the Δ20 degrees. As a consequence, the diamond embedding {0,c,d,0'} is different from the one first constructed by Downey in 1989 in [5]. To obtain this, we will construct c.e. degrees a,b, d.c.e. degrees c > a,d > b and a non-zero ω-c.e. degree e ≤ c,d such that (i) a,b form a minimal pair, (ii) a isolates c, and (iii) b isolates d. From this, we can have that c,d form a minimal pair in the d.c.e. degrees, and Kaddah's; result follows immediately. In our construction, we apply Kaddah's; original idea to make e below both c and d. Our construction allows us to separate the minimal pair argument (a∩b = 0), the splitting of 0′ (c∪d = 0'), and the non-minimal pair of c,d (in the Δ20 degrees), into several parts, to avoid direct conflicts that could be involved if only c,d and e are constructed. We also point out that our construction allows us to make a,b above (and hence c,d) high.